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Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS AND THE TIME VALUE OF MONEY Explanation and Examples M any areas of law require a working knowledge of financial calculations. For example, Tort Law Practice often involves calculating the present value of lost future wages. Family Law Practice involves valuing a stream of future income or a deferred compen- sation plan. Tax Law Practice necessitates an under- standing of the Internal Revenue Code time value of money sections, which themselves require a working knowledge of financial calculations. Inexpensive calculators have alleviated the need for lawyers to understand the actual formulas; however, be- cause the calculators and their respective manuals are often complicated, lawyers lacking an accounting or fi- nance background may shy away from this important area of law. This booklet serves three purposes: 1. It provides a basic explanation - with law- yers as the intended audience - of the use and application of a typical hand-held Calculators Types of Calculations Definitions 2 FINANCIAL CALCULATIONS FOR LAWYERS financial calculator, the Hewlett Packard 10 Bii. 2. It discusses the legal system’s use of financial terminology. 3. It includes workable JavaScript Financial Calculators Calculators that solve most of the prob- lems faced by lawyers. I. USE OF A CALCULATOR My first advice is “read the manual.” Most financial cal- culator manuals explain the various types of calculations and provide understandable examples. This book does not preempt or replace those manuals for other calcula- tors. Rather, it supplements them with explanations and examples geared toward lawyers. For users who rely solely on the JavaScript Finan- cial Calculators included on the attached CD ROM, this booklet serves as the instruction manual. A. TYPES OF CALCULATIONS While financial calculators can compute many things, six types of calculations are fundamental: 1. Present Value of a Sum 2. Future Value of a Sum 3. Present Value of an Annuity Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 3 4. Future Value of an Annuity 5. Sinking Fund 6. Amortization For Tax Lawyers, each of these calculations is rel- evant to one or more Internal Revenue Code provisions. For example, section 7872 - dealing with below market loans - requires the use of Present Value of a Sum and Present Value of an Annuity functions. Sections 1272 and 1274 - dealing with original issue discount loans - involve Amortization. And, section 467 - dealing with prepaid or deferred rent - involves the use of a Sinking Fund. Future Value of an Annuity, as well as Sinking Fund calculations are relevant to deferred compensation and retirement planning. For Family Law and Tort Lawyers, each is also relevant, particularly the Present Value and Sinking Fund calculations. For example, a Tort Lawyer will often need to compute the present value of lost future wages: i.e., the Present Value of an Annuity. A Family Lawyer might similarly need to value a business using the Present Value of an Annuity Calculation. Or, he might utilize a Sinking Fund in computing needed savings for a child’s educa- tion, as part of an agreed marital settlement. General Practitioners and Real Estate Attorneys will find the amortization calculations particularly useful, as they compute the needed payments on a home loan. The Present Value calculators are relevant for contracts requiring advance payments; similarly, the Future Value calculations are relevant for contracts involving deferred Calculators Types of Calculations Definitions 4 FINANCIAL CALCULATIONS FOR LAWYERS payments. Each calculation relies on the same basic formula, involving six factors, with the typical key label. The JavaScript Calculators included on the CD use the words for each function: JavaScript 10Bii Key Key 1. The present value (PV) Present Value 2. The future value (FV) Future Value 3. The interest rate per year (I/YR) Nominal Interest Rate 4. The number of periods or pay ments per year (P/YR) Payments Per Year 5. The amount of each payment (PMT) Payment 6. The number of periods (N) Number of Payments Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 5 In the typical example, five of the six factors are known. The calculator can then easily solve for the sixth. In addition, annuity calculations re- quire a mode setting, indicating whether payments occur at the beginning or the end Mode of a period. B.COMMON DIFFICULTIES Unless the calculator is defective, which is unlikely, it will produce the correct answer if given the correct information. Nevertheless, many users, at one time or another, exclaim “This thing doesn’t work!”1 Usually, they have violated one of the following rules: 1. COMMON DIFFICULTY: First, clear the machine. A calculator knows only what you tell it and it does not forget until you tell it to forget, typically even if you turn off the machine. Thus, be certain to clear all functions and memory when beginning a new calculation. This is particularly important for hand-held calcula- Clear the tors, such as the HP Machine 10Bii: the display shows only one func- tion at a time, creating the risk that the user will not remember to clear all other functions. The 1 In frustration, I’ve said it myself many times. I was, however, always wrong as to that point. Calculators Types of Calculations Definitions 6 FINANCIAL CALCULATIONS FOR LAWYERS JavaScript Calculators included on the attached CD do not have this risk because the display shows all function values at all times. All calculators have a clear key, usually denominated with a C or the word clear. In addition, many calculators have a function key by which merely the last information entered can be cleared, and a different function key by which all information can be cleared. a. HP lOBii Calculator The HP lOBii calculator has three levels for the clear function. 1. The C key - when pressed in unshifted mode - will clear the entire displayed number; however, it leaves the memory intact. See Example 1. 2. The back arrow key will clear single digits, one at a time. See Example 2. 3. The C ALL key when pressed in the “shifted mode” will clear the entire memory, as well as the displayed number. To perform this function on an HP10Bii calculator,2 first press the orange downshift key and then press C ALL. These strokes shift the function to C ALL (clear all) rather than C (clear). Before working a new problem, you should press these keys: 2 The older HP 10B has a CLEAR ALL button instead. Some use green for the shift color. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 7 Do not, however, perform the clear all function in the middle of a problem in which you are comparing alternative values of a particular element. Caution: The HP10Bii C ALL function does not clear either the periods per year (P/YR) or mode (BEG/END). You must change these manually. EXAMPLE 1 (HP 10Bii) If you input 50 + 20 + 30 but intended 50 + 20 + 40, press C erasing the 30 but leaving the 70 in memory. You can then press 40 and = . The display will then read 110. 50 + 20 + 30 40 The display will read 110. Calculators Types of Calculations Definitions 8 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 2 (HP 10Bii) If you input 523 but intended 524, you may use the backward arrow key to erase the 4. Then simply enter the number 3. The display will read 523. In con- trast, the C key will clear the entire number 523. 3 The C ALL function does not reset the number of periods per year. If you change this setting, it will remain - even if you turn off the calculator - until you manually change it or remove the battery. Also, the C ALL function does not change the mode. Thus if you reset the mode from end to begin, or vice versa, it will remain - even if you turn off the calculator- until you reset it manually through the procedure described below or remove the battery. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 9 TIP The JavaScript Calculators are much easier to clear than are typical hand-held machines, thus producing fewer mistakes. b. JavaScript Financial Calculator The JavaScript Financial Calculator has two clear functions: clear all 1. A Clear All button. This will reset all numbers to their default amount. 2. The Backspace key. Using this key on a typical computer keyboard will erase single or multiple digits just as it does with any other program. Because the included JavaScript Calculators dis- play all function values at all times, the risk of a user failing to clear some values - and thus computing a wrong value - is largely eliminated. To test the two "clear" functions, type in a number in the above box. Use the backspace key to erase it. Or, click on the "clear all" button to clear the box. Calculators Types of Calculations Definitions 10 FINANCIAL CALCULATIONS FOR LAWYERS 2. COMMON DIFFICULTY: Set the cash flows with the proper sign. Many, but not all, calcu- lators require that cash flows be directional. This means that one set of cash flows must be positive and the other must be negative. This is true of the HP 10Bii Calculator. It is not true of the JavaScript Calculators on the enclosed CD. For example, in machines such as the HP 10Bii, the present value amount may be expressed as a positive number - a Set the deposit - while the fu- ture value amount will Cash Flows be expressed as a Correctly negative number - a withdrawal. Or, the opposite may be true; however, the present and future values cannot both be posi- tive or both be negative at the same time. On the other hand, some calculators - such as the JavaScript Financial Calcula- tor - eliminate this feature. Hence, be sure to read your own- ers manual. a. HP lOBii Calculator 3 The HP lOBii calculator requires that cash flows be entered with opposite signs. As shown in Example 3, failure to do so will prompt the display no SoLution. A negative number may be entered in two ways. For example, to input the number (1000), first enter the positive number 1000, then press the “plus/minus” key: 3 Many other calculators - such as those manufactured by Texas Instruments - do not require opposite signs between present value and future value. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 11 1000 This will change the sign from positive to negative or back from negative to positive. In the alternative, press the minus sign, the number, and then the equal sign, as follows: 1000 The display will show a negative 1000. EXAMPLE 3 (HP 10Bii) Suppose you want to compute the annual interest rate inherent to a present value of 500, a future value of 1000 and a period of 10 years. The correct answer is 7.177346254. To achieve this, either the 500 or the 1000 must be expressed as a negative number while the other must be positive. To enter 500 as a nega- tive number, press: 500 Calculators Types of Calculations Definitions 12 FINANCIAL CALCULATIONS FOR LAWYERS b. JavaScript Financial Calculator The JavaScript Financial Calculator eliminates the need to input cash-flows directionally. Hence, all numbers may be entered as positive numbers. The calculator then converts them, as appropriate. Thus the enclosed JavaScript Calculators eliminate the second most common difficulty faced by users of hand- held calculators. TIP The JavaScript Calculators eliminate the need for cash- flow inputs and negative num- bers. 3. COMMON DIFFICULTY: Set the mode correctly. Calculations involving annuities, sinking funds and amortizations, require a “mode” setting: either begin mode Set the Mode Correctly Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 13 or end mode. This is true of all calculators, including the HP 10Bii as well as the JavaScript Calculators. Begin mode applies if payments (or deposits or with- drawals) occur at the beginning of each period. End mode applies if payments occur at the end of each period. Typically, a sinking fund uses the begin mode because the depositor wants to begin immediately. Typically, an am- ortization - such as the repayment of a loan - uses the end mode because loan payments do not begin on the date of the loan. Instead, loan payments begin at the end of each period. For example, payments on a car loan typically start one month after the purchase. Using begin mode for a loan amortization generally makes little sense: a payment on the date of the borrowing merely collapses to a lower amount borrowed, resulting in end mode. “Future Value of a Sum” and “Present Value of a Sum” calculations are not affected by the mode setting. a. HP lOBii Calculator To set the mode on an HP lOBii calculator, first press the orange shift key and then press the BEG/END key to operate the mode function. Most calculators are preset at the factory in end mode. Pressing these two keys will change it to begin mode, which the display will note with the word BEGIN. To revert to end mode, press the two keys again. The display will no longer indicate the mode. If you change the setting to begin mode, Calculators Types of Calculations Definitions 14 FINANCIAL CALCULATIONS FOR LAWYERS it will remain, even if you turn off the calculator or utilize the clear all (C ALL) function. To revert to end mode, you must do so manually by repeating the above steps. A common mistake among calculator users involves Begin Mode annuities, sinking funds, or amortizations. Be- cause the HP10Bii display does not indicate End Mode, the user may forget to change the mode to Begin, thus produc- ing significant (but not obvious) incorrect results. The de- fault setting is for End Mode because that is consistent with most amortizations, a common calculation involving Mode. Caution: With the HP10Bii calculator, an indication of Mode setting appears only with Begin Mode. Thus, be careful when com- puting annuities: you might be in End Mode and not know it! The JavaScript Calculators eliminate this risk by always indicating mode. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 15 EXAMPLE 4 (HP 10Bii) (mode illustration) Your child was born today. You would like to accumulate $100,000 when she reaches the age of 18. You ex- pect to earn 6% nominal annual in- terest compounded monthly (after tax). To de- termine how much you must deposit, press: 12 216 6 100,000 The display will read 258.16. The display will read 256.88. Calculators Types of Calculations Definitions 16 FINANCIAL CALCULATIONS FOR LAWYERS However, sinking funds and annuities commonly use Begin Mode, necessitating a different calculator setting. For example, if you were saving for a child’s educa- tion and desired to make monthly deposits, a sinking fund calculation can tell you the necessary monthly deposit to make, depending on the child’s age and the expected inter- est rate. If you were to begin the deposits today, you would use Begin Mode. Or, if you to begin making the deposits at the end of the first month, you would use End Mode. As shown in Example 4, new parents who desire to accumulate $100,000 for their child’s 18th birthday and who expect to earn 6% nominal annual interest compounded monthly, must deposit $258.16 monthly if they begin making deposits at the end of Month 1. Or, they need deposit only $256.88 if they begin immediately. Although the differences may seem slight in this problem, they can be material in many other situations. TIP A Mode setting is necessary only for annu- ities, sinking funds, and amortizations. It does not apply to Present Value or Future Value of a Sum calculations. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 17 b. JavaScript Financial Calculator The JavaScript Financial Calculator has buttons labeled Begin and End to designate the mode. Begin Also, when in Begin Mode, the words begin mode End appear in red. Similarly, in End Mode, the words end mode appear in black. Hence you are unlikely to forget to set the mode correctly. This effectively eliminates the third most common difficulty with using hand-held calculators. 4. COMMON DIFFICULTY: Set the inter- est rate to compound for each payment period. This involves the P/YR button Pressing the orange (some- times green) shift key along with the P/YR key sets the num- ber of payments per year. For example, to set the calculator for quarterly payments, press the following: 4 Set the Interest Payment and Compounding Periods the Same. Calculators Types of Calculations Definitions 18 FINANCIAL CALCULATIONS FOR LAWYERS A basic law of finance is that the compounding period and the payment period must be the same. Thus, if the facts provide for annual payments, the interest rate must be stated as an annual rate. If, instead, the facts provide for semian- nual payments, then the interest rate must be stated as a semiannual rate. Likewise, monthly payments call for a monthly interest rate. Law of Finance Setting the P/YR to the correct amount to corre- The compounding spond with payments re- period and the pay- quires the user to have the correct information. This, ment period must be in turn, necessitates that the same. the interest rate be stated using correct terminology. As a result, common prac- tice involves stating inter- est rates using a nominal annual uncompounded format along with a statement of the compounding period. Caution: Always define the interest rate using correct terminology. Interest Rate Definitions Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 19 If, however, the facts - such as a contract - do not state the interest rate in such a format, you must convert them to a nominal annual uncompounded format or the equivalent. Oth- erwise, no calculator will produce the correct answer. Several scenarios are possible: 1. CONTRACT SCENARIO: The facts given (such as in a contract) may state a “nominal annual interest rate” as well as a “compounding period.” This would be the correct format for stating an inter- est rate. If the compounding period is the same length as the payment period (e.g., semiannual com- pounding and payments every six months), simply enter the stated “nominal annual interest rate” using the I/YR function. a. HP lOBii Calculator A 10% nominal annual interest rate would be entered as: 10 Use of the NOM% function is optional: you could also enter the 10% nominal annual interest rate as: 10 Typically, however, the NOM% key is used for conversion of an effective rate to a nominal rate. It is unnecessary for entry of a given nominal rate. Calculators Types of Calculations Definitions 20 FINANCIAL CALCULATIONS FOR LAWYERS b. JavaScript Financial Calculator The JavaScript Calculator does not have separate I/ YR and NOM% buttons. Thus the 10% nominal annual inter- est rate would be entered simply as 10. Because the compounding period and payments per year function are the same in this scenario (semiannual com- pounding and payments every six months), they are both entered through the P/YR function. This is true both on the HP10Bii and JavaScript Calculators. On the HP10Bii, semiannual payments and semian- nual compounding (as assumed above) would be entered as: 2 On the JavaScript Calculator, semiannual payments and compounding are both entered as 2 in the Payments Per Year box. Payments Per Year 2.00 Each calculator will automatically convert the interest rate to a periodic rate of 5% per period by dividing the I/YR amount by the P/YR amount. This is internal to the calcula- tor and is not displayed on either the HP10Bii or on the JavaScript Calculator (although many other calculators dis- play the periodic rate). Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 21 2. CONTRACT SCENARIO: The facts given (such as in a contract) may state a “nominal annual interest rate” without also stating a “compound- ing period.” This would not be the correct format for stating an interest rate, but is also not unusual. In such a case, the parties should clarify the compounding period before proceeding. If this is not done, most users would probably assume annual compounding (if pay- ments are no more frequent than annual) or com- pounding coextensive with the payment period. With such assumptions, they would proceed as above in scenario one. Caution: Always define the interest rate using correct terminology. Interest Rate Definitions If this assumption is not consistent with one party’s understanding, litigation may result, illustrating the need for clear statement of interest rates. Neither the HP10Bii nor the JavaScript Calculator can solve this problem because it results from a poorly drafted Calculators Types of Calculations Definitions 22 FINANCIAL CALCULATIONS FOR LAWYERS document. The parties must clarify the interest rate or suffer the inevitable confusion and litigation. 3. CONTRACT SCENARIO: The facts given (as in a contract) may state a “nominal annual inter- est rate” as well as a “compounding period.” If the facts also provide for a payment frequency inconsistent with the compounding period, conver- sion of the interest rate would be necessary. a. HP lOBii Calculator For an HP10Bii Calculator, this is a two-step pro- cess. First, this requires conversion of the “nominal annual interest rate” to an “effective rate” and then second, recon- version to a “nominal annual interest rate” with a compound- ing period consistent with the payment period. Box One Caution: Always define the in- terest rate using cor- rect terminology. The JavaScript Inter- est Rate Conversion Calculator is designed to help you do this. Interest Rate Definitions Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 23 illustrates Step One: how to convert a “nominal annual inter- est rate” to an “effective interest rate” using an HP10Bii Cal- culator. Example 5a illustrates Step Two: how to convert the “effective interest rate” to the “nominal annual interest rate compounded semi-annually”using an HP10Bii Calculator. Once the user changes the number of periods per year (P/ YR) to correspond with the payment frequency, the calcula- tor automatically computes the correct nominal annual rate. The user must, however, press the shift and NOM% keys to make this effective. b. JavaScript Financial Calculator Each of the various financial calculators automatically converts a nominal rate to the equivalent effective rate. The user must enter the nominal rate and the calculator displays the effective rate. No buttons need to be pressed. For example, a nominal interest rate of 10% with a compounding period of six months (two payments or periods per year) results in an effective interest rate of 10.25%. 10.00 Nominal Interest Rate 10.25 Effective Interest Rate Number of Years 2.00 Payments Per Year Calculators Types of Calculations Definitions 24 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 5 An “effective interest rate” of 10% with semi-annual payments is not equivalent to a semi-annual rate of 5%. Instead, it is the equivalent of 4.8808848170% semi-an- nually (the periodic rate) or 9.761769636% nomi- nally (compounded semi-annually). Thus, 9.761769636% nominal annual interest, compounded semi-annually is the same as 10.00% nominal interest, compounded annually. In contrast, two $500 payments - one each six months - is not the equivalent of a single $1000 payment. A single payment of $1000 has a future value of $1,100 at an effective interest rate of 10%. Two semi-annual $500 payments at an effective interest rate of 10% have a future value of $1,024.40, assuming an annuity in arears or $1,074.40, assuming an annuity due. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 25 EXAMPLE 5 a (HP 10Bii) (from Box One) √ Re-set the number of payments per year to correspond with the payment period. Press: 2 √ Convert to the nominal Interest Rate compounded semi-annu- ally: The display will read 10.210662664, which is the nominal annual interest rate compounded semi-annually. √ Enter other factors normally: 2 500 √ Press either FV or PV to solve for the desired factor: The display will read -1,077.88319038 (begin mode) or -1025.52665666 (end mode). or The display will read -975.713261795 (begin mode) or -928.319476691 (end mode). If you want to compute both the FV and the PV, you must re-set the alternative values to zero between the computation. For example, if you first compute the FV as above, before pressing PV, enter the following: 0 Calculators Types of Calculations Definitions 26 FINANCIAL CALCULATIONS FOR LAWYERS BOX ONE Convert Nominal Rate to Effective Rate T o convert a nominal annual interest rate compounded periodically to an effective interest rate, enter the following values. This ex- ample assumes a 10% nominal annual interest rate compounded monthly and semi-annual payments of $500. √ Set the number of payments per year to correspond with the stated compounding period. Press: 12 √ Enter the nominal Interest Rate: 10 The nominal rate may be entered using either the I/YR or NOM% func- tions; thus, use of the orange shift key is optional for this function. √ Press the EFF% button: The display will read 10.471306744, which is BOTH the effective an- nual interest rate and the nominal annual interest rate compounded annually. Go to Example 5a for completion of this example. