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					Calculators          Types of Calculations              Definitions




    FINANCIAL CALCULATIONS
         FOR LAWYERS

      AND THE TIME VALUE OF MONEY



             Explanation and Examples




 M
          any areas of law require a working knowledge of
           financial calculations. For example, Tort Law
           Practice often involves calculating the present
 value of lost future wages. Family Law Practice involves
 valuing a stream of future income or a deferred compen-
 sation plan. Tax Law Practice necessitates an under-
 standing of the Internal Revenue Code time value of money
 sections, which themselves require a working knowledge
 of financial calculations.

         Inexpensive calculators have alleviated the need for
 lawyers to understand the actual formulas; however, be-
 cause the calculators and their respective manuals are
 often complicated, lawyers lacking an accounting or fi-
 nance background may shy away from this important area
 of law.

       This booklet serves three purposes:

            1. It provides a basic explanation - with law-
               yers as the intended audience - of the
               use and application of a typical hand-held
Calculators          Types of Calculations                 Definitions
2             FINANCIAL CALCULATIONS FOR LAWYERS




                     financial calculator, the Hewlett Packard
                     10 Bii.

                  2. It discusses the legal system’s use of
                    financial terminology.

                  3. It includes workable JavaScript Financial
    Calculators      Calculators that solve most of the prob-
                     lems faced by lawyers.

       I.     USE OF A CALCULATOR
              My first advice is “read the manual.” Most financial cal-
       culator manuals explain the various types of calculations
       and provide understandable examples. This book does
       not preempt or replace those manuals for other calcula-
       tors. Rather, it supplements them with explanations and
       examples geared toward lawyers.

              For users who rely solely on the JavaScript Finan-
       cial Calculators included on the attached CD ROM, this
       booklet serves as the instruction manual.

              A. TYPES OF CALCULATIONS
              While financial calculators can compute many
       things, six types of calculations are fundamental:

                  1. Present Value of a Sum

                  2. Future Value of a Sum

                  3. Present Value of an Annuity
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 3




           4. Future Value of an Annuity

           5. Sinking Fund

           6. Amortization

       For Tax Lawyers, each of these calculations is rel-
evant to one or more Internal Revenue Code provisions.
For example, section 7872 - dealing with below market
loans - requires the use of Present Value of a Sum and
Present Value of an Annuity functions. Sections 1272
and 1274 - dealing with original issue discount loans -
involve Amortization. And, section 467 - dealing with
prepaid or deferred rent - involves the use of a Sinking
Fund. Future Value of an Annuity, as well as Sinking
Fund calculations are relevant to deferred compensation
and retirement planning.

       For Family Law and Tort Lawyers, each is also
relevant, particularly the Present Value and Sinking Fund
calculations. For example, a Tort Lawyer will often need
to compute the present value of lost future wages: i.e.,
the Present Value of an Annuity. A Family Lawyer might
similarly need to value a business using the Present Value
of an Annuity Calculation. Or, he might utilize a Sinking
Fund in computing needed savings for a child’s educa-
tion, as part of an agreed marital settlement.

        General Practitioners and Real Estate Attorneys
will find the amortization calculations particularly useful,
as they compute the needed payments on a home loan.
The Present Value calculators are relevant for contracts
requiring advance payments; similarly, the Future Value
calculations are relevant for contracts involving deferred
Calculators          Types of Calculations           Definitions
4             FINANCIAL CALCULATIONS FOR LAWYERS




     payments.

            Each calculation relies on the same basic formula,
     involving six factors, with the typical key label. The
     JavaScript Calculators included on the CD use the words
     for each function:

                                                      JavaScript
10Bii Key                                                Key
            1. The present value (PV)
                                                   Present Value

            2. The future value (FV)
                                                    Future Value

            3. The interest rate per year (I/YR)
                                                        Nominal
                                                   Interest Rate
            4. The number of periods or pay
               ments per year (P/YR)                   Payments
                                                        Per Year

            5. The amount of each payment
               (PMT)                                    Payment


            6. The number of periods (N)
                                                      Number of
                                                      Payments
Calculators                    Types of Calculations                                Definitions
                     FINANCIAL CALCULATIONS FOR LAWYERS                                        5




                 In the typical example, five of the six factors
          are known. The calculator can then easily solve
          for the sixth.

                       In addition, annuity calculations re-
                quire a mode setting, indicating whether
                payments occur at the beginning or the end                                  Mode
                of a period.

          B.COMMON DIFFICULTIES
        Unless the calculator is defective, which is unlikely,
it will produce the correct answer if given the correct
information. Nevertheless, many users, at one time or
another, exclaim “This thing doesn’t work!”1 Usually, they
have violated one of the following rules:

     1. COMMON DIFFICULTY: First, clear
the machine. A calculator knows only what you tell it
and it does not forget until you tell it to forget, typically
even if you turn off the machine. Thus, be certain to
clear all functions and memory when beginning a new
                                   calculation. This is
                                   particularly important
                                   for hand-held calcula-
   Clear the                       tors, such as the HP
   Machine                         10Bii: the display
                                   shows only one func-
                                   tion at a time, creating
                                   the risk that the user
will not remember to clear all other functions. The


1 In frustration, I’ve said it myself many times. I was, however, always wrong as to that
point.
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6                FINANCIAL CALCULATIONS FOR LAWYERS




     JavaScript Calculators included on the attached CD do not
     have this risk because the display shows all function values
     at all times.

              All calculators have a clear key, usually denominated
     with a C or the word clear. In addition, many calculators
     have a function key by which merely the last information
     entered can be cleared, and a different function key by which
     all information can be cleared.

                 a. HP lOBii Calculator

            The HP lOBii calculator has three levels for the clear
     function.

             1. The C key - when pressed in unshifted mode - will
                clear the entire displayed number; however, it leaves
                the memory intact. See Example 1.

             2. The back arrow key will clear single digits, one at a
                time. See Example 2.

             3. The C ALL key when pressed in the “shifted
             mode” will clear the entire memory, as well as the
             displayed number.

             To perform this function on an HP10Bii calculator,2
     first press the orange downshift key and then press C
     ALL. These strokes shift the function to C ALL (clear all)
     rather than C (clear). Before working a new problem, you
     should press these keys:



      2
          The older HP 10B has a CLEAR ALL button instead. Some use green for the shift color.
Calculators         Types of Calculations             Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS               7




      Do not, however, perform the clear all function in the
middle of a problem in which you are comparing alternative
values of a particular element.



                         Caution:

                The HP10Bii C ALL function
                  does not clear either the
                 periods per year (P/YR) or
                mode (BEG/END). You must
                   change these manually.




  EXAMPLE 1 (HP 10Bii)

  If you input 50 + 20 + 30 but intended
  50 + 20 + 40, press C erasing the 30
  but leaving the 70 in memory. You can
  then press 40 and = . The display will then read
  110.

  50 + 20 + 30

           40             The display will read 110.
Calculators          Types of Calculations               Definitions
8               FINANCIAL CALCULATIONS FOR LAWYERS




    EXAMPLE 2 (HP 10Bii)

    If you input 523 but intended 524,
    you may use the backward arrow key
    to erase the 4. Then simply enter
    the number 3. The display will read 523. In con-
    trast, the C key will clear the entire number 523.

            3




               The C ALL function does not reset the number of
       periods per year. If you change this setting, it will remain -
       even if you turn off the calculator - until you manually change
       it or remove the battery. Also, the C ALL function does not
       change the mode. Thus if you reset the mode from end to
       begin, or vice versa, it will remain - even if you turn off the
       calculator- until you reset it manually through the procedure
       described below or remove the battery.
Calculators            Types of Calculations                 Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                      9




                                   TIP
                  The JavaScript Calculators are
                   much easier to clear than are
                 typical hand-held machines, thus
                    producing fewer mistakes.



       b. JavaScript Financial Calculator

       The JavaScript Financial Calculator has two clear
functions:

       clear all        1. A Clear All button. This will reset
                        all numbers to their default amount.

                        2. The Backspace key. Using this
               key on a typical computer keyboard will erase
               single or multiple digits just as it does with
               any other program.

       Because the included JavaScript Calculators dis-
play all function values at all times, the risk of a user failing to
clear some values - and thus computing a wrong value - is
largely eliminated.




   To test the two "clear" functions, type in a
 number in the above box. Use the backspace
key to erase it. Or, click on the "clear all" button
                 to clear the box.
Calculators              Types of Calculations                            Definitions
10             FINANCIAL CALCULATIONS FOR LAWYERS




          2. COMMON DIFFICULTY: Set the cash
     flows with the proper sign. Many, but not all, calcu-
     lators require that cash flows be directional. This means that
     one set of cash flows must be positive and the other must be
     negative. This is true of the HP 10Bii Calculator. It is not
     true of the JavaScript Calculators on the enclosed CD.

              For example, in machines such as the HP 10Bii, the
     present value amount
     may be expressed as
     a positive number - a       Set the
     deposit - while the fu-
     ture value amount will      Cash Flows
     be expressed as a           Correctly
     negative number - a
     withdrawal. Or, the
     opposite may be true;
     however, the present and future values cannot both be posi-
     tive or both be negative at the same time. On the other hand,
     some calculators - such as the JavaScript Financial Calcula-
     tor - eliminate this feature. Hence, be sure to read your own-
     ers manual.

              a. HP lOBii Calculator
                                                  3
            The HP lOBii calculator requires that cash flows be
     entered with opposite signs. As shown in Example 3, failure
     to do so will prompt the display no SoLution. A negative
     number may be entered in two ways.

           For example, to input the number (1000), first enter
     the positive number 1000, then press the “plus/minus” key:
     3 Many other calculators - such as those manufactured by Texas Instruments - do not
     require opposite signs between present value and future value.
Calculators         Types of Calculations            Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS              11




 1000

This will change the sign from positive to negative or back
from negative to positive.

      In the alternative, press the minus sign, the number,
and then the equal sign, as follows:


          1000


The display will show a negative 1000.


  EXAMPLE 3 (HP 10Bii)

  Suppose you want to compute the
  annual interest rate inherent to a
  present value of 500, a future value
  of 1000 and a period of 10 years.
  The correct answer is 7.177346254.

  To achieve this, either the 500 or the 1000 must
  be expressed as a negative number while the
  other must be positive. To enter 500 as a nega-
  tive number, press:

  500
Calculators        Types of Calculations              Definitions
12            FINANCIAL CALCULATIONS FOR LAWYERS




           b. JavaScript Financial Calculator

           The JavaScript Financial Calculator eliminates the
     need to input cash-flows directionally. Hence, all numbers
     may be entered as positive numbers. The calculator then
     converts them, as appropriate.

            Thus the enclosed JavaScript Calculators eliminate
     the second most common difficulty faced by users of hand-
     held calculators.



                                TIP
                    The JavaScript Calculators
                    eliminate the need for cash-
                  flow inputs and negative num-
                                bers.




          3. COMMON DIFFICULTY: Set the mode
     correctly. Calculations involving annuities, sinking funds
     and amortizations, require a “mode” setting: either begin mode


                     Set the
                     Mode
                     Correctly
Calculators          Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 13




or end mode. This is true of all calculators, including the HP
10Bii as well as the JavaScript Calculators.

      Begin mode applies if payments (or deposits or with-
drawals) occur at the beginning of each period. End mode
applies if payments occur at the end of each period.

       Typically, a sinking fund uses the begin mode because
the depositor wants to begin immediately. Typically, an am-
ortization - such as the repayment of a loan - uses the end
mode because loan payments do not begin on the date of
the loan. Instead, loan payments begin at the end of each
period. For example, payments on a car loan typically start
one month after the purchase. Using begin mode for a loan
amortization generally makes little sense: a payment on the
date of the borrowing merely collapses to a lower amount
borrowed, resulting in end mode.

       “Future Value of a Sum” and “Present Value of a Sum”
calculations are not affected by the mode setting.
  a. HP lOBii Calculator

      To set the mode on an HP lOBii calculator, first press
the orange shift key and then press the BEG/END key to
operate the mode function.




       Most calculators are preset at the factory in end mode.
Pressing these two keys will change it to begin mode, which
the display will note with the word BEGIN. To revert to end
mode, press the two keys again. The display will no longer
indicate the mode. If you change the setting to begin mode,
Calculators         Types of Calculations                Definitions
14            FINANCIAL CALCULATIONS FOR LAWYERS




     it will remain, even if you turn off the calculator or utilize the
     clear all (C ALL) function. To revert to end mode, you must
     do so manually by repeating the above steps.

            A common mistake among calculator users involves
     Begin Mode annuities, sinking funds, or amortizations. Be-
     cause the HP10Bii display does not indicate End Mode, the
     user may forget to change the mode to Begin, thus produc-
     ing significant (but not obvious) incorrect results. The de-
     fault setting is for End Mode because that is consistent with
     most amortizations, a common calculation involving Mode.


                              Caution:

                 With the HP10Bii calculator,
                 an indication of Mode setting
                appears only with Begin Mode.

                 Thus, be careful when com-
                puting annuities: you might be
                in End Mode and not know it!


                 The JavaScript Calculators
                 eliminate this risk by always
                       indicating mode.
Calculators        Types of Calculations           Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS           15




  EXAMPLE 4 (HP 10Bii)
  (mode illustration)
  Your child was born today. You would
  like to accumulate $100,000 when
  she reaches the age of 18. You ex-
  pect to earn 6% nominal annual in-
  terest compounded monthly (after tax). To de-
  termine how much you must deposit, press:

  12

  216

  6

  100,000

                  The display will read 258.16.

                                 The display will read
                                    256.88.
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16            FINANCIAL CALCULATIONS FOR LAWYERS




     However, sinking funds and annuities commonly use Begin
     Mode, necessitating a different calculator setting.

            For example, if you were saving for a child’s educa-
     tion and desired to make monthly deposits, a sinking fund
     calculation can tell you the necessary monthly deposit to
     make, depending on the child’s age and the expected inter-
     est rate.

           If you were to begin the deposits today, you would
     use Begin Mode. Or, if you to begin making the deposits at
     the end of the first month, you would use End Mode.

           As shown in Example 4, new parents who desire to
     accumulate $100,000 for their child’s 18th birthday and who
     expect to earn 6% nominal annual interest compounded
     monthly, must deposit $258.16 monthly if they begin making
     deposits at the end of Month 1. Or, they need deposit only
     $256.88 if they begin immediately. Although the differences
     may seem slight in this problem, they can be material in
     many other situations.




                                TIP

        A Mode setting is necessary only for annu-
         ities, sinking funds, and amortizations. It
        does not apply to Present Value or Future
                 Value of a Sum calculations.
Calculators         Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   17




      b. JavaScript Financial Calculator

      The JavaScript Financial Calculator has buttons
labeled Begin and End to designate the mode.                    Begin


       Also, when in Begin Mode, the words begin mode
                                                                End
appear in red. Similarly, in End Mode, the words end
mode appear in black. Hence you are unlikely to forget to
set the mode correctly. This effectively eliminates the third
most common difficulty with using hand-held calculators.




      4. COMMON DIFFICULTY: Set the inter-
est rate to compound for each payment period.
This involves the P/YR button Pressing the orange (some-
times green) shift key along with the P/YR key sets the num-
ber of payments per year. For example, to set the calculator
for quarterly payments, press the following:

  4




             Set the Interest
               Payment and
               Compounding
                Periods the Same.
Calculators           Types of Calculations               Definitions
18             FINANCIAL CALCULATIONS FOR LAWYERS




           A basic law of finance is that the compounding period and
        the payment period must be the same. Thus, if the facts
        provide for annual payments, the interest rate must be stated
        as an annual rate. If, instead, the facts provide for semian-
        nual payments, then the interest rate must be stated as a
                                             semiannual rate. Likewise,
                                             monthly payments call for
                                             a monthly interest rate.
        Law of Finance
                                            Setting the P/YR to the
                                         correct amount to corre-
      The compounding                    spond with payments re-
     period and the pay-                 quires the user to have the
                                         correct information. This,
     ment period must be                 in turn, necessitates that
            the same.                    the interest rate be stated
                                         using correct terminology.
                                         As a result, common prac-
                                         tice involves stating inter-
      est rates using a nominal annual uncompounded format along
      with a statement of the compounding period.



                                  Caution:

                          Always define the
                          interest rate using
                         correct terminology.


                          Interest Rate Definitions
Calculators          Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 19




   If, however, the facts - such as a contract - do not state
the interest rate in such a format, you must convert them to a
nominal annual uncompounded format or the equivalent. Oth-
erwise, no calculator will produce the correct answer.

  Several scenarios are possible:

    1. CONTRACT SCENARIO: The facts given (such
    as in a contract) may state a “nominal annual
    interest rate” as well as a “compounding period.”

    This would be the correct format for stating an inter-
    est rate. If the compounding period is the same
    length as the payment period (e.g., semiannual com-
    pounding and payments every six months), simply
    enter the stated “nominal annual interest rate” using
    the I/YR function.

  a. HP lOBii Calculator

    A 10% nominal annual interest rate would be entered as:

    10

    Use of the NOM% function is optional: you could also
enter the 10% nominal annual interest rate as:

    10


Typically, however, the NOM% key is used for conversion of
an effective rate to a nominal rate. It is unnecessary for
entry of a given nominal rate.
Calculators        Types of Calculations              Definitions
20            FINANCIAL CALCULATIONS FOR LAWYERS




       b. JavaScript Financial Calculator

            The JavaScript Calculator does not have separate I/
     YR and NOM% buttons. Thus the 10% nominal annual inter-
     est rate would be entered simply as 10.

            Because the compounding period and payments per
     year function are the same in this scenario (semiannual com-
     pounding and payments every six months), they are both
     entered through the P/YR function. This is true both on the
     HP10Bii and JavaScript Calculators.

           On the HP10Bii, semiannual payments and semian-
     nual compounding (as assumed above) would be entered
     as:

         2


           On the JavaScript Calculator, semiannual payments
     and compounding are both entered as 2 in the Payments
     Per Year box.

     Payments Per Year                                     2.00


            Each calculator will automatically convert the interest
     rate to a periodic rate of 5% per period by dividing the I/YR
     amount by the P/YR amount. This is internal to the calcula-
     tor and is not displayed on either the HP10Bii or on the
     JavaScript Calculator (although many other calculators dis-
     play the periodic rate).
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  21




    2. CONTRACT SCENARIO: The facts given (such
    as in a contract) may state a “nominal annual
    interest rate” without also stating a “compound-
    ing period.”

    This would not be the correct format for stating an
    interest rate, but is also not unusual. In such a case,
    the parties should clarify the compounding period
    before proceeding. If this is not done, most users
    would probably assume annual compounding (if pay-
    ments are no more frequent than annual) or com-
    pounding coextensive with the payment period. With
    such assumptions, they would proceed as above in
    scenario one.



                         Caution:

                Always define the
                interest rate using
               correct terminology.
                Interest Rate Definitions

       If this assumption is not consistent with one party’s
understanding, litigation may result, illustrating the need for
clear statement of interest rates.

      Neither the HP10Bii nor the JavaScript Calculator
can solve this problem because it results from a poorly drafted
Calculators        Types of Calculations              Definitions
22            FINANCIAL CALCULATIONS FOR LAWYERS




     document. The parties must clarify the interest rate or suffer
     the inevitable confusion and litigation.

         3. CONTRACT SCENARIO: The facts given (as
         in a contract) may state a “nominal annual inter-
         est rate” as well as a “compounding period.”

         If the facts also provide for a payment frequency
         inconsistent with the compounding period, conver-
         sion of the interest rate would be necessary.

       a. HP lOBii Calculator

            For an HP10Bii Calculator, this is a two-step pro-
     cess. First, this requires conversion of the “nominal annual
     interest rate” to an “effective rate” and then second, recon-
     version to a “nominal annual interest rate” with a compound-
     ing period consistent with the payment period. Box One



                            Caution:

                    Always define the in-
                    terest rate using cor-
                      rect terminology.

                    The JavaScript Inter-
                     est Rate Conversion
                    Calculator is designed
                     to help you do this.

                    Interest Rate Definitions
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  23




illustrates Step One: how to convert a “nominal annual inter-
est rate” to an “effective interest rate” using an HP10Bii Cal-
culator.

       Example 5a illustrates Step Two: how to convert the
“effective interest rate” to the “nominal annual interest rate
compounded semi-annually”using an HP10Bii Calculator.
Once the user changes the number of periods per year (P/
YR) to correspond with the payment frequency, the calcula-
tor automatically computes the correct nominal annual rate.
The user must, however, press the shift and NOM% keys to
make this effective.

  b. JavaScript Financial Calculator

       Each of the various financial calculators automatically
converts a nominal rate to the equivalent effective rate. The
user must enter the nominal rate and the calculator displays
the effective rate. No buttons need to be pressed.

      For example, a nominal interest rate of 10% with a
compounding period of six months (two payments or periods
per year) results in an effective interest rate of 10.25%.
                                                      10.00
Nominal Interest Rate

                                                      10.25
Effective Interest Rate


Number of Years

                                                        2.00
Payments Per Year
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24            FINANCIAL CALCULATIONS FOR LAWYERS




     EXAMPLE 5

     An “effective interest rate” of 10%
     with semi-annual payments is not
     equivalent to a semi-annual rate of 5%. Instead,
     it is the equivalent of 4.8808848170% semi-an-
     nually (the periodic rate) or 9.761769636% nomi-
     nally (compounded semi-annually).

     Thus, 9.761769636% nominal annual interest,
     compounded semi-annually is the same as 10.00%
     nominal interest, compounded annually.

     In contrast, two $500 payments - one each six
     months - is not the equivalent of a single $1000
     payment.

     A single payment of $1000 has a future value of
     $1,100 at an effective interest rate of 10%.

     Two semi-annual $500 payments at an effective
     interest rate of 10% have a future value of
     $1,024.40, assuming an annuity in arears or
     $1,074.40, assuming an annuity due.
Calculators              Types of Calculations               Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                       25




  EXAMPLE 5 a (HP 10Bii)
  (from Box One)

  √    Re-set the number of payments per year to
  correspond with the payment period. Press:

  2


  √ Convert to the nominal Interest Rate compounded semi-annu-
  ally:
                             The display will read 10.210662664,
                             which is the nominal annual interest
                             rate compounded semi-annually.

  √ Enter other factors normally:

  2                500


  √ Press either FV or PV to solve for the desired factor:

                   The display will read -1,077.88319038 (begin mode)
                   or -1025.52665666 (end mode).

  or

                   The display will read -975.713261795 (begin mode)
                   or -928.319476691 (end mode).

  If you want to compute both the FV and the PV, you must re-set the
  alternative values to zero between the computation. For example, if
  you first compute the FV as above, before pressing PV, enter the
  following:

  0
Calculators               Types of Calculations                 Definitions
26                 FINANCIAL CALCULATIONS FOR LAWYERS




                                  BOX ONE

                   Convert Nominal Rate to
                       Effective Rate

     T  o convert a nominal annual interest rate compounded periodically
        to an effective interest rate, enter the following values. This ex-
     ample assumes a 10% nominal annual interest rate compounded
     monthly and semi-annual payments of $500.

     √ Set the number of payments per year to correspond with the stated
     compounding period. Press:

     12


     √    Enter the nominal Interest Rate:


     10

     The nominal rate may be entered using either the I/YR or NOM% func-
     tions; thus, use of the orange shift key is optional for this function.

     √    Press the EFF% button:



     The display will read 10.471306744, which is BOTH the effective an-
     nual interest rate and the nominal annual interest rate compounded
     annually.

     Go to Example 5a for completion of this example.
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                       27




                             BOX TWO

              Convert Effective Rate to
                  Nominal Rate

  T  o convert an effective annual interest rate to a nominal annual
     interest rate compounded periodically, enter the following val-
  ues. This example assumes an effective rate of 10% and semi-
  annual payments of $500.

  √ Set the number of payments per year to correspond with the
  facts. This uses the P/YR button as a payments per year button,
  its traditional function. Press:

  2


  √ Set the Effective Interest Rate:

  10

  √ Press the NOM% button:



  The display will read 9.761769634, which is the nominal annual
  interest rate compounded semi-annually equivalent to 10% effec-
  tive interest rate or 10% nominal annual interest compounded an-
  nually.


  Go to Example 5 b for completion of this example.
Calculators               Types of Calculations                  Definitions
28                FINANCIAL CALCULATIONS FOR LAWYERS


                   Interest Rate Converter
                          Box Three
  Convert Effective Rate to Nominal Rate and Periodic Rate
               JavaScript Financial Calculator
     PV Annuity                                                    PV Sum
     FV Annuity     Interest Rate Conversion                     Amortization
 Sinking Fund                  Instructions ON OFF                  FV Sum


     Convert Effective         Convert Nominal          Convert Periodic
      Rate to Nominal           Rate to Effective        Rate to Nominal
     Rate and Periodic         Rate and Periodic        Rate and Effective
            Rate                      Rate                     Rate




Nominal Interest Rate         9.7617696340

Periodic Interest Rate        4.8808848170

Effective Interest Rate         10.00000000
                                          2.00
Payments Per Year

                                  clear all




             1. Press "Clear All" prior to working a problem.

             2. Select the appropriate conversion function.

             3. Type appropriate numbers in the white boxes.

             4. The answer will appear in the green boxes.

             5. Place your cursor over terms for a definition of the term.
Calculators               Types of Calculations             Definitions
                FINANCIAL CALCULATIONS FOR LAWYERS                       29


                  Interest Rate Converter
                         Box Four
  Convert Nominal Rate to Effective Rate and Periodic Rate
             JavaScript Financial Calculator
  PV Annuity                                                   PV Sum
   FV Annuity      Interest Rate Conversion                  Amortization
 Sinking Fund                 Instructions ON OFF              FV Sum


   Convert Effective          Convert Nominal       Convert Periodic
    Rate to Nominal            Rate to Effective     Rate to Nominal
   Rate and Periodic          Rate and Periodic     Rate and Effective
          Rate                       Rate                  Rate




Nominal Interest Rate        10.000000000

Periodic Interest Rate        0.833333333

Effective Interest Rate
                             10.4713067441

Payments Per Year                     12.00

                                clear all




                                            TIP
                    This feature - conversion to all three ways of
                  stating interest - is particularly helpful for legal
                documents. A well-drafted document will state both
                 the nominal rate (including the compounding fre-
                 quency) plus the effective rate and periodic rate.
Calculators        Types of Calculations               Definitions
30            FINANCIAL CALCULATIONS FOR LAWYERS




            Also, as shown in Box Four, the Interest Rate Con-
     version Calculator will automatically convert a Nominal Rate
     to both an Effective Rate and a Periodic Rate. This feature
     is particularly helpful for legal documents. A well-drafted
     document will state both the nominal rate (including the com-
     pounding frequency) plus the effective rate.

