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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
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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
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
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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.

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.
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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.
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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.

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
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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
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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,
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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.
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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

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.
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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.
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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
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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

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:
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
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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,

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.
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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 -
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.]

[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.

-1,210.

-1,610.51.

-13,780,612.34.

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

-83,522,265.73.

-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.

-              1,610.51.

1750                             [The +/- key enters the value as a
negative. If you forget to do this, the

Then Change the Future Value to -2000.

2000

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

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.]

[This is the Effective Interest Rate.

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

12               12

[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

[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.]

[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

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

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.]
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.

-909.090909091.

-683.013455365.

-0.079822287.

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

-0.013170141.

-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

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

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
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.

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

[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
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
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.
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]

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.

-9,364.92009173.

-10,906.2959359.

-10,999.2017771.

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

-9,333.22158668

-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.

-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.]

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

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
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.]

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
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
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
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

This is the future value of \$1000 in five years
earning 3% EFF.

4

This is the future value of \$1000 in five years
earning 4% EFF.

7

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
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-
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

4

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-

√    If you have student loans outstanding, this function will

√    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

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

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
stallment payments, in-
TIP                     cluding all interest due and
some principal. Those
From the lender ’s
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,

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-
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
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,
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-
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

The display will
7.000

6.364                                    The display will

“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

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

The display will

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

The display will

The simple yield is        7.778.
The yield is               8.008.
The yield to maturity is   8.769.
Calculators        Types of Calculations           Definitions
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
1.892935887.
The display will
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
5.682226203
The display will
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.
Calculators          Types of Calculations               Definitions
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.”
Calculators            Types of Calculations                  Definitions
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.
Calculators        Types of Calculations              Definitions
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
Calculators          Types of Calculations               Definitions
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

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-

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

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
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
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

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

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

\$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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