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 27 BOX TWO Convert Effective Rate to Nominal Rate T o convert an effective annual interest rate to a nominal annual interest rate compounded periodically, enter the following val- ues. This example assumes an effective rate of 10% and semi- annual payments of $500. √ Set the number of payments per year to correspond with the facts. This uses the P/YR button as a payments per year button, its traditional function. Press: 2 √ Set the Effective Interest Rate: 10 √ Press the NOM% button: The display will read 9.761769634, which is the nominal annual interest rate compounded semi-annually equivalent to 10% effec- tive interest rate or 10% nominal annual interest compounded an- nually. Go to Example 5 b for completion of this example. Calculators Types of Calculations Definitions 28 FINANCIAL CALCULATIONS FOR LAWYERS Interest Rate Converter Box Three Convert Effective Rate to Nominal Rate and Periodic Rate JavaScript Financial Calculator PV Annuity PV Sum FV Annuity Interest Rate Conversion Amortization Sinking Fund Instructions ON OFF FV Sum Convert Effective Convert Nominal Convert Periodic Rate to Nominal Rate to Effective Rate to Nominal Rate and Periodic Rate and Periodic Rate and Effective Rate Rate Rate Nominal Interest Rate 9.7617696340 Periodic Interest Rate 4.8808848170 Effective Interest Rate 10.00000000 2.00 Payments Per Year clear all 1. Press "Clear All" prior to working a problem. 2. Select the appropriate conversion function. 3. Type appropriate numbers in the white boxes. 4. The answer will appear in the green boxes. 5. Place your cursor over terms for a definition of the term. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 29 Interest Rate Converter Box Four Convert Nominal Rate to Effective Rate and Periodic Rate JavaScript Financial Calculator PV Annuity PV Sum FV Annuity Interest Rate Conversion Amortization Sinking Fund Instructions ON OFF FV Sum Convert Effective Convert Nominal Convert Periodic Rate to Nominal Rate to Effective Rate to Nominal Rate and Periodic Rate and Periodic Rate and Effective Rate Rate Rate Nominal Interest Rate 10.000000000 Periodic Interest Rate 0.833333333 Effective Interest Rate 10.4713067441 Payments Per Year 12.00 clear all TIP This feature - conversion to all three ways of stating interest - is particularly helpful for legal documents. A well-drafted document will state both the nominal rate (including the compounding fre- quency) plus the effective rate and periodic rate. Calculators Types of Calculations Definitions 30 FINANCIAL CALCULATIONS FOR LAWYERS Also, as shown in Box Four, the Interest Rate Con- version Calculator will automatically convert a Nominal Rate to both an Effective Rate and a Periodic Rate. This feature is particularly helpful for legal documents. A well-drafted document will state both the nominal rate (including the com- pounding frequency) plus the effective rate. 4. CONTRACT SCENARIO: The facts given (such as in a contract) may state an “effective annual interest rate.” As in the first scenario, this would also be a correct format for stating an interest rate. If the facts also provide for payment frequency other than annual, conversion of the interest rate would be neces- sary. This would require the conversion of the “effective interest rate” to a “nominal annual interest rate” with a com- pounding period consistent with the payment period. a. HP lOBii Calculator Box Two illustrates how to convert an “effective inter- est rate” to a “nominal annual interest rate” using an HP lOBii Calculator. It involves entering the stated effective rate and then using the NOM% key to convert it to the ap- propriate nominal rate for the P/YR already entered. Example 5b illustrates the completion of the problem. As is evident, the process is not difficult; however, it is es- sential: without conversion of the stated interest rate to one corresponding with the payment period, the calculator would provide the wrong answer. This further shows the impor- tance for the proper statement of interest rates in any legal document. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 31 EXAMPLE 5 b (HP 10Bii) (from Box Two) √ Enter other factors normally: 2 500 √ Press either FV or PV to solve for the desired factor: The display will read -1,024.40442409 (end mode) or -1,074.40442408 (begin mode). or The display will read -931.276749168 (end mode) or -976.731294623 (begin mode). If you want to compute both the FV and the PV, you must re-set the alternative values to zero before the alternate computation. For example, if you first compute the FV as above, before press- ing PV, enter the following: 0 Calculators Types of Calculations Definitions 32 FINANCIAL CALCULATIONS FOR LAWYERS Caution: Never interchange interest rate termi- nology: nominal, an- nual, and effective percentage rates are terms of art: use them correctly. b. JavaScript Financial Calculator The Interest Conversion Calculator converts an ef- fective rate to a nominal rate and the corresponding periodic rate. Simply insert the effective rate in the appropriate box (labeled Effective Interest Rate) and insert the number of Payments per Year. Box Three demonstrates how. This calculator will also convert a nominal annual rate to the equivalent effective rate and periodic rate. In addition, it contains a feature to convert a periodic rate to the equiva- lent nominal annual rate and effective rates. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 33 5. CONTRACT SCENARIO: The facts given (such as in a contract) may state an “annual interest rate” or an “annual percentage rate.” This would not be the correct format for stating an interest rate. In such a case, the parties should clarify the terminology before proceeding. If this is not done, most users would probably assume the terms to be the equivalent of an “effective interest rate,” although some may disagree. If this assumption is not consistent with one party’s understanding, litigation may result, illustrat- ing the need for clear statement of interest rates. Neither the HP10Bii nor the JavaScript Calculator can solve this problem because it results from a poorly drafted document. The parties must clarify the interest rate or suffer the inevitable confusion and litigation. To reiterate: if the stated interest compounding period and the payment period are not the same, you must convert the interest rate to an equivalent one using a compounding period identical with that of the payments. Some calculators do this automatically with a feature labeled Iconv. Others require the user to make the computations, which are not difficult. Box One explains how using an HP10Bii. Addi- tional examples of such a conversion appear. Also, as shown in Example 5, if the interest com- pounding period and the payment period are not coexten- sive, you must convert the interest rate to an equivalent rate compounded consistent with the payment period, rather than convert the payment period to the stated compounding pe- riod. Calculators Types of Calculations Definitions 34 FINANCIAL CALCULATIONS FOR LAWYERS Notice that Examples 5a and 5b have different re- sults. This occurs because they begin with different as- sumptions. Example 5a uses the BOX ONE assumption of a 10% effective interest rate. Example 5b uses the BOX TWO assumption of a 10% nominal rate. This further illus- trates that you cannot interchange the different interest rate terminology. 5. COMMON DIFFICULTY: Set the periods per year correctly. Most calculators are factory preset for twelve periods per year. This assumes the common facts of monthly payments and monthly compounding. All calculators, how- ever, can easily be reset. For example, in a problem Set the involving a single annual payment and annual com- Periods pounding, the payments Per Year per period (P/YR) function must be set at one. Correctly a. HP lOBii Calcula- tor The factory default setting of an HP 10Bii is twelve pay- ments per year. To change the factory setting, press the desired number - 1 - then press the orange shift key and then P/YR, to set the payments per year, as follows: 1 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 35 You can check the setting by pressing the orange shift key and then holding down the C ALL (clear all) key: The display will indicate the number of payments per year. This will remain as the setting - even if you turn off the calcu- lator or utilize the clear all (C ALL)function - until you manu- ally reset the payments per year, using the above proce- dure. b. JavaScript Financial Calculator The JavaScript Calculator has the payments per year preset to one. Changing it is simple: merely type the appro- priate number in the space provided. Placing the cursor on the “Periods/Payments Per Year” label provides additional explanation. As illustrated in Example 6, changing the payments per year setting is not always necessary. Some people find it easier to set the PIYR function to one, meaning one pay- ment per period, rather than one payment per year. Then, in entering an interest rate they always enter a periodic rate. In the Problems and Answers section of this booklet, this alternative method is labeled the Periodic Method. It actually represents the math formula used by the calculator, which uses a periodic interest rate rather than an annual rate. For convenience, most calculators permit the entry of an annual, uncompounded rate, which the machine converts to a periodic rate. For example, if a problem calls for ten years of monthly Calculators Types of Calculations Definitions 36 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 6 (HP 10Bii) (calculator method) Compute the present value of an annuity in ar- rears of 1000 per month for ten years at 12% nomi- nal annual interest compounded monthly. Be cer- tain the calculator is set in END Mode. Press the following keys: 12 1000 12 120 The display will read - 69,700.5220314. Did You Notice? Both methods of calculating compound interest use the nominal annual interest (NAI). As a practical matter, any legal document should always de- scribe interest in NAI terms. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 37 EXAMPLE 7 (HP 10Bii) (periodic method) Compute the present value of an annuity in ar- rears of 1000 per month for ten years at 12% nomi- nal annual interest compounded monthly. Be cer- tain the calculator is set in END Mode. Press the following keys: 1 1000 1 120 The display will read - 69,700.5220314. TIP You may enter the various values in any order. For example, you could enter the I/YR prior to entering the PMT. Calculators Types of Calculations Definitions 38 FINANCIAL CALCULATIONS FOR LAWYERS Present Value of an Annuity Calculator EXAMPLE 6 (JavaScript Calculator) Amortization Interest Conversion Present Value of an Annuity Calculator FV Annuity PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 69,700.52 Future Value 0..00 Nominal Interest Rate 12.00 12.6825030132 Effective Interest Rate 10.00 Number of Years 12.00 Payments Per Year 120.00 Number of Payments 1000.00 end Payment mode clear all Did You Notice? The JavaScript Calculators always denotes the mode setting - even in end mode. This will help you avoid costly mistakes which can result from an incorrect setting. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 39 Present Value of an Annuity Calculator EXAMPLE 7 (JavaScript Calculator) Amortization Interest Conversion Present Value of an Annuity Calculator FV Annuity PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 69,700.52 Future Value 0..00 Nominal Interest Rate 1.00 1.0000000000 Effective Interest Rate 120.00 Number of Years 1.00 Payments Per Year 120.00 Number of Payments 1000.00 end Payment mode clear all Calculators Types of Calculations Definitions 40 FINANCIAL CALCULATIONS FOR LAWYERS payments at 12% nominal annual interest, the calculator is indifferent to whether it is told 12 P/YR at 12% interest or 1 P/YR at 1% interest. In either case, the N setting must be 120 to indicate the correct number of payments. Example 6 illustrates the Calculator Method, while Example 7 illustrates the Periodic Method . In the Calculator Method of Example 6, the calcula- tor divides the 12% nominal interest rate by the number of payments per year (12 P/YR) to achieve the correct peri- odic interest rate of 1%. The calculator then uses the peri- odic rate of 1% to execute the formula. Caution: This explanation of the Periodic Method is op- tional. While some us- ers will find it easier, most will prefer the Calculator Method. In the Periodic Method of Example 7, the user tells the calculator the periodic interest rate. The calculator then makes no adjustment before executing the formula. Also, the user sets the payments per year (P/YR) at one. Essen- Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 41 tially, the P/YR function becomes a payments per period function permanently set at one and thus irrelevant. As a practical matter, some people find it easier to remember to enter the correct periodic rate than to remem- ber to enter the correct number of payments per year. They thus set the calculator to one payment per year and leave it at that setting for all calculations. In contrast. people who use the Calculator Method must always remember to set the P/YR function correctly, as it varies from problem to prob- lem. Because most hand held calculators, such as the 4 HP10Bii have no clear way of indicating the P/YR setting, a user can easily forget to reset the amount, resulting in an incorrect answer which may not be obviously incorrect. Whichever method works best for the individual user is the one he or she should use. 4 The HP10Bii indicates the P/YR setting whenever the user presses the C ALL button, a two-step process. While simple, this function is easily forgot- ten or ignored, particularly in problems involving many variables and alterna- tives. In such a problem, use of the C ALL function destroys already input- ted information, requiring the user to start over; hence, in such cases, the user may knowingly not want to double check the P/YR setting. This can result in complacency that itself results in the user forgetting to input the P/ YR amount correclty. Calculators Types of Calculations Definitions 42 FINANCIAL CALCULATIONS FOR LAWYERS 6. COMMON DIFFICULTY: Set the display to the correct number of decimal places. Although most calculations involve dollars and cents and thus two places after the decimal, large numbers and long periods of time can be significantly affected by rounding. While the calcula- tor internally uses twelve places after the decimal for calcula- tions, it displays only the number pursuant to its setting. Be- cause some calculations involve the user writing down or otherwise reusing a computed number, it may be helpful to have the display read the full nine spots after the decimal. a. HP lOBii Calculator As with most calculators, the factory setting of an HP lOBii is for two places after the decimal. To change this to nine - the maximum permanent display - press the following keys: 9 The display will then show nine places after the decimal. To change this to any other number from one to eight, reenter the key strokes, using the desired number of places. To display twelve numbers (with no decimal place) press the orange shift key and DISP. The display will temporarily show twelve digits. b. JavaScript Financial Calculator The JavaScript Calculator is set to display two deci- mal places for most items, although it calculates to ten places. The settings cannot be changed by the user. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 43 Set the Display for the Correct Number of Decimal Places Calculators Types of Calculations Definitions 44 FINANCIAL CALCULATIONS FOR LAWYERS C. CALCULATIONS As explained earlier, six types of calculations are fundamental to a lawyer’s practice: 1. Future Value of a Sum 2. Present Value of a Sum 3. Present Value of an Annuity 4. Future Value of an Annuity 5. Sinking Fund 6. Amortization Separate JavaScript Financial Calculators exist for each type of calculation. 1. Future Value of a Sum This calculation computes the future amount or value of a current deposit. For example, $1,000 deposited today, earning 10% interest compounded annually, will increase to $1,100 in one year. In two years it will increase to $1,210. In five years, it will be $1,610.51 and in 100 years it will be $13,780,612.34. As shown in Example 8, to calculate this, input the five known factors into the calculator and solve for the un- Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 45 EXAMPLE 8:(HP 10Bii) Future Value Compute the Future Value of $1,000 depos- ited today, earning 10% nominal annual inter- est compounded annually. Press the illustrated keys. [Remember to clear the machine!] 1 [Remember to set this. The factory setting of 12 P/YR will produce an error.] 10 1 1000 [You may leave the PMT amount blank: the 0 calculator will assume zero.] The display will read -1,100. [This solves for the Future Value.] Calculators Types of Calculations Definitions 46 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 8a:(HP 10Bii) Future Value Alternate Values Without clearing the calculator, you may change any or all of the variables used in Example 8 to com- pute alternative scenarios. Change the N to 2 and re-press FV to determine the value in two years. Then change N to 5 and again press FV. Do the same with an N of 100. 2 The display will read -1,210. 5 The display will read -1,610.51. 100 The display will read -13,780,612.34. Then Change the interest rate to 12% and 15%, alternatively. 12 The display will read -83,522,265.73. 15 The display will read -1,174,313,450. known sixth factor: the Future Value. a. HP lOBii Calculator Set the Present Value (PV) as 1,000.00. Set the In- Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 47 DID YOU NOTICE? In Example 8a, increasing the interest rate from 10% to 12% caused the Future Value in 100 years to increase more than 6 times! Increasing the interest rate to 15% caused the original Future Value to increase more than 85 times! In Example 8, if you enter the Present Value as negative number, the Future Value solution will be positive number. Remember, the PV and FV must have oposite signs on an HP calculator; however, it makes no difference which is posi- tive and which is negative. The JavaScript Caculator elimi- nates this requirement. Calculators Types of Calculations Definitions 48 FINANCIAL CALCULATIONS FOR LAWYERS Future Value of a Sum Calculator EXAMPLE 8 (JavaScript Calculator) Future Value PV Annuity Interest Conversion FV Annuity Future Value of a Sum Calculator PV Sum Sinking Fund Instructions ON OFF Amortization Mode Begin End Present Value 1,000.00 Future Value 1,100.00 Nominal Interest Rate 10.00 Effective Interest Rate 10.000000000 1.00 Number of Years 1.00 Payments Per Year 1.00 Number of Payments 0.00 Payment clear all Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 49 TIP You may enter the various values in any order. For example, you could enter the Nominal Inter- est Rate prior to entering the Present Value . Also, you may change any of the values in white boxes. Doing so will automatically cause the re- calculation of the other values. TIP The JavaScript Calculators always indicate the number of payments per year setting. This will help prevent errors that may result from failing to set this value correctly. Calculators Types of Calculations Definitions 50 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 8b:(HP 10Bii) Alternate Future Values Solving For an Interest Rate As demonstrated in Example 8a, you may change any or all of the variables used in Example 8 to compute alternative scenarios. As shown in 8a change the N to 5, and re-press FV to determine the value in five years. Then change the FV to - 1,750.00, representing a larger desire future amount. The solution will be the necessary nominal annual interest rate. 5 The display will read - 1,610.51. 1750 [The +/- key enters the value as a negative. If you forget to do this, the display will read No SoLution.] The display will read 11.843. Then Change the Future Value to -2000. 2000 The display will read 14.870. [In each of the alternatives, the amount computed is the nominal annual interest rate, compounded annually, needed to reach the stated Future Value when the Present Value is 1,000: 11.843% and 14.870 %.] Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 51 terest (I/YR) rate per year as 10. Set the Number of Periods per year (P/YR) as 1. Set the Payment (PMT) amount as 0. Set the Number of Periods (N) as 1. The order you input these is irrelevant. Finally, solve for the Future Value (FV) by pressing the FV key. As illustrated in Example 7, the answer will appear as (1,100.00), the negative indicating a withdrawal. b. JavaScript Financial Calculator The JavaScript Calcula- tor operates much the same way as the HP10Bii. Press the FV of a Sum button to open the ap- FV Sum Calculator FV Sum Calculator propriate calculator. You should initially press the Clear All button; how- ever, this is not essential. Enter the various values in the white boxes. Do not change amounts in yellow boxes. The answer will automatically appear in the green FV box. It will appear as a positive number. This part of the calculator will not accept negative numbers.The Payment box will not ac- cept an amount: it must be zero. Placing the cursor over any of the functions will open a pop- TIP You may enter the various values in any order. For example, you could enter the FV prior to entering the N. Calculators Types of Calculations Definitions 52 FINANCIAL CALCULATIONS FOR LAWYERS When Would You Want to Compute the Future Value of a Sum? √ If you deposit money into an account, you can compute what it will worth in the future. √ If population growth rates continue at a constant rate, you can compute the population of an area after a given length of time. √ If your client was owed a specific amount as of a prior date, you can compute what he is owed today. The amount owed originally would be the present value and the amount today would be the future value. The intervening period would be the N. √ If a budget item (PV) increases at a particular rate (I/YR), you can compute the amount for a future period. up box with additional instructions or explanations. Example 8a illustrates how you can change the vari- ous functions on an HP10Bii to compute alternative sce- narios. You may do so without first clearing the registers. You must, however, press the FV key again to display the new answer. The JavaScript Calculator works similarly: change the value of one function and the calculator immediately will Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 53 change the computed answer. You need not repress the FV function key. a. Solving for an interest rate The six variables [PV, FV, I/YR, P/YR, N, and PMT] are a function of each other: change one and the others change, as well The calculator will solve for one change at a time. Often the desired alternative involves a changed inter- est rate [I/YR]. In Example 8, the unknown factor was the Future Value. Once the calculator solved for the Future Value, you might then want to know how the interest rate would change if the Future Value were different. This might occur if you solved initially for a Future Value, as in Example 8b, result- ing in an answer of $1,750 in five year at 11.834% effective interest. You might then be curious regarding the amount of interest you would need to earn if, instead, you were to ac- cumulated $2,000 in five years. The answer, as shown, is 14.870% effective interest. In other words, you would know the Present Value, the Future Value, and the Number of Periods, Payments, and Payments Per Period. You could then solve for the Inter- est Rate. 1. HP lOBii Calculator Example 8b illustrates the process of solving for an interest rate using an HP lOBii Calculator. It uses the factors originally entered in Example 8. It then changes the Future Value from the original solution to alternative amounts. Because the Example 8 Present Value of $1,000 was en- Calculators Types of Calculations Definitions 54 FINANCIAL CALCULATIONS FOR LAWYERS Set the Interest Payment and Compounding Periods the Same. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 55 EXAMPLE 8c:(HP 10Bii) Future Values Compounded Interest Rate As demonstrated in Example 8a, you may change any or all of the variables used in Example 8 to com- pute alternative scenarios. Suppose, instead of the facts given for Example 8, you were told the interest rate was 10% nominal, compounded semi-annually. Enter the values as shown in Example 8, with the following changes: [This sets the number of periods at 2, which is the 2 number of semi-annual period inone year.. [This sets the compounding period as six 2 months: twice per year.] The display will read -1,102.50. The display will read 10.25. [This is the Effective Interest Rate. Then Change the N to 12 and the P/YR to 12. 