         4. CONTRACT SCENARIO: The facts given (such
         as in a contract) may state an “effective annual
         interest rate.”

         As in the first scenario, this would also be a correct
         format for stating an interest rate.

            If the facts also provide for payment frequency other
     than annual, conversion of the interest rate would be neces-
     sary. This would require the conversion of the “effective
     interest rate” to a “nominal annual interest rate” with a com-
     pounding period consistent with the payment period.

         a. HP lOBii Calculator

            Box Two illustrates how to convert an “effective inter-
     est rate” to a “nominal annual interest rate” using an HP
     lOBii Calculator. It involves entering the stated effective
     rate and then using the NOM% key to convert it to the ap-
     propriate nominal rate for the P/YR already entered.

            Example 5b illustrates the completion of the problem.
     As is evident, the process is not difficult; however, it is es-
     sential: without conversion of the stated interest rate to one
     corresponding with the payment period, the calculator would
     provide the wrong answer. This further shows the impor-
     tance for the proper statement of interest rates in any legal
     document.
Calculators               Types of Calculations            Definitions
                  FINANCIAL CALCULATIONS FOR LAWYERS                   31




  EXAMPLE 5 b (HP 10Bii)
  (from Box Two)

      √   Enter other factors normally:

  2                 500


  √ Press either FV or PV to solve for the desired factor:

                     The display will read -1,024.40442409 (end
                     mode) or -1,074.40442408 (begin mode).


  or

                     The display will read -931.276749168 (end mode)
                     or -976.731294623 (begin mode).


  If you want to compute both the FV and the PV, you must re-set
  the alternative values to zero before the alternate computation.
  For example, if you first compute the FV as above, before press-
  ing PV, enter the following:

  0
Calculators        Types of Calculations                Definitions
32            FINANCIAL CALCULATIONS FOR LAWYERS




                              Caution:

                      Never interchange
                     interest rate termi-
                      nology: nominal, an-
                      nual, and effective
                     percentage rates are
                       terms of art: use
                        them correctly.



       b. JavaScript Financial Calculator

             The Interest Conversion Calculator converts an ef-
     fective rate to a nominal rate and the corresponding periodic
     rate. Simply insert the effective rate in the appropriate box
     (labeled Effective Interest Rate) and insert the number of
     Payments per Year. Box Three demonstrates how.

             This calculator will also convert a nominal annual rate
     to the equivalent effective rate and periodic rate. In addition,
     it contains a feature to convert a periodic rate to the equiva-
     lent nominal annual rate and effective rates.
Calculators           Types of Calculations                Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   33




    5. CONTRACT SCENARIO: The facts given (such
    as in a contract) may state an “annual interest
    rate” or an “annual percentage rate.”

    This would not be the correct format for stating an
    interest rate. In such a case, the parties should clarify
    the terminology before proceeding. If this is not done,
    most users would probably assume the terms to be
    the equivalent of an “effective interest rate,” although
    some may disagree.

           If this assumption is not consistent with one
    party’s understanding, litigation may result, illustrat-
    ing the need for clear statement of interest rates.

       Neither the HP10Bii nor the JavaScript Calculator
can solve this problem because it results from a poorly drafted
document. The parties must clarify the interest rate or suffer
the inevitable confusion and litigation.

      To reiterate: if the stated interest compounding period
and the payment period are not the same, you must convert
the interest rate to an equivalent one using a compounding
period identical with that of the payments. Some calculators
do this automatically with a feature labeled Iconv. Others
require the user to make the computations, which are not
difficult. Box One explains how using an HP10Bii. Addi-
tional examples of such a conversion appear.

       Also, as shown in Example 5, if the interest com-
pounding period and the payment period are not coexten-
sive, you must convert the interest rate to an equivalent rate
compounded consistent with the payment period, rather than
convert the payment period to the stated compounding pe-
riod.
Calculators        Types of Calculations              Definitions
34            FINANCIAL CALCULATIONS FOR LAWYERS




             Notice that Examples 5a and 5b have different re-
     sults. This occurs because they begin with different as-
     sumptions. Example 5a uses the BOX ONE assumption of
     a 10% effective interest rate. Example 5b uses the BOX
     TWO assumption of a 10% nominal rate. This further illus-
     trates that you cannot interchange the different interest rate
     terminology.


       5. COMMON DIFFICULTY: Set the periods
     per year correctly. Most calculators are factory preset
     for twelve periods per year. This assumes the common
     facts of monthly payments and monthly compounding.

         All calculators, how-
     ever, can easily be reset.
     For example, in a problem      Set the
     involving a single annual
     payment and annual com-        Periods
     pounding, the payments         Per Year
     per period (P/YR) function
     must be set at one.
                                    Correctly

         a. HP lOBii Calcula-
         tor

        The factory default setting of an HP 10Bii is twelve pay-
     ments per year. To change the factory setting, press the
     desired number - 1 - then press the orange shift key and
     then P/YR, to set the payments per year, as follows:

       1
Calculators           Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                    35




   You can check the setting by pressing the orange shift
key and then holding down the C ALL (clear all) key:



The display will indicate the number of payments per year.
This will remain as the setting - even if you turn off the calcu-
lator or utilize the clear all (C ALL)function - until you manu-
ally reset the payments per year, using the above proce-
dure.

b. JavaScript Financial Calculator

       The JavaScript Calculator has the payments per year
preset to one. Changing it is simple: merely type the appro-
priate number in the space provided. Placing the cursor on
the “Periods/Payments Per Year” label provides additional
explanation.

       As illustrated in Example 6, changing the payments
per year setting is not always necessary. Some people find it
easier to set the PIYR function to one, meaning one pay-
ment per period, rather than one payment per year. Then, in
entering an interest rate they always enter a periodic rate.

        In the Problems and Answers section of this booklet,
this alternative method is labeled the Periodic Method. It
actually represents the math formula used by the calculator,
which uses a periodic interest rate rather than an annual
rate. For convenience, most calculators permit the entry of
an annual, uncompounded rate, which the machine converts
to a periodic rate.

       For example, if a problem calls for ten years of monthly
Calculators             Types of Calculations               Definitions
36               FINANCIAL CALCULATIONS FOR LAWYERS




     EXAMPLE 6 (HP 10Bii)
     (calculator method)

     Compute the present value of an annuity in ar-
     rears of 1000 per month for ten years at 12% nomi-
     nal annual interest compounded monthly. Be cer-
     tain the calculator is set in END Mode. Press the following keys:


     12


     1000


     12


     120

                          The display will read - 69,700.5220314.




                          Did You Notice?

     Both methods of calculating compound interest use the nominal
                         annual interest (NAI).

       As a practical matter, any legal document should always de-
                       scribe interest in NAI terms.
Calculators          Types of Calculations               Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                     37




  EXAMPLE 7 (HP 10Bii)
  (periodic method)

  Compute the present value of an annuity in ar-
  rears of 1000 per month for ten years at 12% nomi-
  nal annual interest compounded monthly. Be cer-
  tain the calculator is set in END Mode. Press the following keys:


   1


   1000


   1


   120

                       The display will read - 69,700.5220314.




                                TIP
         You may enter the various values in any order.

         For example, you could enter the I/YR prior to
                      entering the PMT.
Calculators               Types of Calculations            Definitions
38                FINANCIAL CALCULATIONS FOR LAWYERS


     Present Value of an Annuity Calculator


  EXAMPLE 6
  (JavaScript Calculator)

  Amortization                                             Interest Conversion
                  Present Value of an Annuity Calculator
     FV Annuity                                                PV Sum
  Sinking Fund                Instructions ON OFF              FV Sum



Mode
                              Begin     End
Present Value                     69,700.52
Future Value                            0..00

Nominal Interest Rate                   12.00
                             12.6825030132
Effective Interest Rate
                                        10.00
Number of Years
                                       12.00
Payments Per Year
                                       120.00
Number of Payments
                                      1000.00    end
Payment                                         mode
                                 clear all



                           Did You Notice?
      The JavaScript Calculators always denotes the mode setting -
       even in end mode. This will help you avoid costly mistakes
                which can result from an incorrect setting.
Calculators               Types of Calculations            Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                        39


   Present Value of an Annuity Calculator


  EXAMPLE 7
  (JavaScript Calculator)

  Amortization                                             Interest Conversion
                  Present Value of an Annuity Calculator
   FV Annuity                                                  PV Sum
  Sinking Fund                Instructions ON OFF              FV Sum



Mode
                              Begin     End
Present Value                     69,700.52
Future Value                            0..00

Nominal Interest Rate                    1.00
                             1.0000000000
Effective Interest Rate
                                       120.00
Number of Years
                                         1.00
Payments Per Year
                                       120.00
Number of Payments
                                      1000.00    end
Payment                                         mode
                                 clear all
Calculators        Types of Calculations              Definitions
40            FINANCIAL CALCULATIONS FOR LAWYERS




     payments at 12% nominal annual interest, the calculator is
     indifferent to whether it is told 12 P/YR at 12% interest or 1
     P/YR at 1% interest. In either case, the N setting must be
     120 to indicate the correct number of payments.

          Example 6 illustrates the Calculator Method, while
     Example 7 illustrates the Periodic Method .

             In the Calculator Method of Example 6, the calcula-
     tor divides the 12% nominal interest rate by the number of
     payments per year (12 P/YR) to achieve the correct peri-
     odic interest rate of 1%. The calculator then uses the peri-
     odic rate of 1% to execute the formula.



                             Caution:

                  This explanation of the
                  Periodic Method is op-
                  tional. While some us-
                   ers will find it easier,
                    most will prefer the
                    Calculator Method.


            In the Periodic Method of Example 7, the user tells
     the calculator the periodic interest rate. The calculator then
     makes no adjustment before executing the formula. Also,
     the user sets the payments per year (P/YR) at one. Essen-
Calculators                Types of Calculations                        Definitions
                  FINANCIAL CALCULATIONS FOR LAWYERS                               41




tially, the P/YR function becomes a payments per period
function permanently set at one and thus irrelevant.

       As a practical matter, some people find it easier to
remember to enter the correct periodic rate than to remem-
ber to enter the correct number of payments per year. They
thus set the calculator to one payment per year and leave it
at that setting for all calculations. In contrast. people who
use the Calculator Method must always remember to set
the P/YR function correctly, as it varies from problem to prob-
lem. Because most hand held calculators, such as the         4
HP10Bii have no clear way of indicating the P/YR setting, a
user can easily forget to reset the amount, resulting in an
incorrect answer which may not be obviously incorrect.
Whichever method works best for the individual user is the
one he or she should use.




4
   The HP10Bii indicates the P/YR setting whenever the user presses the C
ALL button, a two-step process. While simple, this function is easily forgot-
ten or ignored, particularly in problems involving many variables and alterna-
tives. In such a problem, use of the C ALL function destroys already input-
ted information, requiring the user to start over; hence, in such cases, the
user may knowingly not want to double check the P/YR setting. This can
result in complacency that itself results in the user forgetting to input the P/
YR amount correclty.
Calculators        Types of Calculations                Definitions
42            FINANCIAL CALCULATIONS FOR LAWYERS




     6. COMMON DIFFICULTY: Set the display to
     the correct number of decimal places. Although
     most calculations involve dollars and cents and thus two places
     after the decimal, large numbers and long periods of time
     can be significantly affected by rounding. While the calcula-
     tor internally uses twelve places after the decimal for calcula-
     tions, it displays only the number pursuant to its setting. Be-
     cause some calculations involve the user writing down or
     otherwise reusing a computed number, it may be helpful to
     have the display read the full nine spots after the decimal.

     a. HP lOBii Calculator

     As with most calculators, the factory setting of an HP lOBii is
     for two places after the decimal. To change this to nine - the
     maximum permanent display - press the following keys:

                                 9


     The display will then show nine places after the decimal. To
     change this to any other number from one to eight, reenter
     the key strokes, using the desired number of places.

     To display twelve numbers (with no decimal place) press the
     orange shift key and DISP. The display will temporarily show
     twelve digits.

     b. JavaScript Financial Calculator

            The JavaScript Calculator is set to display two deci-
     mal places for most items, although it calculates to ten places.
     The settings cannot be changed by the user.
Calculators        Types of Calculations           Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS           43




  Set the Display
  for the Correct
  Number of
  Decimal Places
Calculators         Types of Calculations               Definitions
44            FINANCIAL CALCULATIONS FOR LAWYERS




              C. CALCULATIONS
          As explained earlier, six types of calculations are
     fundamental to a lawyer’s practice:

                 1. Future Value of a Sum

                 2. Present Value of a Sum

                 3. Present Value of an Annuity

                 4. Future Value of an Annuity

                 5. Sinking Fund

                 6. Amortization

                Separate JavaScript Financial Calculators
     exist for each type of calculation.


                   1. Future Value of a Sum
            This calculation computes the future amount or value
     of a current deposit.

             For example, $1,000 deposited today, earning 10%
     interest compounded annually, will increase to $1,100 in one
     year. In two years it will increase to $1,210. In five years, it
     will be $1,610.51 and in 100 years it will be $13,780,612.34.

            As shown in Example 8, to calculate this, input the
     five known factors into the calculator and solve for the un-
Calculators         Types of Calculations                    Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                          45




     EXAMPLE 8:(HP 10Bii)
     Future Value

     Compute the Future Value of $1,000 depos-
     ited today, earning 10% nominal annual inter-
     est compounded annually. Press the illustrated keys.

                                    [Remember to clear the machine!]


     1                               [Remember to set this. The factory
                                     setting of 12 P/YR will produce an
                                     error.]
     10


     1


     1000

                           [You may leave the PMT amount blank: the
     0                     calculator will assume zero.]

                           The display will read -1,100.
                            [This solves for the Future Value.]
Calculators           Types of Calculations              Definitions
46             FINANCIAL CALCULATIONS FOR LAWYERS




     EXAMPLE 8a:(HP 10Bii)
     Future Value
     Alternate Values

     Without clearing the calculator, you may
     change any or all of the variables used in Example 8 to com-
     pute alternative scenarios.

     Change the N to 2 and re-press FV to determine the value
     in two years. Then change N to 5 and again press FV. Do
     the same with an N of 100.

     2                           The display will read
                                 -1,210.

     5                           The display will read
                                 -1,610.51.

     100                         The display will read
                                 -13,780,612.34.

     Then Change the interest rate to 12% and 15%, alternatively.

     12                          The display will read
                                 -83,522,265.73.

     15                          The display will read
                                 -1,174,313,450.



         known sixth factor: the Future Value.

               a. HP lOBii Calculator

                Set the Present Value (PV) as 1,000.00. Set the In-
Calculators         Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  47




                  DID YOU NOTICE?
    In Example 8a, increasing the interest rate from 10% to
     12% caused the Future Value in 100 years to increase
                      more than 6 times!

     Increasing the interest rate to 15% caused the original
          Future Value to increase more than 85 times!




  In Example 8, if you enter the Present Value
  as negative number, the Future Value solution
  will be positive number.


  Remember, the PV and FV must have oposite signs on an
  HP calculator; however, it makes no difference which is posi-
  tive and which is negative. The JavaScript Caculator elimi-
  nates this requirement.
Calculators               Types of Calculations         Definitions
48                FINANCIAL CALCULATIONS FOR LAWYERS


           Future Value of a Sum Calculator


  EXAMPLE 8
  (JavaScript Calculator)
  Future Value
     PV Annuity                                        Interest Conversion
   FV Annuity
                   Future Value of a Sum Calculator        PV Sum
  Sinking Fund               Instructions ON OFF        Amortization



Mode
                              Begin      End
Present Value                         1,000.00

Future Value                          1,100.00

Nominal Interest Rate                    10.00

Effective Interest Rate      10.000000000
                                          1.00
Number of Years
                                         1.00
Payments Per Year
                                         1.00
Number of Payments
                                          0.00
Payment

                                 clear all
Calculators         Types of Calculations                Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 49




                         TIP
    You may enter the various values in any order.
   For example, you could enter the Nominal Inter-
    est Rate prior to entering the Present Value .

    Also, you may change any of the values in white
    boxes. Doing so will automatically cause the re-
             calculation of the other values.




                         TIP
    The JavaScript Calculators always indicate the
    number of payments per year setting. This will
   help prevent errors that may result from failing to
               set this value correctly.
Calculators                Types of Calculations                    Definitions
50             FINANCIAL CALCULATIONS FOR LAWYERS




     EXAMPLE 8b:(HP 10Bii)
         Alternate Future Values

      Solving For an Interest Rate

     As demonstrated in Example 8a, you may change any or all
     of the variables used in Example 8 to compute alternative
     scenarios.

     As shown in 8a change the N to 5, and re-press FV to
     determine the value in five years. Then change the FV to -
     1,750.00, representing a larger desire future amount. The
     solution will be the necessary nominal annual interest rate.

     5                                The display will read
                       -              1,610.51.

     1750                             [The +/- key enters the value as a
                                      negative. If you forget to do this, the
                                      display will read No SoLution.]
                                      The display will read 11.843.


     Then Change the Future Value to -2000.

     2000


                                       The display will read
                                       14.870.
     [In each of the alternatives, the amount computed is the nominal annual
     interest rate, compounded annually, needed to reach the stated Future
     Value when the Present Value is 1,000: 11.843% and 14.870 %.]
Calculators         Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                51




terest (I/YR) rate per year as 10. Set the Number of Periods
per year (P/YR) as 1. Set the Payment (PMT) amount as 0.
Set the Number of Periods (N) as 1. The order you input
these is irrelevant.

      Finally, solve for the Future Value (FV) by pressing
the FV key. As illustrated in Example 7, the answer will
appear as (1,100.00), the negative indicating a withdrawal.

      b. JavaScript Financial Calculator

       The JavaScript Calcula-
tor operates much the same way
as the HP10Bii. Press the FV
of a Sum button to open the ap-        FV Sum Calculator
                                       FV Sum Calculator
propriate calculator.

       You should initially press the Clear All button; how-
ever, this is not essential. Enter the various values in the
white boxes. Do not change amounts in yellow boxes. The
answer will automatically appear in the green FV box. It will
appear as a positive number. This part of the calculator will
not accept negative numbers.The Payment box will not ac-
cept an amount: it must be zero.

Placing the cursor over any of the functions will open a pop-

                                TIP
           You may enter the various values in any order.

            For example, you could enter the FV prior to
                         entering the N.
Calculators             Types of Calculations              Definitions
52               FINANCIAL CALCULATIONS FOR LAWYERS




              When Would You Want to Compute the
                    Future Value of a Sum?

     √   If you deposit money into an account, you can compute
     what it will worth in the future.

     √     If population growth rates continue at a constant rate, you
     can compute the population of an area after a given length of
     time.

     √    If your client was owed a specific amount as of a prior
     date, you can compute what he is owed today. The amount
     owed originally would be the present value and the amount
     today would be the future value. The intervening period would
     be the N.

     √    If a budget item (PV) increases at a particular rate (I/YR),
     you can compute the amount for a future period.



         up box with additional instructions or explanations.

               Example 8a illustrates how you can change the vari-
         ous functions on an HP10Bii to compute alternative sce-
         narios. You may do so without first clearing the registers.
         You must, however, press the FV key again to display the
         new answer.

                The JavaScript Calculator works similarly: change
         the value of one function and the calculator immediately will
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  53




change the computed answer. You need not repress the FV
function key.

       a.     Solving for an interest rate

       The six variables [PV, FV, I/YR, P/YR, N, and PMT]
are a function of each other: change one and the others
change, as well The calculator will solve for one change at a
time. Often the desired alternative involves a changed inter-
est rate [I/YR].

        In Example 8, the unknown factor was the Future
Value. Once the calculator solved for the Future Value, you
might then want to know how the interest rate would change
if the Future Value were different. This might occur if you
solved initially for a Future Value, as in Example 8b, result-
ing in an answer of $1,750 in five year at 11.834% effective
interest. You might then be curious regarding the amount of
interest you would need to earn if, instead, you were to ac-
cumulated $2,000 in five years. The answer, as shown, is
14.870% effective interest.

       In other words, you would know the Present Value,
the Future Value, and the Number of Periods, Payments,
and Payments Per Period. You could then solve for the Inter-
est Rate.

              1. HP lOBii Calculator

       Example 8b illustrates the process of solving for an
interest rate using an HP lOBii Calculator. It uses the
factors originally entered in Example 8. It then changes the
Future Value from the original solution to alternative amounts.
Because the Example 8 Present Value of $1,000 was en-
Calculators        Types of Calculations           Definitions
54            FINANCIAL CALCULATIONS FOR LAWYERS




      Set the Interest Payment and
              Compounding
            Periods the Same.
Calculators         Types of Calculations                      Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                            55




         EXAMPLE 8c:(HP 10Bii)
               Future Values
          Compounded Interest Rate

     As demonstrated in Example 8a, you may
     change any or all of the variables used in Example 8 to com-
     pute alternative scenarios.

     Suppose, instead of the facts given for Example 8, you
     were told the interest rate was 10% nominal, compounded
     semi-annually. Enter the values as shown in Example 8,
     with the following changes:
                     [This sets the number of periods at 2, which is the
     2               number of semi-annual period inone year..

                                 [This sets the compounding period as six
     2                           months: twice per year.]


                               The display will read -1,102.50.


                                 The display will read 10.25.
                                   [This is the Effective Interest Rate.

     Then Change the N to 12 and the P/YR to 12.

     12               12


                              The display will read -1.104.713.


                                 The display will read 10.471.
                                 [This is the Effective Interest Rate.
Calculators               Types of Calculations         Definitions
56                FINANCIAL CALCULATIONS FOR LAWYERS


          Future Value of a Sum Calculator


  EXAMPLE 8c
  (JavaScript Calculator)
  Future Value
     PV Annuity                                        Interest Conversion
     FV Annuity
                   Future Value of a Sum Calculator        PV Sum
  Sinking Fund               Instructions ON OFF        Amortization



Mode
                              Begin      End
Present Value                         1,000.00

Future Value                          1,102.50

Nominal Interest Rate                    10.00

Effective Interest Rate      10.2500000000
                                          1.00
Number of Years
                                         2.00
Payments Per Year
                                         2.00
Number of Payments
                                          0.00
Payment

                                 clear all


                          Did you notice?
      On the JavaScript Calculator: changing the 1 Payment Per
         Year to “2” and later to “12” is all you need do. The
       HP10Bii requires pressing 16 buttons to accomplish the
                          same calculations.
Calculators               Types of Calculations       Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                   57


         Future Value of a Sum Calculator


  EXAMPLE 8c
  (JavaScript Calculator)
  Future Value
   PV Annuity                                         Interest Conversion
   FV Annuity
                  Future Value of a Sum Calculator        PV Sum
  Sinking Fund                Instructions ON OFF      Amortization



Mode
                              Begin      End
Present Value                         1,000.00

Future Value                          1,104.71

Nominal Interest Rate                    10.00

Effective Interest Rate      10.4713067441
                                          1.00
Number of Years
                                        12.00
Payments Per Year
                                        12.00
Number of Payments
                                          0.00
Payment

                                 clear all
Calculators        Types of Calculations               Definitions
58            FINANCIAL CALCULATIONS FOR LAWYERS




     tered as a positive number, the alternative Future Values must
     be negative numbers.

                   2. JavaScript Financial Calculator

           The JavaScript Financial calculator is not currently
     designed to solve for an unknown interest rate.

           b. Compounded Interest Rates

             One of the cardinal rules of financial calculations is
     that the interest rate period and the payment period must be
     the same. For example, if payments - either of principal or
     interest - occur semiannually, then the interest rate must also
     be stated as a semiannual rate. Or, the payments occur quar-
     terly, the rate must also be expressed as a quarterly rate.
     This is also true if the interest merely compounds more fre-
     quently than once per year, but is paid at maturity: the inter-
     est rate and the P/YR must be consistent.

             Example 8c illustrates the use of compounded inter-
     est in a Future Value problem. It assumes an interest rate of
     10% nominal annual interest compounded semiannually, which
     is the equivalent of 10.25% nominal annual interest com-
     pounded annually. This is also the effective rate of interest.




                      Law of Economics
                       As Interest Rates
                       Increase, Future
                       Values Increase.
Calculators         Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                59




       Notice that the calculated Future Value is higher than
that calculated in Example 8: this occurred because the
interest rate also increased. Hence a Law of Economics: As
Interest Rates Increase, Future Values Increase.

              c. Converting an interest rate

       A rule that bears repeating is this: the interest rate
period and the payment period must be the same. This is
required by the formula and is true of all calculators.

       As explained in a prior section, you can work prob-
lems involving compound interest in two ways:

      1. The Calculator Method

      2. The Periodic Method.

       The Calculator Method sets the P/YR at the number of
payments per year. It sets the I/YR at the nominal annual
interest rate to be compounded at a rate consistent with the



                 The Calculator
                 Method is
                 easy to use;
                 however, it
                 requires some exact-
                 ness on the HP10Bii:
                 you must use both the
                 EFF% and NOM%
Calculators         Types of Calculations                 Definitions
60            FINANCIAL CALCULATIONS FOR LAWYERS




     P/YR. It sets the N at the total number of periods (not the
     number of years).