12 12 The display will read -1.104.713. The display will read 10.471. [This is the Effective Interest Rate. Calculators Types of Calculations Definitions 56 FINANCIAL CALCULATIONS FOR LAWYERS Future Value of a Sum Calculator EXAMPLE 8c (JavaScript Calculator) Future Value PV Annuity Interest Conversion FV Annuity Future Value of a Sum Calculator PV Sum Sinking Fund Instructions ON OFF Amortization Mode Begin End Present Value 1,000.00 Future Value 1,102.50 Nominal Interest Rate 10.00 Effective Interest Rate 10.2500000000 1.00 Number of Years 2.00 Payments Per Year 2.00 Number of Payments 0.00 Payment clear all Did you notice? On the JavaScript Calculator: changing the 1 Payment Per Year to “2” and later to “12” is all you need do. The HP10Bii requires pressing 16 buttons to accomplish the same calculations. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 57 Future Value of a Sum Calculator EXAMPLE 8c (JavaScript Calculator) Future Value PV Annuity Interest Conversion FV Annuity Future Value of a Sum Calculator PV Sum Sinking Fund Instructions ON OFF Amortization Mode Begin End Present Value 1,000.00 Future Value 1,104.71 Nominal Interest Rate 10.00 Effective Interest Rate 10.4713067441 1.00 Number of Years 12.00 Payments Per Year 12.00 Number of Payments 0.00 Payment clear all Calculators Types of Calculations Definitions 58 FINANCIAL CALCULATIONS FOR LAWYERS tered as a positive number, the alternative Future Values must be negative numbers. 2. JavaScript Financial Calculator The JavaScript Financial calculator is not currently designed to solve for an unknown interest rate. b. Compounded Interest Rates One of the cardinal rules of financial calculations is that the interest rate period and the payment period must be the same. For example, if payments - either of principal or interest - occur semiannually, then the interest rate must also be stated as a semiannual rate. Or, the payments occur quar- terly, the rate must also be expressed as a quarterly rate. This is also true if the interest merely compounds more fre- quently than once per year, but is paid at maturity: the inter- est rate and the P/YR must be consistent. Example 8c illustrates the use of compounded inter- est in a Future Value problem. It assumes an interest rate of 10% nominal annual interest compounded semiannually, which is the equivalent of 10.25% nominal annual interest com- pounded annually. This is also the effective rate of interest. Law of Economics As Interest Rates Increase, Future Values Increase. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 59 Notice that the calculated Future Value is higher than that calculated in Example 8: this occurred because the interest rate also increased. Hence a Law of Economics: As Interest Rates Increase, Future Values Increase. c. Converting an interest rate A rule that bears repeating is this: the interest rate period and the payment period must be the same. This is required by the formula and is true of all calculators. As explained in a prior section, you can work prob- lems involving compound interest in two ways: 1. The Calculator Method 2. The Periodic Method. The Calculator Method sets the P/YR at the number of payments per year. It sets the I/YR at the nominal annual interest rate to be compounded at a rate consistent with the The Calculator Method is easy to use; however, it requires some exact- ness on the HP10Bii: you must use both the EFF% and NOM% Calculators Types of Calculations Definitions 60 FINANCIAL CALCULATIONS FOR LAWYERS P/YR. It sets the N at the total number of periods (not the number of years). The Periodic Method sets the P/YR at one. It sets the I/YR at the periodic interest rate. It also sets the N at the total number of periods. The Periodic Method is consistent with the undelying math formula. Some users find it easier than the Calcula- tor Method. Most do not. Both methods require the use of the nominal annual interest (NAI) rate. The Calculator method uses the actual NAI as the I/YR. The Periodic Method requires the user to divide the NAI by the number of periods in one year and then to enter the result as the I/YR. A common problem involves the misstatement of the interest rate. Because both methods use the NAI, the inter- est rate must be stated in that format or converted to it. Luck- ily, this is not really much of a problem because the typical calculator can easily convert an interest rate to an equivalent rate for another period, including a NAI. Many calculators Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 61 EXAMPLE 9: (HP 10Bii) Periodic Method 1000 [This renders the P/YR button su- perfluous: it becomes a periods per 1 period button.] 2 4.8808848 [This function is not necessary: the calculator will assume 0 if nothing is entered.] 0 The display will read -1,099.99999964. [This rounds to -1,100, which is the original value plus 10% interest, proving that 4.88% semi-annual interset is the equivalent of 10% annual.] do this automatically; or, you can do it manually. Pages 23 to 34 explain how to convert an interest rate into an equivalent rate. The simplest method is to use the JavaScript Interest Conversion Calculator, which automati- cally converts among nominal annual rates, effective rates, and equivalent periodic rates. Pages 132 to 170 define the various terms involving interest rates. Calculators Types of Calculations Definitions 62 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 9a: (HP 10Bii) Periodic Method 1000 [This renders the P/YR button su- 1 perfluous: it becomes a periods per period button.] [This is 2 because that is the number of six month 2 periods in one year]. [You must multiple 1000 times 1.1 (thus adding the 10% interest). Do this in your 1,100 head or by using the calculator.] The display will read 4.880884817. [Multiply the displayed amount by 2 to produce 9.761769634% nominal annual intereest compounded semi-annually.] Did You Notice? Set the Cash In the Periodic Flows Correctly: Method the I/YR key PV and FV must functions as a peri- have opposite odic interest key with the PIYR key being signs - one positive and set to one and thus one negative. rendered superfluous. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 63 EXAMPLE 9b:(HP 10Bii) Calculator Method Interest Rate Conversion Convert 10% effecitve interest to the equivalent nominal annual interest compounded semi-annually. [This is 2 because that is the number of 2 six month periods in one year]. 10 The display will read 9.761669634. [Divide the displayed amount by 2 to produce 4.880884817% periodic interest semi-annu- ally. This division function is not necessary: the calculator does it automatically.] Then proceed normally as in Example 9c. Do not clear the calculator! TIP Most current users prefer the Calculator Method. The Periodic Method is consistent with the math formula and thus may be more intuitive to those users accus- tomed to working without electronic calculators. Calculators Types of Calculations Definitions 64 FINANCIAL CALCULATIONS FOR LAWYERS Future Value of a Sum Calculator EXAMPLE 9 (JavaScript Calculator) Future Value PV Annuity Interest Conversion FV Annuity Future Value of a Sum Calculator PV Sum Sinking Fund Instructions ON OFF Amortization Mode Begin End Present Value 1,000.00 Future Value 1,100.00 Nominal Interest Rate 4.8808858 Effective Interest Rate 4.880885800 2.00 Number of Years 1.00 Payments Per Year 2.00 Number of Payments 0.00 Payment clear all Did you notice? The JavaScript Calculator required the entry of 4 numbers. The HP10Bii requires pressing 12 buttons to accomplish the same calculation. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 65 Interest Rate Converter EXAMPLE 9b (JavaScript Calculator) Future Value PV Annuity PV Sum FV Annuity Interest Rate Conversion Amortization Sinking Fund Instructions ON OFF FV Sum Convert Effective Convert Nominal Convert Periodic Rate to Nominal Rate to Effective Rate to Nominal Rate and Periodic Rate and Periodic Rate and Effective Rate Rate Rate 9.7617696340 Press this Nominal Interest Rate button first. Periodic Interest Rate 4.8808848170 Effective Interest Rate 10.000000000 Payments Per Year 2.00 Enter these clear all numbers. Next, copy the nominal rate and enter it into the appropriate white box on the Future Value of A Sum Calculator. Calculators Types of Calculations Definitions 66 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 9c:(HP 10Bii) Calculator Method After converting the 10% effective rate to the equivalent nominal rate, press the following keys. Do not use the C ALL function. 1000 2 The display will read -1,100. As explained earlier in Example 4, 4.8808848% in- terest, paid semiannually is the equivalent of 10% interest, paid annually. This is true because the interest paid during the first six month period will itself earn interest of 4.88% during the second six month period. Stated precisely, 9.761769634% nominal annual interest compounded semi- annually is an effective interest rate of 10%. To prove this, set PV as 1,000.00, I/YR as 4.8808848, P/YR as 1, PMT as 0, and N as 2. Press FV. The answer is (1,099.99999964), which is the equivalent of (1,100.00). Hence, $1,000 earning 10% interest annually produces the same result as $1,000 earnings 4.88% semiannually. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 67 Example 9 thus demonstrates that the I/YR key can function as an interest per period key and the P/YR key can be a number of periods per period key, rather than number of periods per year. Use of the keys in this manner is con- sistent with the mathematical formula for financial calcula- tions, which requires a periodic rate. This is the Periodic Method. Another version of Example 9 converts a known 10% effective interest rate into a NAI rate compounded semian- nually. Example 9a illustrates this conversion. This method works the same for the HP10Bii Calculator and the JavaScript Calculator. The HP10Bii provides an alternate - and simpler - method utilizing the calculator’s built-in conversion function. Enter 10.00 as the effective interest rate using the EFF% function, then solve for the nominal interest rate, using the NOM% key. Dividing the result by 2 would produce the semi- annual periodic rate. Examples 9b and 9c illustrate this method. This is the Calculator Method. The HP10Bii calculator requires the user to convert an effective interest rate to a nominal rate, either manually (as illustrated in Example 9) or by using the calculator (as illustrated in Example 9b). In other words, you may not merely enter the effective interest rate (EFF%) and then pro- ceed: you must actually convert the rate, as illustrated above. The HP10Bii calculator will accept the entry of a nominal rate either by use of the I/YR button or the NOM% button; however, it will not accept entry of an effective rate by use of the EFF% button unless the user then presses the NOM% button. Calculators Types of Calculations Definitions 68 FINANCIAL CALCULATIONS FOR LAWYERS 2. Present Value of a Sum This calculation computes the present value of a fu- ture amount. For example, $1,100 in one year, discounted at 10% interest compounded annually has a present value of $1,000.00 today. Thus, if you owe $1,100 one year from now, you should be able to pay off the obligation with only $1000, assuming the appropriate interest rate is 10% nomi- nal annual interest compounded annually. In comparison, $1,100 in two years has a present value of $909.09. Discounting $1,100 for five years produces a present value of $683.01. Discounting it for 100 years at 10% produces a present value of $0.0798 - just under eight cents! Thus if you owed $1,100 one hundred years from now, you should be able to satisfy the debt with merely eight cents (assuming the constant 10% nominal annual interest). As shown in Example 10, to calculate the present value of a sum, input the five known factors into the calcula- tor and solve for the unknown sixth factor, the present value. a. HP lOBii Calculator First, set the Future Value (FV) as 1,100.00. Set the Interest (I/YR) rate per year as 10. Set the Number of Peri- ods per year (P/YR) as 1. Set the Payment (PMT) amount as 0. Set the Number of Periods (N) as 1. The order you input these is irrelevant. Finally, solve for the Present Value (PV) by pressing the PV key. As illustrated in Example 10, the answer will appear as (1,000.00), the negative indicating a current de- posit. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 69 EXAMPLE 10:(HP 10Bii) Present Value Compute the Present Value of $1,100 owed one year from now, at 10% nominal annual in- terest compounded annually. Press the illustrated keys. [Remember to clear the machine!] 1 [Remember to set this. The factory setting of 12 P/YR will produce an error.] 10 1 1100 [You may leave the PMT amount blank: the 0 calculator will assume zero.] The display will read -1,000. Calculators Types of Calculations Definitions 70 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 10a:(HP 10Bii) Present Value Alternate Values Without clearing the calculator, you may change any or all of the variables used in Example 10 to compute alternative scenarios. Change the N to 2 and re-press FV to determine the present value of the amount two years hence. Then change N to 5 and again press FV. Do the same with an N of 100. 2 The display will read -909.090909091. 5 The display will read -683.013455365. 100 The display will read -0.079822287. Then Change the interest rate to 12% and 15%, alternatively. 12 The display will read -0.013170141. 15 The display will read -0.000936718. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 71 DID YOU NOTICE? In Example 10a, increasing the interest rate from 10% to 12% caused the Present Value 100 years earlier to de- crease more than 6 times! Increasing the interest rate to 15% produces a Present Value of less than 1/100th of one cent! In other words, a penny invested at 15% will produce more than $11,000 in 100 years. Calculators Types of Calculations Definitions 72 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 10b:(HP 10Bii) Present Value Solving For Interest Rate Compute the interest rate inherent to a Present Value of $9,000, a Future Value of $10,000, and a period of two years. Press the illustrated keys. [This can be set at any number; how- 1 ever, the resulting rate must then be interpreted correctly, as shown in Example 10d.] 2 10,000 9,000 The display will read 5.409255339. Thus the Seller offers a discount of 5.41% nominal an- nual interest compounded annually for advance pay- ments. Which Interest Rate is Higher? The two interest rates are not comparable without conversion to a common compounding period. The 10b rate of 5.41% is an effective rate of 5.41% (compounded annually, the effective and nominal rates are the same). The 10c rate of 5.35% is an effective rate of 5.49% (see Example 10d). Thus the 10c rate is better for the customer! Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 73 EXAMPLE 10c:HP10Bii) Present Value Solving For Interest Rate Compute the interest rate inherent to a Present Value of $26,250, a Future Value of $30,000, and a period of thirty years. [This can be set at any number; how- 12 ever, the resulting rate must then be interpreted correctly, as shown in 30 Example 10d.] 30,000 26,250 The display will read 5.353160450. Thus the Seller offers a discount of 5.35% nominal an- nual interest compounded monthly for advance pay- ments. Did You Notice? The 5.35% rate is larger than the 5.41% rate! The two interest rates are not comparable without conversion to a common compounding period. Calculators Types of Calculations Definitions 74 FINANCIAL CALCULATIONS FOR LAWYERS Laws of Economics 1. As Interest Rates Decrease, Present Values Increase. 2. As Interest Rates Increase, Present Values Decrease. Did You Notice? The above Laws of Economics are consistent with common sense: 1. The lower the discount for paying in advance, the more you must pay. 2. The bigger the discount for paying in advance, the less you must pay. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 75 When Would You Want to Compute the Present Value of a Sum? √ If you owe money in the future, you can compute what it equals in current value. √ If you want a discount for an advance payment for goods or services, you would compute the present value of the future obligation. √ If you know the amount you need at a future time - such as for retirement or college entrance - you can compute the present value needed to produce that future amount. TIP If a service provider will accept less money for an advance payment, you can calculate the discount rate being used. Calculators Types of Calculations Definitions 76 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 10d:(HP10Bii) Present Value Comparing Interest Rates The 10c display read 5.353160450. Convert it to an Effective Rate so that it can be compared to other Examples. The display will read 5.486474725. Thus the 5.35% nominal annual interest compounded monthly is an effective rate of 5.49%. Comparing the 5.35% rate to the Example 10b 5.41% rate is like comparing apples to oranges: it makes no sense. The two rates must be converted to the same compounding pe- riod to be comparable. Instead, the 10b rate can be converted to a nominal annual rate compounded monthly. The 10b display reads 5.409255339. To Convert it to a monthly rate press: 12 24 The disiplay will read 5.279606096. [Thus 5.41% compounded annually is equivalent to 5.28% compounded monthly. Both are an effective rate of 5.41%] Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 77 Caution: When comparing interest rates, always translate them to comparable compounding periods. b. JavaScript Financial Calculator The JavaScript Calculator operates much the same way as the HP10Bii. Press the PV of a Sum button to open the appropriate calculator. The Payment box will not accept an amount. You should Sum Calculator PV Sum Calculator initially press the Clear All but- ton; however, this is not essen- tial. Enter the various values in the white boxes. Do not change amounts in yellow boxes.The answer will automatically appear in the green PV box. It will appear as a positive number. This part of the calculator will not accept negative numbers.The Payment box will not ac- cept an amount. Placing the cursor over any of the functions will open a pop-up box with additional instructions or explanations. Calculators Types of Calculations Definitions 78 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 10 (JavaScript Calculator) Present Value PV Annuity Interest Conversion Present Value of a Sum Calculator Amortization FV Annuity Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 1,000.00 Future Value 1,100.00 Nominal Interest Rate 10.00 10.000000000 Effective Interest Rate 1.00 Number of Years 1.00 Payments Per Year 1.00 Number of Payments 0.00 Payment clear all Did you notice? Unlike the HP10Bii, the JavaScript Calculator does not require the entry of negative numbers. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 79 EXAMPLE 10 (JavaScript Calculator) Present Value Alternate Values PV Annuity Interest Conversion Present Value of a Sum Calculator Amortization FV Annuity Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 0.00 0.00093671753426 Future Value 1,100.00 15.00 Nominal Interest Rate 15.000000000 Effective Interest Rate 100.00 Number of Years 1.00 Payments Per Year 1.00 Number of Payments 0.00 Payment clear all The Present Value is set to show only two decimal places. But, if you click on the green box, it will show 14 decimal places. Calculators Types of Calculations Definitions 80 FINANCIAL CALCULATIONS FOR LAWYERS a. Solving for an interest rate Remember: the six variables [PV, FV, I/YR, P/YR, N, and PMT] are a function of each other: change one and the others change, as well The calculator will solve for one change at a time. Often the desired alternative involves a changed interest rate [I/YR]. In Example 10, the unknown factor was the Present Value. Once the calculator solved for the Present Value, you might then want to know how the interest rate would change if the Present Value were different. In other words, you would know the Present Value, the Future Value, and the Number of Periods, Payments, and Payments Per Period. You could then solve for the Interest Rate. Why would you want to know this? Perhaps you need to pay $10,000 two years from now and you know that can pay $9000 today to satisfy the obligation. You could solve for the interest rate to determine what rate the seller/service provider is using. Or, you may represent the seller/service provider. You might know that he is willing to accept $26,250 today for a future amount due of $30,000 in thirty months. In Example 10, if you enter the Future Value as negative 1,100, the Present Value solution will be positive 1,000. Remember, the PV and FV must have oposite signs on an HP calculator; however, it makes no difference which is posi- tive and which is negative. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 81 Examples 10b and 10c illustrates how you would deter- mine the correct discount interest rate in each case. 1. HP lOBii Calculator Example 10b and 10c illustrate the process of solv- ing for an interest rate. In 10b, the Present Value is entered as a positive number; thus, the Future Value must be entered as a negative. Similarly, in 10c, the Present Value is en- tered as a negative number; thus, the Future Value must be entered as a positive. Example 10d emphasizes the need to translate inter- est rates to a common compounding period for purposes of comparison. Without such a translation, comparison makes no sense. As shown in the examples, 5.35% is greater than 5.41%, but only because the two are compounded differ- ently. If one party to a transaction understands this point, but the other does not, the one who understands can easily manipulate the other. Failure to understand the effect of com- pounding will result in the making of wrong choices. 2. JavaScript Financial Calculator The JavaScript Financial Calculators are not currently designed to solve for an unknown interest rate. Calculators Types of Calculations Definitions 82 FINANCIAL CALCULATIONS FOR LAWYERS 3. Present Value of an Annuity a. End Mode - An Annuity in Arrears This calculation computes the present value of a se- ries of equal payments made at the end of regular intervals, earning a constant interest rate. For example, $1,000 de- posited at the end of each year for ten years, earning 10% interest compounded annually, has a present value today of $6,144.57. Similarly, $6,144.57 deposited today, earning 10% interest compounded annu- ally will produce a fund from which $1000 could be withdrawn for ten consecu- Caution: tive years, beginning one year from today. To compute an an- nuity in arrears, the This might be used to compute the payoff calculator must be amount for a loan or to in End Mode. value lottery winnings. The display will not 1. HP lOBii Calcu- indicate end mode. lator Thus press the shift and mode keys only As illustrated in Ex- if the display indi- ample 11, input the five known factors into the HP cates Begin Mode. lOBii calculator and solve for the unknown sixth fac- tor, the present value. First, set the Payment (PMT) as 1,000.00. Set the Interest (I/YR) rate per period as 10. Set the Number of Periods per year (P/YR) as 1. Set the Num- ber of Periods (N) as 10. Set the Future Value (FV) as 0. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 83 EXAMPLE 11: (HP10Bii) Present Value of an Annuity in Arrears Compute the Present Value of $1,000 to be received annu- ally, beginning one year from now, at 10% nominal annual interest compounded annually. Press the illustrated keys. [Remember to clear the machine!] 1 [Remember to set this. The factory setting of 12 P/YR will produce an error.] 10 10 1000 0 [You may leave the FV amount blank: the calculator will assume zero.] [Enter this only if the calculator is in Begin Mode.] 6144.56710570 Calculators Types of Calculations Definitions 84 FINANCIAL CALCULATIONS FOR LAWYERS You must also remember to set the calculator in End Mode, if it is not already in End Mode. Do so by pressing the shift key and the BEG/END key. End Mode tells the calculator that the first payment will be made one year from today and each successive payment will occur at the end of each following period. In contrast, in Begin Mode each pay- ment occurs at the beginning of each period. If the display reads Begin, the calculator is in Begin Mode. If, instead, it does not have any words, it is in End Mode and you must not change this for this problem. Solve for the Present Value (PV) by pressing the PV key. The answer will appear as (6,144.56710570), the nega- tive indicating a required deposit necessary to generate the level annuity. The Future Value is zero because at the end of ten years, no money would remain in the account. 