            The Periodic Method sets the P/YR at one. It sets the
     I/YR at the periodic interest rate. It also sets the N at the total
     number of periods.


                  The Periodic
                  Method is
                  consistent
                  with the
                  undelying math formula.
                  Some users find it
                  easier than the Calcula-
                  tor Method. Most do
                  not.




            Both methods require the use of the nominal annual
     interest (NAI) rate. The Calculator method uses the actual
     NAI as the I/YR. The Periodic Method requires the user to
     divide the NAI by the number of periods in one year and
     then to enter the result as the I/YR.

              A common problem involves the misstatement of the
     interest rate. Because both methods use the NAI, the inter-
     est rate must be stated in that format or converted to it. Luck-
     ily, this is not really much of a problem because the typical
     calculator can easily convert an interest rate to an equivalent
     rate for another period, including a NAI. Many calculators
Calculators          Types of Calculations                       Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                              61




  EXAMPLE 9: (HP 10Bii)
  Periodic Method


  1000
                                   [This renders the P/YR button su-
                                   perfluous: it becomes a periods per
  1                                period button.]



  2


  4.8808848
                   [This function is not necessary: the calculator will
                   assume 0 if nothing is entered.]
  0

                  The display will read -1,099.99999964.
                   [This rounds to -1,100, which is the original value plus
                   10% interest, proving that 4.88% semi-annual interset
                   is the equivalent of 10% annual.]




do this automatically; or, you can do it manually.

       Pages 23 to 34 explain how to convert an interest rate
into an equivalent rate. The simplest method is to use the
JavaScript Interest Conversion Calculator, which automati-
cally converts among nominal annual rates, effective rates,
and equivalent periodic rates. Pages 132 to 170 define the
various terms involving interest rates.
Calculators         Types of Calculations                      Definitions
62            FINANCIAL CALCULATIONS FOR LAWYERS




     EXAMPLE 9a: (HP 10Bii)
     Periodic Method


     1000

                                [This renders the P/YR button su-
     1                          perfluous: it becomes a periods per
                                period button.]

                    [This is 2 because that is the number of six month
     2              periods in one year].

                                [You must multiple 1000 times 1.1 (thus
                                adding the 10% interest). Do this in your
     1,100
                                head or by using the calculator.]

                    The display will read 4.880884817.
                    [Multiply the displayed amount by 2 to produce
                    9.761769634% nominal annual intereest compounded
                    semi-annually.]




                                                  Did You Notice?

     Set the Cash
                                                In the Periodic
     Flows Correctly:                           Method the I/YR key
     PV and FV must                             functions as a peri-
     have opposite                              odic interest key with
                                                the PIYR key being
     signs - one positive and                   set to one and thus
     one negative.                              rendered superfluous.
Calculators          Types of Calculations                      Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                             63




  EXAMPLE 9b:(HP 10Bii)
   Calculator Method
  Interest Rate Conversion

  Convert 10% effecitve interest to the equivalent nominal annual
  interest compounded semi-annually.

                             [This is 2 because that is the number of
  2                          six month periods in one year].



  10

                           The display will read
                           9.761669634.
                            [Divide the displayed amount by 2 to produce
                            4.880884817% periodic interest semi-annu-
                            ally. This division function is not necessary:
                            the calculator does it automatically.]
  Then proceed normally as in Example 9c. Do not clear
  the calculator!




                                TIP
       Most current users prefer the Calculator Method. The
        Periodic Method is consistent with the math formula
       and thus may be more intuitive to those users accus-
         tomed to working without electronic calculators.
Calculators               Types of Calculations         Definitions
64                FINANCIAL CALCULATIONS FOR LAWYERS


          Future Value of a Sum Calculator


  EXAMPLE 9
  (JavaScript Calculator)
  Future Value
     PV Annuity                                        Interest Conversion
   FV Annuity
                   Future Value of a Sum Calculator        PV Sum
  Sinking Fund               Instructions ON OFF        Amortization



Mode
                              Begin      End
Present Value                         1,000.00

Future Value                          1,100.00

Nominal Interest Rate            4.8808858

Effective Interest Rate       4.880885800
                                          2.00
Number of Years
                                         1.00
Payments Per Year
                                         2.00
Number of Payments
                                          0.00
Payment

                                 clear all


                          Did you notice?
      The JavaScript Calculator required the entry of 4 numbers.
       The HP10Bii requires pressing 12 buttons to accomplish
                       the same calculation.
Calculators               Types of Calculations                     Definitions
                FINANCIAL CALCULATIONS FOR LAWYERS                             65


                  Interest Rate Converter


  EXAMPLE 9b
  (JavaScript Calculator)
  Future Value
  PV Annuity                                                          PV Sum
   FV Annuity      Interest Rate Conversion                         Amortization
 Sinking Fund                 Instructions ON OFF                     FV Sum


   Convert Effective          Convert Nominal          Convert Periodic
    Rate to Nominal            Rate to Effective        Rate to Nominal
   Rate and Periodic          Rate and Periodic        Rate and Effective
          Rate                       Rate                     Rate




                             9.7617696340           Press this
Nominal Interest Rate
                                                    button first.
Periodic Interest Rate       4.8808848170

Effective Interest Rate
                             10.000000000

Payments Per Year                       2.00
                                                               Enter these
                                clear all                       numbers.




            Next, copy the nominal rate and enter it into the
            appropriate white box on the Future Value of A
                            Sum Calculator.
Calculators            Types of Calculations             Definitions
66              FINANCIAL CALCULATIONS FOR LAWYERS




     EXAMPLE 9c:(HP 10Bii)
     Calculator Method

     After converting the 10% effective
     rate to the equivalent nominal rate, press the
     following keys. Do not use the C ALL function.

     1000


     2


                     The display will read -1,100.




                 As explained earlier in Example 4, 4.8808848% in-
         terest, paid semiannually is the equivalent of 10% interest,
         paid annually. This is true because the interest paid during
         the first six month period will itself earn interest of 4.88%
         during the second six month period. Stated precisely,
         9.761769634% nominal annual interest compounded semi-
         annually is an effective interest rate of 10%.

                To prove this, set PV as 1,000.00, I/YR as 4.8808848,
         P/YR as 1, PMT as 0, and N as 2. Press FV. The answer
         is (1,099.99999964), which is the equivalent of (1,100.00).
         Hence, $1,000 earning 10% interest annually produces the
         same result as $1,000 earnings 4.88% semiannually.
Calculators           Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   67




       Example 9 thus demonstrates that the I/YR key can
function as an interest per period key and the P/YR key can
be a number of periods per period key, rather than number
of periods per year. Use of the keys in this manner is con-
sistent with the mathematical formula for financial calcula-
tions, which requires a periodic rate. This is the Periodic
Method.

       Another version of Example 9 converts a known 10%
effective interest rate into a NAI rate compounded semian-
nually. Example 9a illustrates this conversion. This method
works the same for the HP10Bii Calculator and the
JavaScript Calculator.

       The HP10Bii provides an alternate - and simpler -
method utilizing the calculator’s built-in conversion function.
Enter 10.00 as the effective interest rate using the EFF%
function, then solve for the nominal interest rate, using the
NOM% key. Dividing the result by 2 would produce the semi-
annual periodic rate. Examples 9b and 9c illustrate this
method. This is the Calculator Method.

        The HP10Bii calculator requires the user to convert
an effective interest rate to a nominal rate, either manually
(as illustrated in Example 9) or by using the calculator (as
illustrated in Example 9b). In other words, you may not
merely enter the effective interest rate (EFF%) and then pro-
ceed: you must actually convert the rate, as illustrated above.
The HP10Bii calculator will accept the entry of a nominal
rate either by use of the I/YR button or the NOM% button;
however, it will not accept entry of an effective rate by use of
the EFF% button unless the user then presses the NOM%
button.
Calculators        Types of Calculations              Definitions
68            FINANCIAL CALCULATIONS FOR LAWYERS




                   2. Present Value of a Sum
           This calculation computes the present value of a fu-
     ture amount. For example, $1,100 in one year, discounted at
     10% interest compounded annually has a present value of
     $1,000.00 today. Thus, if you owe $1,100 one year from
     now, you should be able to pay off the obligation with only
     $1000, assuming the appropriate interest rate is 10% nomi-
     nal annual interest compounded annually.

            In comparison, $1,100 in two years has a present value
     of $909.09. Discounting $1,100 for five years produces a
     present value of $683.01. Discounting it for 100 years at
     10% produces a present value of $0.0798 - just under eight
     cents! Thus if you owed $1,100 one hundred years from
     now, you should be able to satisfy the debt with merely eight
     cents (assuming the constant 10% nominal annual interest).

            As shown in Example 10, to calculate the present
     value of a sum, input the five known factors into the calcula-
     tor and solve for the unknown sixth factor, the present value.

           a. HP lOBii Calculator

            First, set the Future Value (FV) as 1,100.00. Set the
     Interest (I/YR) rate per year as 10. Set the Number of Peri-
     ods per year (P/YR) as 1. Set the Payment (PMT) amount
     as 0. Set the Number of Periods (N) as 1. The order you
     input these is irrelevant.

            Finally, solve for the Present Value (PV) by pressing
     the PV key. As illustrated in Example 10, the answer will
     appear as (1,000.00), the negative indicating a current de-
     posit.
Calculators         Types of Calculations                   Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                          69




     EXAMPLE 10:(HP 10Bii)
      Present Value

     Compute the Present Value of $1,100 owed
     one year from now, at 10% nominal annual in-
     terest compounded annually. Press the illustrated keys.

                                     [Remember to clear the machine!]


     1                               [Remember to set this. The factory
                                     setting of 12 P/YR will produce an
                                     error.]
     10


     1


     1100

                            [You may leave the PMT amount blank: the
     0
                            calculator will assume zero.]
                           The display will read -1,000.
Calculators         Types of Calculations               Definitions
70            FINANCIAL CALCULATIONS FOR LAWYERS




     EXAMPLE 10a:(HP 10Bii)
     Present Value
     Alternate Values

     Without clearing the calculator, you may
     change any or all of the variables used in Example 10 to
     compute alternative scenarios.

     Change the N to 2 and re-press FV to determine the present
     value of the amount two years hence. Then change N to 5
     and again press FV. Do the same with an N of 100.

     2                          The display will read
                                -909.090909091.

     5                          The display will read
                                -683.013455365.

     100                        The display will read
                                -0.079822287.

     Then Change the interest rate to 12% and 15%, alternatively.

     12                         The display will read
                                -0.013170141.

     15                         The display will read
                                -0.000936718.
Calculators        Types of Calculations           Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS              71




                  DID YOU NOTICE?
   In Example 10a, increasing the interest rate from 10% to
    12% caused the Present Value 100 years earlier to de-
                 crease more than 6 times!

    Increasing the interest rate to 15% produces a Present
    Value of less than 1/100th of one cent! In other words,
   a penny invested at 15% will produce more than $11,000
                         in 100 years.
Calculators             Types of Calculations                    Definitions
72               FINANCIAL CALCULATIONS FOR LAWYERS




        EXAMPLE 10b:(HP 10Bii)
                  Present Value
            Solving For Interest Rate
        Compute the interest rate inherent to a Present
        Value of $9,000, a Future Value of $10,000,
        and a period of two years. Press the illustrated keys.
                                       [This can be set at any number; how-
        1                              ever, the resulting rate must then be
                                       interpreted correctly, as shown in
                                       Example 10d.]
        2


        10,000


        9,000

                            The display will read 5.409255339.

        Thus the Seller offers a discount of 5.41% nominal an-
        nual interest compounded annually for advance pay-
        ments.




                 Which Interest Rate is Higher?

      The two interest rates are not comparable without conversion to a
                        common compounding period.

     The 10b rate of 5.41% is an effective rate of 5.41% (compounded
     annually, the effective and nominal rates are the same). The 10c
     rate of 5.35% is an effective rate of 5.49% (see Example 10d).

            Thus the 10c rate is better for the customer!
Calculators          Types of Calculations                    Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                            73




       EXAMPLE 10c:HP10Bii)
             Present Value
       Solving For Interest Rate
     Compute the interest rate inherent to a Present
     Value of $26,250, a Future Value of $30,000,
     and a period of thirty years.
                                    [This can be set at any number; how-
     12                             ever, the resulting rate must then be
                                    interpreted correctly, as shown in
     30                             Example 10d.]



     30,000


     26,250

                         The display will read 5.353160450.

     Thus the Seller offers a discount of 5.35% nominal an-
     nual interest compounded monthly for advance pay-
     ments.




                         Did You Notice?

                The 5.35% rate is larger
                 than the 5.41% rate!
                  The two interest rates are not comparable
                without conversion to a common compounding
                                    period.
Calculators        Types of Calculations           Definitions
74            FINANCIAL CALCULATIONS FOR LAWYERS




              Laws of Economics

        1. As Interest Rates Decrease,
        Present Values Increase.

        2. As Interest Rates Increase,
        Present Values Decrease.




                              Did You Notice?
                         The above Laws of Economics are
                         consistent with common sense:

                         1. The lower the discount for
                         paying in advance, the more you
                         must pay.

                         2. The bigger the discount for
                         paying in advance, the less you
                         must pay.
Calculators         Types of Calculations             Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   75




          When Would You Want to Compute the
                Present Value of a Sum?

  √   If you owe money in the future, you can compute what it
  equals in current value.

  √    If you want a discount for an advance payment for goods
  or services, you would compute the present value of the future
  obligation.

  √    If you know the amount you need at a future time - such
  as for retirement or college entrance - you can compute the
  present value needed to produce that future amount.




                             TIP

                If a service provider will
               accept less money for an
              advance payment, you can
                       calculate the
               discount rate being used.
Calculators         Types of Calculations                   Definitions
76            FINANCIAL CALCULATIONS FOR LAWYERS




          EXAMPLE 10d:(HP10Bii)
               Present Value
          Comparing Interest Rates

     The 10c display read 5.353160450. Convert
     it to an Effective Rate so that it can be compared to other
     Examples.

                           The display will read 5.486474725.


     Thus the 5.35% nominal annual interest compounded monthly
     is an effective rate of 5.49%.

     Comparing the 5.35% rate to the Example 10b 5.41% rate
     is like comparing apples to oranges: it makes no sense. The
     two rates must be converted to the same compounding pe-
     riod to be comparable.

     Instead, the 10b rate can be converted to a nominal annual
     rate compounded monthly. The 10b display reads
     5.409255339. To Convert it to a monthly rate press:

     12


     24

                   The disiplay will read 5.279606096.
                     [Thus 5.41% compounded annually is equivalent to
                     5.28% compounded monthly. Both are an effective rate
                     of 5.41%]
Calculators         Types of Calculations             Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                77




               Caution:

       When comparing
        interest rates,
   always translate them to
   comparable compounding
            periods.




      b. JavaScript Financial Calculator

      The JavaScript Calculator operates much the same
way as the HP10Bii. Press the PV of a Sum button to open
the appropriate calculator.

         The Payment box will not
accept an amount. You should               Sum Calculator
                                       PV Sum Calculator
initially press the Clear All but-
ton; however, this is not essen-
tial. Enter the various values in
the white boxes. Do not change amounts in yellow boxes.The
answer will automatically appear in the green PV box. It will
appear as a positive number. This part of the calculator will
not accept negative numbers.The Payment box will not ac-
cept an amount.

      Placing the cursor over any of the functions will open
a pop-up box with additional instructions or explanations.
Calculators               Types of Calculations         Definitions
78                FINANCIAL CALCULATIONS FOR LAWYERS




  EXAMPLE 10
  (JavaScript Calculator)
  Present Value
     PV Annuity                                         Interest Conversion
                  Present Value of a Sum Calculator      Amortization
   FV Annuity
  Sinking Fund                Instructions ON OFF           FV Sum



Mode
                              Begin      End
Present Value                      1,000.00
Future Value                          1,100.00

Nominal Interest Rate                   10.00
                             10.000000000
Effective Interest Rate
                                          1.00
Number of Years
                                         1.00
Payments Per Year
                                          1.00
Number of Payments
                                          0.00
Payment

                                 clear all


                          Did you notice?
         Unlike the HP10Bii, the JavaScript Calculator does not
                 require the entry of negative numbers.
Calculators               Types of Calculations            Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                        79




  EXAMPLE 10
  (JavaScript Calculator)
  Present Value Alternate Values
  PV Annuity                                               Interest Conversion
                 Present Value of a Sum Calculator          Amortization
   FV Annuity
  Sinking Fund                Instructions ON OFF              FV Sum



Mode
                              Begin      End
Present Value                            0.00       0.00093671753426

Future Value                          1,100.00
                                        15.00
Nominal Interest Rate
                             15.000000000
Effective Interest Rate
                                       100.00
Number of Years
                                         1.00
Payments Per Year
                                         1.00
Number of Payments
                                         0.00
Payment

                                 clear all


    The Present Value is set to
    show only two decimal places.
    But, if you click on the green
    box, it will show 14 decimal
    places.
Calculators            Types of Calculations               Definitions
80              FINANCIAL CALCULATIONS FOR LAWYERS




                a.    Solving for an interest rate

                 Remember: the six variables [PV, FV, I/YR, P/YR, N,
         and PMT] are a function of each other: change one and the
         others change, as well The calculator will solve for one change
         at a time. Often the desired alternative involves a changed
         interest rate [I/YR].

                In Example 10, the unknown factor was the Present
         Value. Once the calculator solved for the Present Value, you
         might then want to know how the interest rate would change
         if the Present Value were different. In other words, you
         would know the Present Value, the Future Value, and the
         Number of Periods, Payments, and Payments Per Period.
         You could then solve for the Interest Rate.

                Why would you want to know this? Perhaps you need
         to pay $10,000 two years from now and you know that can
         pay $9000 today to satisfy the obligation. You could solve
         for the interest rate to determine what rate the seller/service
         provider is using. Or, you may represent the seller/service
         provider. You might know that he is willing to accept $26,250
         today for a future amount due of $30,000 in thirty months.


     In Example 10, if you enter the Future Value
     as negative 1,100, the Present Value solution
     will be positive 1,000.


     Remember, the PV and FV must have oposite signs on an
     HP calculator; however, it makes no difference which is posi-
     tive and which is negative.
Calculators          Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 81




Examples 10b and 10c illustrates how you would deter-
mine the correct discount interest rate in each case.

              1. HP lOBii Calculator

        Example 10b and 10c illustrate the process of solv-
ing for an interest rate. In 10b, the Present Value is entered
as a positive number; thus, the Future Value must be entered
as a negative. Similarly, in 10c, the Present Value is en-
tered as a negative number; thus, the Future Value must be
entered as a positive.

       Example 10d emphasizes the need to translate inter-
est rates to a common compounding period for purposes of
comparison. Without such a translation, comparison makes
no sense.

       As shown in the examples, 5.35% is greater than
5.41%, but only because the two are compounded differ-
ently. If one party to a transaction understands this point,
but the other does not, the one who understands can easily
manipulate the other. Failure to understand the effect of com-
pounding will result in the making of wrong choices.

              2. JavaScript Financial Calculator

      The JavaScript Financial Calculators are not currently
designed to solve for an unknown interest rate.
Calculators         Types of Calculations           Definitions
82            FINANCIAL CALCULATIONS FOR LAWYERS




     3. Present Value of an Annuity
              a.   End Mode - An Annuity in Arrears

             This calculation computes the present value of a se-
     ries of equal payments made at the end of regular intervals,
     earning a constant interest rate. For example, $1,000 de-
     posited at the end of each year for ten years, earning 10%
     interest compounded annually, has a present value today of
     $6,144.57. Similarly, $6,144.57 deposited today, earning 10%
     interest compounded annu-
     ally will produce a fund
     from which $1000 could be
     withdrawn for ten consecu-             Caution:
     tive years, beginning one
     year from today.                  To compute an an-
                                     nuity in arrears, the
            This might be used
     to compute the payoff            calculator must be
     amount for a loan or to             in End Mode.
     value lottery winnings.
                                     The display will not
              1. HP lOBii Calcu-      indicate end mode.
     lator                           Thus press the shift
                                     and mode keys only
            As illustrated in Ex-
                                      if the display indi-
     ample 11, input the five
     known factors into the HP        cates Begin Mode.
     lOBii calculator and solve
     for the unknown sixth fac-
     tor, the present value. First, set the Payment (PMT) as
     1,000.00. Set the Interest (I/YR) rate per period as 10. Set
     the Number of Periods per year (P/YR) as 1. Set the Num-
     ber of Periods (N) as 10. Set the Future Value (FV) as 0.
Calculators        Types of Calculations                     Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                            83




     EXAMPLE 11: (HP10Bii)
     Present Value of an Annuity in
     Arrears

     Compute the Present Value of $1,000 to be received annu-
     ally, beginning one year from now, at 10% nominal annual
     interest compounded annually. Press the illustrated keys.

                                 [Remember to clear the machine!]


     1                              [Remember to set this. The factory
                                    setting of 12 P/YR will produce an
                                    error.]
     10


     10


     1000


     0                      [You may leave the FV amount blank: the
                            calculator will assume zero.]

                                 [Enter this only if the calculator is in
                                 Begin Mode.]


                                6144.56710570
Calculators         Types of Calculations               Definitions
84            FINANCIAL CALCULATIONS FOR LAWYERS




           You must also remember to set the calculator in End
     Mode, if it is not already in End Mode. Do so by pressing
     the shift key and the BEG/END key. End Mode tells the
     calculator that the first payment will be made one year from
     today and each successive payment will occur at the end of
     each following period. In contrast, in Begin Mode each pay-
     ment occurs at the beginning of each period. If the display
     reads Begin, the calculator is in Begin Mode. If, instead, it
     does not have any words, it is in End Mode and you must not
     change this for this problem.

             Solve for the Present Value (PV) by pressing the PV
     key. The answer will appear as (6,144.56710570), the nega-
     tive indicating a required deposit necessary to generate the
     level annuity. The Future Value is zero because at the end of
     ten years, no money would remain in the account.

                2. JavaScript Financial Calculator

            The JavaScript Calculator operates much the same
     way as the HP10Bii. Press the PV of an Annuity button to
     open the appropriate
     calculator.
                                   PV Annuity Calculator
                                    PV Annuity Calculator
             The FV box will
     not accept an amount.
     You should initially
     press the Clear All button; however, this is not essential. Enter
     the various values in the white boxes. Do not change amounts
     in yellow boxes.The answer will automatically appear in the
     green PV box. It will appear as a positive number. This part
     of the calculator will not accept negative numbers. You must
     be certain to press the End Mode button. If the calculator is
Calculators          Types of Calculations               Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                      85




           When Would You Want to Compute the
               Present Value of an Annuity?

           √    If you owe money at regular intervals in the fu-
           ture, you can compute what it equals in current value.
           This would tell the “pay-off” amount. This could be
  useful in a Family Law case to compute the value of a lump-
  sum award to pay-off an alimony obligation.

  √    If you want a discount for an advance payment for goods
  or services which will be provided at regular intervals, you would
  compute the present value of the future obligation. This might
  involve an insurance contract or rent or a contract for the pro-
  vision of utilities or some similar product.

  √    In a tort case, the victim may have lost future wages or
  suffer regular future medical expenses. The present value of
  such an amount would be the tort-feasor’s obligation.

  √     You might have won a state lottery. The present value of
  the future payments would be the alternative amount that might
  be elected.

  √    You might need to compute the value of a bond or similar
  financial instrument. The regular interest payments would be
  an annuity. The present value of them added to the present
  value of the final payment (the present value of a sum) would
  be the current value of the bond.
Calculators               Types of Calculations            Definitions
86               FINANCIAL CALCULATIONS FOR LAWYERS




EXAMPLE 11
(JavaScript Calculator)
Present Value
 of an Annuity in Arrears
  Amortization                                             Interest Conversion
                  Present Value of an Annuity Calculator
   FV Annuity                                                  PV Sum
  Sinking Fund                Instructions ON OFF              FV Sum



Mode
                              Begin     End
Present Value                         6,144.57

Future Value                             0.00

Nominal Interest Rate
                                        10.00
                             10.000000000
Effective Interest Rate
                                        10.00
Number of Years
                                         1.00
Payments Per Year
                                        10.00
Number of Payments
                                                 end
                                   1,000.00
Payment                                          mode

                                 clear all


                          Did you notice?
       The JavaScript Annuity Calculators always indicate the
      mode. This can help prevent resulting errors. Notice the
     substantial difference in present values between end mode
                           and begin mode.
Calculators               Types of Calculations            Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                        87




EXAMPLE 12
(JavaScript Calculator)
Present Value
 of an Annuity Due
  Amortization                                             Interest Conversion
                  Present Value of an Annuity Calculator
   FV Annuity                                                  PV Sum
  Sinking Fund                Instructions ON OFF              FV Sum



Mode
                              Begin      End
Present Value                         6,759.02

Future Value                             0.00

Nominal Interest Rate                    10.00
                             10.000000000
Effective Interest Rate
                                        10.00
Number of Years
                                         1.00
Payments Per Year
                                        10.00
Number of Payments
                                                 begin
                                   1,000.00
Payment                                          mode

                                 clear all



          Click on Begin Mode to con-
          vert Example 10 to Example
          11 and on End Mode to con-
          vert back.
Calculators         Types of Calculations           Definitions
88            FINANCIAL CALCULATIONS FOR LAWYERS




     in Begin Mode, you will obtain the wrong answer.

           Placing the cursor over any of the functions will open
     a pop-up box with additional instructions or explanations.

              b.   Begin Mode: An Annuity Due

             This calculation computes the present value of a se-
     ries of equal payments
     made at the begin-
     nings of regular inter-
     vals, earning a con-                Caution:
     stant interest rate. For
     example, $1,000 de-          To compute an annuity
     posited at the begin-       due, the calculator must
     ning of each year for
     ten years, earning
                                     be in Begin Mode.
     10% interest com-
     pounded annually, has       The HP10Bii display will
     a present value today        not indicate end mode.
     of $6,759.02. The             Thus press the shift
     amount exceeds that of       and mode keys only if
     the above calculation         the display does not
     because the first pay-        indicate Begin Mode.
     ment here is made to-
     day, whereas in the
     End Mode (Annuity in        The JavaScript Calcula-
     Arrears), the first pay-      tors do not have this
     ment is not made until              difficulty.
     one year from now.
     Similarly, $6,759.02
     deposited today, earn-
     ing 10% interest compounded annually will produce a fund
     from which $1000 could be withdrawn for ten consecutive
     years, beginning today.
Calculators        Types of Calculations                    Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                          89




     EXAMPLE 12: (HP10Bii)
     Present Value of an Annuity
     Due

     Compute the Present Value of $1,000 to be received annu-
     ally, beginning today, at 10% nominal annual interest com-
     pounded annually. Press the illustrated keys.

                                   [Remember to clear the machine!]



                                      [Remember to set this. The factory
     1                                setting of 12 P/YR will produce an
                                      error.]

     10


     10


     1000
                            [You may leave the FV amount blank: the cal-
     0                      culator will assume zero.]
                                    [Enter this only if the machine is in End
                                    Mode]


                            The display will read -6,759.02.
Calculators             Types of Calculations                 Definitions
90              FINANCIAL CALCULATIONS FOR LAWYERS




                1. HP lOBii Calculator

                As illustrated in Example 12, input the five known
         factors into the calculator and solve for the unknown sixth
         factor, the present value. First, set the calculator in Begin
         Mode. Then, set the Payment (PMT) as 1,000.00. Set the



                        Did You Notice?
     With annuities, the farther into the future the payment, the less
     significant the present value. Hence, the Example 12 ten-year
     annuity had a PV of $6,759 compared to a twenty-year annuity
     PV of $9,364: a large increase from 10 to 20 years. But, the fifty-
     year annuity has a value of $10,906, a much smaller increase,
     even though the period lenth more than doubled! And, the 100-
     year annuity has a PV of $10,999, an insigificant increase over a
     fifty year annuity! Compute the PV of a 1000-year annuity. It has
     a PV of $11,000: only 80cents more than a 100-year annuity. Thus
     the extra 900 years of payments are almost worthless in present
     value terms.



                     Laws of Economics

     1. As Interest Rates Decrease, Present Values In-
     crease.

     2. As Interest Rates Increase, Present Values De-
     crease.
Calculators         Types of Calculations                Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                     91




     EXAMPLE 12a:(HP 10Bii)
     Present Value of an Annuity
     Alternate Periods and Interest

     Without clearing the calculator, you may change any or all of
     the variables used in Example 12 to compute alternative
     scenarios.

     Change the N to 20 and re-press PV to determine the
     present value of the amount if paid for twenty years. Then
     change N to 50 and again press PV. Do the same with an N
     of 100.

     20                          The display will read
                                 -9,364.92009173.

     50                          The display will read
                                 -10,906.2959359.

     100                         The display will read
                                 -10,999.2017771.

     Then Change the interest rate to 12% and 15%, alternatively.

     12                          The display will read
                                 -9,333.22158668

     15                          The display will read
                                 -7,666.66013803.
Calculators               Types of Calculations            Definitions
92                FINANCIAL CALCULATIONS FOR LAWYERS




EXAMPLE 12a
(JavaScript Calculator)
Present Value
 of an Annuity Due Alternate Periods
and Interest
  Amortization                                             Interest Conversion
                  Present Value of an Annuity Calculator
     FV Annuity                                                PV Sum
  Sinking Fund                Instructions ON OFF              FV Sum

Mode                          Begin      End

Present Value
                                      7,666.66

Future Value
                                          0.00
                                         15.00
Nominal Interest Rate
                             15.000000000
Effective Interest Rate
                                       100.00
Number of Years
                                          1.00
Payments Per Year
                                        10.00
Number of Payments                               begin
                                   1,000.00
                                                 mode
Payment
                                 clear all
Calculators          Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                    93




Interest (I/YR) rate per period as 10. Set the Number of
Periods per year (P/YR) as 1. Set the Number of Periods
(N) as 10. Set the Future Value (FV) as 0. Solve for the
Present Value (PV) by pressing the PV key. The answer will
appear as (6,144.56710570), the negative indicating a re-
quired deposit necessary to generate the level annuity. The
Future Value is zero because at the end often years, no
money would remain in the account.

      2. JavaScript Financial Calculator

        The JavaScript Calculator operates much the same
way as the HP10Bii. Press
the PV of an Annuity button
to open the appropriate cal- PVPV an Annuity Calculator
                                      of Annuity Calculator
culator. The FV box will not
accept an amount. You
should initially press the Clear
All button; however, this is not essential.

     Fill in the various values in the white boxes. The answer
will automatically appear in the green PV box. The answer
will appear as a positive number. This part of the calculator
will not accept negative numbers.

    Also, you must be certain to press the Begin Mode
button. The calculator will show the words begin mode in         Begin
red letters when in this mode. If the calculator is in End
Mode, you will obtain the wrong answer. To help prevent er-
rors, the calculator will always indicate the mode. To com-
pute alternative mode answers, merely click on End Mode
and the answer will automatically appear. Placing the cursor
over any of the functions will open a pop-up box with addi-
tional instructions or explanations.
Calculators          Types of Calculations                Definitions
94            FINANCIAL CALCULATIONS FOR LAWYERS




     EXAMPLE 12b: (HP10Bii)
     Present Value of an Annuity
     Due: shift of mode

     After working Example 12 and without clearing the machine
     the display will read -6,759.02. You can shift to End mode
     (rather than Begin) to determine the alternative present value
     if the payments begin one year from now rather than today.


                                     The display will read
                                     -6,144.57.


     You must press PV to tell the machine to re-compute.
Calculators           Types of Calculations            Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS               95




4. Future Value of an Annuity
       a. End Mode: An Annuity in Arrears

       This calculation computes the future value of a series
of equal payments made at the end of regular intervals, earn-
ing a constant interest rate. For example, $1,000 deposited
at the end of each year for ten years earning 10% interest
compounded annually, has a fu-
ture value in ten years of
$15,937.42.
                                             Caution:
       This calculation is particu-
larly helpful in planning for retire-
                                         To compute an an-
ment or saving for a child’s edu-
cation.                                 nuity in arrears, the
                                         calculator must be
       1. HP lOBii Calculator               in End Mode.

       As illustrated in Example        The display will not
13, input the five known factors into   indicate end mode.
the calculator and solve for the       Thus press the shift
unknown sixth factor, the present
                                        and mode keys only
value. First, set the Payment
(PMT) as 1,000.00. Set the Inter-        if the display indi-
est (I/YR) rate per period as 10.        cates Begin Mode.
Set the Number of Periods per
year (P/YR) as 1. Set the Number
of Periods (N) as 10. Set the
Present Value (PV) as 0. Solve for the Future Value (FV) by
pressing the PV key. The answer will appear as
(15,937.4246010), the negative indicating withdrawal pos-
sible after the ten deposits. The Present Value is zero be-
cause at the beginning, no money has yet been deposited.
Calculators        Types of Calculations                     Definitions
96            FINANCIAL CALCULATIONS FOR LAWYERS




     EXAMPLE 13: (HP10Bii)
     Future Value of an Annuity in
     Arrears
     Compute the Future Value of $1,000 to be deposited annu-
     ally, beginning one year from now, at 10% nominal annual
     interest compounded annually. Press the illustrated keys.

                                 [Remember to clear the machine!]

                                   [Remember to set this. The factory
     1                             setting of 12 P/YR will produce an
                                   error.]

     10


     10


     1000

                            [You may leave the PV amount blank: the
     0
                            calculator will assume zero.]


                                [Enter this only if the calculator is in
                                Begin Mode.]


                            The display will read -15,937.42.
Calculators          Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                      97




           When Would You Want to Compute the
                Future Value of an Annuity?

  √    If you save money for a retirement plan at regular inter-
  vals, you can compute what it will be worth in the future.

  √     If you save money for a child’s education at regular inter-
  vals, you can compute what the fund will be worth in the future.




                            Caution:

         The future value of an annuity is stated
         in future dollars, which are not compa-
                 rable to current values.

           Thus the answer might not be useful;
           however, two methods can be used to
          convert the answer to a useful number:

            1. Convert the amount to a present
                           value.
         2. Modify the interest rate to reflect a
           “real rate of return” rather than the
                   actual predicted rate.
Calculators            Types of Calculations              Definitions
98              FINANCIAL CALCULATIONS FOR LAWYERS




               Do not forget to set the calculator in End Mode if you
         changed it for the prior example.

               2. JavaScript Financial Calculator

             The JavaScript Calculator operates much the same
                                       way as the HP10Bii. Press
                                       the FV of an Annuity button
 FVFV an Annuity Calculator to open the appropriate cal-
    of Annuity Calculator
                                       culator. The PV box will not
                                       accept an amount. You
                                       should initially press the Clear
     All button; however, this is not essential.

                Fill in the various values in the white boxes. Do not
         change the amounts in the yellow boxes. The answer will
         automatically appear in the green FV box. The answer will
         appear as a positive number. This part of the calculator will
         not accept negative numbers.

                You must be certain to press the End Mode button.
           The calculator will show the words begin mode in red let-
 End
           ters when in begin mode and the words end mode in black
           letters when in end mode. If the calculator is in Begin
 Begin
           Mode, you will obtain the wrong answer. To compute alter-
           native mode answers, merely click on Begin Mode and the
           answer will automatically appear.

               Placing the cursor over any of the functions will open
         a pop-up box with additional instructions or explanations.
Calculators               Types of Calculations           Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                       99




EXAMPLE 12
(JavaScript Calculator)
Future Value
 of an Annuity In Arears
  PV Annuity                                              Interest Conversion
                  Future Value of an Annuity Calculator
  Amortization                                                PV Sum
  Sinking Fund                Instructions ON OFF             FV Sum


Mode
                              Begin     End
Present Value                            0.00

Future Value                      15,937.42

Nominal Interest Rate
                                       10.00

Effective Interest Rate
                             10.000000000
                                       10.00
Number of Years
                                         1.00
Payments Per Year
                                       10.00
Number of Payments
                                                 end
                                      1000.00
Payment                                         mode

                                 clear all
Calculators         Types of Calculations              Definitions
100           FINANCIAL CALCULATIONS FOR LAWYERS




              b. Begin Mode: An Annuity Due
              This calculation computes the future value of a series
      of equal payments made at the beginnings of regular inter-
      vals, earning a constant interest rate. For example, $1,000
      deposited at the beginning of each year for ten years, earn-
      ing 10% interest compounded annually, has a future value in
      ten years of $15,937.42. The amount exceeds that of the
      prior calculation (an annuity in arrears) because the first
      payment here is made to-
      day and thus earns inter-
      est beginning today,
      whereas in the End Mode              Caution:
      (Annuity in Arrears), the
      first payment is not made       To compute and
      until one year from now.         annuity due, the
            1. HP lOBii Cal-         calculator must be in
      culator                             Begin Mode.

             As illustrated in       The display will not
      Example 14, input the           indicate end mode.
      five known factors into the
                                     Thus press the shift
      calculator and solve for
      the unknown sixth factor,      and mode keys only
      the present value. First,       if the display does
      set the calculator in Be-        not indicate Begin
      gin Mode. Then, set the                Mode.
      Payment (PMT) as
      1,000.00. Set the Interest
      (I/YR) rate per period as
      10. Set the Number of Periods per year (P/YR) as 1. Set the
      Number of Periods (N) as 10. Set the Present Value (PV) as
      0. Solve for the Future Value (FV) by pressing the FV key.
Calculators         Types of Calculations                     Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                            101




     EXAMPLE 14: (HP10Bii)
     Future Value of an Annuity Due

     Compute the Future Value of $1,000 to be deposited annu-
     ally, beginning today, at 10% nominal annual interest com-
     pounded annually. Press the illustrated keys.

                                 [Remember to clear the machine!]

                                    [Remember to set this. The factory
     1                              setting of 12 P/YR will produce an
                                    error.]

     10


     10


     1000


     0                       [You may leave the PV amount blank: the
                             calculator will assume zero.]


                                 [Enter this only if the calculator is in
                                 End Mode.]


                            The display will read -17,531.17.
Calculators            Types of Calculations             Definitions
102             FINANCIAL CALCULATIONS FOR LAWYERS




         The answer will appear as (17,531.1670611), the negative
         indicating a required deposit necessary to generate the level
         annuity. The Present Value is zero because at the begin-
         ning, the account contains no money: the purpose of the
         computation is to determine the necessary deposits.

               Do not forget to set the calculator in Begin Mode if
         you changed it for the prior example.

               2. JavaScript Financial Calculator

           The JavaScript Calculator operates much the same
     way as the HP10Bii. Press the FV of an Annuity button to
                                  open the appropriate calcu-
                                  lator. The PV box will not ac-
                                  cept an amount. You should
   of Annuity Calculator
 FVFV an Annuity Calculator
                                  initially press the Clear All
                                  button; however, this is not
                                  essential.

                Fill in the various values in the white boxes. Do not
         change the amounts in the yellow boxes. The answer will
         automatically appear in the green FV box. The answer will
         appear as a positive number. This part of the calculator will
         not accept negative numbers.

                 You must be certain to press the Begin Mode button.
 Begin
            The calculator will show the words begin mode in red let-
            ters when in begin mode. If the calculator is in End Mode,
           you will obtain the wrong answer. To compute alternative
 End
           mode answers, merely click on End Mode and the answer
           will automatically appear.      Placing the cursor over any
           of the functions will open a pop-up box with additional in-
         structions or explanations.
Calculators               Types of Calculations           Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                     103




EXAMPLE 12
(JavaScript Calculator)
Future Value
 of an Annuity Due
  PV Annuity                                              Interest Conversion
                  Future Value of an Annuity Calculator
  Amortization                                                PV Sum
  Sinking Fund                Instructions ON OFF             FV Sum


Mode
                              Begin     End
Present Value                            0.00

Future Value                      17,351.17

Nominal Interest Rate
                                       10.00

Effective Interest Rate
                             10.000000000
                                       10.00
Number of Years
                                         1.00
Payments Per Year
                                       10.00
Number of Payments
                                                begin
                                      1000.00
Payment                                         mode

                                 clear all
Calculators          Types of Calculations              Definitions
104           FINANCIAL CALCULATIONS FOR LAWYERS




             3. Converting the Annuity Future Value to
             a more useful number.

              The Future Value of An Annuity calculation provides
      an answer that may not be useful - and may actually be
      misleading. While this is true of any future value calculation,
      it is particularly true when an annuity is involved.

             For example, a simple Future Value of an Amount cal-
      culation presents an answer in terms of future dollars at an
      assumed interest rate. Example 8a illustrated that $1000
      today, invested at 10% effective interest, is the equivalent of
      $1,100 in one year, $1,210 in two years, $1,610.51 in five
      years, and $13,780,612.34 in 100 years. Those “future dol-
      lars” are not the same as present dollars because they nor-
      mally include inflation effects, as well as risk and liquidity
      components.

             Typical interest rates include three factors:

             1. inflation
             2. risk
             3. liquidity



                     Did you notice?
      People charge interest for three fundamental reasons:

   1. to compensate for inflation
   2. to compensate for risk
   3. to compensate for a lack of liquidity
Calculators           Types of Calculations                 Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                      105




        The inflation factor involves a prediction of the general
economy, while the risk factor involves a prediction of the
individual borrower’s reliability. The liquidity factor is more
fixed - typically at three to four percent.

        At one level, this observation states the obvious be-
cause that is the essence of a future value calculation. At
this level it is not normally misunderstood because the calcu-
                                     lation requires the user in-
                                     put the equivalent present
                                     value. Hence, anyone per-
            Caution:                 forming the calculation
                                     knows that $1,610.51 in
    What the user knows              five years is the equivalent
                                     of $1,000 today.
      to be true - that
    present and future                      When a future value
   values are equivalent            calculation involves an an-
    - is not really true.           nuity, however, the effects
                                    of inflation on the interest
                                    rate are less apparent.

                                             But, at another level,
                                     what the user knows to be
                                    true - that the present and
                                    future values are equivalent
                                    - is not really true. Putting
                                    aside the inherent uncer-
    Thus Future Values              tainly of future inflation and
    are often more use-             risk - and thus the chance
     ful when computed              of substantial errors due to
    without the inflation           bad predictions - the num-
       or risk factors.             bers have a built-in differ-
                                    ence: the liquidity element.
Calculators             Types of Calculations                 Definitions
106             FINANCIAL CALCULATIONS FOR LAWYERS




         Interest rates typically include a “real” factor, i.e., what people
         charge for the use of money when inflation and risk are both
         zero. Over time, this averages about three to four percent,
         with recent evidence suggesting the four percent figure to be
         more accurate.

             As a result, future values are often more useful when
        computed without the inflation or risk factors. This permits
                                    them to be compared to current
                                    present values. It eliminates
                                    much of the risk of bad predic-
           Caution:                 tions of the future inflation rate.
                                    But, even such computations, re-
      A Future Value is not         quire some translation.
     the equivalent of a
                                               For example, in Example
    Present Value if the                8a, a $1,000 present value is the
   interest rate includes               equivalent of a $1,610.51 future
      a risk component.                 value in five years at an as-




                                            Because the future
         sumed 10% effective in-
         terest rate. However, the           value includes the
         two numbers are not re-             earning resulting
         ally interchangeable even          from risk taking, it
         if the predicted 10% ef-            includes an extra
         fective interest proves to         component beyond
         be accurate. All the com-          what inflation com-
         putation tells us is that         pensation will provide.
         $1,000 invested today at
Calculators          Types of Calculations                Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   107




10% effective interest will yield an account containing
$1,610.51 in terms of future buying power in five years.

        The future value calculation, however, tells us nothing
about what that future value would purchase today: it cer-
tainly would not purchase $1,000 of value, but, instead, would
purchase significantly more. This is true because the
$1,610.51 includes compensation both for the risk and the
liquidity factors.




                   Did you notice?
                   The future value calcula-
                     tion tells us nothing
                    about what the future
                    value would purchase
                             today.


        If the risk factor is accurate, some instances will pro-
duce significantly less than $1,610.51 because they will in-
volve partial or full default. The future value computation
traditionally includes the risk component because interest
rates include it; however, the calculator result essentially as-
sumes the risk becomes zero in reality: the result actually
earns the full assumed interest and the original principle re-
mains. This may often be accurate; however, it will not al-
ways be accurate.

       Also, because the result includes the earning result-
ing from risk taking, it includes an extra component beyond
what inflation will provide.
Calculators           Types of Calculations               Definitions
108           FINANCIAL CALCULATIONS FOR LAWYERS




      The liquidity factor                  The liquidity factor of in-
      of interest also re-           terest also results in extra real
                                     value in a Future Value result.
       sults in extra real
                                     Overtime, considering large
       value beyond what             numbers of accounts, investors
      inflation compensa-            will earn approximately four per-
        tion will provide.           cent real return. Another way
                                     of stating this is: investors earn
                                      enough to compensate for ex-


       pected inflation plus
       expected risk plus an           General Law of
       additional four percent.              Finance
       Because the market
       sometimes wrongly es-
       timates future inflation             From a macro eco-
       and risk, the actual re-     nomic viewpoint, investors
       turn for any particular      earn enough to compensate
       account may be sig-          for expected inflation plus ex-
       nificantly different from    pected risk plus an additional
       four percent. Never-         four percent.
       theless, all things be-
       ing equal, the four per-
       cent figure is a useful, consistent, conservative, predictable
       real return.

             Example 15 illustrates several modifications to the
       Example 8 calculation. These should provide more useful
       information.

             Example 8 involved computing the Future Value of
       $1,000 in five years using an effective interest rate of ten per
Calculators         Types of Calculations                    Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                       109




     EXAMPLE 15: (HP10Bii)
     Modified Future Values
     Compute the Future Value in five years of
     $1,000 at various modified interest rates. Press the illus-
     trated keys.

     3


     5


     1000



     0

                      The display will read -1,159.27.
                     This is the future value of $1000 in five years
                     earning 3% EFF.

     4

                      The display will read -1,216.65.
                     This is the future value of $1000 in five years
                     earning 4% EFF.

     7

                      The display will read -1,402.55.
                     This is the future value of $1000 in five years
                     earning 7% EFF.
Calculators               Types of Calculations        Definitions
110              FINANCIAL CALCULATIONS FOR LAWYERS


        Future Value of a Sum Calculator


  EXAMPLE 15
  (JavaScript Calculator)
  Future Value Modifications
   PV Annuity                                         Interest Conversion
   FV Annuity
                  Future Value of a Sum Calculator        PV Sum
  Sinking Fund               Instructions ON OFF       Amortization



Mode
                              Begin      End
Present Value                         1,000.00

Future Value                          1,159.27

Nominal Interest Rate                    3.00

Effective Interest Rate      3.0000000000
                                          5.00
Number of Years
                                         1.00
Payments Per Year
                                         5.00
Number of Payments
                                          0.00
Payment

                                 clear all
Calculators               Types of Calculations       Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                 111


       Future Value of a Sum Calculator


  EXAMPLE 15
  (JavaScript Calculator)
  Future Value Modifications
   PV Annuity                                         Interest Conversion
   FV Annuity
                  Future Value of a Sum Calculator        PV Sum
  Sinking Fund                Instructions ON OFF      Amortization



Mode
                              Begin      End
Present Value                         1,000.00

Future Value                          1,216.65

Nominal Interest Rate                    4.00

Effective Interest Rate         4.00000000
                                          5.00
Number of Years
                                         1.00
Payments Per Year
                                         5.00
Number of Payments
                                          0.00
Payment

                                 clear all




    Change the Nominal Interest Rate to 7% and
     the colored boxes will change as well. The
               Future Value becomes
                                       1,402.55
Calculators           Types of Calculations                Definitions
112            FINANCIAL CALCULATIONS FOR LAWYERS




        cent. The examples illustrate the correct computation based
        on the ten per cent assumption; however, the assumption
        itself is likely unrealistic. While an investor may indeed earn
        a nominal 10% interest over a five year period, it is unlikely
        to represent a “real” rate of return: i.e., the purchasing power
        will not likely increase by 10% per annum.

                                               One useful modification
                                       involves adjusting the interest
   One useful modification             rate to exclude the inflation fac-
    involves adjusting the             tor. Assuming the 10% effec-
   interest rate to exclude            tive interest figure comprised
      the inflation factor.            four percent for liquidity and
                                       three percent for risk, it would
      Example 15 illustrates           also have comprised three per-
       this by using the 7%            cent for expected inflation.
      effective interest rate .        Thus seven percent would be
                                       the expected real interest rate
                                       (assuming the risk element
                                       proves unnecessary: i.e., the
                                       borrower does not fail).

                At seven percent interest, $1,000 invested today would
        yield $1,402.55 in five years. Those would be uninflated
        dollars and thus would reflect the same purchasing power as
        $1,402.55 would have today. In nominal terms, the future
        account will include $1,610.51 future dollars, which will pur-
        chase what $1,402.55 present dollars would purchase today.
        This is, as are all future value computations, a prediction and
        it is based on the assumption that expected inflation will be
        3%. Actual inflation, in hindsight, will almost always be a
        different amount . . . greater or lesser than the expected
        amount. Thus the prediction is no better than the assump-
        tion.
Calculators          Types of Calculations             Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  113




       Another useful modification involves adjusting the in-
terest rate to include only the four percent liquidity factor.
This would be the most conservative modification, with the
use of three percent being the more conservative of that. At
three percent effective interest, $1,000 today would yield
$1,159.27 in five years. At four percent, it would yield
$1,216.65. As above, these
would reflect what the ex-
pected future account of           Another useful modifica-
$1,610.51 would purchase             tion involves adjusting
today: significantly more than        the interest rate to in-
merely $1,000, but signifi-         clude only the four per-
cantly less than the full              cent liquidity factor.
$1,610.51.
                                    Example 15 illustrates
         The difference be-            this by using the 4%
tween the two modifications           effective interest rate.
involves the risk factor. In a
micro-economic sense, the
account may very well earn
the full three percent risk com-
ponent, yielding the full $1,402.55 purchasing power. But, in
a macroeconomic sense, it will not. If the risk component is
correctly set, some accounts will earn it, while others will
earn nothing or even face default, resulting in a loss of the
original $1,000 as well as the risk factor interest. Hence, if
the investor has a sufficient number of investments suffi-
ciently diversified, some will earn the full risk component,
others will earn part of it and some will fail. Net, investors
should earn nothing but the liquidity factor plus the inflation
factor. In reality, the U.S. economy seems to overstate the
risk component of investment, resulting in historically con-
sistent returns greater than inflation plus three or four per-
Calculators           Types of Calculations               Definitions
114           FINANCIAL CALCULATIONS FOR LAWYERS




                                         cent. Perhaps some of the
                                         reason involves overstated in-
        General Law of                   flation expectations coupled
      Finance (re-stated)                with overstated risk.

   If the investor has a sufficient              An investor wanting to
   number of investments suffi-          modify a future value calcu-
   ciently diversified, some will        lation to a useful figure would
   earn the full risk component,         thus likely want to use an
   others will earn part of it and       interest rate greater than
   some will fail. Net, investors        three percent and probably
   should earn nothing but the li-       greater than four percent,
   quidity factor plus the inflation     but probably not much
   factor, of which only the liquid-     greater, unless he is particu-
   ity factor represents a “real” re-    larly optimistic about the
   turn.                                 market overstating risks.
                                         This should generate a real-
                                         istic picture of the buying
                                         power of a future account.