2. JavaScript Financial Calculator The JavaScript Calculator operates much the same way as the HP10Bii. Press the PV of an Annuity button to open the appropriate calculator. PV Annuity Calculator PV Annuity Calculator The FV box will not accept an amount. You should initially press the Clear All button; however, this is not essential. Enter the various values in the white boxes. Do not change amounts in yellow boxes.The answer will automatically appear in the green PV box. It will appear as a positive number. This part of the calculator will not accept negative numbers. You must be certain to press the End Mode button. If the calculator is Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 85 When Would You Want to Compute the Present Value of an Annuity? √ If you owe money at regular intervals in the fu- ture, you can compute what it equals in current value. This would tell the “pay-off” amount. This could be useful in a Family Law case to compute the value of a lump- sum award to pay-off an alimony obligation. √ If you want a discount for an advance payment for goods or services which will be provided at regular intervals, you would compute the present value of the future obligation. This might involve an insurance contract or rent or a contract for the pro- vision of utilities or some similar product. √ In a tort case, the victim may have lost future wages or suffer regular future medical expenses. The present value of such an amount would be the tort-feasor’s obligation. √ You might have won a state lottery. The present value of the future payments would be the alternative amount that might be elected. √ You might need to compute the value of a bond or similar financial instrument. The regular interest payments would be an annuity. The present value of them added to the present value of the final payment (the present value of a sum) would be the current value of the bond. Calculators Types of Calculations Definitions 86 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 11 (JavaScript Calculator) Present Value of an Annuity in Arrears Amortization Interest Conversion Present Value of an Annuity Calculator FV Annuity PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 6,144.57 Future Value 0.00 Nominal Interest Rate 10.00 10.000000000 Effective Interest Rate 10.00 Number of Years 1.00 Payments Per Year 10.00 Number of Payments end 1,000.00 Payment mode clear all Did you notice? The JavaScript Annuity Calculators always indicate the mode. This can help prevent resulting errors. Notice the substantial difference in present values between end mode and begin mode. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 87 EXAMPLE 12 (JavaScript Calculator) Present Value of an Annuity Due Amortization Interest Conversion Present Value of an Annuity Calculator FV Annuity PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 6,759.02 Future Value 0.00 Nominal Interest Rate 10.00 10.000000000 Effective Interest Rate 10.00 Number of Years 1.00 Payments Per Year 10.00 Number of Payments begin 1,000.00 Payment mode clear all Click on Begin Mode to con- vert Example 10 to Example 11 and on End Mode to con- vert back. Calculators Types of Calculations Definitions 88 FINANCIAL CALCULATIONS FOR LAWYERS in Begin Mode, you will obtain the wrong answer. Placing the cursor over any of the functions will open a pop-up box with additional instructions or explanations. b. Begin Mode: An Annuity Due This calculation computes the present value of a se- ries of equal payments made at the begin- nings of regular inter- vals, earning a con- Caution: stant interest rate. For example, $1,000 de- To compute an annuity posited at the begin- due, the calculator must ning of each year for ten years, earning be in Begin Mode. 10% interest com- pounded annually, has The HP10Bii display will a present value today not indicate end mode. of $6,759.02. The Thus press the shift amount exceeds that of and mode keys only if the above calculation the display does not because the first pay- indicate Begin Mode. ment here is made to- day, whereas in the End Mode (Annuity in The JavaScript Calcula- Arrears), the first pay- tors do not have this ment is not made until difficulty. one year from now. Similarly, $6,759.02 deposited today, earn- ing 10% interest compounded annually will produce a fund from which $1000 could be withdrawn for ten consecutive years, beginning today. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 89 EXAMPLE 12: (HP10Bii) Present Value of an Annuity Due Compute the Present Value of $1,000 to be received annu- ally, beginning today, at 10% nominal annual interest com- pounded annually. Press the illustrated keys. [Remember to clear the machine!] [Remember to set this. The factory 1 setting of 12 P/YR will produce an error.] 10 10 1000 [You may leave the FV amount blank: the cal- 0 culator will assume zero.] [Enter this only if the machine is in End Mode] The display will read -6,759.02. Calculators Types of Calculations Definitions 90 FINANCIAL CALCULATIONS FOR LAWYERS 1. HP lOBii Calculator As illustrated in Example 12, input the five known factors into the calculator and solve for the unknown sixth factor, the present value. First, set the calculator in Begin Mode. Then, set the Payment (PMT) as 1,000.00. Set the Did You Notice? With annuities, the farther into the future the payment, the less significant the present value. Hence, the Example 12 ten-year annuity had a PV of $6,759 compared to a twenty-year annuity PV of $9,364: a large increase from 10 to 20 years. But, the fifty- year annuity has a value of $10,906, a much smaller increase, even though the period lenth more than doubled! And, the 100- year annuity has a PV of $10,999, an insigificant increase over a fifty year annuity! Compute the PV of a 1000-year annuity. It has a PV of $11,000: only 80cents more than a 100-year annuity. Thus the extra 900 years of payments are almost worthless in present value terms. Laws of Economics 1. As Interest Rates Decrease, Present Values In- crease. 2. As Interest Rates Increase, Present Values De- crease. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 91 EXAMPLE 12a:(HP 10Bii) Present Value of an Annuity Alternate Periods and Interest Without clearing the calculator, you may change any or all of the variables used in Example 12 to compute alternative scenarios. Change the N to 20 and re-press PV to determine the present value of the amount if paid for twenty years. Then change N to 50 and again press PV. Do the same with an N of 100. 20 The display will read -9,364.92009173. 50 The display will read -10,906.2959359. 100 The display will read -10,999.2017771. Then Change the interest rate to 12% and 15%, alternatively. 12 The display will read -9,333.22158668 15 The display will read -7,666.66013803. Calculators Types of Calculations Definitions 92 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 12a (JavaScript Calculator) Present Value of an Annuity Due Alternate Periods and Interest Amortization Interest Conversion Present Value of an Annuity Calculator FV Annuity PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 7,666.66 Future Value 0.00 15.00 Nominal Interest Rate 15.000000000 Effective Interest Rate 100.00 Number of Years 1.00 Payments Per Year 10.00 Number of Payments begin 1,000.00 mode Payment clear all Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 93 Interest (I/YR) rate per period as 10. Set the Number of Periods per year (P/YR) as 1. Set the Number of Periods (N) as 10. Set the Future Value (FV) as 0. Solve for the Present Value (PV) by pressing the PV key. The answer will appear as (6,144.56710570), the negative indicating a re- quired deposit necessary to generate the level annuity. The Future Value is zero because at the end often years, no money would remain in the account. 2. JavaScript Financial Calculator The JavaScript Calculator operates much the same way as the HP10Bii. Press the PV of an Annuity button to open the appropriate cal- PVPV an Annuity Calculator of Annuity Calculator culator. The FV box will not accept an amount. You should initially press the Clear All button; however, this is not essential. Fill in the various values in the white boxes. The answer will automatically appear in the green PV box. The answer will appear as a positive number. This part of the calculator will not accept negative numbers. Also, you must be certain to press the Begin Mode button. The calculator will show the words begin mode in Begin red letters when in this mode. If the calculator is in End Mode, you will obtain the wrong answer. To help prevent er- rors, the calculator will always indicate the mode. To com- pute alternative mode answers, merely click on End Mode and the answer will automatically appear. Placing the cursor over any of the functions will open a pop-up box with addi- tional instructions or explanations. Calculators Types of Calculations Definitions 94 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 12b: (HP10Bii) Present Value of an Annuity Due: shift of mode After working Example 12 and without clearing the machine the display will read -6,759.02. You can shift to End mode (rather than Begin) to determine the alternative present value if the payments begin one year from now rather than today. The display will read -6,144.57. You must press PV to tell the machine to re-compute. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 95 4. Future Value of an Annuity a. End Mode: An Annuity in Arrears This calculation computes the future value of a series of equal payments made at the end of regular intervals, earn- ing a constant interest rate. For example, $1,000 deposited at the end of each year for ten years earning 10% interest compounded annually, has a fu- ture value in ten years of $15,937.42. Caution: This calculation is particu- larly helpful in planning for retire- To compute an an- ment or saving for a child’s edu- cation. nuity in arrears, the calculator must be 1. HP lOBii Calculator in End Mode. As illustrated in Example The display will not 13, input the five known factors into indicate end mode. the calculator and solve for the Thus press the shift unknown sixth factor, the present and mode keys only value. First, set the Payment (PMT) as 1,000.00. Set the Inter- if the display indi- est (I/YR) rate per period as 10. cates Begin Mode. Set the Number of Periods per year (P/YR) as 1. Set the Number of Periods (N) as 10. Set the Present Value (PV) as 0. Solve for the Future Value (FV) by pressing the PV key. The answer will appear as (15,937.4246010), the negative indicating withdrawal pos- sible after the ten deposits. The Present Value is zero be- cause at the beginning, no money has yet been deposited. Calculators Types of Calculations Definitions 96 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 13: (HP10Bii) Future Value of an Annuity in Arrears Compute the Future Value of $1,000 to be deposited annu- ally, beginning one year from now, at 10% nominal annual interest compounded annually. Press the illustrated keys. [Remember to clear the machine!] [Remember to set this. The factory 1 setting of 12 P/YR will produce an error.] 10 10 1000 [You may leave the PV amount blank: the 0 calculator will assume zero.] [Enter this only if the calculator is in Begin Mode.] The display will read -15,937.42. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 97 When Would You Want to Compute the Future Value of an Annuity? √ If you save money for a retirement plan at regular inter- vals, you can compute what it will be worth in the future. √ If you save money for a child’s education at regular inter- vals, you can compute what the fund will be worth in the future. Caution: The future value of an annuity is stated in future dollars, which are not compa- rable to current values. Thus the answer might not be useful; however, two methods can be used to convert the answer to a useful number: 1. Convert the amount to a present value. 2. Modify the interest rate to reflect a “real rate of return” rather than the actual predicted rate. Calculators Types of Calculations Definitions 98 FINANCIAL CALCULATIONS FOR LAWYERS Do not forget to set the calculator in End Mode if you changed it for the prior example. 2. JavaScript Financial Calculator The JavaScript Calculator operates much the same way as the HP10Bii. Press the FV of an Annuity button FVFV an Annuity Calculator to open the appropriate cal- of Annuity Calculator culator. The PV box will not accept an amount. You should initially press the Clear All button; however, this is not essential. Fill in the various values in the white boxes. Do not change the amounts in the yellow boxes. The answer will automatically appear in the green FV box. The answer will appear as a positive number. This part of the calculator will not accept negative numbers. You must be certain to press the End Mode button. The calculator will show the words begin mode in red let- End ters when in begin mode and the words end mode in black letters when in end mode. If the calculator is in Begin Begin Mode, you will obtain the wrong answer. To compute alter- native mode answers, merely click on Begin Mode and the answer will automatically appear. Placing the cursor over any of the functions will open a pop-up box with additional instructions or explanations. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 99 EXAMPLE 12 (JavaScript Calculator) Future Value of an Annuity In Arears PV Annuity Interest Conversion Future Value of an Annuity Calculator Amortization PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 0.00 Future Value 15,937.42 Nominal Interest Rate 10.00 Effective Interest Rate 10.000000000 10.00 Number of Years 1.00 Payments Per Year 10.00 Number of Payments end 1000.00 Payment mode clear all Calculators Types of Calculations Definitions 100 FINANCIAL CALCULATIONS FOR LAWYERS b. Begin Mode: An Annuity Due This calculation computes the future value of a series of equal payments made at the beginnings of regular inter- vals, earning a constant interest rate. For example, $1,000 deposited at the beginning of each year for ten years, earn- ing 10% interest compounded annually, has a future value in ten years of $15,937.42. The amount exceeds that of the prior calculation (an annuity in arrears) because the first payment here is made to- day and thus earns inter- est beginning today, whereas in the End Mode Caution: (Annuity in Arrears), the first payment is not made To compute and until one year from now. annuity due, the 1. HP lOBii Cal- calculator must be in culator Begin Mode. As illustrated in The display will not Example 14, input the indicate end mode. five known factors into the Thus press the shift calculator and solve for the unknown sixth factor, and mode keys only the present value. First, if the display does set the calculator in Be- not indicate Begin gin Mode. Then, set the Mode. Payment (PMT) as 1,000.00. Set the Interest (I/YR) rate per period as 10. Set the Number of Periods per year (P/YR) as 1. Set the Number of Periods (N) as 10. Set the Present Value (PV) as 0. Solve for the Future Value (FV) by pressing the FV key. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 101 EXAMPLE 14: (HP10Bii) Future Value of an Annuity Due Compute the Future Value of $1,000 to be deposited annu- ally, beginning today, at 10% nominal annual interest com- pounded annually. Press the illustrated keys. [Remember to clear the machine!] [Remember to set this. The factory 1 setting of 12 P/YR will produce an error.] 10 10 1000 0 [You may leave the PV amount blank: the calculator will assume zero.] [Enter this only if the calculator is in End Mode.] The display will read -17,531.17. Calculators Types of Calculations Definitions 102 FINANCIAL CALCULATIONS FOR LAWYERS The answer will appear as (17,531.1670611), the negative indicating a required deposit necessary to generate the level annuity. The Present Value is zero because at the begin- ning, the account contains no money: the purpose of the computation is to determine the necessary deposits. Do not forget to set the calculator in Begin Mode if you changed it for the prior example. 2. JavaScript Financial Calculator The JavaScript Calculator operates much the same way as the HP10Bii. Press the FV of an Annuity button to open the appropriate calcu- lator. The PV box will not ac- cept an amount. You should of Annuity Calculator FVFV an Annuity Calculator initially press the Clear All button; however, this is not essential. Fill in the various values in the white boxes. Do not change the amounts in the yellow boxes. The answer will automatically appear in the green FV box. The answer will appear as a positive number. This part of the calculator will not accept negative numbers. You must be certain to press the Begin Mode button. Begin The calculator will show the words begin mode in red let- ters when in begin mode. If the calculator is in End Mode, you will obtain the wrong answer. To compute alternative End mode answers, merely click on End Mode and the answer will automatically appear. Placing the cursor over any of the functions will open a pop-up box with additional in- structions or explanations. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 103 EXAMPLE 12 (JavaScript Calculator) Future Value of an Annuity Due PV Annuity Interest Conversion Future Value of an Annuity Calculator Amortization PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 0.00 Future Value 17,351.17 Nominal Interest Rate 10.00 Effective Interest Rate 10.000000000 10.00 Number of Years 1.00 Payments Per Year 10.00 Number of Payments begin 1000.00 Payment mode clear all Calculators Types of Calculations Definitions 104 FINANCIAL CALCULATIONS FOR LAWYERS 3. Converting the Annuity Future Value to a more useful number. The Future Value of An Annuity calculation provides an answer that may not be useful - and may actually be misleading. While this is true of any future value calculation, it is particularly true when an annuity is involved. For example, a simple Future Value of an Amount cal- culation presents an answer in terms of future dollars at an assumed interest rate. Example 8a illustrated that $1000 today, invested at 10% effective interest, is the equivalent of $1,100 in one year, $1,210 in two years, $1,610.51 in five years, and $13,780,612.34 in 100 years. Those “future dol- lars” are not the same as present dollars because they nor- mally include inflation effects, as well as risk and liquidity components. Typical interest rates include three factors: 1. inflation 2. risk 3. liquidity Did you notice? People charge interest for three fundamental reasons: 1. to compensate for inflation 2. to compensate for risk 3. to compensate for a lack of liquidity Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 105 The inflation factor involves a prediction of the general economy, while the risk factor involves a prediction of the individual borrower’s reliability. The liquidity factor is more fixed - typically at three to four percent. At one level, this observation states the obvious be- cause that is the essence of a future value calculation. At this level it is not normally misunderstood because the calcu- lation requires the user in- put the equivalent present value. Hence, anyone per- Caution: forming the calculation knows that $1,610.51 in What the user knows five years is the equivalent of $1,000 today. to be true - that present and future When a future value values are equivalent calculation involves an an- - is not really true. nuity, however, the effects of inflation on the interest rate are less apparent. But, at another level, what the user knows to be true - that the present and future values are equivalent - is not really true. Putting aside the inherent uncer- Thus Future Values tainly of future inflation and are often more use- risk - and thus the chance ful when computed of substantial errors due to without the inflation bad predictions - the num- or risk factors. bers have a built-in differ- ence: the liquidity element. Calculators Types of Calculations Definitions 106 FINANCIAL CALCULATIONS FOR LAWYERS Interest rates typically include a “real” factor, i.e., what people charge for the use of money when inflation and risk are both zero. Over time, this averages about three to four percent, with recent evidence suggesting the four percent figure to be more accurate. As a result, future values are often more useful when computed without the inflation or risk factors. This permits them to be compared to current present values. It eliminates much of the risk of bad predic- Caution: tions of the future inflation rate. But, even such computations, re- A Future Value is not quire some translation. the equivalent of a For example, in Example Present Value if the 8a, a $1,000 present value is the interest rate includes equivalent of a $1,610.51 future a risk component. value in five years at an as- Because the future sumed 10% effective in- terest rate. However, the value includes the two numbers are not re- earning resulting ally interchangeable even from risk taking, it if the predicted 10% ef- includes an extra fective interest proves to component beyond be accurate. All the com- what inflation com- putation tells us is that pensation will provide. $1,000 invested today at Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 107 10% effective interest will yield an account containing $1,610.51 in terms of future buying power in five years. The future value calculation, however, tells us nothing about what that future value would purchase today: it cer- tainly would not purchase $1,000 of value, but, instead, would purchase significantly more. This is true because the $1,610.51 includes compensation both for the risk and the liquidity factors. Did you notice? The future value calcula- tion tells us nothing about what the future value would purchase today. If the risk factor is accurate, some instances will pro- duce significantly less than $1,610.51 because they will in- volve partial or full default. The future value computation traditionally includes the risk component because interest rates include it; however, the calculator result essentially as- sumes the risk becomes zero in reality: the result actually earns the full assumed interest and the original principle re- mains. This may often be accurate; however, it will not al- ways be accurate. Also, because the result includes the earning result- ing from risk taking, it includes an extra component beyond what inflation will provide. Calculators Types of Calculations Definitions 108 FINANCIAL CALCULATIONS FOR LAWYERS The liquidity factor The liquidity factor of in- of interest also re- terest also results in extra real value in a Future Value result. sults in extra real Overtime, considering large value beyond what numbers of accounts, investors inflation compensa- will earn approximately four per- tion will provide. cent real return. Another way of stating this is: investors earn enough to compensate for ex- pected inflation plus expected risk plus an General Law of additional four percent. Finance Because the market sometimes wrongly es- timates future inflation From a macro eco- and risk, the actual re- nomic viewpoint, investors turn for any particular earn enough to compensate account may be sig- for expected inflation plus ex- nificantly different from pected risk plus an additional four percent. Never- four percent. theless, all things be- ing equal, the four per- cent figure is a useful, consistent, conservative, predictable real return. Example 15 illustrates several modifications to the Example 8 calculation. These should provide more useful information. Example 8 involved computing the Future Value of $1,000 in five years using an effective interest rate of ten per Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 109 EXAMPLE 15: (HP10Bii) Modified Future Values Compute the Future Value in five years of $1,000 at various modified interest rates. Press the illus- trated keys. 3 5 1000 0 The display will read -1,159.27. This is the future value of $1000 in five years earning 3% EFF. 4 The display will read -1,216.65. This is the future value of $1000 in five years earning 4% EFF. 7 The display will read -1,402.55. This is the future value of $1000 in five years earning 7% EFF. Calculators Types of Calculations Definitions 110 FINANCIAL CALCULATIONS FOR LAWYERS Future Value of a Sum Calculator EXAMPLE 15 (JavaScript Calculator) Future Value Modifications PV Annuity Interest Conversion FV Annuity Future Value of a Sum Calculator PV Sum Sinking Fund Instructions ON OFF Amortization Mode Begin End Present Value 1,000.00 Future Value 1,159.27 Nominal Interest Rate 3.00 Effective Interest Rate 3.0000000000 5.00 Number of Years 1.00 Payments Per Year 5.00 Number of Payments 0.00 Payment clear all Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 111 Future Value of a Sum Calculator EXAMPLE 15 (JavaScript Calculator) Future Value Modifications PV Annuity Interest Conversion FV Annuity Future Value of a Sum Calculator PV Sum Sinking Fund Instructions ON OFF Amortization Mode Begin End Present Value 1,000.00 Future Value 1,216.65 Nominal Interest Rate 4.00 Effective Interest Rate 4.00000000 5.00 Number of Years 1.00 Payments Per Year 5.00 Number of Payments 0.00 Payment clear all Change the Nominal Interest Rate to 7% and the colored boxes will change as well. The Future Value becomes 1,402.55 Calculators Types of Calculations Definitions 112 FINANCIAL CALCULATIONS FOR LAWYERS cent. The examples illustrate the correct computation based on the ten per cent assumption; however, the assumption itself is likely unrealistic. While an investor may indeed earn a nominal 10% interest over a five year period, it is unlikely to represent a “real” rate of return: i.e., the purchasing power will not likely increase by 10% per annum. One useful modification involves adjusting the interest One useful modification rate to exclude the inflation fac- involves adjusting the tor. Assuming the 10% effec- interest rate to exclude tive interest figure comprised the inflation factor. four percent for liquidity and three percent for risk, it would Example 15 illustrates also have comprised three per- this by using the 7% cent for expected inflation. effective interest rate . Thus seven percent would be the expected real interest rate (assuming the risk element proves unnecessary: i.e., the borrower does not fail). At seven percent interest, $1,000 invested today would yield $1,402.55 in five years. Those would be uninflated dollars and thus would reflect the same purchasing power as $1,402.55 would have today. In nominal terms, the future account will include $1,610.51 future dollars, which will pur- chase what $1,402.55 present dollars would purchase today. This is, as are all future value computations, a prediction and it is based on the assumption that expected inflation will be 3%. Actual inflation, in hindsight, will almost always be a different amount . . . greater or lesser than the expected amount. Thus the prediction is no better than the assump- tion. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 113 Another useful modification involves adjusting the in- terest rate to include only the four percent liquidity factor. This would be the most conservative modification, with the use of three percent being the more conservative of that. At three percent effective interest, $1,000 today would yield $1,159.27 in five years. At four percent, it would yield $1,216.65. As above, these would reflect what the ex- pected future account of Another useful modifica- $1,610.51 would purchase tion involves adjusting today: significantly more than the interest rate to in- merely $1,000, but signifi- clude only the four per- cantly less than the full cent liquidity factor. $1,610.51. Example 15 illustrates The difference be- this by using the 4% tween the two modifications effective interest rate. involves the risk factor. In a micro-economic sense, the account may very well earn the full three percent risk com- ponent, yielding the full $1,402.55 purchasing power. But, in a macroeconomic sense, it will not. If the risk component is correctly set, some accounts will earn it, while others will earn nothing or even face default, resulting in a loss of the original $1,000 as well as the risk factor interest. Hence, if the investor has a sufficient number of investments suffi- ciently diversified, some will earn the full risk component, others will earn part of it and some will fail. Net, investors should earn nothing but the liquidity factor plus the inflation factor. In reality, the U.S. economy seems to overstate the risk component of investment, resulting in historically con- sistent returns greater than inflation plus three or four per- Calculators Types of Calculations Definitions 114 FINANCIAL CALCULATIONS FOR LAWYERS cent. Perhaps some of the reason involves overstated in- General Law of flation expectations coupled Finance (re-stated) with overstated risk. If the investor has a sufficient An investor wanting to number of investments suffi- modify a future value calcu- ciently diversified, some will lation to a useful figure would earn the full risk component, thus likely want to use an others will earn part of it and interest rate greater than some will fail. Net, investors three percent and probably should earn nothing but the li- greater than four percent, quidity factor plus the inflation but probably not much factor, of which only the liquid- greater, unless he is particu- ity factor represents a “real” re- larly optimistic about the turn. market overstating risks. This should generate a real- istic picture of the buying power of a future account. But, the U.S. economy seems to overstate the risk component of invest- ment, resulting in his- torically consistent re- turns greater than infla- tion plus four percent. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 115 Thus, an investor wanting to modify a future value calcuation to a useful figure would thus likely want to use an interest rate greater than three percent and probably greater than four percent, but probably not much greater. A third - and arguably simpler - modification would take the computed future value and discount it to the present by either the inflation component or by the sum of the inflation and risk components. These calculations will not yield the same answers as the above modifications; however, they will approximate the same results. The differences arise because this modification involves discounting both the inflated dollars as well as the dollars reflecting real interest. Considering the inherent un- certainty in pre- dicting future in- terest, however, A third modification would the inaccuracy of discount the future value by this modification either the inflation compo- may not be impor- nent or by the sum of the tant. inflation and risk compo- nents, excluding the liquidity factor. Example 15 illustrates this by using the 7% effective interest. Calculators Types of Calculations Definitions 116 FINANCIAL CALCULATIONS FOR LAWYERS Caution Modifying a Fu- ture Value of an Annuity Modifying a Future Value of an calculation is a bit more Annuity calcuation is more com- complicated than doing plicated than modifying a simple so for a simple Future Future Value of a Sum calcula- Value of a Sum calcula- tion. tion. This is true because the Annuity involves not only a present sum but also a series of future sums. Unless they, too, are modi- fied, the answer may not be fully useful. Example 16 illus- trates a useful method. For ex- ample, to deter- TIP mine what is needed for a fu- Not only must you modify the in- ture event - such terest rate, but you must also as a child’s edu- modify the payments. cation, one cannot know future costs. However, one might conclude that present costs will inflate to become fu- ture costs and that the inflation component will generally ap- proximate the inflation component of interest. As such, it falls out of the calculation. One can then determine how much would be needed today, in terms of present dollars, to purchase the needed future item. Perhaps, for example, a child newly graduated from high school would need $100,000 today. If so, one might conclude that education and living costs will rise by the ex- Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 117 pected inflation rate such that $100,000 discounted at a non- inflated interest rate would yield the amount needed to yield the purchasing power of $100,000 today. Whether the dis- count rate should be three percent, four percent, or some- thing greater than four percent depends on the user ’s own beliefs about market and economic risks. The discount calculation would either involve the com- putation of the Present Value of a Sum or a Sinking Fund, as explained later. Because the Sinking Fund is merely the reflection of an Annuity, it is relevant now to the annuity dis- cussion. The resulting sinking fund payment would tell the investor the amount to deposit each period for the desired number of periods to yield the expected purchasing power, as shown in Example 16a. Without further modification, however, the account will inevitably prove to be insufficient. That is because the omit- ted inflation factor must be put back in. Hence, the annuity would no longer be a level annuity of a constant payment. Instead, each payment would need to increase by the imme- diately past periodic inflation rate. This will yield a future account with the approximate purchasing power of the origi- nally needed $100,000 for the hypothetical current gradu- ate. Example 16 also illustrates the calculations, but from the perspective of an annuity calculation rather than a sink- ing fund. The initial Future Value of an Annuity calculation computes that $2,000 invested annually at 10% effective in- terest, beginning today, for eighteen years would yield ap- proximately $100,000. The answer, however, is not useful because one does not know what $100,000 will purchase in eighteen years. The user, however, indeed knows what Calculators Types of Calculations Definitions 118 FINANCIAL CALCULATIONS FOR LAWYERS Sinking Fund Calculator EXAMPLE 16a (JavaScript Calculator) Sinking Fund Calculations PV Annuity Interest Conversion FV Annuity Sinking Fund Calculator PV Sum Amortization Instructions ON OFF FV Sum Mode Begin End Present Value 0.00 Future Value 100,000.00 Nominal Interest Rate 10.00 10.00 Effective Interest Rate 18.00 Number of Years 1.00 Payments Per Year 18.00 Number of Payments 1,993.66 begin Payment mode clear all A Future Value of 100,000 in 18 years, using a Nominal Interest Rate of 10%, compounded annually requires annual deposits of $1,993.66 beginning today. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 119 Sinking Fund Calculator EXAMPLE 16a (JavaScript Calculator) Sinking Fund Alternatives PV Annuity Interest Conversion FV Annuity Sinking Fund Calculator PV Sum Amortization Instructions ON OFF FV Sum Mode Begin End Present Value 0.00 Future Value 100,000.00 Nominal Interest Rate 4.00 7.00 10.00 Effective Interest Rate 18.00 Number of Years 1.00 Payments Per Year 18.00 Number of Payments 3,749.36 begin 2,748.84 Payment mode clear all To yield the same Future Value, but using a more useful Nominal Rate of 4%, com- pounded annualy would require annual deposits of $3,749.36 beginning today. At an optimistic 7% NAI, the deposit need only be 2,748.84. Calculators Types of Calculations Definitions 120 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 16: (HP10Bii) Modified Future Value of an Annuity Compute the Future Value in eighteen years of $2,000 de- posited annually, beginning today, at 10% nominal annual interest compounded annually. 10 18 0 2000 The display will read -100,318.18. 4 The display will read -3,761.29. 7 The display will read -2,757.59. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 121 $100,000 would purchase today. If that amount is sufficient, the user need only modify the interest rate to exclude the inflation and risk factors or at least the inflation factor. The more conservative approach would be to use three or four percent and the more liberal (optimistic) approach would be to use seven percent. The resulting payment amount is the more useful and realistic annuity needed to yield the desired result. This, too, however, must be modified to put back the inflation component. Thus, the investor would want to deposit something between $2,748.84 and $3,749.36 an- nually with each annual payment increasing by the inflation rate for the prior year. This should yield an account with the approximate purchasing power of $100,000 today. Did you Notice? The Future Value of an Annuity computation did not produce the precise numbers as given by the Sinking Fund computation. This results because the Annuity reflects an annual payment of $2,000, which yields $100,318.18. The Sinking Fund begins with a Future Value of only $100,000.00 and thus produces a slightly smaller payment amount. Calculators Types of Calculations Definitions 122 FINANCIAL CALCULATIONS FOR LAWYERS 5. Amortization This calculation solves for the amount of the regular payment needed, at a stated interest rate and period, to pay off a present value. This is the opposite of the calculation involving the Present Value of an Annuity. For example, if you were to borrow $100,000 today and agreed to make 360 equal monthly payments at an inter- est rate of eight percent nominal annual interest, each pay- ment would need to be $733.76, as illustrated in Example 17. Amortization schedules typically involve the end mode because loan payments generally occur at the end of each period, rather than at the beginning. For example, if you were to borrow money to purchase a new car, the first pay- ment on the loan would not occur until one month from now. While that might be the beginning of a month, it is the end of the first month since the purchase, necessitating the use of end mode. TIP Amortization schedules typically involve the end mode because loan payments generally occur at the end of each period, rather than at the beginning. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 123 When Would You Want Use the Amortiza- tion Function? √ If you want to purchase a home, this function will deter- mine your monthly loan payments. √ If you have student loans outstanding, this function will determine your monthly payments. √ If you need to re-finance a loan or to combine various credit card obligations, this function will compute the monthly payments. 1. HP l0 Bii Calculator To perform the amortization function using the HP I0 Bii calculator, first compute the amount of the payments as explained above: set the calculator in end mode, set the Present Value (PV)as 100,000, the Future Value (FV) as 0, the Interest Rate per Year (I/YR) as 8, the Number of Pay- ments Per Year (P/YR) as 12, and the Number of Payments (N) as 360. Then solve for the amount of the Payment (PMT). The displayed answer will be -733.764573879, the negative indicating the payment. Thus the necessary payment is $733.76, which includes both the interest and principal. Calculators Types of Calculations Definitions 124 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 17: (HP10Bii) Amortization Compute the monthly payment needed to pay- off a loan of $100,000 in 30 years at a nominal annual inter- est rate of eight percent. 8 12 360 0 100,000 The display will read -733.76 2. JavaScript Financial Calculator The JavaScript Calculator operates much the same way as the HP10Bii. Press the Amortization Calculator button to open the correct Amortization Calculator End Mode Amortization calculator. Insert the nomi- nal annual interest rate, the number of periods per year, the number of years, and the present value (the loan amount) in the appropriate while boxes. Do not change the amounts in any yellow boxes. The answer will appear in the green Payment box. Begin Mode Amortization Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 125 End Mode Amortization EXAMPLE 16a (JavaScript Calculator) Amortization PV Annuity Interest Conversion FV Annuity Amortization Calculator PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 100,000.00 Future Value 0.00 Nominal Interest Rate 8.00 8.2999506808 Effective Interest Rate 30.00 Number of Years 12.00 Payments Per Year 360.00 Number of Payments end 733.76 Payment mode clear all If , instead, you were to use begin mode, the re- quired payment would be only 728.91. The amount is less Begin because the first payment would be made today rather than one month from now, as with the end mode calcula- tion. Note that in both cases, the Future Value is zero End because the entire loan is then paid off. Calculators Types of Calculations Definitions 126 FINANCIAL CALCULATIONS FOR LAWYERS End Mode Amortization Begin Mode Amortization 3. Amortization Schedule The above calculation computes the payment needed to amortize the present value. Most users, however, will also want to know the remaining balance owed after each pay- ment, as well as the portion of each payment comprising interest and principle. A list of these amount is called an amortization schedule. a. HP l0 Bii Calculator Press the orange shift key and then the AMORT key. This shifts to the calculator’s Amortization function, rather than the Future Value (FV) function. The display will read PER 1-12. This indicates Periods 1 through 12. Next, press the key indicating the equal sign. The display will read Prin, and -835.36, indicating the principal included in the pay- ments for period 1 through 12. Press the equal sign key again. The display will then read INT and -7,969.81, which is the amount of the periods 1 through 12interest. Next, press the equal sign key again. The display will read BAL and 99,164.63, indicating the balance of principal owed after periods 1 through 12. Next, press the orange shift key and then the AMORT key. The display will read PER 13-24. Repeat the above process using the equal sign key to display the applicable figures for periods 13 through 24. Then again press the orange shift key and then the AMORT key. The display will read PER 25-36. Repeat the above process using the equal sign key to display the appli- cable figures for periods 25 through 36. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 127 Then again press the orange shift key and then the AMORT key, along with the equal sign key. Repeat this process for all thirty years. b. JavaScript Financial Calculator The JavaScript Calculator operates much more sim- ply than does the HP 10Bii. It automatically provides a sched- ule for up to 360 payments, detailing the payment amount, the interest, the principal paid, and the remaining principal amount. End Mode Amortization Begin Mode Amortization Calculators Types of Calculations Definitions 128 FINANCIAL CALCULATIONS FOR LAWYERS Sinking Fund Calculator 6. Sinking Fund This calculation solves for the amount of the regular deposit needed, at a stated interest rate and period, to accu- mulate a future value. This is the opposite of the calculation involving the Future Value of an Annuity. For example, if you wanted to accumulate $25,000 in ten years and were willing to make ten equal annual depos- its, beginning today, at an annual interest rate of ten percent, each deposit would need to be $1,426.03. Beginning one year from now, the necessary deposits would be $1,568.63. Sinking Fund schedules often involve the begin mode because savings plan deposits often begin at the inception of the plan, which would be the beginning of the first period. The end mode calculation, however, may also be used. When Would You Want to Compute a Sink- ing Fund? √ If you need to save for a child’s education and know the amount needed. √ If you need to save for retirement and know the amount needed. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 129 TIP Sinking Fund schedules typically involve the begin mode because the depositor wishes to begin im- mediately. Sinking Fund Calculator EXAMPLE 18: (HP10Bii) Sinking Fund Compute the annual payment, beginning to- day, needed to accumulate $25,000 in 10 years at a nomi- nal annual interest rate of ten percent. 10 1 10 0 25,000 The display will read -1,426.03. Calculators Types of Calculations Definitions 130 FINANCIAL CALCULATIONS FOR LAWYERS 1. HP lOBii Calculator As illustrated in Example 18, set the calculator in begin mode, set the Present Value (PV) as 0, the Future Value (FV) as 25,000, the Interest Rate per Year (I/YR) as 10, the Number of Payments Per Year (P/YR) as 1, and the Number of Payments (N) as 10. Then solve for the amount of the Payment (PMT). The displayed answer will be -1,426.03, the negative indicating the deposit. Thus the ten necessary deposits are each $1,426.03. With accumulated interest the fund will equal $25,000 in ten years. 2. JavaScript Financial Calculator The JavaScript Calculator operates similarly. Press the Sinking Fund Calculator button to open the correct cal- culator. Insert the nominal annual inter- Sinking Fund Calculator Sinking Fund Calculator est rate, the number of periods per year, the number of years, and the future value in the appropriate white boxes. Do not change any amounts in the yellow boxes. The answer will appear in the green Payment box. Placing the cursor over any of the functions will open a pop-up box with additional instructions or explanations. Begin Be sure to use the correct mode: begin mode for payments beginning immediately and end mode for payments beginning at the end of the first pe- End riod. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 131 Sinking Fund Calculator EXAMPLE 18 (JavaScript Calculator) Sinking Fund PV Annuity Interest Conversion FV Annuity Sinking Fund Calculator PV Sum Amortization Instructions ON OFF FV Sum Mode End Begin Present Value 0.00 Future Value 25,000.00 Nominal Interest Rate 10.00 10.00 Effective Interest Rate 10.00 Number of Years 1.00 Payments Per Year 10.00 Number of Payments 1,426.03 begin 1,568.63 Payment mode end clear all mode Press End for end mode and the Payment amount will be 1,568.63. It is higher because the first payment would not be made until one year from now - the end of the first period. Calculators Types of Calculations Definitions 132 FINANCIAL CALCULATIONS FOR LAWYERS II. DEFINITIONS A. Interest Rate Standing alone, the term interest rate has no useful meaning. Instead, it requires one or more modifiers to indi- cate the period and frequency of compounding. Four differ- ent descriptions of interest are common. Each has its own Caution Standing alone, the term interest rate has no useful meaning. appropriate use; thus, no description is correct or incorrect: they simply have different meanings and uses. 1. Nominal annual interest rate. Sometimes called the “stated interest rate” or “coupon rate” this is the periodic interest rate times the number of periods per year. It is sometimes abbreviated as NAI. An interest rate of one percent per month produces a nominal annual interest rate of twelve percent per year, com- pounded monthly. To be correctly stated, it requires the full description of 12% NAI, compounded monthly. The NAI lan- guage denotes it as a nominal rate and the “compounded monthly” denotes the number of periods per year. Without Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 133 those two modifiers, the statement of 12% per year has little meaning. The NAI number is necessary for calculations involv- ing multiple periods per year because the inverse of the NAI equation is also true: the periodic interest rate equals the nominal rate divided by the number of periods per year. All calculators work with a periodic rate because the interest compounding period and the payment period must be the same. Thus, calculations involving multiple annual payments require the calculation of a periodic rate, which itself re- quires the use of the nominal rate. Whenever the interest compounds annually, the nomi- nal annual interest rate will equal the effective interest rate. However, whenever the interest compounds more often than annually, the nominal annual interest rate will be less than the effective rate. This can result in some persons being misled TIP Whenever the interest compounds more often than annually, the nominal annual interest rate will be less than the effective rate. by the statement of an interest rate. For example, a document may refer to a nominal rate of 10%, while later providing for monthly compounding. The effective interest rate would be 10.471306744%. A reader who does not appreciate the difference between the two rates (effective and nominal) - and their uses - may mistakenly Calculators Types of Calculations Definitions 134 FINANCIAL CALCULATIONS FOR LAWYERS visualize a lower rate of interest for the transaction than is accurate. While the statement of the 10% nominal rate com- pounded monthly would be correct, it is thus also easily mis- understood. Hence, a well-drafted document will provide the NAI rate (along with the compounding frequency), the peri- TIP A well-drafted document will provide the NAI rate (along with the compounding frequency), the periodic rate, plus the equivalent effective rate. odic rate, plus the equivalent effective rate. Boxes One and Two on pages 18 and 22 illustrate the conversion of an interest rate from nominal to effective or effective to nominal, using an HP 10Bii calculator. The JAVAScript Financial Calculator automatically converts the nominal rate to the effective rate. It also in- cludes an interest conversion calculator which converts an effective rate back to the equivalent nominal rate and peri- odic rate. Example 19 illustrates the amortization of a $100,000 student loan at 7.5% nominal annual interest with monthly payments for ten years beginning today. The necessary payment is $1,179.64. Using end mode, the necessary pay- Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 135 EXAMPLE 19 (JavaScript Calculator) Amortization PV Annuity Interest Conversion FV Annuity Amortization Calculator PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 100,000.00 Future Value 0.00 Nominal Interest Rate 7.50 7.7632598856 Effective Interest Rate 10.00 Number of Years 12.00 Payments Per Year 120.00 Number of Payments begin 1,179.64 Payment mode clear all DID YOU NOTICE? The JavaScript Calculator automatically computes the Effective Interest Rate. Calculators Types of Calculations Definitions 136 FINANCIAL CALCULATIONS FOR LAWYERS ment would be $1,187.02. For a problem involving an amortization, terminology is particularly important. The loan document will certainly state a nominal annual interest rate. It likely will not refer to a compounding period because interest on a loan does not normally compound: that is because it is paid regularly, along with partial principle payments. Most likely, the loan docu- ment also does not state the effective rate, once again be- cause the interest does not compound as a feature of the loan - at least not between the lender and borrower. Thus, in the case of Example 19, the document would not state the effective rate of 7.763%. If the debtor wants to know how much interest he is paying, the answer should refer to the 7.5% nominal figure. But, if the debtor wants to compare the loan to a savings account (perhaps to decide whether to payoff the loan with TIP A typical loan agreement will not state the ef- fective interest rate. It will state the nominal rate, which is a lower number and thus will ap- pear - to a novice - to involve lower interest. It will also state the annual percentage rate (APR) as required by federal law. The APR, however, is not the same as the effective rate and will always be lower than the effective rate if pay- ments are more frequent than annual. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 137 TIP Interest on a typical loan does not compound. Nevertheless, it has compounding effects: the lender’s source of funds compounds, as does the account receiving the payments. Similarly, the borrower’s cost of capital (and thus source of payments) compounds. existing investments), he would need to know the effective rate. Although interest on the loan does not strictly com- pound, it nevertheless had compounding effects because of its monthly payment nature. The lender receives each month’s interest and invests it somewhere, undoubtedly earning a com- pounded return. Also, the borrower pays the monthly inter- est with funds that, but for the payment, otherwise would be invested at a compounded return. Hence the effective rate reflects the true economic cost of the loan. In the example, unless the borrower was earning more than 7.763% on existing funds, he would be better off paying off the loan early (all other factors being equal). If he did not know about effective interest rates - and how to compute them - he would likely erroneously use a 7.5% figure for the loan to investment comparison. 2 . Periodic Interest Rate. This is the amount of interest per period. Any cal- culation involving multiple payments per year requires the use of a periodic rate. Well-drafted legal documents will state a peri- Calculators Types of Calculations Definitions 138 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 20 (JavaScript Calculator) Interest Conversion PV Annuity PV Sum FV Annuity Interest Rate Conversion Amortization Sinking Fund Instructions ON OFF FV Sum Convert Effective Convert Nominal Convert Periodic Rate to Nominal Rate to Effective Rate to Nominal Rate and Periodic Rate and Periodic Rate and Effective Rate Rate Rate Nominal Interest Rate 19.200000000 Periodic Interest Rate 1.5 Effective Interest Rate 20.98300406509 Payments Per Year 12.00 clear all Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 139 odic rate, as well as the equivalent nominal annual rate and the equivalent effective annual rate. For example, a periodic rate of I % per month is the equivalent of a nominal annual rate of 12%, compounded monthly and an effective annual rate of 12.682503013%. The periodic rate is necessary for any calculations. In addition, it is useful if the transactions involves less than a full year. Example 20 demonstrates the conversion of a Periodic Rate of 1.6% per month. It equals 19.2% nominal interest and 20.983% effective interest. Such a rate might be used on a credit card. The nominal rate will likely be disclosed, as might the annual percentage rate (which will be the same as the nominal). However, the lender will not likely disclose the effective rate, which is significantly higher than the nominal rate. Instructions 1. Press "Clear All" prior to working a problem. 2. Select the appropriate conversion function. 3. Type appropriate numbers in the white boxes. 4. The answer will appear in the green boxes. 5. Place your cursor over blue terms for a definition of the term. Calculators Types of Calculations Definitions 140 FINANCIAL CALCULATIONS FOR LAWYERS 3. Effective Interest Rate. This term has the same general meaning as the annual percentage yield or the yield to maturity and a similar mean- ing to the term internal rate of return. The four similar terms, however, have their own uses and are not precisely inter- changeable. a. Deposits. For original deposits, with no withdrawals, each of the four terms will be the same. The effective interest rate will be the annual compounded rate of interest: the actual amount of interest earned for a particular year divided by the amount on deposit at the beginning of the year. Financial institutions generally quote this rate com- pounded for the appropriate number of periods for an entire year. This is the most useful number for purposes of com- paring one deposit with another. For example, one financial institution may offer 10% nomi- nal annual interest compounded semiannually, while another Effective Rate Formula py effective rate = 100 1+ ( pr 100 ) -1 ) pr = periodic rate py = payments per year Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 141 offers 9.9% nominal annual interest compounded quarterly, and a third offers 9.8% compounded daily. A comparison of those three rates is difficult because of the differing com- pounding periods. Stating each in terms of an effective an- nual rate eliminates any confusion. The first institution is ac- tually offering 10.25% effective interest. The second is offering 10.273639392% effec- TIP tive interest, slightly more than the first even Because the nominal annual in- those it offers a lower terest rate for a deposit is lower nominal rate. The third than the effective interest rate, institution is offering I financial institutions will often 0.294827704, more still prominently quotethe effective even though it offers interest rate or annual percent- the lowest of the three age yield on a deposit. They nominal rates. may most prominently quote the lower nominal rate for a loan. The nominal an- Nominal Rate Formula 1 py nominal rate = 100py ((1+ eff 100 ) -1) eff = effective rate py = payments per year Calculators Types of Calculations Definitions 142 FINANCIAL CALCULATIONS FOR LAWYERS nual interest rate for a deposit will always be lower than the effective interest rate. As a result, financial institutions will often quote, in the most prominent language, the effective interest rate or annual percentage yield on a deposit. Accounts which have occasional withdrawals or addi- tional deposits will have the same effective interest rate and annual percentage yield or yield to maturity; however, they may have a different internal rate of return. Sales of a debt instrument subsequent to issue and prior to maturity - or offers to sell it -may result in a different yield to maturity and internal rate of return, because of the changing present value as a result of market forces. b. Loans. Discount loans with no pay- ments prior to maturity and no points have an effective inter- est rate equal both to the annual percentage yield and the nominal annual rate. They also have an annual percentage rate equal to the nominal rate. Installment loans and loans with points, however, have differing effective interest rates, nominal rates, and annual percentage rates. The effective rate on an installment loan with no points will be the interest rate that would accrue annually if the interest on the loan compounded. In actuality, TIP interest on an installment loan without negative am- Installment loans and ortization does not com- loans with points have pound; instead, the install- differing effective ments pay the interest due interest rates, nominal plus, usually, a portion of rates, and annual per- the principal. As a result, centage rates. no interest is charged on Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 143 interest. In a sense, the effective interest rate on such a loan is not representative of reality: while the effective rate is a compounded rate, the actual interest on the loan does not compound. The effective rate reflects what would happen if the interest compounded. In reality, the interest does compound, though not specifically with regard to the loan instrument. This is true both from the standpoint of the lender and the borrower. From the lender’s viewpoint, he receives in- stallment payments, in- TIP cluding all interest due and some principal. Those From the lender ’s amounts do not earn ad- viewpoint, interest ditional interest from this compounds. Thus a borrower with regard to lender should compare this loan; however, the an installment loan’s lender must do something effective rate to his with the funds. If depos- cost of funds. ited or loaned elsewhere, they will earn additional interest. Caution: If expended, they will free up Comparing either the other funds which can earn in- nominal or annual per- terest, or they will reduce the centage rate of an in- need for borrowing, which will stallment loan to the reduce other interest costs. Thus, lender’s cost of capital effectively, the funds earn inter- would be misleading. est for the entire year (unless the Calculators Types of Calculations Definitions 144 FINANCIAL CALCULATIONS FOR LAWYERS applicable currency is stuffed in a mattress or some other unproductive investment). Stating the uncompounded peri- odic rate on the particular loan as a compounded effective rate reflects the reality that the funds will earn interest from some source for the entire year. From the borrower’s viewpoint, he makes installment payments, including all interest due and some principal. As a result, he does not owe additional interest on those funds to that lender with regard to that loan; however, the borrower must have a source of funds to make the payments. That source TIP of funds itself has a cost, which reflects its own interest rate. If From the lender ’s he uses other available funds to viewpoint, interest make the payments, the borrower compounds. Thus a is then unable to earn interest lender should compare elsewhere on those funds. Or, if an installment loan’s ef- he borrows the funds to make the fective rate to his cost payments, the borrower must pay of funds. additional interest on such addi- tionally borrowed funds. Thus, effectively, the funds cost interest for the entire year Caution: (unless the borrower steals or prints the cur- Comparing either the rency, it has a cost). Stat- nominal or annual per- ing the uncompounded centage rate of an in- periodic rate on the par- stallment loan to the ticular loan as a com- borrower’s savings ac- pounded effective rate re- counts would be mis- flects the reality that the leading. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 145 Caution: BORROWERS Both the nominal and the annual percent- age rate [APR] on an installment loan will always be lower than the effective rate. Thus, financial institutions rarely, if ever, prominently disclose or otherwise adver- tise the effective rate of a loan. funds cost interest for the entire year. The nominal annual interest rate and the annual percent- age rate on an installment loan will always be lower than the effective rate. As a result, financial institutions rarely, if ever, prominently disclose or otherwise advertise the effective rate of a loan. This contrasts with their eagerness to advertise the effective rate on a deposit. Federal law does not require dis- closure of the effective rate. In fact it expressly requires promi- nent disclosure of the annual percentage rate [APR],3 which is always a lower number on an installment loan (and which does not reflect the above described reality). Also, federal law expressly permits the disclosure of a 3 15 U.S.C. § 1637 (for open ended credit); § 1638 (for other credit transac- tions). 4 12C.F.R.§226.1i. Calculators Types of Calculations Definitions “Comparative Index of Credit Cost’4 which has some char- acteristics of an effective rate, but which also is inevitably lower than the effective rate of interest. 4. Annual Percentage Rate. In credit transactions not involving points or some other fees, the annual percentage rate equals TIP the nominal annual interest rate. However, transactions involving The APR is a partially points and some other fees have compounded rate: it an annual percentage rate which reflects the nominal reflects both the nominal rate and rate with the “points” the compounded amortized effect amortized over the of the points or other fees. Dis- stated life of the loan. closure of this rate is required by federal law for most credit DID YOU NOTICE? The APR [annual percentage rate] equals the NAI [nominal annual interest rate] for an in- stallment loan with no points. The APR is higher than the NAI for an install- ment loan with points. The EFF [effective interest rate] is higher than both the APR and the NAI for an install- ment loan (regardless whether it has points). Calculators Types of Calculations Definitions transactions. It is typically ab- breviated as the APR. TIP In credit parlance, a “point” is equal to one percent of A “point” is equal to the principal amount loaned. one percent of the prin- Thus on a $100,000 loan, one cipal amount loaned. point equals $1000 and two points equals $2000. On a $200,000 loan, one point equals $2000 and two points equals $4000. Institutions charge points for three general reasons: First, the points - which are actually discounted inter- est - are not reflected in the nominal annual interest rate. As a result, the nominal rate is understated. While a lender must prominently disclose the annual percentage rate, which re- flects the points, it can do so along with disclosure of the nominal rate. Thus, lenders hope borrowers will visualize the nominal rate as the true rate, rather than the more accurate and higher - and sometimes less prominent - A.P.R. or the Caution: Because points are not reflected in the nominal rate, the nominal rate is always understated in an installment loan with points. most accurate and highest - and almost certainly undisclosed - effective rate. Calculators Types of Calculations Definitions 148 FINANCIAL CALCULATIONS FOR LAWYERS DID YOU NOTICE? Lenders charge points for three general reasons: 1. Points allow them to understate the nominal interest rate. 2. Points are generally tax deductible by the borrower. 3. Points are non-refundable, resulting in an excess return if the loan is paid-off early (as most home loans are). Caution: Two of the three rea- sons for points favor the lender. TIP Borrowers should If the present value of be very cautious the tax advantage with points. outweighs the ex- cess cost of early payoff, points are good for the borrower. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 149 Second, the points - if at- TIP tributable to a home loan - are generally deductible by the bor- Points paid on a loan rower for federal income tax pur- for a primary resi- poses. As a result, borrowers dence are generally may benefit from having more of deductible for federal the interest deductible in the first income tax purposes. year of the loan. Third, the points are almost always nonrefundable. They are paid - either with separate funds or by being withheld from the loan proceeds - at the time of the loan transaction: If the loan is outstanding for its entire term, the points are effec- tively paid periodically over the life of the loan. However, if the borrower pays the loan prematurely, he must pay all re- maining principal, unreduced by the points. For example, a $100,000 loan with two points is the equivalent of a $98,000 loan because just as soon as the borrower receives the $100,000 he must pay back $2000 as points. Nevertheless, the borrower is immediately liable for the entire $100,000 loan principal, even if he were to repay the loan the next day. Caution: Points are almost always non-refundable. Thus, if you expect to pay off the loan early (e.g., when you sell the house to buy a new house), con- sider avoiding points. Calculators Types of Calculations Definitions 150 FINANCIAL CALCULATIONS FOR LAWYERS As a practical matter, most TIP home loans are paid early be- cause they contain a “due on Most home mortgage sale” clause, accelerating loans are paid off long them whenever the underly- before the original matu- ing security changes hands. rity date of fifteen or thirty Many purchasers of resi- years. dential property sell the prop- erty - and thus pay off the re- This occurs because the spective loan early - prior to borrower sells the house, the end of the original loan pays off the loan and buys term. As a result, the lender a new house with a new earns an extraordinary inter- loan. est rate - higher even than the original effective interest rate. Often, much higher. To compute the annual percentage rate on an install- ment loan with points, follow these steps: 1. Amortize the loan and record the payment. 2. Subtract the points from the principal. 3. Input the Step 2 amount as the new princi- pal, the payment amount, the term, the pay- ment frequency and then solve for the inter- est rate. The Step 3 interest rate is the Annual Percentage Rate. To convert it to the effective rate, use the Interest Conversion Calculator: simply input the APR as the nominal rate and the calculator will automatically convert it. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 151 Example 21 illustrates computation of an Annual Per- centage Rate using an HP 10Bii calculator. Example 21a then illustrates the conversion of the APR to an effective rate. Although the JavaScript Financial Calculator is not currently designed to solve for a missing interest rate, it will do so with a simple process of interpolation. It then auto- matically converts the interpolated rate to an effective rate. EXAMPLE 21: (HP10Bii) Annual Percentage Rate Compute the annual percentate rate on a home loan of $200,000, a thirty year term, monthly payments, a nominal annual interest rate of 7.5% and three points. 7.5 12 30 200,000 The display will read -1,398.43 . 197,000 The display will read 7.655055419 . [This is the APR] Calculators Types of Calculations Definitions 152 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 21a: (HP10Bii) Conversion of Annual Percent- age Rate to Effective Rate Covert an APR of 7.655055419% with monthly payments to the equivalent effective rate. 7.655055419 12 The display will read 7.929432144 . [This is the EFF.] Example 22 illustrate the interpolation method. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 153 EXAMPLE 21 (JavaScript Calculator) APR Interpolation PV Annuity Interest Conversion FV Annuity Amortization Calculator PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 200,000.00 Future Value 0.00 Nominal Interest Rate 7.50 7.7632598856 Effective Interest Rate 30.00 Number of Years 12.00 Payments Per Year 360.00 Number of Payments end 1,398.43 Payment mode clear all Step One for Interpolation Method: amortize the loan Calculators Types of Calculations Definitions 154 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 21 (JavaScript Calculator) APR Interpolation PV Annuity Interest Conversion FV Annuity Amortization Calculator PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 197,000.00 Future Value 0.00 Nominal Interest Rate 7.50 7.70 7.7632598856 Effective Interest Rate 30.00 Number of Years 12.00 Payments Per Year 360.00 Number of Payments 1,377.45 1,404.53 Payment clear all Step Two for Interpolation Method: re-amortize the loan, using the true prinipal amount (subtract the points from the stated principal). Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 155 EXAMPLE 21 (JavaScript Calculator) APR Interpolation PV Annuity Interest Conversion FV Annuity Amortization Calculator PV Sum Sinking Fund Instructions ON OFF FV Sum Mode Begin End Present Value 197,000.00 Future Value 0.00 Nominal Interest Rate 7.65 7.655 7.7632598856 Effective Interest Rate 30.00 Number of Years 12.00 Payments Per Year 360.00 Number of Payments 1,397.74 1,398.42 Payment clear all Step Three for Interpolation Method: change the interest rate until the payment amount equals the Step One amortization pay- ment of 1,398.43. Hint: start with the already computed effective rate rounded to one deci- mal place. This may take several tries. Calculators Types of Calculations Definitions 156 FINANCIAL CALCULATIONS FOR LAWYERS B. Other Important Financial Terms Several other important terms arise in relation to inter- est rates. They include: a. simple yield b. yield c. yield to maturity (YTM) d. internal rate of return (IRR). The first two of these terms - simple yield and yield - are not terms of art: their Caution precise definitions may vary from user to The terms, simple yield user. Thus anyone yield, are not terms of art. using either of them in a legal context should Users should provide or provide a precise defi- demand a precise defini- nition. Likewise, any- tion in any legal docu- one coming across ment. them in a legal context should demand a pre- cise definition. The lat- ter two terms - yield to maturity and internal rate of return - have generally accepted, precise meanings. a. Simple Yield. This is an easy-to-compute, but imprecise measure of the return on a debt instrument. As Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 157 illustrated in Example 23, It is the nominal annual interest divided by the current market price of the instrument. This computation would change constantly, as the mar- ket value of the instrument changed. The ease of computa- tion justifies the use of the figure. It is, however, an inferior EXAMPLE 23: (HP10Bii) Simple Yield Compute the simple yield of a $1,000 (face amount) bond paying 7% nominal annuals interest, paid quar- terly. Assume that it sells, alternatively, for $900, $1000, and $1,100. 70 = 7.778% simple yield 900 70 = 7.000% simple yield 1000 70 = 6.364% simple yield 1100 “Simple Yield” Formula: nominal annual interest current market price Calculators Types of Calculations Definitions 158 FINANCIAL CALCULATIONS FOR LAWYERS measure of the true return on the bond or similar investment. The actual yield for a stated period or the yield to maturity would be more accurate and thus more useful. Some users may interchange this term with the slightly different term “yield.” Others might compound the quarterly payment to generate a more precise calculation. Neither use is wrong - they are merely different. As cautioned above, if someone uses the term “simple yield,” request a definition. b. Yield. This measure of the return on a debt in- strument is sometimes interchanged with the slightly different term TIP “simple yield.” More commonly, however, it An instrument’s “yield” dif- constitutes the actual fers from its “simple yield on an instrument yield” in two ways: for a stated period of time, as a function of 1. The yield is a function the purchase price. of purchase price rather Thus it would divide than market price. any periodic interest payment by the pur- 2. The yield is a com- chase price and then pounded rather than convert it to an annual simple (uncompounded rate). rate, compounding the periodic rate for the number of periods. Example 24 illustrates the computation of a Yield. This measure of the instrument differs from the “simple yield” in two respects. First, it adds the compounding fea- Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 159 EXAMPLE 24: (HP10Bii) Yield Compute the yield of a $1,000 (face amount) bond paying 7% nominal annuals interest, paid quarterly. Assume that it sells, alternatively, for $900, $1000, and $1,100. Step One: compute the simple yield, per Example 23. Step Two: convert the simple yields to comparable effec tive rates. 