                                              But, the U.S.
                                            economy seems
                                       to overstate the risk
                                     component of invest-
                                     ment, resulting in his-
                                     torically consistent re-
                                    turns greater than infla-
                                     tion plus four percent.
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  115




     Thus, an investor wanting to modify a future value
     calcuation to a useful figure would thus likely want
     to use an interest rate greater than three percent
      and probably greater than four percent, but
           probably not much greater.




                A third - and arguably simpler - modification
        would take the computed future value and discount it
       to the present by either the inflation component or by
      the sum of the inflation and risk components. These
     calculations will not yield the same answers as the above
    modifications; however, they will approximate the same
   results. The differences arise because this modification
  involves discounting both the inflated dollars as well as the
 dollars reflecting real interest. Considering the inherent un-
                                             certainty in pre-
                                             dicting future in-
                                             terest, however,
     A third modification would              the inaccuracy of
    discount the future value by             this modification
     either the inflation compo-             may not be impor-
      nent or by the sum of the              tant.
    inflation and risk compo-
   nents, excluding the liquidity
               factor.

    Example 15 illustrates this
     by using the 7% effective
             interest.
Calculators         Types of Calculations              Definitions
116           FINANCIAL CALCULATIONS FOR LAWYERS




              Caution                            Modifying a Fu-
                                          ture Value of an Annuity
   Modifying a Future Value of an         calculation is a bit more
   Annuity calcuation is more com-        complicated than doing
   plicated than modifying a simple       so for a simple Future
   Future Value of a Sum calcula-         Value of a Sum calcula-
   tion.                                  tion. This is true because
                                          the Annuity involves not
                                          only a present sum but
                                          also a series of future
                                 sums. Unless they, too, are modi-
       fied, the answer may not be fully useful. Example 16 illus-
       trates a useful method.

             For     ex-
      ample, to deter-                        TIP
      mine what is
      needed for a fu-          Not only must you modify the in-
      ture event - such         terest rate, but you must also
      as a child’s edu-         modify the payments.
      cation, one cannot
      know future costs.
      However, one
      might conclude that present costs will inflate to become fu-
      ture costs and that the inflation component will generally ap-
      proximate the inflation component of interest. As such, it
      falls out of the calculation. One can then determine how
      much would be needed today, in terms of present dollars, to
      purchase the needed future item.

            Perhaps, for example, a child newly graduated from
      high school would need $100,000 today. If so, one might
      conclude that education and living costs will rise by the ex-
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   117




pected inflation rate such that $100,000 discounted at a non-
inflated interest rate would yield the amount needed to yield
the purchasing power of $100,000 today. Whether the dis-
count rate should be three percent, four percent, or some-
thing greater than four percent depends on the user ’s own
beliefs about market and economic risks.

       The discount calculation would either involve the com-
putation of the Present Value of a Sum or a Sinking Fund, as
explained later. Because the Sinking Fund is merely the
reflection of an Annuity, it is relevant now to the annuity dis-
cussion. The resulting sinking fund payment would tell the
investor the amount to deposit each period for the desired
number of periods to yield the expected purchasing power,
as shown in Example 16a.

        Without further modification, however, the account will
inevitably prove to be insufficient. That is because the omit-
ted inflation factor must be put back in. Hence, the annuity
would no longer be a level annuity of a constant payment.
Instead, each payment would need to increase by the imme-
diately past periodic inflation rate. This will yield a future
account with the approximate purchasing power of the origi-
nally needed $100,000 for the hypothetical current gradu-
ate.

        Example 16 also illustrates the calculations, but from
the perspective of an annuity calculation rather than a sink-
ing fund. The initial Future Value of an Annuity calculation
computes that $2,000 invested annually at 10% effective in-
terest, beginning today, for eighteen years would yield ap-
proximately $100,000. The answer, however, is not useful
because one does not know what $100,000 will purchase in
eighteen years. The user, however, indeed knows what
Calculators               Types of Calculations       Definitions
118              FINANCIAL CALCULATIONS FOR LAWYERS


                 Sinking Fund Calculator


  EXAMPLE 16a
  (JavaScript Calculator)
  Sinking Fund Calculations
  PV Annuity                                          Interest Conversion
   FV Annuity      Sinking Fund Calculator                PV Sum
  Amortization                Instructions ON OFF         FV Sum



Mode
                              Begin    End
Present Value                           0.00

Future Value                    100,000.00

Nominal Interest Rate                 10.00
                                       10.00
Effective Interest Rate
                                      18.00
Number of Years
                                        1.00
Payments Per Year
                                       18.00
Number of Payments
                                   1,993.66 begin
Payment                                     mode

                                 clear all




      A Future Value of 100,000 in 18 years, using a
      Nominal Interest Rate of 10%, compounded
      annually requires annual deposits of $1,993.66
                      beginning today.
Calculators               Types of Calculations       Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                 119


                 Sinking Fund Calculator


  EXAMPLE 16a
  (JavaScript Calculator)
  Sinking Fund Alternatives
  PV Annuity                                          Interest Conversion
   FV Annuity      Sinking Fund Calculator                PV Sum
  Amortization                Instructions ON OFF         FV Sum



Mode
                              Begin    End
Present Value                           0.00

Future Value                    100,000.00

Nominal Interest Rate                   4.00                   7.00
                                       10.00
Effective Interest Rate
                                      18.00
Number of Years
                                        1.00
Payments Per Year
                                       18.00
Number of Payments
                                   3,749.36 begin        2,748.84
Payment                                     mode

                                 clear all



   To yield the same Future Value, but using a more
            useful Nominal Rate of 4%, com-
   pounded annualy would require annual deposits of
    $3,749.36 beginning today. At an optimistic 7%
        NAI, the deposit need only be 2,748.84.
Calculators         Types of Calculations                Definitions
120           FINANCIAL CALCULATIONS FOR LAWYERS




      EXAMPLE 16: (HP10Bii)
      Modified Future Value of an
      Annuity
      Compute the Future Value in eighteen years of $2,000 de-
      posited annually, beginning today, at 10% nominal annual
      interest compounded annually.

      10


      18                     0


      2000


                      The display will read -100,318.18.


      4


                      The display will read -3,761.29.


      7               The display will read -2,757.59.
Calculators          Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 121




$100,000 would purchase today. If that amount is sufficient,
the user need only modify the interest rate to exclude the
inflation and risk factors or at least the inflation factor.

       The more conservative approach would be to use three
or four percent and the more liberal (optimistic) approach
would be to use seven percent. The resulting payment amount
is the more useful and realistic annuity needed to yield the
desired result. This, too, however, must be modified to put
back the inflation component. Thus, the investor would want
to deposit something between $2,748.84 and $3,749.36 an-
nually with each annual payment increasing by the inflation
rate for the prior year. This should yield an account with the
approximate purchasing power of $100,000 today.




                  Did you Notice?
   The Future Value of an Annuity computation did not
   produce the precise numbers as given by the Sinking
   Fund computation. This results because the Annuity
   reflects an annual payment of $2,000, which yields
   $100,318.18. The Sinking Fund begins with a Future
   Value of only $100,000.00 and thus produces a slightly
   smaller payment amount.
Calculators         Types of Calculations              Definitions
122           FINANCIAL CALCULATIONS FOR LAWYERS




      5. Amortization
             This calculation solves for the amount of the regular
      payment needed, at a stated interest rate and period, to pay
      off a present value. This is the opposite of the calculation
      involving the Present Value of an Annuity.

             For example, if you were to borrow $100,000 today
      and agreed to make 360 equal monthly payments at an inter-
      est rate of eight percent nominal annual interest, each pay-
      ment would need to be $733.76, as illustrated in Example
      17.

              Amortization schedules typically involve the end mode
      because loan payments generally occur at the end of each
      period, rather than at the beginning. For example, if you
      were to borrow money to purchase a new car, the first pay-
      ment on the loan would not occur until one month from now.
      While that might be the beginning of a month, it is the end of
      the first month since the purchase, necessitating the use of
      end mode.




                               TIP
                  Amortization schedules typically
                  involve the end mode because
                  loan payments generally occur
                  at the end of each period, rather
                  than at the beginning.
Calculators         Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                123




           When Would You Want Use the Amortiza-
           tion Function?

  √   If you want to purchase a home, this function will deter-
  mine your monthly loan payments.

  √    If you have student loans outstanding, this function will
  determine your monthly payments.

  √    If you need to re-finance a loan or to combine various
  credit card obligations, this function will compute the monthly
  payments.



      1. HP l0 Bii Calculator

       To perform the amortization function using the HP I0
Bii calculator, first compute the amount of the payments as
explained above: set the calculator in end mode, set the
Present Value (PV)as 100,000, the Future Value (FV) as 0,
the Interest Rate per Year (I/YR) as 8, the Number of Pay-
ments Per Year (P/YR) as 12, and the Number of Payments
(N) as 360. Then solve for the amount of the Payment (PMT).
The displayed answer will be -733.764573879, the negative
indicating the payment. Thus the necessary payment is
$733.76, which includes both the interest and principal.
Calculators            Types of Calculations                Definitions
124             FINANCIAL CALCULATIONS FOR LAWYERS




        EXAMPLE 17: (HP10Bii)
        Amortization
        Compute the monthly payment needed to pay-
        off a loan of $100,000 in 30 years at a nominal annual inter-
        est rate of eight percent.

        8                  12


        360                        0


        100,000


                                 The display will read -733.76




               2. JavaScript Financial Calculator

                The JavaScript Calculator operates much the same
                                        way as the HP10Bii. Press
                                        the Amortization Calculator
                                        button to open the correct
       Amortization Calculator
      End Mode Amortization             calculator. Insert the nomi-
                                        nal annual interest rate, the
                                        number of periods per year,
        the number of years, and the present value (the loan amount)
        in the appropriate while boxes. Do not change the amounts
        in any yellow boxes. The answer will appear in the green
        Payment box.

                                       Begin Mode Amortization
Calculators               Types of Calculations        Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                  125


                      End Mode Amortization



  EXAMPLE 16a
  (JavaScript Calculator)
  Amortization
  PV Annuity                                           Interest Conversion
   FV Annuity       Amortization Calculator                PV Sum
  Sinking Fund                Instructions ON OFF          FV Sum



Mode
                              Begin    End
Present Value                    100,000.00

Future Value                            0.00

Nominal Interest Rate                   8.00
                             8.2999506808
Effective Interest Rate
                                       30.00
Number of Years
                                       12.00
Payments Per Year
                                      360.00
Number of Payments
                                                 end
                                      733.76
Payment                                         mode

                                 clear all



       If , instead, you were to use begin mode, the re-
quired payment would be only 728.91. The amount is less          Begin
because the first payment would be made today rather
than one month from now, as with the end mode calcula-
tion. Note that in both cases, the Future Value is zero           End
because the entire loan is then paid off.
Calculators         Types of Calculations              Definitions
126           FINANCIAL CALCULATIONS FOR LAWYERS


  End Mode Amortization               Begin Mode Amortization

            3. Amortization Schedule

             The above calculation computes the payment needed
      to amortize the present value. Most users, however, will also
      want to know the remaining balance owed after each pay-
      ment, as well as the portion of each payment comprising
      interest and principle. A list of these amount is called an
      amortization schedule.

                    a. HP l0 Bii Calculator

             Press the orange shift key and then the AMORT key.
      This shifts to the calculator’s Amortization function, rather
      than the Future Value (FV) function. The display will read
      PER 1-12. This indicates Periods 1 through 12. Next, press
      the key indicating the equal sign. The display will read Prin,
      and -835.36, indicating the principal included in the pay-
      ments for period 1 through 12. Press the equal sign key
      again. The display will then read INT and -7,969.81, which is
      the amount of the periods 1 through 12interest.

            Next, press the equal sign key again. The display will
      read BAL and 99,164.63, indicating the balance of principal
      owed after periods 1 through 12.

             Next, press the orange shift key and then the AMORT
      key. The display will read PER 13-24. Repeat the above
      process using the equal sign key to display the applicable
      figures for periods 13 through 24.

             Then again press the orange shift key and then the
      AMORT key. The display will read PER 25-36. Repeat the
      above process using the equal sign key to display the appli-
      cable figures for periods 25 through 36.
Calculators          Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 127




      Then again press the orange shift key and then the
AMORT key, along with the equal sign key. Repeat this
process for all thirty years.

      b. JavaScript Financial Calculator

       The JavaScript Calculator operates much more sim-
ply than does the HP 10Bii. It automatically provides a sched-
ule for up to 360 payments, detailing the payment amount,
the interest, the principal paid, and the remaining principal
amount.




                  End Mode Amortization


                Begin Mode Amortization
Calculators          Types of Calculations               Definitions
128           FINANCIAL CALCULATIONS FOR LAWYERS


              Sinking Fund Calculator

             6. Sinking Fund
             This calculation solves for the amount of the regular
      deposit needed, at a stated interest rate and period, to accu-
      mulate a future value. This is the opposite of the calculation
      involving the Future Value of an Annuity.

              For example, if you wanted to accumulate $25,000 in
      ten years and were willing to make ten equal annual depos-
      its, beginning today, at an annual interest rate of ten percent,
      each deposit would need to be $1,426.03. Beginning one
      year from now, the necessary deposits would be $1,568.63.

             Sinking Fund schedules often involve the begin mode
      because savings plan deposits often begin at the inception of
      the plan, which would be the beginning of the first period.
      The end mode calculation, however, may also be used.




           When Would You Want to Compute a Sink-
           ing Fund?

  √  If you need to save for a child’s education and know the
  amount needed.

  √   If you need to save for retirement and know the amount
  needed.
Calculators         Types of Calculations            Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                129




                            TIP
              Sinking Fund schedules typically
              involve the begin mode because
              the depositor wishes to begin im-
              mediately.




              Sinking Fund Calculator

     EXAMPLE 18: (HP10Bii)
     Sinking Fund
     Compute the annual payment, beginning to-
     day, needed to accumulate $25,000 in 10 years at a nomi-
     nal annual interest rate of ten percent.

     10                1


     10                 0


     25,000


                            The display will read -1,426.03.
Calculators          Types of Calculations             Definitions
130            FINANCIAL CALCULATIONS FOR LAWYERS




               1. HP lOBii Calculator

              As illustrated in Example 18, set the calculator in
      begin mode, set the Present Value (PV) as 0, the Future
      Value (FV) as 25,000, the Interest Rate per Year (I/YR) as
      10, the Number of Payments Per Year (P/YR) as 1, and the
      Number of Payments (N) as 10. Then solve for the amount of
      the Payment (PMT). The displayed answer will be -1,426.03,
      the negative indicating the deposit. Thus the ten necessary
      deposits are each $1,426.03. With accumulated interest the
      fund will equal $25,000 in ten years.

               2. JavaScript Financial Calculator

             The JavaScript Calculator operates similarly. Press
      the Sinking Fund Calculator button to open the correct cal-
                                             culator. Insert the
                                             nominal annual inter-
         Sinking Fund Calculator
           Sinking Fund Calculator           est rate, the number of
                                             periods per year, the
                                             number of years, and
                                             the future value in the
      appropriate white boxes. Do not change any amounts in the
      yellow boxes. The answer will appear in the green Payment
      box. Placing the cursor over any of the functions will open a
      pop-up box with additional instructions or explanations.

       Begin          Be sure to use the correct mode: begin mode
                for payments beginning immediately and end mode
                for payments beginning at the end of the first pe-
       End      riod.
Calculators               Types of Calculations       Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                 131


                 Sinking Fund Calculator


  EXAMPLE 18
  (JavaScript Calculator)
  Sinking Fund
  PV Annuity                                          Interest Conversion
   FV Annuity      Sinking Fund Calculator                PV Sum
  Amortization                Instructions ON OFF         FV Sum



Mode                                   End
                              Begin
Present Value                           0.00

Future Value                      25,000.00

Nominal Interest Rate                 10.00
                                       10.00
Effective Interest Rate
                                      10.00
Number of Years
                                        1.00
Payments Per Year
                                       10.00
Number of Payments
                                   1,426.03 begin        1,568.63
Payment                                     mode
                                                               end
                                 clear all                    mode




      Press End       for end mode and the Payment
    amount will be 1,568.63. It is higher because the
     first payment would not be made until one year
         from now - the end of the first period.
Calculators         Types of Calculations              Definitions
132           FINANCIAL CALCULATIONS FOR LAWYERS




                        II. DEFINITIONS

              A. Interest Rate

             Standing alone, the term interest rate has no useful
      meaning. Instead, it requires one or more modifiers to indi-
      cate the period and frequency of compounding. Four differ-
      ent descriptions of interest are common. Each has its own


                                 Caution
         Standing alone, the term interest rate has
                    no useful meaning.




      appropriate use; thus, no description is correct or incorrect:
      they simply have different meanings and uses.

                     1. Nominal annual interest rate.
             Sometimes called the “stated interest rate” or “coupon
      rate” this is the periodic interest rate times the number of
      periods per year. It is sometimes abbreviated as NAI.

            An interest rate of one percent per month produces a
      nominal annual interest rate of twelve percent per year, com-
      pounded monthly. To be correctly stated, it requires the full
      description of 12% NAI, compounded monthly. The NAI lan-
      guage denotes it as a nominal rate and the “compounded
      monthly” denotes the number of periods per year. Without
Calculators          Types of Calculations                Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   133




those two modifiers, the statement of 12% per year has little
meaning.

       The NAI number is necessary for calculations involv-
ing multiple periods per year because the inverse of the NAI
equation is also true: the periodic interest rate equals the
nominal rate divided by the number of periods per year. All
calculators work with a periodic rate because the interest
compounding period and the payment period must be the
same. Thus, calculations involving multiple annual payments
require the calculation of a periodic rate, which itself re-
quires the use of the nominal rate.

       Whenever the interest compounds annually, the nomi-
nal annual interest rate will equal the effective interest rate.
However, whenever the interest compounds more often than
annually, the nominal annual interest rate will be less than the
effective rate. This can result in some persons being misled


                             TIP
     Whenever the interest compounds more often than
    annually, the nominal annual interest rate will be less
                   than the effective rate.



by the statement of an interest rate.

       For example, a document may refer to a nominal rate
of 10%, while later providing for monthly compounding. The
effective interest rate would be 10.471306744%. A reader
who does not appreciate the difference between the two rates
(effective and nominal) - and their uses - may mistakenly
Calculators         Types of Calculations              Definitions
134           FINANCIAL CALCULATIONS FOR LAWYERS




      visualize a lower rate of interest for the transaction than is
      accurate.

            While the statement of the 10% nominal rate com-
      pounded monthly would be correct, it is thus also easily mis-
      understood. Hence, a well-drafted document will provide the
      NAI rate (along with the compounding frequency), the peri-


                                   TIP
         A well-drafted document will provide the NAI
         rate (along with the compounding frequency),
         the periodic rate, plus the equivalent effective
         rate.



      odic rate, plus the equivalent effective rate.

             Boxes One and Two on pages 18 and 22 illustrate the
      conversion of an interest rate from nominal to effective or
      effective to nominal, using an HP 10Bii calculator.

             The JAVAScript Financial Calculator automatically
      converts the nominal rate to the effective rate. It also in-
      cludes an interest conversion calculator which converts an
      effective rate back to the equivalent nominal rate and peri-
      odic rate.

            Example 19 illustrates the amortization of a $100,000
      student loan at 7.5% nominal annual interest with monthly
      payments for ten years beginning today. The necessary
      payment is $1,179.64. Using end mode, the necessary pay-
Calculators               Types of Calculations         Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                   135




  EXAMPLE 19
  (JavaScript Calculator)
  Amortization
  PV Annuity                                            Interest Conversion
   FV Annuity       Amortization Calculator                 PV Sum
  Sinking Fund                Instructions ON OFF           FV Sum



Mode
                              Begin    End
Present Value                    100,000.00

Future Value                            0.00

Nominal Interest Rate                   7.50
                             7.7632598856
Effective Interest Rate
                                       10.00
Number of Years
                                       12.00
Payments Per Year
                                      120.00
Number of Payments
                                                begin
                                   1,179.64
Payment                                         mode

                                 clear all




                        DID YOU NOTICE?
   The JavaScript Calculator automatically computes
   the Effective Interest Rate.
Calculators          Types of Calculations                 Definitions
136           FINANCIAL CALCULATIONS FOR LAWYERS




      ment would be $1,187.02.
             For a problem involving an amortization, terminology
      is particularly important. The loan document will certainly
      state a nominal annual interest rate. It likely will not refer to a
      compounding period because interest on a loan does not
      normally compound: that is because it is paid regularly, along
      with partial principle payments. Most likely, the loan docu-
      ment also does not state the effective rate, once again be-
      cause the interest does not compound as a feature of the
      loan - at least not between the lender and borrower. Thus, in
      the case of Example 19, the document would not state the
      effective rate of 7.763%.

              If the debtor wants to know how much interest he is
      paying, the answer should refer to the 7.5% nominal figure.
      But, if the debtor wants to compare the loan to a savings
      account (perhaps to decide whether to payoff the loan with



                                    TIP
         A typical loan agreement will not state the ef-
         fective interest rate. It will state the nominal
         rate, which is a lower number and thus will ap-
         pear - to a novice - to involve lower interest. It
         will also state the annual percentage rate (APR)
         as required by federal law. The APR, however,
         is not the same as the effective rate and will
         always be lower than the effective rate if pay-
         ments are more frequent than annual.
Calculators           Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                    137




                             TIP
   Interest on a typical loan does not compound.
   Nevertheless, it has compounding effects: the
   lender’s source of funds compounds, as does
   the account receiving the payments. Similarly,
   the borrower’s cost of capital (and thus source
   of payments) compounds.


existing investments), he would need to know the effective
rate. Although interest on the loan does not strictly com-
pound, it nevertheless had compounding effects because of
its monthly payment nature. The lender receives each month’s
interest and invests it somewhere, undoubtedly earning a com-
pounded return. Also, the borrower pays the monthly inter-
est with funds that, but for the payment, otherwise would be
invested at a compounded return. Hence the effective rate
reflects the true economic cost of the loan.

        In the example, unless the borrower was earning more
than 7.763% on existing funds, he would be better off paying
off the loan early (all other factors being equal). If he did not
know about effective interest rates - and how to compute
them - he would likely erroneously use a 7.5% figure for the
loan to investment comparison.

              2 . Periodic Interest Rate.
       This is the amount of interest per period. Any cal-
culation involving multiple payments per year requires the
use of a periodic rate.

              Well-drafted legal documents will state a peri-
Calculators               Types of Calculations             Definitions
138             FINANCIAL CALCULATIONS FOR LAWYERS




  EXAMPLE 20
  (JavaScript Calculator)
  Interest Conversion
  PV Annuity                                                   PV Sum
   FV Annuity      Interest Rate Conversion                  Amortization
 Sinking Fund                 Instructions ON OFF              FV Sum


   Convert Effective          Convert Nominal       Convert Periodic
    Rate to Nominal            Rate to Effective     Rate to Nominal
   Rate and Periodic          Rate and Periodic     Rate and Effective
          Rate                       Rate                  Rate




Nominal Interest Rate        19.200000000
Periodic Interest Rate                   1.5

Effective Interest Rate
                             20.98300406509

Payments Per Year                     12.00

                                clear all
Calculators              Types of Calculations             Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                    139




odic rate, as well as the equivalent nominal annual rate and
the equivalent effective annual rate. For example, a periodic
rate of I % per month is the equivalent of a nominal annual
rate of 12%, compounded monthly and an effective annual
rate of 12.682503013%. The periodic rate is necessary for
any calculations. In addition, it is useful if the transactions
involves less than a full year.

              Example 20 demonstrates the conversion of a
Periodic Rate of 1.6% per month. It equals 19.2% nominal
interest and 20.983% effective interest. Such a rate might
be used on a credit card. The nominal rate will likely be
disclosed, as might the annual percentage rate (which will
be the same as the nominal). However, the lender will not
likely disclose the effective rate, which is significantly higher
than the nominal rate.



                          Instructions
  1. Press "Clear All" prior to working a problem.

  2. Select the appropriate conversion function.

  3. Type appropriate numbers in the white boxes.

  4. The answer will appear in the green boxes.

  5. Place your cursor over blue terms for a definition of the
  term.
Calculators         Types of Calculations                 Definitions
140           FINANCIAL CALCULATIONS FOR LAWYERS




                      3. Effective Interest Rate.
              This term has the same general meaning as the annual
      percentage yield or the yield to maturity and a similar mean-
      ing to the term internal rate of return. The four similar terms,
      however, have their own uses and are not precisely inter-
      changeable.

                          a. Deposits. For original deposits, with
      no withdrawals, each of the four terms will be the same. The
      effective interest rate will be the annual compounded rate of
      interest: the actual amount of interest earned for a particular
      year divided by the amount on deposit at the beginning of
      the year. Financial institutions generally quote this rate com-
      pounded for the appropriate number of periods for an entire
      year. This is the most useful number for purposes of com-
      paring one deposit with another.