4 4 The display will 7.778 read 8.0078 The display will 7.000 read 7.1859 6.364 The display will read 6.5175 “Yield” Formula: compounded annual interest purchase price Calculators Types of Calculations Definitions 160 FINANCIAL CALCULATIONS FOR LAWYERS ture, when appropriate; hence the yields are greater when the payment period is less than one year. Second, it does not change constantly as the market price of the instrument changes; instead, it is fixed by the purchase or issue price of the instrument (depending on whose viewpoint is involved). Also, although the above definition describes a num- ber which is more accurate and hence more useful than the number described as a “simple yield,” this term - yield - does not present a true measure of an instrument’s return. Two inaccuracies are inherent. One, it relies on interest compounding, when, in fact, as far as the instrument is concerned the interest is paid and thus does not compound. This is less criticism of the calcula- tion - and more mere observation, however, because that feature is inevitable. For yields to be useful, they must generally be com- parable to those of other instruments. For this to be possible, they must be based on a common standard - such as the year. Instruments which pay interest annually will thus present an accurate yield. In contrast, instruments which pay at pe- riods other than a year will never present an accurate annual yield: it violates the fourth rule stated earlier: the payment period and the compounding period must be the same. For instruments that pay interest other than annually, an annual yield will never be precise because it inherently requires an assumption that the interest paid continued to earn interest at the same internal rate. While useful, such an assumption is not perfect. As long as users understand this feature, the calculation of a yield can be very useful and generally accurate. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 161 EXAMPLES 23-24 (JavaScript Calculator) Simple Yield and Yield PV Annuity Interest Conversion Yield Calculator Amortization FV Annuity Sinking Fund Instructions ON OFF FV Sum Periodic Interest Payment $17.50 Market Price $1000.00 Simple Yield 7.000000000 Periodic Yield 1.750000000 Yield 7.1859031289 Payments Per Year 4 clear all Calculators Types of Calculations Definitions 162 FINANCIAL CALCULATIONS FOR LAWYERS A second inaccuracy of the “yield” calculation involves its failure to consider the impact of changing values, i.e, changing market interest rates. Another way of stating this somewhat obvious point is that the term - as defined above - ignores market discounts and premiums. While the point of the calculation is simply to look at paid returns for a particu- lar period - and thus it accomplishes what its definition con- strains it to do, the calculation nevertheless risks presenting a significantly inaccurate picture. For example, an instrument sold at a premium will have the same “yield,” regardless of its life. In contrast, it will have a higher “yield to maturity” the longer the period until maturity. Similarly, an instrument sold at a discount will have the same “yield,” regardless of its life, although it will have a lower “yield to maturity” the longer the period until maturity. Despite some inherent inaccuracies of its own, the “yield to maturity” calculation presents the most accurate and useful picture of a debt instrument. Hence a comparison of the yields of two in- TIP struments, ignoring the terms of the instru- ments, might (though The “yield to maturity” cal- not necessarily) culation presents the present a small, or most accurate and useful even largely distorted picture of a debt instru- picture. A comparison ment. of yields that consid- ers the terms would in- deed be mostly accurate; however, it would also be a com- parison of ”yields to maturity” and thus, by definition, not a Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 163 comparison of mere yields. 5 C. Yield to Maturity. This is the most accu- rate measure of the return on a debt instrument. Comparable to - and sometimes interchanged with either the “effective interest rate” or the “internal rate of return” - it considers the instrument’s actual cash flows. Thus it is the most realistic measure of an instrument’s return. As illustrated in Example 25, the yield to maturity calculation amortizes the premium or discount element of the issue price over the life of the instrument. This is more useful than the mere “yield” which, as defined above, ignores the premium or discount. Nevertheless, the yield to maturity cal- culation is subject to at least two potential inaccuracies. First, it as- Caution sumes - as does the effective interest rate - The Yield to Maturity cal- that any payments culation assumes that all continue to earn or interest received will con- cost the same constant tinue to earn the same interest rate. This is un- constant interest rate. likely to be accurate; This assumption is un- nevertheless, because likely to prove precisely no investor has a crys- accurate. tal ball with which to determine future in- 5 Surprisingly, I have found some disagreement regarding the mean- ing of “yield to maturity.” Most authorities define the term as I do. However, at least one book defines it differently. Joel G. Siegel and Jae K. Shim, ACCOUNTING HANDBOOK 2d Ed., Barrons 1995 at 700 (providing a formula using an arithmetic rather than geometric com- pounding of interest). Calculators Types of Calculations Definitions 164 FINANCIAL CALCULATIONS FOR LAWYERS vestment returns, such an assumption is the best possible. It also permits realistic comparisons between instruments. Nev- ertheless, it can result in some misunderstandings and thus should be fully understood. The assumption that all returns are reinvested at the same rate, while necessary mathematically, can cause mis- understanding. An investor might assume that the two instru- ments in Example 26 are interchangeable because they have the same original cost and the same yield to maturity. Because they have different cash flows, however, they are Caution comparable only with the above assumption, which may The Yield to Maturity cal- - or may not - be realistic. culation assumes the in- strument will be outstand- The second potential ing for its entire scheduled inaccuracy involving the yield life. Because many instru- to maturity calculation involves ments are called or paid the assumption that the instru- early, this assumption is ment will be outstanding for its often incorrect, rending entire expected life. This, the YTM inaccurate. again, is a necessary as- sumption: to input a future maturity value one must know the future date. Because no crystal balls exist to foretell the future, the assumption be- comes necessary that the instrument will continue to be out- standing for its entire scheduled life and will make all sched- uled payments. Many instruments, however, have a put or call feature under which either the maker or purchaser- or both - may offer or demand payment early, respectively. In Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 165 EXAMPLE 25: Yield to Maturity A bond paying 7.0% nominal annual interest, paid quar- terly - $70.00 per year or $17.50 per quarter - and sold at par would have a yield to maturity of 7.186%, regard- less of its life. To compute this figure, set the P/YR as 4, the I/YR as 7.0 and solve for the effective interest rate, which will be 7.186. 4 7 The display will The display will read 7.186. read 7.186. The simple yield is 7.000. The yield is 7.186. The yield to maturity is 7.186 Calculators Types of Calculations Definitions 166 FINANCIAL CALCULATIONS FOR LAWYERS EXAMPLE 25a: Yield to Maturity If, instead, the bond sold for $900.00 and were outstand- ing for two years, it would have a yield to maturity of 13.365%. To compute this figure, set the P/YR as 4, the PMT as 17.50, the N as 8, the PV as (900), and the FV as 1000. Then solve for the I/YR, which will be 12.743 and also for the effective interest rate, which will be 13.365 . 17.50 8 900 The display will 1000 read 12.743. The display will read 13.365. The simple yield is 7.778. The yield is 8.008. The yield to maturity is 13.365. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 167 EXAMPLE 25b: Yield to Maturity If, instead, the bond sold for $900.00 and were outstand- ing for ten years, it would have a yield to maturity of 13.365%. To compute this figure, set the P/YR as 4, the PMT as 17.50, the N as 40, the PV as (900), and the FV as 1000. Then solve for the I/YR, which will be 8.494 and also for the effective interest rate, which will be 8.769 . 17.50 40 900 The display will 1000 read 8.494. The display will read 8.769. The simple yield is 7.778. The yield is 8.008. The yield to maturity is 8.769. 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 1000 read 1.892935887. The display will read 1.906415353. The simple yield is 6.364. The yield is 6.517 The yield to maturity is 1.906. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 169 EXAMPLE 25d: Yield to Maturity If, instead, the bond sold for $1100.00 and were outstand- ing for ten years, it would have a yield to maturity of 13.365%. To compute this figure, set the P/YR as 4, the PMT as 17.50, the N as 8, the PV as (1100), and the FV as 1000. Then solve for the I/YR, which will be 12.743 and also for the effective interest rate, which will be 13.365 . 17.50 40 1100 The display will 1000 read 5.682226203 The display will read 5.804455792. The simple yield is 6.364 The yield is 6.517 The yield to maturity is 5.804 Calculators Types of Calculations Definitions 170 FINANCIAL CALCULATIONS FOR LAWYERS such cases, the statement of a yield to maturity should note the assumption regarding maturity. d. Internal Rate of Return. This is the effec- tive interest rate at which the initial investment equals the present value of all future cash flows. If all cash flows are level and in the same direction, this computation is relatively simple and essentially parallels the computation of a yield to maturity. Uneven cash flows - and particularly those which change direction - present computational difficulties. Most calculators actually use a trial and error approach because the formula can be extremely complex. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 171 Example 26 Comparison of Instruments with the Same Yield Instrument One with a face value of $100,000 is- sued for $96,000, paying 10% nominal annual inter- est for five years will have a yield to maturity of 11 .084585%. Instrument Two involves $96,000 invested at 11.084585 nominal annual interest compounded an- nually for five years will generate $162,382.87. It, too, has a yield to maturity of 11.084585%. But, Instrument One will generate approximately $162,382.87 only if each $10,000 interest payment is itself reinvested at 11.084585% (the extra $1.01 is due to a rounding error). Calculators Types of Calculations Definitions 172 FINANCIAL CALCULATIONS FOR LAWYERS INTEREST RATES FOR USE IN LEGAL CALCULATIONS EXPERT TESTIMONY BY ECONOMISTS, ACCOUNTANTS, AND OTHERS By definition, all financial calculations involving the time value of money require the use of an interest rate. The choice of the applicable interest rate is typically the most important factor in a legal valuation: small variations in the rate can have large consequences in the computed amount. The choice of rate is also arguably the least understood factor, the most subjective factor, and the one which varies the most in the testimony of “experts.” Example 27 illustrates how the choice of the interest rate can be the most important factor in a wrongful death matter. In a personal injury or wrongful death case involving future lost wages, the plaintiff will be entitled to the present value of the future loss. In addition to liability, the plaintiff must prove: TIP The choice of the interest rate is the most im- portant factor in a legal valuation of personal in- juries. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 173 Example 27 The Effect of Interest Rates on Wrong- ful Death Valuations John died with a work expectancy of 40 years. He was then earning $100,000 per year. Undiscounted, the loss would be $4,000,000 - $100,000 per year for 40 years. However, because the present value would be paid by the torfeasor, the court or parties would discount the 40-year annuity to the present. Often, a plaintiff’s expert will testify that the appropriate dis- count rate is one percent. Using that number, the present value would be $3,283,468.61. Often, a defense expert will testify that the appropriate dis- count rate is 7.5%, or some similar number reflecting conser- vative, long-term investment returns. Using that number, the present value would be $1,259,440.87 Properly analyzed, the correct discount rate should be be- tween four and four and one-half percent. Using those num- bers, the present value would be between $1,840,158.44 and $1,979,277.39. DID YOU NOTICE? The argument of the interest rate involves more than $2,000,000 - by far the largest single factor in the case. Calculators Types of Calculations Definitions 174 FINANCIAL CALCULATIONS FOR LAWYERS √ the number of years of the loss, which equates with the N function √ the annual amount of loss, which equates with the PMT function √ the frequency of the lost , which equates with the P/YR function √ the timing of the loss, which equates with the Mode function √ the interest rate, which equates with the I/Yr func- tion Typically, the number of years will be easily determined and may even involve a stipulation: it would be based on work or life expectancy. The annual loss will be a question of fact, but may be debated only within a range of numbers based on the plaintiffs income experience and the value of his or her personal services and consumption. The frequency and timing of the loss affects the calculation; however, they are largely inconsequential: a timing change will cause a corresponding number of years change, and a frequency change will cause a corresponding interest rate change. Caution The choice of the interest rate is the most sub- jective factor in valuaing a personal injury case. It is one over which experts sharply disagree. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 175 Example 27 The Effect of Changing Life Effectancy on Wrongful Death Valuations Using the facts of Example 27, suppose the parties could not decide the correct work expectancy for John. Using the Plaintiff’s interest rate of 1%, adding ten years to the work-life expectancy (increasing it to 50) changes the present value to $3,919,611.75 - an increase of $636,143.14 or 19.37%. Thus a very large - 25% - increase in work-life expectancy results, at a low interest rate, to a large increase in present value. Using the Defense’s interset rate of 7.5%, adding ten years to the work-life expectancy changes the present value to $1,297,481.16 - an increase of $38,040.29 or 3%. Thus a very large increase in work-life expectancy results, at higher interest rates to a very modest increase in present value. Using the 4.5% suggested rate and the 50 year term, the present value would be $1,976,200.78, an increase of $136,042.34 or 7.4% Calculators Types of Calculations Definitions 176 FINANCIAL CALCULATIONS FOR LAWYERS The choice of the appropriate interest rate, however, will have both a large impact and little case-specific eviden- tiary support. Indeed much evidence exists regarding inter- est rate; however, very little of it has anything to do with a specific matter. DID YOU NOTICE? Defense could concede a 50, rather than 40, year work-life expectancy if Plaintiff would concede a 4.5%, rather than 4% discount rate. This is true because the present value of an annuity of $100,000 per year for 50 years discounted at 4.5% is less than the present value of an annuity of 100,000 per year for 40 years discounted at 4%. The 1/2% change in interest is more significant than the ten year increase in the length of the annuity! Why Do People Change Interest? To understand how an economist, accountant, or other expert chooses the appropriate interest rate, consider why people charge interest. They do so for four reasons: 1. To compensate for inflation. In times of low inflation, this factor is small, while in times of Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 177 high expected inflation, this factor is correspondingly high. The key word here is “expected” inflation. Interest rates are often compared to past inflation - a interesting comparison that shows the market’s ability or in- ability to predict. This comparison, however, is not itself im- portant for the prediction of future interest rates, which con- sider only future inflation. Past inflation, to the extent it pre- dicts the future, is relevant; however, it must be understood as a predictor of a factor and not a factor itself. Some financial Instruments, are not subject to discount for this factor: those which bear interest in an amount that fully compensates for expected inflation. In contrast, non- interest or low-interest bearing notes must be discounted to reflect their insufficient interest. This has very little to do with the instrument’s marketability or the solvency of the obligor; instead, it merely reflects the nature of the contract. Because all instruments must ultimately produce an adequate return on investment, the market will price them to do so. Three types of examples illustrate the relevance of this factor for valuations important in legal matters: one tax ex- ample, one personal injury example, and one family law ex- ample. Example 28 illustrates a tax law example. For federal tax purposes, income must be recognized upon the receipt of a “cash equivalent.” The rule applies to all cash method taxpayers and to most accrual method taxpayers. Courts gen- erally define a “cash equivalent” as an unconditional promise to pay, of a solvent obligor, assignable, not subject to setoff, readily marketable, and subject to a discount not substan- 6 tially greater than the prevailing market rate. 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 QUESTIONS & ANSWERS Future Value of a Sum Calculator 1 a. Compute the future value of $1,000.00 in 10 years at 6% nominal annual interest compounded annually. $ 1,790.85 To calculate the above amount, clear the register and set the Future Value of a Sum Calculator with the following values: Mode Begin End irrelevant Present Value 1,000.00 Future Value 1,790.85 computed by calculator Nominal Interest Rate 6.00 Effective Interest Rate 6.00 computed by calculator Number of Years 10.00 Paymernts Per Year 1.00 Number of Payments 10.00 computed by calculator Payment 0.00 Calculators Types of Calculations Definitions 188 FINANCIAL CALCULATIONS FOR LAWYERS Future Value of a Sum Calculator Future Value of a Sum Calculator 1 b. Compute the future value of $1,000 in 10 years at 6% nominal annual interest compounded semiannually. $ 1,806.11 To calculate the above amount, clear the register and set the Future Value of a Sum Calculator with the following values: Mode Begin End irrelevant Present Value 1,000.00 Future Value 1,806.11 computed by calculator Nominal Interest Rate 6.00 Effective Interest Rate 6.09 computed by calculator Number of Years 10.00 Paymernts Per Year 2.00 Number of Payments 20.00 computed by calculator Payment 0.00 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 189 Future Value of a Sum Calculator 1 c. Compute the future value of $1,000 in 10 years at 6% nominal annual interest compounded monthly. $ 1,819.40 To calculate the above amount, clear the register and set the Future Value of a Sum Calculator with the following values: Mode Begin End irrelevant Present Value 1,000.00 Future Value 1,819.41 computed by calculator Nominal Interest Rate 6.00 Effective Interest Rate 6.09 computed by calculator Number of Years 10.00 Paymernts Per Year 12.00 Number of Payments 120.00 computed by calculator Payment 0.00 Calculators Types of Calculations Definitions 190 FINANCIAL CALCULATIONS FOR LAWYERS Future Value of a Sum Calculator 1 d. Compute the future value of $1,000 in 10 years at 6% nominal annual interest compounded daily. $ 1,822.027707 (assuming 360 days per year, 30 per month,) or $1,822.028955 (assuming 365 days per year) To calculate the above amount, clear the register and set the Future Value of a Sum Calculator with the following values: Mode Begin End irrelevant Present Value 1,000.00 Future Value 1,822.03 computed by calculator Nominal Interest Rate 6.00 Effective Interest Rate 6.18 computed by calculator Number of Years 10.00 Paymernts Per Year 365.00 Number of Payments 3,650.00 computed by calculator Payment 0.00 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 191 Present Value of a Sum Calculator 2 a. Compute the present value of $100,000.00 to be received 18 years from today at 6% nominal annual in- terest compounded annually. $ 35,034.38 To calculate the above amount, clear the register and set the Present Value of a Sum Calculator with the following values: Mode Begin End irrelevant Present Value 35,034.38 computed by calculator Future Value 100,000.00 Nominal Interest Rate 6.00 Effective Interest Rate 6.00 computed by calculator Number of Years 18.00 Paymernts Per Year 1.00 Number of Payments 18.00 computed by calculator Payment 0.00 Calculators Types of Calculations Definitions 192 FINANCIAL CALCULATIONS FOR LAWYERS Present Value of a Sum Calculator 2 b. Compute the present value of $100,000.00 to be received 18 years from today at 6% nominal annual interest compounded semiannually. $ 34,503.24 To calculate the above amount, clear the register and set the Present Value of a Sum Calculator with the following values: Mode Begin End irrelevant Present Value 34,503.24 computed by calculator Future Value 100,000.00 Nominal Interest Rate 6.00 Effective Interest Rate 6.09 computed by calculator Number of Years 18.00 Paymernts Per Year 2.00 Number of Payments 36.00 computed by calculator Payment 0.00 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 193 Present Value of a Sum Calculator 2 c. Compute the present value of $100,000.00 to be received 18 years from today at 6% nominal annual interest compounded monthly. $ 34,051.06 To calculate the above amount, clear the register and set the Present Value of a Sum Calculator with the following values: Mode Begin End irrelevant Present Value 34,051.06 computed by calculator Future Value 100,000.00 Nominal Interest Rate 6.00 Effective Interest Rate 6.09 computed by calculator Number of Years 18.00 Paymernts Per Year 12.00 Number of Payments 216.00 computed by calculator Payment 0.00 Calculators Types of Calculations Definitions 194 FINANCIAL CALCULATIONS FOR LAWYERS Present Value of a Sum Calculator 2 d. Compute the present value of $100,000 to be re- ceived 18 years from today at 6% nominal annual inter- est compounded daily. $ 33,962.6087222 (360 day convention) $ 33,962.5668597 (365 day convention) To calculate the above amount, clear the register and set the Present Value of a Sum Calculator with the following values: Mode Begin End irrelevant Present Value 33,962.57 computed by calculator Future Value 100,000.00 Nominal Interest Rate 6.00 Effective Interest Rate 6.09 computed by calculator Number of Years 18.00 Paymernts Per Year 12.00 Number of Payments 216.00 computed by calculator Payment 0.00 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 195 Did you notice? √ As the compounding frequency increases (from annual to daily), in problems a to d, the Present Value of a Future Sum decreases. This occurs because the more frequent compounding results in a higher effective interest rate (EFF). √ Thus, as interest rates increase, present values decrease. For example, if you increase the interest rate in problem (d) to 12% nomi- nal annual interest compounded daily, the present value drops to $11,536.61: thus, double the interest and the present value drops by two-thirds! √ In contrast, as interest rates increase, fu- ture values also increase. Calculators Types of Calculations Definitions 196 FINANCIAL CALCULATIONS FOR LAWYERS Future Value of an Annuity Calculator 3 a. Compute the future value of $1,000 to be paid annually for ten years, the first payment due at the be- ginning of the period (an annuity due). Use a nominal annual interest rate of 6% compounded annually. $ 13,971.64 (annuity due) To calculate the above amount, clear the register and set the Future Value of an Annuity Calculator with the following values: Begin Mode Present Value 0.00 Future Value 13,971.64 computed by calculator Nominal Interest Rate 6.00 Effective Interest Rate 6.00 computed by calculator Number of Years 10.00 Paymernts Per Year 1.00 Number of Payments 10.00 computed by calculator Payment 1,000.00 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 197 Future Value of an Annuity Calculator 3 b. Compute the future value of $1,000 to be paid an- nually for ten years, the first payment due at the end of the period (an annuity in arrears). Use a nominal annual interest rate of 6% compounded annually. $13,180.79 (annuity in arrears) To calculate the above amount, clear the register and set the Future Value of an Annuity Calculator with the following values: End Mode Present Value 0.00 Future Value 13,180.79 computed by calculator Nominal Interest Rate 6.00 Effective Interest Rate 6.00 computed by calculator Number of Years 10.00 Paymernts Per Year 1.00 Number of Payments 10.00 computed by calculator Payment 1,00.00 Calculators Types of Calculations Definitions 198 FINANCIAL CALCULATIONS FOR LAWYERS Future Value of an Annuity Calculator 3 c. Compute the future value of $1,000 to be paid annually for ten years, the first payment due at the end of the period (an annuity in arrears). Use a nominal annual interest rate of 6% compounded semi-annually. $ 13,236.64 (annuity in arrears) To calculate the above amount, clear the register and set the Future Value of an Annuity Calculator with the following values: Mode End Present Value 0.00 Future Value 13,236.64 computed by calculator Nominal Interest Rate 6.09 Effective Interest Rate 6.09 computed by calculator Number of Years 10.00 Paymernts Per Year 1.00 Number of Payments 10.00 computed by calculator Payment 1,000.00 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 199 If you change problem (c) to an an- nuity due (Begin Mode), the present value is $14,042.75 To prove this, you need not clear the regis- ter: merely change the mode to Begin. BE CAREFUL: In Problem 3c, you cannot convert the annuity from ten $1,000 payments to twenty $500 payments: that would produce an incorrect answer. By splitting the payments in two, you would accelerate one-half of them, resulting in more interest earned. Because that is not consistent with the given facts, it would not answer the ques- tion asked. You must, instead, convert the interest rate from a nominal annual rate compounded semi-annually to an equivalent effective interest rate. Use the Interest Rate Conversion Calculator to do this. Interest Rate Conversion Calculator Calculators Types of Calculations Definitions 200 FINANCIAL CALCULATIONS FOR LAWYERS PV Annuity PV Sum FV Annuity Interest Rate Conversion Amortization Sinking Fund Instructions ON OFF FV Sum Convert Effective Convert Nominal Convert Periodic Rate to Nominal Rate to Effective Rate to Nominal Rate and Periodic Rate and Periodic Rate and Effective Rate Rate Rate Nominal Interest Rate 6.00 3.00 Click here for the Periodic Interest Rate correct converter for Effective Interest Rate 6.09 Problem 3c. Payments Per Year 2.00 clear all For Problem 3c, convert the 6.00% nominal interest rate, compounded semi-annually to an effective rate of 6.09% . Use this as the nominal rate, compounded annually. TIP: Remember, an effective rate is the Did you Notice: equivalent of a nominal rate com- Problem 4a computes pounded annually. the Present Value of Lottery Winnings. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 201 Present Value of an Annuity Calculator 4 a. Compute the present value of $50,000 to be paid annually for twenty years, the first payment due at the beginning of the period (an annuity due). Use a nominal annual interest rate of 6% compounded annually. $ 607,905.82 (annuity due) To calculate the above amount, clear the register and set the Present Value of an Annuity Calculator with the following values: Mode Begin Present Value 607,905.82 computed by calculator Future Value 0.00 Nominal Interest Rate 6.00 Effective Interest Rate 6.00 computed by calculator Number of Years 20.00 Paymernts Per Year 1.00 Number of Payments 20.00 computed by calculator Payment 50,000.00 Calculators Types of Calculations Definitions 202 FINANCIAL CALCULATIONS FOR LAWYERS Present Value of an Annuity Calculator 4 b. Compute the present value of $1000 to be paid monthly for ten years, the first payment due at the beginning of the period (an annuity due). Use a nominal annual interest rate of 6% compounded annually. $ 90,523.82 (annuity due) To calculate the above amount, clear the register and set the Present Value of an Annuity Calculator with the following values: Begin Mode Present Value 90,523.82 computed by calculator Future Value 0.00 Nominal Interest Rate 6.00 Effective Interest Rate 6.00 computed by calculator Number of Years 10.00 Paymernts Per Year 12.00 Number of Payments 120.00 computed by calculator Payment 1,000.00 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 203 Did you notice? If you reduce the interest rate in problem 4a to 3%, the present value rises to $766,189.96. If you reduce it further to 2%, the present value rises to $833,923.10. Hence an important rela- tionship of interest rates and present values: ♦ As the interest rate decreases, the present value increases - and vice versa. If you change problem 4 b to an annutity in arrears, the present value is $ 90,073.45. To prove this, you need not clear the register: merely change the mode to End. BE CAREFUL: In Problem 4b, you cannot convert the annuity from twelve $1,000 payments per to one $12,000 payment: that would produce an incorrect answer. By combining the payments, you would accelerate 11/12ths of them, resulting in more interest earned. You must, instead, convert the interest rate from a nominal annual rate compounded annually to an equivalent nominal annual rate compounded monthly. A cardinal rule of annuties is: ♦ The payment period and the compounding period must be the Calculators Types of Calculations Definitions 204 FINANCIAL CALCULATIONS FOR LAWYERS Sinking Fund Calculator 5 a. You need $100,000 18 years from now for your child’s education. Assume a nominal annual interest rate of 6% compounded monthly. How much must you de- posit each month. Assume alternatively an annuity due and an annuity in arrears. Ignore any income tax con- sequences or inflation. $ 256.88 (annuity due) $ 258.16 (annuity in arrears) To calculate the above amount, clear the register and set the Sinking Fund Calculator with the following values: Mode Begin Present Value 0.00 Future Value 100,000.00 Nominal Interest Rate 6.00 Effective Interest Rate 6.17 computed by calculator Number of Years 18.00 Paymernts Per Year 12.00 Number of Payments 216.00 computed by calculator Payment 256.88 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 205 Did You Notice? If you start saving when the child is one month old (an annuity in arrears) each monthly deposit must be approximately $1.28 greater to achieve the same $100,000 future value. TIP: To compute the annuity in arrears, simply click on End the end mode button. The answer will appear automatically. Did you Notice: Problem 5a computes the needed monthly saving for a child’s education. Calculators Types of Calculations Definitions 206 FINANCIAL CALCULATIONS FOR LAWYERS Sinking Fund Calculator 5 b. Suppose you have no idea what you will need for your child in 18 years for his or her education. However, you estimate that if your child were 18 years old now, you would feel comfortable having $100,000 in savings . You do not know what the inflation rate will be for education costs; however, you estimate that it will average an amount close to the inflation rate for the general economy. You also do not know future tax rates; however, you assume they will remain stable. How much must you deposit each month to attain your goal. Assume alternatively an annuity due and an annuity in arrears. This problem Illustrates real life issues. While we cannot accurately predict future costs, we can accurately determine present costs. Hence we begin with the $ 100,000 present cost of college. This number must be gauged for the individual: some need more and some need less, depending on family expectations. But, at least the number is based on reality - current costs - rather than a guess of costs 18 years into the future. The following analysis also ignores the earnings the fund would earn during matriculation. A more complex calculation, however, would consider this factor. Next we must acknowledge that college costs will Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 207 increase over the years. In many recent years, this has been more rapid than the general inflation rate - at least for the portion of college costs attributable to tuition and books. Living expenses, however, are more likely to increase, on average, at the general inflation rate. Also, in some years tuition and books have not increased by the full inflation rate. For any Did you Notice: Problem 5b is a better way to com- pute the needed monthly saving for a child’s education. The 4a method is commonly used, but flawed. savings plan, the saver must determine whether the general inflation component of interest rates represents the future inflation rate of the item (here education) for which he or she is saving. If so, then it becomes appropriate to ignore the inflation component of interest as well as the inflation component of the future costs. If not, then the saver must adjust the expected cost by any differential. A reasonable conclusion might be that the inflation component of general interest rates will be - over the next 18 years - comparable to the inflationary increase in college education costs, including tuition, books, and living expenses. Calculators Types of Calculations Definitions 208 FINANCIAL CALCULATIONS FOR LAWYERS Next recall that people pay interest for three reasons: expected inflation, risk, and liquidity (the borrower pays extra for the convenience of cash while the lender charges for the inconvenience of not having cash). Recently - with the advent of inflation adjusted bonds, government issues have tended to yield approximately 3.5% over the inflation rate (and sometimes up to 4% or slightly more). Economists call this the “real” or “ underlying” interest rate - the portion paid and charged for the liquidity factor. In the short run, the market may estimate future Inflation incorrectly and thus pay more or less than this amount after actual inflation is considered. For example, if the market expects 2% inflation, government bonds should yield approximatly 5.5 percent. But, if inflation ultimately turns out to be 1%, then the bonds will have yielded a “real” rate of 4.5 percent. Or, if inflation ultimately turns out to be 4%, then the government bonds will have yielded TIP: a “real” rate of only 1.5 percent. The best conservative pre-tax effective In the short run, interest rate to use for such mistakes occur; a college savings plan however, in the longer is approximately 3.5%. run, they tend to level out and the “real” rate tends to be about 3.5 percent. The 18 year period for which the problem contemplates saving is sufficiently long so as to justify the following conclusion: the underlying interest rate will be approximatly 3.5%. Naturally, the savings account can earn more than this if the saver take risks; however, the Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 209 account can then also lose value. Individual savers can benefit or lose from risk taking. The overall market, however, can ignore it because, at least in general, for each winner, there is a loser. Probably you do not want to invest a child’s education fund in a very risky instrument. Thus, you might reasonably conclude that you can earn, conservatively TIP: 3.5% after inflation or liberally 5.5% after inflation (mortgage A more risky, but backed securities or high arguably realistic pre- grade equities or mid- tax effective interest grade corporate bonds rate to use for a each currently yield college savings plan is closer to 8 percent). approximately 5.5%. Next, taxes cannot be ignored. A child under the age of 14 must pay taxes at his parent’s marginal rate, which currently is likely 14%, 28%, 31 %, or 38% (ignoring state and local income taxes). A child 14 or older must pay taxes at his own rate, which currently is likely 14%. These numbers ignore the child’s modest personal exemption Caution: and any standard deduction. Short term government bonds - at Do not forget to the election of the anticipate the impact taxpayer - produce of taxes. either tax-deferred or Calculators Types of Calculations Definitions 210 FINANCIAL CALCULATIONS FOR LAWYERS tax free income; hence, we can ignore the tax consequences of them. However, they also pay low rates. State and local bonds produce no taxable income; however, they can be risky, at least in the short run. Mortgage backed securities, equity investments, and corporate bonds produce taxable income - some of which can be deferred and some of which is subject to a maximum 20% rate. But, they also can be risky, particularly in the short-run. Thus, as the time approaches for the needed funds, you may want to shift from such investments into ones containing less market risk. Another real factor to consider are intangible taxes imposed by some states - such as Florida - on some investments. Or, you might consider placing all or part of the investment in a section 529 plan, which (with some limitations) is exempt from tax when used for qualified education expenses. The bottom line is that you might realistically yield 3.0% after inflation, risk, and taxes in a very conservative investment. Or, you might yield 5.0% after inflation, risk, and taxes in an acceptably risky investment. Or, perhaps you might yield 7.0% after inflation, risk, and taxes in a more aggressive investment. More is probably TIP: unrealistic, especially considering the need to Realistic after-tax become more effective rates for a conservative as the child college savings plan are approaches college age. 3.0% (conservative), 5.0% (moderate) and 7.0% With the above (aggressive. assumptions and conclusions - each of Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 211 which is realistic - we can now work the problem. In the conservative assumption the required monthly payment would be: $351.12 (annuity in arrears) or $350.26 (annuity due). To calculate the above amounts, clear the register and set the Sinking Fund Calculator with the following values: Mode Begin End Present Value 0.00 Future Value 100,000.00 Nominal Interest Rate 2.96 Effective Interest Rate 2.99 computed by calculator Number of Years 18.00 Paymernts Per Year 12.00 Number of Payments 216.00 computed by Payment End 351.12 calculator Begin 350.26 Calculators Types of Calculations Definitions 212 FINANCIAL CALCULATIONS FOR LAWYERS In the moderate assumption the required monthly payment would be: $289.61 (annuity in arrears) or $288.43 (annuity due). To calculate the above amount, clear the register and set the Sinking Fund Calculator with the following values: Mode Begin End Present Value 0.00 Future Value 100,000.00 Nominal Interest Rate 4.89 Effective Interest Rate 5.00 computed by calculator Number of Years 18.00 Paymernts Per Year 12.00 Number of Payments 216.00 computed by Payment End 289.61 calculator Begin 288.43 Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 213 In the aggressive assumption the required monthly payment would be: $237.70 (annuity in arrears) or $236.37 (annuity due). To calculate the above amount, clear the register and set the Sinking Fund Calculator with the following values: Mode Begin End Present Value 0.00 Future Value 100,000.00 Nominal Interest Rate 6.78 Effective Interest Rate 6.99 computed by calculator Number of Years 18.00 Paymernts Per Year 12.00 Number of Payments 216.00 computed by Payment End 237.70 calculator Begin 236.37 Calculators Types of Calculations Definitions 214 FINANCIAL CALCULATIONS FOR LAWYERS Thus, with liberal assumptions, you must save $236.37 per month from the day the child was born to generate, in eighteen years, an amount that would be the future equivalent of $100,000 today, regardless of the future inflation rate. With the alternative conservative and moderate assumptions, you must save at least $350.26 or $283.43 per month. Even these amounts, however, will not generate a fund equal to the future value of $100,000. That is true because they are uninflated. To compensate, you must increase the Caution: You must annually increase the savings amount by the actual inflation rate. amount of monthly savings by the actual inflation rate. To do this, simply multiple the monthly amount each year by one plus the reported increase in the Consumer Price Index. For example, if, during the first year of saving the inflation rate is 3.0%, the various savings amounts for the second year must be $243.46 (aggressive model), $291.93 (moderate model) or $360.77 (conservative model). These numbers are 1.03 times $236.37 (aggressive), $283.43 (moderate) and $350.26 (conservative). Continue to increase them annually. The resulting Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 215 fund in 18 years should then be sufficient, Caution: assuming the original $100,000 figure was If you make the accuarate and the aggressvie correct model was assumptions, but chosen (aggressive, only achieve the moderate, or conservative conservative). If, however, you used the earnings, your fund aggressive model to will be short. compute the original savings amount, but you can only achieve the conservative model level of earnings,your fund will be insufficient. If, in the alternative, you estimate that college costs will increase on the average 2% faster than the general Inflation rate, then you must save the equivalent of $142,824.62 in 18 years. To compute this number,use the Future Value of a Sum Calculator. Insert $100,000 for the Present Value, 2.0 for the nominal annual interest rate, 18 for the number of years, and solve for the Future Value. Using this number and our conservative conclusions, we must save $337.59 (aggressive model), $411.95 (moderate model), or $500.21 (conservative model) per month, starting the day the child is born. These numbers would then need to be adjusted annually by the actual inflation rate. Calculators Types of Calculations Definitions 216 FINANCIAL CALCULATIONS FOR LAWYERS PV Annuity Interest Conversion Future Value of a Sum Calculator Amortization FV Annuity Sinking Fund Instructions ON OFF PV Sum Mode Begin End Present Value 142,824.62 Future Value 100,000 Nominal Interest Rate 2.00 Effective Interest Rate 2.00 18.00 Number of Years 1.00 Payments Per Year 18.00 Number of Payments 0.00 Payment clear all While none of these answers is precise, they give you realistic estimates without your having to estimate future inflation or future interest rates. Did you Notice: Problem 6a computes the required payments on a home loan. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 217 Amortization Schedule 6 a. You borrowed $150,000.00 from your grandmother to purchase a new home. She said that she “wants to earn an annual percentage rate of 7% on her money.” How much will your monthly payments be on a thirty year loan? $ 997.95 (7.0% APR with no points) To calculate the above amount, clear the register and set the Amortization Calculator with the following values: End Mode Present Value 150,000.00 Future Value 0.00 Nominal Interest Rate 7.00 Effective Interest Rate 7.23 computed by calculator Number of Years 30.00 Paymernts Per Year 12.00 Number of Payments 360.00 computed by Calculators Types of Calculations Definitions 218 FINANCIAL CALCULATIONS FOR LAWYERS CAUTION: Private lenders often misuse loan terminology. CAUTION: You must determine whether the lender wants to charge the stated interest rate or earn the stated rate. The two are not the same! Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 219 Payment 997.95 calculator Grandmother, however, stated that she “wanted to earn a 7.0% APR on her money.” This involves a misuse of the term APR, which is a nominal rate computed with points amortized over the life of a loan. The misuse of terminology, however, is commonplace. The term “annual percentage rate” is commonly used because of federal lending disclosure requirements. Unfortunately, it is also commonly used when it is inapplicable. An investor does not earn an “APR”; rather, an investor earns an annual yield or an effective interest rate. A lender, in contrast, charges an APR; however, the lender “earns” an effective rate. Hence, the problem cannot accurately be worked without more information. The above solution is the TIP: If the lender wants to “charge” 7.00%, he probably is comparing the rate to commercial loans. Thus use 7.00% nominal annual interest at the appropriate compounding period. most literal constuction of her terminology; however, to be certain, better information is needed. Did Grandmother want to “charge” 7.0% APR or did Calculators Types of Calculations Definitions 220 FINANCIAL CALCULATIONS FOR LAWYERS she want to “yield” 7.0%? Grandmother may not have realized the technical meaning accorded that term. If she was measuring her desired interest by comparing it to comparable loan offerings by local financial institutions, then she probably intended the term’s technical meaning. If so, the above calculation would be correct. The above calculation assumed that Grandmother wanted to “yield” 7.0% on her money. Quite possibly, she measured her desired interest by comparing it to what she could earn in a certificate of deposit or other investment, then she probably intended to describe her desired annual TIP: If the lender wants to “earn ” 7.00%, he probably is comparing the rate to savings accounts. Thus use 7.00% effective interest, converted to the appropriate nominal rate and compounding period. Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 221 percentage yield, or effective interest rate. If so, the following would be the correct analysis: $ 976.39 (7.0% EFF or APY with no points) To calculate the above amount, clear the register and set the End Amortization Calculator with the following values: Mode Present Value 150,000.00 Future Value 0.00 Nominal Interest Rate 6.785 Effective Interest Rate 7.00 computed by calculator Number of Years 30.00 Paymernts Per Year 12.00 Number of Payments 360.00 computed Calculators Types of Calculations Definitions 222 FINANCIAL CALCULATIONS FOR LAWYERS PV Annuity PV Sum FV Annuity Interest Rate Conversion Amortization Sinking Fund Instructions ON OFF FV Sum Convert Effective Convert Nominal Convert Periodic Rate to Nominal Rate to Effective Rate to Nominal Rate and Periodic Rate and Periodic Rate and Effective Rate Rate Rate Click here for the Nominal Interest Rate 6.78497 correct converter for Periodic Interest Rate 0.565 Problem 6a. Effective Interest Rate 7.00 12.00 Payments Per Year clear all by Payment 976.39 calculator To work this version of problem 6a, you must convert the 7.00% effective rate to the comparable nominal annual rate compounded monthly. To do so, use the Interest Rate Conversion Calculator. 6 b. Compute the same amount, however, do it for a fifteen year loan. $ 1,348.24 (7.0% APR with no points) $ 1,330.28 (7.0% EFF or APY with no points) Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 223 Note that in each Instance, an increase in the payments of approximately 35%, results in a decrease in the term of 50%, going from a 30-year to a 15-year loan. From the opposite perspective, going from a 15-year to a 30-year loan, if the term doubles, the payments drop merely by approximately 25%. To calculate the above amounts, you need not clear the register for the Amortization Calculator. Instead, merely change the Number of Years to 15. The new payment will appear automatically. TIP: Increasing the payments by approximately 35% corresponds to a decrease in the term of 50%! Stated differently, double the term and the payment drops by onely one-fourth! Calculators Types of Calculations Definitions 224 FINANCIAL CALCULATIONS FOR LAWYERS c. You owe $100,000 for student loans, at an APR of 8%. The loan bears no interest until six months after graduation. You have the option of paying off the loan over 10, 15, 20, or 30 years. What will be the amount of your payments? $1,205.24 (10 year payoff) $ 949.32 (15 year payoff) $ 830.90 (20 year payoff) $ 728.91 (30 year payoff) To calculate the above amounts, clear the register and set the Amortization Calculator with the following values: Mode Begin Present Value 100,000.00 Future Value 0.00 Nominal Interest Rate 8.00 Effective Interest Rate 7.00 computed by calculator Number of Years 10.00 Paymernts Per Year 12.00 Number of Payments 120.00 computed by Payment 1,205.24 calculator Calculators Types of Calculations Definitions FINANCIAL CALCULATIONS FOR LAWYERS 225 PresentValueCalculator Future Value Calculator Present Value of an Annuity Calculator Future Value of an Annuity Calculator Amortization Calculator (Begin Mode) Amortization Calculator (End Mode) Sinking Fund Calculator Interest Rate Converter Yield Calculator

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