        For example, one financial institution may offer 10% nomi-
      nal annual interest compounded semiannually, while another



                    Effective Rate Formula
                                                       py
         effective rate = 100 1+        (       pr
                                               100
                                                      )     -1   )

        pr = periodic rate
        py = payments per year
Calculators          Types of Calculations                  Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   141




offers 9.9% nominal annual interest compounded quarterly,
and a third offers 9.8% compounded daily. A comparison of
those three rates is difficult because of the differing com-
pounding periods. Stating each in terms of an effective an-
nual rate eliminates any confusion. The first institution is ac-
tually offering 10.25%
effective interest. The
second is offering
10.273639392% effec-                         TIP
tive interest, slightly
more than the first even      Because the nominal annual in-
those it offers a lower       terest rate for a deposit is lower
nominal rate. The third       than the effective interest rate,
institution is offering I     financial institutions will often
0.294827704, more still       prominently quotethe effective
even though it offers         interest rate or annual percent-
the lowest of the three       age yield on a deposit. They
nominal rates.                may most prominently quote the
                              lower nominal rate for a loan.
        The nominal an-



                Nominal Rate Formula
                                                        1
                                                     py
   nominal rate = 100py             ((1+      eff
                                             100    )       -1)

 eff = effective rate
 py = payments per year
Calculators          Types of Calculations                Definitions
142           FINANCIAL CALCULATIONS FOR LAWYERS




      nual interest rate for a deposit will always be lower than the
      effective interest rate. As a result, financial institutions will
      often quote, in the most prominent language, the effective
      interest rate or annual percentage yield on a deposit.

             Accounts which have occasional withdrawals or addi-
      tional deposits will have the same effective interest rate and
      annual percentage yield or yield to maturity; however, they
      may have a different internal rate of return. Sales of a debt
      instrument subsequent to issue and prior to maturity - or
      offers to sell it -may result in a different yield to maturity and
      internal rate of return, because of the changing present value
      as a result of market forces.

                         b. Loans. Discount loans with no pay-
      ments prior to maturity and no points have an effective inter-
      est rate equal both to the annual percentage yield and the
      nominal annual rate. They also have an annual percentage
      rate equal to the nominal rate. Installment loans and loans
      with points, however, have differing effective interest rates,
      nominal rates, and annual percentage rates.

         The effective rate on an installment loan with no points will
      be the interest rate that
      would accrue annually if
      the interest on the loan
      compounded. In actuality,                  TIP
      interest on an installment
      loan without negative am-        Installment loans and
      ortization does not com-         loans with points have
      pound; instead, the install-     differing      effective
      ments pay the interest due       interest rates, nominal
      plus, usually, a portion of      rates, and annual per-
      the principal. As a result,      centage rates.
      no interest is charged on
Calculators           Types of Calculations                Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                    143




interest. In a sense, the effective interest rate on such a loan
is not representative of reality: while the effective rate is a
compounded rate, the actual interest on the loan does not
compound.

       The effective rate reflects what would happen if the
interest compounded. In reality, the interest does compound,
though not specifically with regard to the loan instrument.
This is true both from the standpoint of the lender and the
borrower.

                                            From the lender’s
                                      viewpoint, he receives in-
                                      stallment payments, in-
              TIP                     cluding all interest due and
                                      some principal. Those
   From the lender ’s
                                      amounts do not earn ad-
   viewpoint, interest
                                      ditional interest from this
   compounds. Thus a
                                      borrower with regard to
   lender should compare
                                      this loan; however, the
   an installment loan’s
                                      lender must do something
   effective rate to his
                                      with the funds. If depos-
   cost of funds.
                                      ited or loaned elsewhere,



they will earn additional interest.
                                                Caution:
If expended, they will free up
                                         Comparing either the
other funds which can earn in-
                                         nominal or annual per-
terest, or they will reduce the
                                         centage rate of an in-
need for borrowing, which will
                                         stallment loan to the
reduce other interest costs. Thus,
                                         lender’s cost of capital
effectively, the funds earn inter-
                                         would be misleading.
est for the entire year (unless the
Calculators          Types of Calculations               Definitions
144           FINANCIAL CALCULATIONS FOR LAWYERS




      applicable currency is stuffed in a mattress or some other
      unproductive investment). Stating the uncompounded peri-
      odic rate on the particular loan as a compounded effective
      rate reflects the reality that the funds will earn interest from
      some source for the entire year.

              From the borrower’s viewpoint, he makes installment
      payments, including all interest due and some principal. As a
      result, he does not owe additional interest on those funds to
      that lender with regard to that loan; however, the borrower
                                   must have a source of funds to
                                   make the payments. That source
              TIP                  of funds itself has a cost, which
                                   reflects its own interest rate. If
   From the lender ’s              he uses other available funds to
   viewpoint, interest             make the payments, the borrower
   compounds. Thus a               is then unable to earn interest
   lender should compare           elsewhere on those funds. Or, if
   an installment loan’s ef-       he borrows the funds to make the
   fective rate to his cost        payments, the borrower must pay
   of funds.                       additional interest on such addi-
                                   tionally borrowed funds. Thus,


      effectively, the funds cost
      interest for the entire year           Caution:
      (unless the borrower
      steals or prints the cur-       Comparing either the
      rency, it has a cost). Stat-    nominal or annual per-
      ing the uncompounded            centage rate of an in-
      periodic rate on the par-       stallment loan to the
      ticular loan as a com-          borrower’s savings ac-
      pounded effective rate re-      counts would be mis-
      flects the reality that the     leading.
Calculators               Types of Calculations                       Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                             145




                              Caution:
                        BORROWERS

       Both the nominal and the annual percent-
       age rate [APR] on an installment loan will
       always be lower than the effective rate.

       Thus, financial institutions rarely, if ever,
       prominently disclose or otherwise adver-
       tise the effective rate of a loan.



funds cost interest for the entire year.

     The nominal annual interest rate and the annual percent-
age rate on an installment loan will always be lower than the
effective rate. As a result, financial institutions rarely, if ever,
prominently disclose or otherwise advertise the effective rate
of a loan. This contrasts with their eagerness to advertise the
effective rate on a deposit. Federal law does not require dis-
closure of the effective rate. In fact it expressly requires promi-
nent disclosure of the annual percentage rate [APR],3 which
is always a lower number on an installment loan (and which
does not reflect the above described reality).

     Also, federal law expressly permits the disclosure of a
3
  15 U.S.C. § 1637 (for open ended credit); § 1638 (for other credit transac-
tions).
4
  12C.F.R.§226.1i.
Calculators         Types of Calculations              Definitions




      “Comparative Index of Credit Cost’4 which has some char-
      acteristics of an effective rate, but which also is inevitably
      lower than the effective rate of interest.

                    4. Annual Percentage Rate.
                               In credit transactions not involving
                                  points or some other fees, the
                                  annual percentage rate equals
              TIP                 the nominal annual interest rate.
                                  However, transactions involving
   The APR is a partially         points and some other fees have
   compounded rate: it            an annual percentage rate which
   reflects the nominal           reflects both the nominal rate and
   rate with the “points”         the compounded amortized effect
   amortized over the             of the points or other fees. Dis-
   stated life of the loan.       closure of this rate is required
                                  by federal law for most credit



                    DID YOU NOTICE?
       The APR [annual percentage rate] equals the
       NAI [nominal annual interest rate] for an in-
       stallment loan with no points.

       The APR is higher than the NAI for an install-
       ment loan with points.

       The EFF [effective interest rate] is higher
       than both the APR and the NAI for an install-
       ment loan (regardless whether it has points).
Calculators          Types of Calculations               Definitions




transactions. It is typically ab-
breviated as the APR.
                                                  TIP
       In credit parlance, a
“point” is equal to one percent of     A “point” is equal to
the principal amount loaned.           one percent of the prin-
Thus on a $100,000 loan, one           cipal amount loaned.
point equals $1000 and two points
equals $2000. On a $200,000
loan, one point equals $2000 and
two points equals $4000.

       Institutions charge points for three general reasons:

       First, the points - which are actually discounted inter-
est - are not reflected in the nominal annual interest rate. As
a result, the nominal rate is understated. While a lender must
prominently disclose the annual percentage rate, which re-
flects the points, it can do so along with disclosure of the
nominal rate. Thus, lenders hope borrowers will visualize the
nominal rate as the true rate, rather than the more accurate
and higher - and sometimes less prominent - A.P.R. or the



                        Caution:
   Because points are not reflected in the nominal
   rate, the nominal rate is always understated in
   an installment loan with points.



most accurate and highest - and almost certainly undisclosed
- effective rate.
Calculators        Types of Calculations           Definitions
148           FINANCIAL CALCULATIONS FOR LAWYERS




                  DID YOU NOTICE?
   Lenders charge points for three general reasons:

      1. Points allow them to understate the nominal
         interest rate.

      2. Points are generally tax deductible by the
        borrower.

      3. Points are non-refundable, resulting in an
        excess return if the loan is paid-off early (as
        most home loans are).




         Caution:
   Two of the three rea-
   sons for points favor
   the lender.                               TIP
   Borrowers should                  If the present value of
   be very cautious                  the tax advantage
   with points.                      outweighs the ex-
                                     cess cost of early
                                   payoff, points are good
                                   for the borrower.
Calculators            Types of Calculations             Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                 149




       Second, the points - if at-                TIP
tributable to a home loan - are
generally deductible by the bor-       Points paid on a loan
rower for federal income tax pur-      for a primary resi-
poses. As a result, borrowers          dence are generally
may benefit from having more of        deductible for federal
the interest deductible in the first   income tax purposes.
year of the loan.

       Third, the points are almost
always nonrefundable. They are
paid - either with separate funds or by being withheld from
the loan proceeds - at the time of the loan transaction: If the
loan is outstanding for its entire term, the points are effec-
tively paid periodically over the life of the loan. However, if
the borrower pays the loan prematurely, he must pay all re-
maining principal, unreduced by the points. For example, a
$100,000 loan with two points is the equivalent of a $98,000
loan because just as soon as the borrower receives the
$100,000 he must pay back $2000 as points. Nevertheless,
the borrower is immediately liable for the entire $100,000
loan principal, even if he were to repay the loan the next day.


                             Caution:
   Points are almost always non-refundable.

   Thus, if you expect to pay off the loan early (e.g.,
   when you sell the house to buy a new house), con-
   sider avoiding points.
Calculators          Types of Calculations                Definitions
150           FINANCIAL CALCULATIONS FOR LAWYERS




                                       As a practical matter, most
               TIP                     home loans are paid early be-
                                       cause they contain a “due on
   Most home mortgage                  sale” clause, accelerating
   loans are paid off long             them whenever the underly-
   before the original matu-           ing security changes hands.
   rity date of fifteen or thirty      Many purchasers of resi-
   years.                              dential property sell the prop-
                                       erty - and thus pay off the re-
   This occurs because the             spective loan early - prior to
   borrower sells the house,           the end of the original loan
   pays off the loan and buys          term. As a result, the lender
   a new house with a new              earns an extraordinary inter-
   loan.                               est rate - higher even than the
                                       original effective interest rate.
                                       Often, much higher.

            To compute the annual percentage rate on an install-
      ment loan with points, follow these steps:

              1. Amortize the loan and record the payment.

              2. Subtract the points from the principal.

              3. Input the Step 2 amount as the new princi-
                 pal, the payment amount, the term, the pay-
                 ment frequency and then solve for the inter-
                 est rate.

             The Step 3 interest rate is the Annual Percentage Rate.
      To convert it to the effective rate, use the Interest Conversion
      Calculator: simply input the APR as the nominal rate and the
      calculator will automatically convert it.
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   151




        Example 21 illustrates computation of an Annual Per-
centage Rate using an HP 10Bii calculator. Example 21a
then illustrates the conversion of the APR to an effective rate.

      Although the JavaScript Financial Calculator is not
currently designed to solve for a missing interest rate, it will
do so with a simple process of interpolation. It then auto-
matically converts the interpolated rate to an effective rate.



    EXAMPLE 21: (HP10Bii)
    Annual Percentage Rate
    Compute the annual percentate rate on a home
    loan of $200,000, a thirty year term, monthly payments, a
    nominal annual interest rate of 7.5% and three points.

    7.5                 12


    30                                  200,000



                       The display will read -1,398.43 .


    197,000


                        The display will read 7.655055419 .
                                          [This is the APR]
Calculators         Types of Calculations               Definitions
152           FINANCIAL CALCULATIONS FOR LAWYERS




      EXAMPLE 21a: (HP10Bii)
      Conversion of Annual Percent-
      age Rate to Effective Rate
      Covert an APR of 7.655055419% with monthly payments
      to the equivalent effective rate.

      7.655055419



      12




                        The display will read 7.929432144 .
                                           [This is the EFF.]




      Example 22 illustrate the interpolation method.
Calculators               Types of Calculations        Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                  153




  EXAMPLE 21
  (JavaScript Calculator)
  APR Interpolation
  PV Annuity                                           Interest Conversion
   FV Annuity       Amortization Calculator                PV Sum
  Sinking Fund                Instructions ON OFF          FV Sum



Mode
                              Begin    End
Present Value                    200,000.00

Future Value                            0.00

Nominal Interest Rate                   7.50
                             7.7632598856
Effective Interest Rate
                                       30.00
Number of Years
                                       12.00
Payments Per Year
                                      360.00
Number of Payments
                                                 end
                                   1,398.43
Payment                                         mode

                                 clear all



  Step One for Interpolation Method:
                      amortize the loan
Calculators               Types of Calculations         Definitions
154              FINANCIAL CALCULATIONS FOR LAWYERS




  EXAMPLE 21
  (JavaScript Calculator)
  APR Interpolation
  PV Annuity                                            Interest Conversion
   FV Annuity       Amortization Calculator                 PV Sum
  Sinking Fund                Instructions ON OFF           FV Sum



Mode
                              Begin    End
Present Value                    197,000.00

Future Value                            0.00

Nominal Interest Rate                   7.50           7.70
                             7.7632598856
Effective Interest Rate
                                       30.00
Number of Years
                                       12.00
Payments Per Year
                                      360.00
Number of Payments
                                   1,377.45         1,404.53
Payment

                                 clear all




          Step Two for Interpolation Method:
          re-amortize the loan, using the true prinipal
          amount (subtract the points from the stated
                           principal).
Calculators               Types of Calculations          Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                   155




  EXAMPLE 21
  (JavaScript Calculator)
  APR Interpolation
  PV Annuity                                            Interest Conversion
   FV Annuity       Amortization Calculator                 PV Sum
  Sinking Fund                Instructions ON OFF           FV Sum



Mode
                              Begin    End
Present Value                    197,000.00

Future Value                            0.00

Nominal Interest Rate                   7.65          7.655
                             7.7632598856
Effective Interest Rate
                                       30.00
Number of Years
                                       12.00
Payments Per Year
                                      360.00
Number of Payments
                                   1,397.74         1,398.42
Payment

                                 clear all


 Step Three for Interpolation Method:
  change the interest rate until the payment
amount equals the Step One amortization pay-
 ment of 1,398.43. Hint: start with the already
 computed effective rate rounded to one deci-
    mal place. This may take several tries.
Calculators          Types of Calculations               Definitions
156           FINANCIAL CALCULATIONS FOR LAWYERS




              B. Other Important Financial Terms

             Several other important terms arise in relation to inter-
      est rates. They include:

              a. simple yield

              b. yield

              c. yield to maturity (YTM)

              d. internal rate of return (IRR).

             The first two of
      these terms - simple
      yield and yield - are
      not terms of art: their              Caution
      precise definitions
      may vary from user to       The terms, simple yield
      user. Thus anyone           yield, are not terms of art.
      using either of them in
      a legal context should      Users should provide or
      provide a precise defi-     demand a precise defini-
      nition. Likewise, any-      tion in any legal docu-
      one coming across           ment.
      them in a legal context
      should demand a pre-
      cise definition. The lat-
      ter two terms - yield to maturity and internal rate of return -
      have generally accepted, precise meanings.

                   a. Simple Yield. This is an easy-to-compute,
      but imprecise measure of the return on a debt instrument. As
Calculators           Types of Calculations                Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                     157




illustrated in Example 23, It is the nominal annual interest
divided by the current market price of the instrument.

       This computation would change constantly, as the mar-
ket value of the instrument changed. The ease of computa-
tion justifies the use of the figure. It is, however, an inferior


      EXAMPLE 23: (HP10Bii)
      Simple Yield
      Compute the simple yield of a $1,000 (face
      amount) bond paying 7% nominal annuals interest, paid quar-
      terly. Assume that it sells, alternatively, for $900, $1000,
      and $1,100.

       70 = 7.778% simple yield
       900

       70 = 7.000% simple yield
      1000


       70 = 6.364% simple yield
      1100




             “Simple Yield” Formula:

                 nominal annual interest
                  current market price
Calculators         Types of Calculations               Definitions
158           FINANCIAL CALCULATIONS FOR LAWYERS




      measure of the true return on the bond or similar investment.
      The actual yield for a stated period or the yield to maturity
      would be more accurate and thus more useful.

             Some users may interchange this term with the slightly
      different term “yield.” Others might compound the quarterly
      payment to generate a more precise calculation. Neither use
      is wrong - they are merely different. As cautioned above, if
      someone uses the term “simple yield,” request a definition.

              b.    Yield. This measure of the return on a debt in-
                                            strument is sometimes
                                            interchanged with the
                                            slightly different term
                      TIP                   “simple yield.” More
                                            commonly, however, it
         An instrument’s “yield” dif-       constitutes the actual
         fers from its “simple              yield on an instrument
         yield” in two ways:                for a stated period of
                                            time, as a function of
         1. The yield is a function         the purchase price.
         of purchase price rather           Thus it would divide
         than market price.                 any periodic interest
                                            payment by the pur-
         2. The yield is a com-
                                            chase price and then
         pounded rather than
                                            convert it to an annual
         simple (uncompounded
         rate).                             rate, compounding the
                                            periodic rate for the
                                            number of periods.
                                            Example 24 illustrates
      the computation of a Yield.

              This measure of the instrument differs from the “simple
      yield” in two respects. First, it adds the compounding fea-
Calculators        Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 159




     EXAMPLE 24: (HP10Bii)
     Yield
     Compute the yield of a $1,000 (face amount)
     bond paying 7% nominal annuals interest, paid quarterly.
     Assume that it sells, alternatively, for $900, $1000, and
     $1,100.

     Step One: compute the simple yield, per Example 23.

     Step Two: convert the simple yields to comparable effec
               tive rates.

     4                            4

                                              The display will
     7.778                                    read 8.0078

                                              The display will
     7.000
                                              read 7.1859

     6.364                                    The display will
                                              read 6.5175




                 “Yield” Formula:

             compounded annual interest
                  purchase price
Calculators         Types of Calculations               Definitions
160           FINANCIAL CALCULATIONS FOR LAWYERS




      ture, when appropriate; hence the yields are greater when
      the payment period is less than one year. Second, it does
      not change constantly as the market price of the instrument
      changes; instead, it is fixed by the purchase or issue price
      of the instrument (depending on whose viewpoint is involved).

            Also, although the above definition describes a num-
      ber which is more accurate and hence more useful than the
      number described as a “simple yield,” this term - yield - does
      not present a true measure of an instrument’s return. Two
      inaccuracies are inherent.

              One, it relies on interest compounding, when, in fact,
      as far as the instrument is concerned the interest is paid and
      thus does not compound. This is less criticism of the calcula-
      tion - and more mere observation, however, because that
      feature is inevitable.

              For yields to be useful, they must generally be com-
      parable to those of other instruments. For this to be possible,
      they must be based on a common standard - such as the
      year. Instruments which pay interest annually will thus present
      an accurate yield. In contrast, instruments which pay at pe-
      riods other than a year will never present an accurate annual
      yield: it violates the fourth rule stated earlier: the payment
      period and the compounding period must be the same.

             For instruments that pay interest other than annually,
      an annual yield will never be precise because it inherently
      requires an assumption that the interest paid continued to
      earn interest at the same internal rate. While useful, such an
      assumption is not perfect. As long as users understand this
      feature, the calculation of a yield can be very useful and
      generally accurate.
Calculators           Types of Calculations           Definitions
                 FINANCIAL CALCULATIONS FOR LAWYERS                 161




  EXAMPLES 23-24
  (JavaScript Calculator)
  Simple Yield and Yield
  PV Annuity                                          Interest Conversion
                            Yield Calculator           Amortization
  FV Annuity
 Sinking Fund               Instructions ON OFF           FV Sum




Periodic Interest Payment          $17.50

Market Price                    $1000.00

Simple Yield                 7.000000000

Periodic Yield               1.750000000

Yield                       7.1859031289

Payments Per Year
                                           4

                               clear all
Calculators          Types of Calculations               Definitions
162           FINANCIAL CALCULATIONS FOR LAWYERS




              A second inaccuracy of the “yield” calculation involves
      its failure to consider the impact of changing values, i.e,
      changing market interest rates. Another way of stating this
      somewhat obvious point is that the term - as defined above -
      ignores market discounts and premiums. While the point of
      the calculation is simply to look at paid returns for a particu-
      lar period - and thus it accomplishes what its definition con-
      strains it to do, the calculation nevertheless risks presenting
      a significantly inaccurate picture.

             For example, an instrument sold at a premium will have
      the same “yield,” regardless of its life. In contrast, it will
      have a higher “yield to maturity” the longer the period until
      maturity. Similarly, an instrument sold at a discount will have
      the same “yield,” regardless of its life, although it will have a
      lower “yield to maturity” the longer the period until maturity.

             Despite some inherent inaccuracies of its own, the
      “yield to maturity” calculation presents the most accurate
      and useful picture of a
      debt instrument.
      Hence a comparison
      of the yields of two in-                TIP
      struments, ignoring the
      terms of the instru-
      ments, might (though         The “yield to maturity” cal-
      not     necessarily)         culation presents the
      present a small, or          most accurate and useful
      even largely distorted       picture of a debt instru-
      picture. A comparison        ment.
      of yields that consid-
      ers the terms would in-
      deed be mostly accurate; however, it would also be a com-
      parison of ”yields to maturity” and thus, by definition, not a
Calculators            Types of Calculations                   Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                        163




comparison of mere yields.
                                        5
               C. Yield to Maturity. This is the most accu-
rate measure of the return on a debt instrument. Comparable
to - and sometimes interchanged with either the “effective
interest rate” or the “internal rate of return” - it considers the
instrument’s actual cash flows. Thus it is the most realistic
measure of an instrument’s return.

       As illustrated in Example 25, the yield to maturity
calculation amortizes the premium or discount element of the
issue price over the life of the instrument. This is more useful
than the mere “yield” which, as defined above, ignores the
premium or discount. Nevertheless, the yield to maturity cal-
culation is subject to at least two potential inaccuracies.

                                                   First, it as-
             Caution                        sumes - as does the
                                            effective interest rate -
    The Yield to Maturity cal-              that any payments
    culation assumes that all               continue to earn or
    interest received will con-             cost the same constant
    tinue to earn the same                  interest rate. This is un-
    constant interest rate.                 likely to be accurate;
    This assumption is un-                  nevertheless, because
    likely to prove precisely               no investor has a crys-
    accurate.                               tal ball with which to
                                             determine future in-

5
 Surprisingly, I have found some disagreement regarding the mean-
ing of “yield to maturity.” Most authorities define the term as I do.
However, at least one book defines it differently. Joel G. Siegel and
Jae K. Shim, ACCOUNTING HANDBOOK 2d Ed., Barrons 1995 at 700
(providing a formula using an arithmetic rather than geometric com-
pounding of interest).
Calculators          Types of Calculations                Definitions
164           FINANCIAL CALCULATIONS FOR LAWYERS




      vestment returns, such an assumption is the best possible. It
      also permits realistic comparisons between instruments. Nev-
      ertheless, it can result in some misunderstandings and thus
      should be fully understood.

           The assumption that all returns are reinvested at the
     same rate, while necessary mathematically, can cause mis-
     understanding. An investor might assume that the two instru-
     ments in Example 26 are interchangeable because they
     have the same original cost and the same yield to maturity.
                                   Because they have different
                                   cash flows, however, they are
          Caution                  comparable only with the
                                   above assumption, which may
   The Yield to Maturity cal-      - or may not - be realistic.
   culation assumes the in-
   strument will be outstand-                  The second potential
   ing for its entire scheduled         inaccuracy involving the yield
   life. Because many instru-           to maturity calculation involves
   ments are called or paid             the assumption that the instru-
   early, this assumption is            ment will be outstanding for its
   often incorrect, rending             entire expected life. This,
   the YTM inaccurate.                  again, is a necessary as-
                                        sumption: to input a future
                                        maturity value one must know
                                        the future date. Because no
      crystal balls exist to foretell the future, the assumption be-
      comes necessary that the instrument will continue to be out-
      standing for its entire scheduled life and will make all sched-
      uled payments. Many instruments, however, have a put or
      call feature under which either the maker or purchaser- or
      both - may offer or demand payment early, respectively. In
Calculators        Types of Calculations             Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS               165




      EXAMPLE 25:
      Yield to Maturity


  A bond paying 7.0% nominal annual interest, paid quar-
  terly - $70.00 per year or $17.50 per quarter - and sold
  at par would have a yield to maturity of 7.186%, regard-
  less of its life.

  To compute this figure, set the P/YR as 4, the I/YR as 7.0
  and solve for the effective interest rate, which will be
  7.186.

  4                          7

                                              The display will
                                    The display will
                                              read 7.186.
                                    read 7.186.


  The simple yield is      7.000.
  The yield is             7.186.
  The yield to maturity is 7.186
Calculators         Types of Calculations            Definitions
166           FINANCIAL CALCULATIONS FOR LAWYERS




      EXAMPLE 25a:
      Yield to Maturity



  If, instead, the bond sold for $900.00 and were outstand-
  ing for two years, it would have a yield to maturity of
  13.365%. To compute this figure, set the P/YR as 4, the
  PMT as 17.50, the N as 8, the PV as (900), and the FV as
  1000. Then solve for the I/YR, which will be 12.743 and
  also for the effective interest rate, which will be 13.365 .

  17.50              8             900


                                The display will
  1000                          read 12.743.


                                The display will
                                read 13.365.


  The simple yield is       7.778.
  The yield is              8.008.
  The yield to maturity is 13.365.
Calculators        Types of Calculations            Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                167




     EXAMPLE 25b:
     Yield to Maturity



  If, instead, the bond sold for $900.00 and were outstand-
  ing for ten years, it would have a yield to maturity of
  13.365%. To compute this figure, set the P/YR as 4, the
  PMT as 17.50, the N as 40, the PV as (900), and the FV as
  1000. Then solve for the I/YR, which will be 8.494 and
  also for the effective interest rate, which will be 8.769 .

  17.50              40           900


                                The display will
  1000                          read 8.494.


                                The display will
                                read 8.769.


  The simple yield is        7.778.
  The yield is               8.008.
  The yield to maturity is   8.769.
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168           FINANCIAL CALCULATIONS FOR LAWYERS




      EXAMPLE 25c:
      Yield to Maturity



  If, instead, the bond sold for $1,100.00 and were out-
  standing for two years, it would have a yield to maturity
  of 1.906%. To compute this figure, set the P/YR as 4, the
  PMT as 17.50, the N as 8, the PV as (1100), and the FV as
  1000. Then solve for the I/YR, which will be 1. and also
  for the effective interest rate, which will be 13.365 .

  17.50              8            1100


                               The display will
  1000                         read
                               1.892935887.
                               The display will
                               read
                               1.906415353.

  The simple yield is        6.364.
  The yield is               6.517
  The yield to maturity is   1.906.
Calculators         Types of Calculations            Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 169




     EXAMPLE 25d:
     Yield to Maturity



  If, instead, the bond sold for $1100.00 and were outstand-
  ing for ten years, it would have a yield to maturity of
  13.365%. To compute this figure, set the P/YR as 4, the
  PMT as 17.50, the N as 8, the PV as (1100), and the FV as
  1000. Then solve for the I/YR, which will be 12.743 and
  also for the effective interest rate, which will be 13.365 .

  17.50              40              1100


                                The display will
  1000                          read
                                5.682226203
                                The display will
                                read
                                5.804455792.

  The simple yield is        6.364
  The yield is               6.517
  The yield to maturity is   5.804
Calculators         Types of Calculations               Definitions
170           FINANCIAL CALCULATIONS FOR LAWYERS




      such cases, the statement of a yield to maturity should note
      the assumption regarding maturity.

                     d. Internal Rate of Return. This is the effec-
      tive interest rate at which the initial investment equals the
      present value of all future cash flows. If all cash flows are
      level and in the same direction, this computation is relatively
      simple and essentially parallels the computation of a yield to
      maturity. Uneven cash flows - and particularly those which
      change direction - present computational difficulties. Most
      calculators actually use a trial and error approach because
      the formula can be extremely complex.
Calculators        Types of Calculations            Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS           171




                       Example 26

    Comparison of Instruments with
           the Same Yield

  Instrument One with a face value of $100,000 is-
  sued for $96,000, paying 10% nominal annual inter-
  est for five years will have a yield to maturity of 11
  .084585%.

  Instrument Two involves $96,000 invested at
  11.084585 nominal annual interest compounded an-
  nually for five years will generate $162,382.87. It,
  too, has a yield to maturity of 11.084585%.

  But, Instrument One will generate approximately
  $162,382.87 only if each $10,000 interest payment is
  itself reinvested at 11.084585% (the extra $1.01 is
  due to a rounding error).
Calculators           Types of Calculations                Definitions
172           FINANCIAL CALCULATIONS FOR LAWYERS




           INTEREST RATES FOR USE IN
              LEGAL CALCULATIONS

         EXPERT TESTIMONY BY ECONOMISTS,
                    ACCOUNTANTS, AND OTHERS

            By definition, all financial calculations involving the time
      value of money require the use of an interest rate. The choice
      of the applicable interest rate is typically the most important
      factor in a legal valuation: small variations in the rate can
      have large consequences in the computed amount. The choice
      of rate is also arguably the least understood factor, the most
      subjective factor, and the one which varies the most in the
      testimony of “experts.”

           Example 27 illustrates how the choice of the interest
      rate can be the most important factor in a wrongful death
      matter.

           In a personal injury or wrongful death case involving
      future lost wages, the plaintiff will be entitled to the present
      value of the future loss. In addition to liability, the plaintiff
      must prove:


                                     TIP
          The choice of the interest rate is the most im-
          portant factor in a legal valuation of personal in-
          juries.
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                   173




                            Example 27

  The Effect of Interest Rates on Wrong-
           ful Death Valuations

  John died with a work expectancy of 40 years. He was then
  earning $100,000 per year.

  Undiscounted, the loss would be $4,000,000 - $100,000 per
  year for 40 years. However, because the present value would
  be paid by the torfeasor, the court or parties would discount
  the 40-year annuity to the present.

  Often, a plaintiff’s expert will testify that the appropriate dis-
  count rate is one percent. Using that number, the present
  value would be $3,283,468.61.

  Often, a defense expert will testify that the appropriate dis-
  count rate is 7.5%, or some similar number reflecting conser-
  vative, long-term investment returns. Using that number, the
  present value would be $1,259,440.87

  Properly analyzed, the correct discount rate should be be-
  tween four and four and one-half percent. Using those num-
  bers, the present value would be between $1,840,158.44 and
  $1,979,277.39.


                   DID YOU NOTICE?
  The argument of the interest rate involves more than $2,000,000
  - by far the largest single factor in the case.
Calculators          Types of Calculations             Definitions
174           FINANCIAL CALCULATIONS FOR LAWYERS




          √ the number of years of the loss, which equates
              with the N function

          √ the annual amount of loss, which equates with the
              PMT function

          √ the frequency of the lost , which equates with the
              P/YR function

          √ the timing of the loss, which equates with the Mode
              function

          √ the interest rate, which equates with the I/Yr func-
              tion

           Typically, the number of years will be easily determined
      and may even involve a stipulation: it would be based on
      work or life expectancy. The annual loss will be a question of
      fact, but may be debated only within a range of numbers
      based on the plaintiffs income experience and the value of
      his or her personal services and consumption. The frequency
      and timing of the loss affects the calculation; however, they
      are largely inconsequential: a timing change will cause a
      corresponding number of years change, and a frequency
      change will cause a corresponding interest rate change.


                               Caution
        The choice of the interest rate is the most sub-
        jective factor in valuaing a personal injury case.
        It is one over which experts sharply disagree.
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  175




                            Example 27

  The Effect of Changing Life Effectancy
      on Wrongful Death Valuations

  Using the facts of Example 27, suppose the parties could not
  decide the correct work expectancy for John.

  Using the Plaintiff’s interest rate of 1%, adding ten years to the
  work-life expectancy (increasing it to 50) changes the present
  value to $3,919,611.75 - an increase of $636,143.14 or 19.37%.

  Thus a very large - 25% - increase in work-life expectancy
  results, at a low interest rate, to a large increase in present
  value.

  Using the Defense’s interset rate of 7.5%, adding ten years to
  the work-life expectancy changes the present value to
  $1,297,481.16 - an increase of $38,040.29 or 3%.

  Thus a very large increase in work-life expectancy results, at
  higher interest rates to a very modest increase in present value.

  Using the 4.5% suggested rate and the 50 year term, the
  present value would be $1,976,200.78, an increase of
  $136,042.34 or 7.4%
Calculators          Types of Calculations                Definitions
176           FINANCIAL CALCULATIONS FOR LAWYERS




            The choice of the appropriate interest rate, however,
      will have both a large impact and little case-specific eviden-
      tiary support. Indeed much evidence exists regarding inter-
      est rate; however, very little of it has anything to do with a
      specific matter.



                   DID YOU NOTICE?
  Defense could concede a 50, rather than 40, year work-life
  expectancy if Plaintiff would concede a 4.5%, rather than 4%
  discount rate.

  This is true because the present value of an annuity of $100,000
  per year for 50 years discounted at 4.5% is less than the
  present value of an annuity of 100,000 per year for 40 years
  discounted at 4%.

  The 1/2% change in interest is more significant than the
  ten year increase in the length of the annuity!




                Why Do People Change Interest?

            To understand how an economist, accountant, or other
      expert chooses the appropriate interest rate, consider why
      people charge interest. They do so for four reasons:

              1. To compensate for inflation.

      In times of low inflation, this factor is small, while in times of
Calculators            Types of Calculations                 Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                      177




high expected inflation, this factor is correspondingly high.
The key word here is “expected” inflation.

         Interest rates are often compared to past inflation - a
interesting comparison that shows the market’s ability or in-
ability to predict. This comparison, however, is not itself im-
portant for the prediction of future interest rates, which con-
sider only future inflation. Past inflation, to the extent it pre-
dicts the future, is relevant; however, it must be understood
as a predictor of a factor and not a factor itself.

        Some financial Instruments, are not subject to discount
for this factor: those which bear interest in an amount that
fully compensates for expected inflation. In contrast, non-
interest or low-interest bearing notes must be discounted to
reflect their insufficient interest. This has very little to do with
the instrument’s marketability or the solvency of the obligor;
instead, it merely reflects the nature of the contract. Because
all instruments must ultimately produce an adequate return
on investment, the market will price them to do so.

       Three types of examples illustrate the relevance of this
factor for valuations important in legal matters: one tax ex-
ample, one personal injury example, and one family law ex-
ample.

        Example 28 illustrates a tax law example. For federal
tax purposes, income must be recognized upon the receipt
of a “cash equivalent.” The rule applies to all cash method
taxpayers and to most accrual method taxpayers. Courts gen-
erally define a “cash equivalent” as an unconditional promise
to pay, of a solvent obligor, assignable, not subject to setoff,
readily marketable, and subject to a discount not substan-
                                                6
tially greater than the prevailing market rate.
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178           FINANCIAL CALCULATIONS FOR LAWYERS




              Instruments which property reflect expected inflation
      will not be subjected to a discount for this factor. In contrast,
      those which insufficiently reflect expected inflation will be
      subject to this discount. The factor itself, however, has noth-
      ing to do with whether the instrument is a cash equivalent,
      the relevant factor instead being “risk discount.” As a result,
      per Example 28, the extent a particular note must be dis-
      counted to reflect the expected inflation factor should be ir-
      relevant in determining whether it is a cash equivalent.

              Example 29 illustrates a personal injury example. In
      a personal injury case, the “expected future inflation” factor
      would require an estimate of inflation that will occur over the
      remaining work-life or life of the injured person, depending
      on which period determines the loss. This speculative factor
      would enter the valuation computation twice: once as a fac-
      tor in estimating future income and again as a discount fac-
      tor. The two instances thus almost cancel each other, mak-
      ing the inflation factor irrelevant.

             As demonstrated in Example 29, the cancellation is
      not exact. Thus, the theoretically correct liquidity discount of
      4.5% must itself be reduced by the expected inflation rate.
      In times of no expected inflation, 4.5% remains the liquidity
      discount. For expected inflation of 5%, the liquidity discount
      becomes 4.275% and for expected inflation of 10%, the li-
      quidity discount becomes 4.05%.

             Example 30 illustrates a family law example. In a
      family law case settling alimony or other liabilities, the factor
      would depend on the period for which the payments would
      otherwise be made. Computing the present value of the fu-
      ture obligations would provide a number with which the mat-
      ter could be settled completely. To reach this number, the
Calculators             Types of Calculations                    Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                          179




                                Example 28

     The Effect of the Inflation Factor on
       the Cash Equivalence Doctrine


  Taxpayer received two notes. Note A pays 8% interest and Note B
  pays no interest. This occurs at a time when expected inflation over
  the remaining term of the notes is 4%. Assume that the appropriate
  “risk factor” attributable to Note A is 10% and to Note B is 4%.

  Note A includes sufficient interest to compensate for future inflation
  and liquidity. To value it correctly, it must be discounted by the risk
  factor: 10%.

  Note B pays no interest. To value it correctly, it must be discounted by
  all three factors: expected inflation, liquidity, and risk: a total of 12%.

  Comparing the two, Note A is subject to a discount rate of 10% while
  Note B is subject to a discount rate of 12%. Viewed simply, Note B
  would appear to be more heavily discounted. But that is incorrect.
  The first 8% of discount on Note B (for expected inflation and liquidity)
  merely places it on the same terms as Note A - that discount has
  nothing to do with the creditworthiness of the maker. Only the addi-
  tional “risk discount” matters.

  Thus, in evaluating whether the instruments constitute “cash equiva-
  lents,” a recipient would look only at the 10% and 4% risk factors ap-
  plied to each. The legal question would be whether the risk discount
  was “significantly greater than the prevailing market rate.”
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180            FINANCIAL CALCULATIONS FOR LAWYERS




                              Example 29

      The Effect of the Inflation Factor on a
              Personal Injury Case


  Plaintiff, who was earning $100,000 per year, was injured such that he
  cannot work for three years. Expected inflation over the next three
  years is five percent per year.

  The correct measure of the loss involves computing the present value
  of a three year annuity equal to the lost future income. Ignoring the
  possibility of productivity increases, the expected loss would be
  $100,000 for year one, $105,000 for year two, and $110,250 for year
  three. Assuming (to simplify this example) that all wages are paid at
  the beginning of each year, the present value of the loss would be
  $100,000 for year one, $95,890.41 for year two, and $91,949.71 for
  year three. The total would be $287,840.12. The discount rate would
  be 9.5%: 5% for expected inflation and 4.5% for liquidity.

  Ignoring inflation, the wages would remain stable at $100,000 per year.
  The present value of a three-year annuity of $100,000, at 4.5% nominal
  annual interest would be $287,266.78.

  The difference in the two computations occurs because the present
  value must actually earn enough to compensate for both inflation and
  liquidity. With an inflation factor of 5%, the liquidity factor becomes
  overstated by 5% of 4.5% or .225%.

  Thus a good approximation would use a liquidity discount of slightly
  less than 4.5%: the greater expected inflation, the lesser the discount.
Calculators             Types of Calculations                    Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                          181




                                Example 30

   The Effect of the Inflation Factor on a
             Family Law Case


  Wife has agreed to alimony equal to $2,000 per month for the remain-
  der of her life (or until she remarries). To settle the amount with a lump
  sum, the parties may wish to compute the present value of the $2,000
  monthly annuity.

  The period would be the shorter of three periods: husband’s life, wife’s
  life and wife’s life until she remarries. The discount rate would involve
  the total of expected inflation, liquidity, and risk (the risk that husband
  could not or would not pay).

  Handled property, the risk factor can be ignored. This results because
  the parties could secure the husband’s obligations with a life insur-
  ance policy on his life, reducing the risk of non-payment to essentially
  zero.

  The inflation factor remains important. Presumably, the parties con-
  sidered future inflation in settling the amount of the contracted pay-
  ments. By not including an inflation adjustment they either concluded
  that husband’s obligation lessenned as time passed, or they relied on
  wife’s ability to seek a modification for changes resulting from inflation.

  In either event, because the future amounts include consideration of
  the inflation factor, the discount rate would also include the inflation
  factor.
Calculators          Types of Calculations                Definitions
182           FINANCIAL CALCULATIONS FOR LAWYERS




      discount rate would likely include the expected inflation fac-
      tor as well as the liquidity factor, but possibly not the risk
      factor. Example 30 explains the reasoning behind this analy-
      sis.

             2. To compensate for liquidity.

             Historically, people charge approximately 3% to 3.5%
      interest in times of no inflation and cases of no risk. This is to
      compensate lenders for their lack of liquidity. Human beings
      expect interest as compensation simply for giving up the use
      of money, even if inflation and risk are zero.

             As with the expected inflation factor, an instrument
      which reflects this liquidity factor will not be subject to a dis-
      count for the factor. In contrast, an instrument that bears no
      interest will be subject to a discount reflecting liquidity, in
      addition to a discount reflecting the other two factors.

              Some experts may argue that this factor varies from
      time to time, or even that it has somehow shifted to a higher
      or lower number. Long term evidence suggests considerable
      stability in this, a largely sociological factor. Short term evi-
      dence may reflect periods of excessive interest rates, caus-
      ing some commentators to suggest that the factor has some-
      how shifted upward. A closer look, however, indicates the
      market’s inability to predict future inflation accurately over
      the short, or even mid term.

             Chart One shows Real Interest Rates for the period
      1985 to 2001, based on the difference between short term
      government rates and actual inflation. Yields were high dur-
      ing the period 1983 to 1987, tending to indicate market and
      Federal reserve over-predictions of inflation. In contrast rates
      were very low in 1992 to 1993 and again in 2001, periods
Calculators         Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS               183




                       Chart One
                   Real Interest Rates
   Source: Federal Reserve Bank of St. Louis. Money Trends,
                     02/20/02 at page 8.
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184           FINANCIAL CALCULATIONS FOR LAWYERS




                              Chart Two
                      Inflation Protected Yields
        Source: Federal Reserve Bank of St. Louis. Money Trends,
                          02/20/02 at page 11.
Calculators           Types of Calculations                 Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                    185




during which the Federal Funds Rate was unusually low, co-
inciding with brief economic recessions. Over time, however,
riskfree short term yields’ tended to range between two and
four percent.

        Chart Two illustrates actual yields on various govern-
ment issued inflation-protected securities. Such instruments
were not widely available until the mid-1990’s. Prior to that
time, economists routinely argued about the appropriate “real”
or liquidity factor for interest. Since, 1997, however, inves-
tors can protect against both expected inflation and risk. This
is possible because the instruments adjust their payments
for immediately prior period inflation figures. Because they
are government backed, the risk element is essentially zero.
Although the yields on such instruments have varied over
time, the trend is highly consistent with the historic 3.5%
liquidity factor.

       In personal injury, wrongful death, family law, and other
legal matters, three and one-half percent is probably the most
appropriate number for this factor, although a colorable case
could be made for a variation of up to one-half of one per-
cent.

        3. To compensate for risk.

         This factor - unique to the maker - reflects the
creditor’s impression of the risk of default. Highly solvent
makers are subject to little, if any, discount for this factor. In
contrast, insolvent debtors will be subject to a very high risk
interest factor.

       In personal injury, wrongful death, family law, and other
legal matters, either zero or up to one percent is probably the
most appropriate number for this factor. Arguably, plaintiff’s
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186           FINANCIAL CALCULATIONS FOR LAWYERS




      should invest any award in a low risk instrument such that
      this factor is irrelevant. To the extent an award recipient can
      earn a greater return by accepting additional risk, he can
      lose a portion of his principal.

             In a free market, the risk of loss and the opportunity
      for an excess return will cancel each other over time. Empiri-
      cal and anecdotal evidence, however, suggest that many per-
      sons earn excess returns, particularly in the late 1990’s.
      Hence some small discount factor for risk may be appropri-
      ate in most legal matters. But any amount much greater than
      one percent is highly speculative, especially during short
      periods.

              4.To compensate for market risk.

               This factor is arguably a component of factors 2 or
      3- liquidity and risk. It compensates the creditor for the risk
      of his experiencing temporary fluctuations in market liquidity
      or valuation of a particular instrument. Generally, the larger
      the market for a given instrument, the less the risk of price
      fluctuations unrelated to the main three factors; neverthe-
      less, at any given point, short term fluctuations occur in any
      market, justifying this additional factor of interest.

             In personal injury, wrongful death, family law, and other
      legal matters, an amount close to zero is probably the most
      appropriate number for this factor. The longer the term, the
      lesser the risk as price and liquidity fluctuations tend to level
      overtime. In addition, the larger the sums involved, the greater
      the ability of the investor to hedge against unanticipated fluc-
      tuations through the use of varying investments or maturity
      dates.
Calculators          Types of Calculations               Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                 187



   Future Value of a Sum Calculator
          FINANCIAL CALCULATOR
                QUESTIONS & ANSWERS

     Future Value of a Sum Calculator

1 a. Compute the future value of $1,000.00 in 10 years
at 6% nominal annual interest compounded annually.

                          $ 1,790.85

To calculate the above amount, clear the register and set
the Future Value of a Sum Calculator with the following values:

Mode                   Begin     End            irrelevant

Present Value              1,000.00

Future Value               1,790.85             computed
                                              by calculator
Nominal Interest Rate           6.00

Effective Interest Rate         6.00            computed
                                              by calculator
Number of Years                10.00

Paymernts Per Year              1.00

Number of Payments             10.00            computed
                                              by calculator
Payment                         0.00
Calculators          Types of Calculations              Definitions
188           FINANCIAL CALCULATIONS FOR LAWYERS


         Future Value of a Sum Calculator

           Future Value of a Sum Calculator


      1 b. Compute the future value of $1,000 in 10 years at
      6% nominal annual interest compounded semiannually.

                                $ 1,806.11

      To calculate the above amount, clear the register and set
      the Future Value of a Sum Calculator with the following values:

      Mode                 Begin      End              irrelevant

      Present Value                1,000.00

      Future Value                 1,806.11           computed
                                                    by calculator
      Nominal Interest Rate            6.00

      Effective Interest Rate          6.09           computed
                                                    by calculator
      Number of Years                10.00

      Paymernts Per Year               2.00

      Number of Payments             20.00            computed
                                                    by calculator
      Payment                          0.00
Calculators          Types of Calculations               Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                 189




     Future Value of a Sum Calculator


1 c. Compute the future value of $1,000 in 10 years at
6% nominal annual interest compounded monthly.

                            $ 1,819.40

To calculate the above amount, clear the register and set
the Future Value of a Sum Calculator with the following values:

Mode                   Begin     End             irrelevant

Present Value               1,000.00

Future Value                1,819.41            computed
                                              by calculator
Nominal Interest Rate           6.00

Effective Interest Rate         6.09            computed
                                              by calculator
Number of Years                10.00

Paymernts Per Year             12.00

Number of Payments             120.00           computed
                                              by calculator
Payment                         0.00
Calculators          Types of Calculations              Definitions
190           FINANCIAL CALCULATIONS FOR LAWYERS




           Future Value of a Sum Calculator

      1 d. Compute the future value of $1,000 in 10 years at
      6% nominal annual interest compounded daily.

      $ 1,822.027707 (assuming 360 days per year, 30 per
       month,) or $1,822.028955 (assuming 365 days per
       year)

      To calculate the above amount, clear the register and set
      the Future Value of a Sum Calculator with the following values:

      Mode                   Begin      End           irrelevant

      Present Value                  1,000.00

      Future Value                   1,822.03         computed
                                                    by calculator
      Nominal Interest Rate             6.00

      Effective Interest Rate           6.18          computed
                                                    by calculator
      Number of Years                   10.00

      Paymernts Per Year              365.00

      Number of Payments             3,650.00         computed
                                                    by calculator
      Payment                           0.00
Calculators          Types of Calculations                Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                  191




   Present Value of a Sum Calculator

2 a. Compute the present value of $100,000.00 to be
received 18 years from today at 6% nominal annual in-
terest compounded annually.

                           $ 35,034.38

To calculate the above amount, clear the register and set
the Present Value of a Sum Calculator with the following values:

Mode                   Begin     End             irrelevant

Present Value               35,034.38            computed
                                               by calculator
Future Value                100,000.00

Nominal Interest Rate            6.00

Effective Interest Rate          6.00            computed
                                               by calculator
Number of Years                18.00

Paymernts Per Year               1.00

Number of Payments              18.00            computed
                                               by calculator
Payment                          0.00
Calculators          Types of Calculations               Definitions
192           FINANCIAL CALCULATIONS FOR LAWYERS




         Present Value of a Sum Calculator

      2 b. Compute the present value of $100,000.00 to be
      received 18 years from today at 6% nominal annual
      interest compounded semiannually.

                                 $ 34,503.24

      To calculate the above amount, clear the register and set
      the Present Value of a Sum Calculator with the following values:

      Mode                   Begin      End            irrelevant

      Present Value             34,503.24              computed
                                                     by calculator
      Future Value             100,000.00

      Nominal Interest Rate            6.00

      Effective Interest Rate          6.09            computed
                                                     by calculator
      Number of Years                18.00

      Paymernts Per Year               2.00

      Number of Payments              36.00            computed
                                                     by calculator
      Payment                          0.00
Calculators          Types of Calculations               Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                  193




   Present Value of a Sum Calculator

2 c. Compute the present value of $100,000.00 to be
received 18 years from today at 6% nominal annual
interest compounded monthly.

                              $ 34,051.06

To calculate the above amount, clear the register and set
the Present Value of a Sum Calculator with the following values:

Mode                  Begin       End            irrelevant

Present Value             34,051.06              computed
                                         by calculator
Future Value             100,000.00

Nominal Interest Rate            6.00

Effective Interest Rate          6.09            computed
                                         by calculator
Number of Years                  18.00

Paymernts Per Year               12.00

Number of Payments              216.00           computed
                                         by calculator
Payment                           0.00
Calculators          Types of Calculations               Definitions
194           FINANCIAL CALCULATIONS FOR LAWYERS




         Present Value of a Sum Calculator

      2 d. Compute the present value of $100,000 to be re-
      ceived 18 years from today at 6% nominal annual inter-
      est compounded daily.

          $ 33,962.6087222 (360 day convention)

          $ 33,962.5668597 (365 day convention)

      To calculate the above amount, clear the register and set
      the Present Value of a Sum Calculator with the following values:

      Mode               Begin      End                irrelevant

      Present Value              33,962.57            computed
                                                   by calculator
      Future Value             100,000.00

      Nominal Interest Rate           6.00

      Effective Interest Rate         6.09            computed
                                                  by calculator
      Number of Years                18.00

      Paymernts Per Year              12.00

      Number of Payments            216.00            computed
                                                  by calculator
      Payment                          0.00
Calculators        Types of Calculations            Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS           195




  Did you notice?
  √ As the compounding frequency increases
  (from annual to daily), in problems a to d, the
  Present Value of a Future Sum decreases.

            This occurs because the more
     frequent compounding results in a
     higher effective interest rate (EFF).

  √ Thus, as interest rates increase, present
  values decrease.

            For example, if you increase the
     interest rate in problem (d) to 12% nomi-
     nal annual interest compounded daily,
     the present value drops to $11,536.61:
     thus, double the interest and the present
     value drops by two-thirds!

   √ In contrast, as interest rates increase, fu-
  ture values also increase.
Calculators          Types of Calculations            Definitions
196           FINANCIAL CALCULATIONS FOR LAWYERS




           Future Value of an Annuity Calculator

      3 a. Compute the future value of $1,000 to be paid
      annually for ten years, the first payment due at the be-
      ginning of the period (an annuity due). Use a nominal
      annual interest rate of 6% compounded annually.

                      $ 13,971.64 (annuity due)

       To calculate the above amount, clear the register and set
      the Future Value of an Annuity Calculator with the following
      values:
                                   Begin
      Mode

      Present Value                  0.00

      Future Value              13,971.64           computed
                                                 by calculator
      Nominal Interest Rate          6.00

      Effective Interest Rate        6.00           computed
                                                 by calculator
      Number of Years               10.00

      Paymernts Per Year             1.00

      Number of Payments            10.00           computed
                                                 by calculator
      Payment                     1,000.00
Calculators         Types of Calculations             Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS              197




     Future Value of an Annuity Calculator


3 b. Compute the future value of $1,000 to be paid an-
nually for ten years, the first payment due at the end of
the period (an annuity in arrears). Use a nominal annual
interest rate of 6% compounded annually.

              $13,180.79 (annuity in arrears)

To calculate the above amount, clear the register and set
the Future Value of an Annuity Calculator with the following
values:
                             End
Mode

Present Value                 0.00

Future Value              13,180.79          computed
                                          by calculator
Nominal Interest Rate          6.00

Effective Interest Rate        6.00          computed
                                          by calculator
Number of Years               10.00

Paymernts Per Year             1.00

Number of Payments            10.00          computed
                                          by calculator
Payment                     1,00.00
Calculators          Types of Calculations            Definitions
198           FINANCIAL CALCULATIONS FOR LAWYERS




           Future Value of an Annuity Calculator

      3 c. Compute the future value of $1,000 to be paid
      annually for ten years, the first payment due at the end
      of the period (an annuity in arrears). Use a nominal
      annual interest rate of 6% compounded semi-annually.

                   $ 13,236.64 (annuity in arrears)

      To calculate the above amount, clear the register and set
      the Future Value of an Annuity Calculator with the following
      values:

      Mode                          End

      Present Value                 0.00

      Future Value              13,236.64          computed
                                                by calculator
      Nominal Interest Rate          6.09

      Effective Interest Rate        6.09          computed
                                                by calculator
      Number of Years               10.00

      Paymernts Per Year             1.00

      Number of Payments            10.00          computed
                                                by calculator
      Payment                     1,000.00
Calculators         Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS               199




        If you change problem (c) to an an-
        nuity due (Begin Mode), the present
        value is $14,042.75

        To prove this, you need not clear the regis-
        ter: merely change the mode to Begin.




  BE CAREFUL:

  In Problem 3c,            you cannot convert
  the annuity from ten $1,000 payments to twenty
  $500 payments: that would produce an incorrect
  answer. By splitting the payments in two, you
  would accelerate one-half of them, resulting in more
  interest earned. Because that is not consistent
  with the given facts, it would not answer the ques-
  tion asked.

  You must, instead, convert the interest rate from a
  nominal annual rate compounded semi-annually to
  an equivalent effective interest rate.




   Use the Interest Rate Conversion Calculator to do this.

    Interest Rate Conversion Calculator
Calculators               Types of Calculations               Definitions
200             FINANCIAL CALCULATIONS FOR LAWYERS




  PV Annuity                                                    PV Sum
   FV Annuity      Interest Rate Conversion                   Amortization
 Sinking Fund                 Instructions ON OFF               FV Sum


   Convert Effective          Convert Nominal        Convert Periodic
    Rate to Nominal            Rate to Effective      Rate to Nominal
   Rate and Periodic          Rate and Periodic      Rate and Effective
          Rate                       Rate                   Rate




Nominal Interest Rate                  6.00

                                        3.00         Click here for the
Periodic Interest Rate
                                                    correct converter for
Effective Interest Rate                 6.09            Problem 3c.

Payments Per Year                       2.00

                                clear all


        For Problem 3c, convert the 6.00% nominal interest rate,
        compounded semi-annually to an effective rate of 6.09% .
        Use this as the nominal rate, compounded annually.


           TIP:
     Remember, an
   effective rate is the                Did you Notice:
     equivalent of a
   nominal rate com-                        Problem 4a computes
   pounded annually.                         the Present Value of
                                              Lottery Winnings.
Calculators         Types of Calculations              Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS               201




   Present Value of an Annuity Calculator

4 a. Compute the present value of $50,000 to be paid
annually for twenty years, the first payment due at the
beginning of the period (an annuity due). Use a nominal
annual interest rate of 6% compounded annually.

                $ 607,905.82 (annuity due)

To calculate the above amount, clear the register and set
the Present Value of an Annuity Calculator with the following
values:

Mode                         Begin


Present Value           607,905.82            computed
                                           by calculator
Future Value                   0.00

Nominal Interest Rate          6.00

Effective Interest Rate        6.00           computed
                                           by calculator
Number of Years               20.00

Paymernts Per Year             1.00

Number of Payments             20.00          computed
                                           by calculator
Payment                    50,000.00
Calculators          Types of Calculations            Definitions
202           FINANCIAL CALCULATIONS FOR LAWYERS




         Present Value of an Annuity Calculator

      4 b. Compute the present value of $1000 to be paid
      monthly for ten years, the first payment due at the
      beginning of the period (an annuity due). Use a nominal
      annual interest rate of 6% compounded annually.

                      $ 90,523.82 (annuity due)

      To calculate the above amount, clear the register and set
      the Present Value of an Annuity Calculator with the following
      values:

                                   Begin
      Mode

      Present Value             90,523.82           computed
                                                 by calculator
      Future Value                   0.00

      Nominal Interest Rate          6.00

      Effective Interest Rate        6.00           computed
                                                 by calculator
      Number of Years                10.00

      Paymernts Per Year             12.00

      Number of Payments            120.00          computed
                                                 by calculator
      Payment                     1,000.00
Calculators         Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                 203




  Did you notice?
  If you reduce the interest rate in problem 4a to 3%, the present
  value rises to $766,189.96. If you reduce it further to 2%, the
  present value rises to $833,923.10. Hence an important rela-
  tionship of interest rates and present values:

  ♦ As the interest rate decreases, the present value
  increases - and vice versa.

  If you change problem 4 b to an annutity in arrears, the
  present value is $ 90,073.45.

  To prove this, you need not clear the register: merely change
  the mode to End.

  BE CAREFUL:
  In Problem 4b,                 you cannot convert the
  annuity from twelve $1,000 payments per to one
  $12,000 payment: that would produce an incorrect answer.
  By combining the payments, you would accelerate 11/12ths of
  them, resulting in more interest earned. You must, instead,
  convert the interest rate from a nominal annual rate compounded
  annually to an equivalent nominal annual rate compounded
  monthly.

  A cardinal rule of annuties is:

           ♦ The payment period and the
           compounding period must be the
Calculators          Types of Calculations          Definitions
204           FINANCIAL CALCULATIONS FOR LAWYERS




          Sinking Fund Calculator
      5 a. You need $100,000 18 years from now for your
      child’s education. Assume a nominal annual interest rate
      of 6% compounded monthly. How much must you de-
      posit each month. Assume alternatively an annuity due
      and an annuity in arrears. Ignore any income tax con-
      sequences or inflation.

      $ 256.88 (annuity due)      $ 258.16 (annuity in arrears)

      To calculate the above amount, clear the register and set
      the Sinking Fund Calculator with the following values:

      Mode                          Begin


      Present Value                  0.00

      Future Value              100,000.00

      Nominal Interest Rate          6.00

      Effective Interest Rate        6.17         computed
                                               by calculator
      Number of Years                18.00

      Paymernts Per Year             12.00

      Number of Payments           216.00         computed
                                               by calculator
      Payment                       256.88
Calculators        Types of Calculations            Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS           205




  Did You Notice?
  If you start saving when the child is one month
  old (an annuity in arrears) each monthly
  deposit must be approximately $1.28 greater
  to achieve the same $100,000 future value.




                           TIP:
          To compute the annuity in
               arrears, simply click on
           End the end mode button. The
               answer     will  appear
          automatically.




                 Did you Notice:
      Problem 5a computes the needed
    monthly saving for a child’s education.
Calculators         Types of Calculations               Definitions
206           FINANCIAL CALCULATIONS FOR LAWYERS




          Sinking Fund Calculator
      5 b. Suppose you have no idea what you will need for
      your child in 18 years for his or her education. However,
      you estimate that if your child were 18 years old now,
      you would feel comfortable having $100,000 in savings .

           You do not know what the inflation rate will be for
      education costs; however, you estimate that it will
      average an amount close to the inflation rate for the
      general economy. You also do not know future tax rates;
      however, you assume they will remain stable.

            How much must you deposit each month to attain
      your goal. Assume alternatively an annuity due and an
      annuity in arrears.

           This problem Illustrates real life issues. While we cannot
      accurately predict future costs, we can accurately determine
      present costs. Hence we begin with the $ 100,000 present
      cost of college. This number must be gauged for the individual:
      some need more and some need less, depending on family
      expectations. But, at least the number is based on reality -
      current costs - rather than a guess of costs 18 years into the
      future.

          The following analysis also ignores the earnings the fund
      would earn during matriculation. A more complex calculation,
      however, would consider this factor.

              Next we must acknowledge that college costs will
Calculators           Types of Calculations                Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS                    207




increase over the years. In many recent years, this has been
more rapid than the general inflation rate - at least for the
portion of college costs attributable to tuition and books. Living
expenses, however, are more likely to increase, on average,
at the general inflation rate. Also, in some years tuition and
books have not increased by the full inflation rate. For any



                  Did you Notice:

   Problem 5b is a better way to com-
   pute the needed monthly saving for
           a child’s education.

      The 4a method is commonly used, but flawed.



savings plan, the saver must determine whether the general
inflation component of interest rates represents the future
inflation rate of the item (here education) for which he or she
is saving. If so, then it becomes appropriate to ignore the
inflation component of interest as well as the inflation
component of the future costs. If not, then the saver must
adjust the expected cost by any differential.

      A reasonable conclusion might be that the inflation
component of general interest rates will be - over the next 18
years - comparable to the inflationary increase in college
education costs, including tuition, books, and living expenses.
Calculators          Types of Calculations               Definitions
208           FINANCIAL CALCULATIONS FOR LAWYERS




              Next recall that people pay interest for three reasons:
      expected inflation, risk, and liquidity (the borrower pays extra
      for the convenience of cash while the lender charges for the
      inconvenience of not having cash). Recently - with the advent
      of inflation adjusted bonds, government issues have tended
      to yield approximately 3.5% over the inflation rate (and
      sometimes up to 4% or slightly more). Economists call this
      the “real” or “ underlying” interest rate - the portion paid and
      charged for the liquidity factor.

             In the short run, the market may estimate future Inflation
      incorrectly and thus pay more or less than this amount after
      actual inflation is considered. For example, if the market
      expects 2% inflation, government bonds should yield
      approximatly 5.5 percent. But, if inflation ultimately turns out
      to be 1%, then the bonds will have yielded a “real” rate of 4.5
      percent. Or, if inflation
      ultimately turns out to be
      4%, then the government
      bonds will have yielded
                                                 TIP:
      a “real” rate of only 1.5
      percent.                       The best conservative
                                     pre-tax        effective
              In the short run,      interest rate to use for
      such mistakes occur;           a college savings plan
      however, in the longer         is approximately 3.5%.
      run, they tend to level out
      and the “real” rate tends
      to be about 3.5 percent. The 18 year period for which the
      problem contemplates saving is sufficiently long so as to
      justify the following conclusion: the underlying interest rate
      will be approximatly 3.5%. Naturally, the savings account
      can earn more than this if the saver take risks; however, the
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  209




account can then also lose value. Individual savers can benefit
or lose from risk taking. The overall market, however, can
ignore it because, at least in general, for each winner, there
is a loser.

       Probably you do not want to invest a child’s education
fund in a very risky instrument. Thus, you might reasonably
                                    conclude that you can
                                    earn, conservatively
              TIP:                  3.5% after inflation or
                                    liberally 5.5% after
                                    inflation (mortgage
  A more risky, but                 backed securities or high
  arguably realistic pre-           grade equities or mid-
  tax effective interest            grade corporate bonds
  rate to use for a                 each currently yield
  college savings plan is           closer to 8 percent).
  approximately 5.5%.
                                        Next, taxes cannot
                                   be ignored. A child under
the age of 14 must pay taxes at his parent’s marginal rate,
which currently is likely 14%, 28%, 31 %, or 38% (ignoring
state and local income taxes). A child 14 or older must pay
taxes at his own rate, which currently is likely 14%. These
numbers ignore the
child’s       modest
personal exemption                Caution:
and any standard
deduction. Short term
government bonds - at
                                Do not forget to
the election of the         anticipate the impact
taxpayer - produce                 of taxes.
either tax-deferred or
Calculators         Types of Calculations               Definitions
210           FINANCIAL CALCULATIONS FOR LAWYERS




      tax free income; hence, we can ignore the tax consequences
      of them. However, they also pay low rates. State and local
      bonds produce no taxable income; however, they can be
      risky, at least in the short run. Mortgage backed securities,
      equity investments, and corporate bonds produce taxable
      income - some of which can be deferred and some of which
      is subject to a maximum 20% rate. But, they also can be
      risky, particularly in the short-run. Thus, as the time
      approaches for the needed funds, you may want to shift
      from such investments into ones containing less market risk.
      Another real factor to consider are intangible taxes imposed
      by some states - such as Florida - on some investments.

             Or, you might consider placing all or part of the
      investment in a section 529 plan, which (with some limitations)
      is exempt from tax when used for qualified education expenses.

             The bottom line is that you might realistically yield
      3.0% after inflation, risk, and taxes in a very conservative
      investment. Or, you might yield 5.0% after inflation, risk,
      and taxes in an acceptably risky investment. Or, perhaps
      you might yield 7.0% after inflation, risk, and taxes in a more
                                            aggressive investment.
                                            More is probably
                    TIP:                    unrealistic, especially
                                            considering the need to
              Realistic after-tax           become              more
             effective rates for a          conservative as the child
          college savings plan are          approaches college age.
         3.0% (conservative), 5.0%
            (moderate) and 7.0%                  With the above
                 (aggressive.               assumptions           and
                                            conclusions - each of
Calculators         Types of Calculations            Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS             211




which is realistic - we can now work the problem.

In the conservative assumption the required monthly payment
would be:

 $351.12 (annuity in arrears) or $350.26 (annuity due).

To calculate the above amounts, clear the register and set
the Sinking Fund Calculator with the following values:

Mode                 Begin      End


Present Value                  0.00

Future Value              100,000.00

Nominal Interest Rate          2.96

Effective Interest Rate        2.99          computed
                                          by calculator
Number of Years                18.00

Paymernts Per Year             12.00

Number of Payments           216.00           computed
                                                  by
Payment              End      351.12          calculator
                    Begin     350.26
Calculators          Types of Calculations          Definitions
212           FINANCIAL CALCULATIONS FOR LAWYERS




      In the moderate assumption the required monthly payment
      would be:

       $289.61 (annuity in arrears) or $288.43 (annuity due).

      To calculate the above amount, clear the register and set
      the Sinking Fund Calculator with the following values:

      Mode                  Begin      End

      Present Value                  0.00

      Future Value              100,000.00

      Nominal Interest Rate          4.89

      Effective Interest Rate        5.00        computed
                                                by calculator
      Number of Years                18.00

      Paymernts Per Year             12.00

      Number of Payments            216.00         computed
                                                       by
      Payment              End      289.61         calculator
                           Begin    288.43
Calculators          Types of Calculations          Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS           213




In the aggressive assumption the required monthly payment
would be:

 $237.70 (annuity in arrears) or $236.37 (annuity due).

To calculate the above amount, clear the register and set
the Sinking Fund Calculator with the following values:

Mode               Begin       End


Present Value                   0.00

Future Value               100,000.00

Nominal Interest Rate           6.78

Effective Interest Rate         6.99       computed
                                          by calculator
Number of Years                 18.00

Paymernts Per Year              12.00

Number of Payments            216.00        computed
                                                by
Payment            End        237.70        calculator
                   Begin      236.37
Calculators         Types of Calculations              Definitions
214           FINANCIAL CALCULATIONS FOR LAWYERS




             Thus, with liberal assumptions, you must save $236.37
      per month from the day the child was born to generate, in
      eighteen years, an amount that would be the future equivalent
      of $100,000 today, regardless of the future inflation rate.
      With the alternative conservative and moderate assumptions,
      you must save at least $350.26 or $283.43 per month.

             Even these amounts, however, will not generate a fund
      equal to the future value of $100,000. That is true because
      they are uninflated. To compensate, you must increase the



                           Caution:
            You must annually increase the
         savings amount by the actual inflation
                        rate.


      amount of monthly savings by the actual inflation rate. To do
      this, simply multiple the monthly amount each year by one
      plus the reported increase in the Consumer Price Index.
      For example, if, during the first year of saving the inflation
      rate is 3.0%, the various savings amounts for the second
      year must be $243.46 (aggressive model), $291.93 (moderate
      model) or $360.77 (conservative model). These numbers
      are 1.03 times $236.37 (aggressive), $283.43 (moderate)
      and $350.26 (conservative).

              Continue to increase them annually. The resulting
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  215




                                     fund in 18 years should
                                     then be sufficient,
        Caution:                     assuming the original
                                     $100,000 figure was
       If you make the               accuarate and the
          aggressvie                 correct model was
      assumptions, but               chosen (aggressive,
      only achieve the               moderate,             or
         conservative                conservative).        If,
                                     however, you used the
    earnings, your fund
                                     aggressive model to
         will be short.
                                     compute the original
                                     savings amount, but you
                                     can only achieve the
                                     conservative model level
of earnings,your fund will be insufficient.

        If, in the alternative, you estimate that college costs
will increase on the average 2% faster than the general
Inflation rate, then you must save the equivalent of
$142,824.62 in 18 years. To compute this number,use the
Future Value of a Sum Calculator.

   Insert $100,000 for the Present Value, 2.0 for the nominal
annual interest rate, 18 for the number of years, and solve
for the Future Value. Using this number and our conservative
conclusions, we must save $337.59 (aggressive model),
$411.95 (moderate model), or $500.21 (conservative model)
per month, starting the day the child is born. These numbers
would then need to be adjusted annually by the actual inflation
rate.
Calculators               Types of Calculations          Definitions
216              FINANCIAL CALCULATIONS FOR LAWYERS




  PV Annuity                                            Interest Conversion
                  Future Value of a Sum Calculator       Amortization
   FV Annuity
  Sinking Fund                Instructions ON OFF           PV Sum



Mode
                              Begin     End
Present Value                   142,824.62

Future Value                          100,000

Nominal Interest Rate                    2.00

Effective Interest Rate
                                         2.00
                                        18.00
Number of Years
                                         1.00
Payments Per Year
                                        18.00
Number of Payments
                                         0.00
Payment

                                 clear all


                While none of these answers is precise, they give
        you realistic estimates without your having to estimate future
        inflation or future interest rates.



                            Did you Notice:
                 Problem 6a computes the required
                     payments on a home loan.
Calculators         Types of Calculations             Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS            217




     Amortization Schedule

6 a. You borrowed $150,000.00 from your grandmother
to purchase a new home. She said that she “wants to
earn an annual percentage rate of 7% on her money.”
How much will your monthly payments be on a thirty
year loan?

           $ 997.95 (7.0% APR with no points)

To calculate the above amount, clear the register and set
the Amortization Calculator          with the following
values:                     End


Mode

Present Value           150,000.00

Future Value                  0.00

Nominal Interest Rate          7.00

Effective Interest Rate        7.23          computed
                                          by calculator
Number of Years               30.00

Paymernts Per Year            12.00

Number of Payments           360.00           computed
                                                by
Calculators        Types of Calculations           Definitions
218           FINANCIAL CALCULATIONS FOR LAWYERS




                       CAUTION:
               Private lenders often misuse loan
                          terminology.




                 CAUTION:
            You must determine
        whether the lender wants to
         charge the stated interest
        rate or earn the stated rate.

         The two are not the same!
Calculators          Types of Calculations               Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                  219




Payment                        997.95           calculator

         Grandmother, however, stated that she “wanted to earn
a 7.0% APR on her money.” This involves a misuse of the
term APR, which is a nominal rate computed with points
amortized over the life of a loan. The misuse of terminology,
however, is commonplace. The term “annual percentage
rate” is commonly used because of federal lending disclosure
requirements. Unfortunately, it is also commonly used when
it is inapplicable.

        An investor does not earn an “APR”; rather, an investor
earns an annual yield or an effective interest rate. A lender,
in contrast, charges an APR; however, the lender “earns” an
effective rate. Hence, the problem cannot accurately be
worked without more information. The above solution is the



                            TIP:
   If the lender wants to “charge” 7.00%, he
   probably is comparing the rate to
   commercial loans. Thus use 7.00% nominal
   annual interest at the appropriate
   compounding period.


most literal constuction of her terminology; however, to be
certain, better information is needed.

       Did Grandmother want to “charge” 7.0% APR or did
Calculators          Types of Calculations               Definitions
220           FINANCIAL CALCULATIONS FOR LAWYERS




      she want to “yield” 7.0%? Grandmother may not have realized
      the technical meaning accorded that term. If she was
      measuring her desired interest by comparing it to comparable
      loan offerings by local financial institutions, then she probably
      intended the term’s technical meaning. If so, the above
      calculation would be correct.

            The above calculation assumed that Grandmother
      wanted to “yield” 7.0% on her money. Quite possibly, she
      measured her desired interest by comparing it to what she
      could earn in a certificate of deposit or other investment,
      then she probably intended to describe her desired annual




                                   TIP:
         If the lender wants to “earn ” 7.00%, he
         probably is comparing the rate to savings
         accounts. Thus use 7.00% effective
         interest, converted to the appropriate
         nominal rate and compounding period.
Calculators          Types of Calculations               Definitions
               FINANCIAL CALCULATIONS FOR LAWYERS               221




percentage yield, or effective interest rate.

 If so, the following would be the correct analysis:

       $ 976.39 (7.0% EFF or APY with no points)


To calculate the above                   amount, clear the
register and set the            End      Amortization
Calculator with the following            values:

Mode

Present Value             150,000.00

Future Value                    0.00

Nominal Interest Rate           6.785

Effective Interest Rate          7.00              computed
                                                by calculator
Number of Years                 30.00

Paymernts Per Year              12.00

Number of Payments              360.00            computed
Calculators               Types of Calculations              Definitions
222              FINANCIAL CALCULATIONS FOR LAWYERS




  PV Annuity                                                    PV Sum
   FV Annuity      Interest Rate Conversion                   Amortization
 Sinking Fund                 Instructions ON OFF               FV Sum


   Convert Effective          Convert Nominal        Convert Periodic
    Rate to Nominal            Rate to Effective      Rate to Nominal
   Rate and Periodic          Rate and Periodic      Rate and Effective
          Rate                       Rate                   Rate




                                                      Click here for the
Nominal Interest Rate              6.78497
                                                     correct converter for
Periodic Interest Rate                0.565              Problem 6a.

Effective Interest Rate                 7.00
                                      12.00
Payments Per Year

                                clear all

                                                                by
       Payment                              976.39          calculator

       To work this version of problem 6a, you must convert the
       7.00% effective rate to the comparable nominal annual rate
       compounded monthly. To do so, use the Interest Rate
       Conversion Calculator.

        6 b. Compute the same amount, however, do it for a
       fifteen year loan.

                   $ 1,348.24 (7.0% APR with no points)

                $ 1,330.28 (7.0% EFF or APY with no points)
Calculators         Types of Calculations              Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS                223




     Note that in each Instance, an increase in the payments
of approximately 35%, results in a decrease in the term of
50%, going from a 30-year to a 15-year loan. From the
opposite perspective, going from a 15-year to a 30-year loan,
if the term doubles, the payments drop merely by
approximately 25%.

       To calculate the above amounts, you need not clear
the register for the Amortization Calculator. Instead, merely
change the Number of Years to 15. The new payment will
appear automatically.




                          TIP:
          Increasing the payments by
      approximately 35% corresponds to a
         decrease in the term of 50%!

     Stated differently, double the term and
    the payment drops by onely one-fourth!
Calculators           Types of Calculations               Definitions
224             FINANCIAL CALCULATIONS FOR LAWYERS




      c. You owe $100,000 for student loans, at an APR of
      8%. The loan bears no interest until six months after
      graduation. You have the option of paying off the loan
      over 10, 15, 20, or 30 years. What will be the amount of
      your payments?

                      $1,205.24 (10 year payoff)
                      $ 949.32 (15 year payoff)
                      $ 830.90 (20 year payoff)
                      $ 728.91 (30 year payoff)
      To calculate the above amounts, clear the
      register and set the Amortization Calculator with the following
      values:

      Mode                            Begin

      Present Value             100,000.00

      Future Value                    0.00

      Nominal Interest Rate           8.00

      Effective Interest Rate         7.00           computed
                                                  by calculator
      Number of Years                10.00

      Paymernts Per Year             12.00

      Number of Payments             120.00           computed
                                                         by
      Payment                        1,205.24        calculator
Calculators        Types of Calculations           Definitions
              FINANCIAL CALCULATIONS FOR LAWYERS          225



                PresentValueCalculator


                Future Value Calculator


              Present Value of an Annuity
                      Calculator

              Future Value of an Annuity
                      Calculator

          Amortization Calculator (Begin
                      Mode)

       Amortization Calculator (End Mode)


                Sinking Fund Calculator


                Interest Rate Converter


                    Yield Calculator

				
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