A Textual Explanation

NAVIGATION INSTRUCTIONS GLOSSARY FINANCIAL CALCULATIONS FOR LAWYERS LECTURES INDEX INTRODUCTION PRESENT VALUE OF A SUM FUTURE VALUE OF A SUM SINKING FUND AMORTIZATION WITH CHART PRESENT VALUE OF AN ANNUITY FUTURE VALUE OF AN ANNUITY INTEREST CONVERSION AMORTIZATION SIMPLE CALCULATOR YIELD CALCULATOR TAX CALCULATORS SOLVING FOR AN INTEREST RATE FINANCIAL CALCULATIONS FOR LAWYERS M A Textual Explanation 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 compensation plan. Tax Law Practice necessitates an understanding 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, because the calculators and their respective manuals are often complicated, lawyers lacking an accounting or finance background may shy away from this important area of law This booklet serves three purposes: 1. It provides a basic explanation - with lawyers as the intended audience - of the use and application of a typical hand-held financial calculator, the Hewlett Packard 10 Bii. 2. It discusses the legal system s use of financial terminology. 3. It includes workable Electronic Financial Calculators that solve most of the problems faced by lawyers. I. INTRODUCTION TO THE USE OF A CALCULATOR My first advice is read the manual. Most financial calculator manuals explain the various types of calculations and provide understandable examples. This book does not preempt or replace those manuals for other calculators. Rather, it supplements them with explanations and examples geared toward lawyers. Steven J. Willis 2009. All Rights Reserved. 1

For users who rely solely on the Electronic Financial Calculators provided with this Course, this booklet serves as the instruction manual. A. WHY LAWYERS NEED TO KNOW FINANCE While financial calculators can compute many things, six types of calculations are fundamental: 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 For Tax Lawyers, each of these calculations is relevant 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 education, as part of an agreed marital settlement. He might also want to compute the present value of a proposed alimony stream to determine the alternative amount of lump sum alimony. 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 Steven J. Willis 2009. All Rights Reserved. 2

advance payments; similarly, the Future Value calculations are relevant for contracts involving deferred payments. B. TYPES OF CALCULATORS Three types of calculators are commonly available: 1. Simple 2. Financial 3. Scientific Simple. Every lawyer probably owns a Simple Calculator and uses it frequently. These inexpensive machines add, subtract, multiply, and divide. They often compute square roots as well as several other common calculations. While in the 1970's such machines were very expensive costing hundreds of dollars (in terms of 2010 prices), they are now insignificant in price. While useful, they are not helpful in most financial calculations. Scientific. Lawyers typically do not use scientific calculators. These machines compute various trigonometric functions such as sine, cosine, tangent, and cotangent. Useful for engineers, these calculations do not arise often in the practice of law. Be careful in purchasing a handheld calculator, as many of the most readily available are scientific calculators. Because these are useful in high school and college math courses, most households have one. A typical Scientific Calculator, however, will not solve for common Financial functions. Financial. The subject of this Course is Financial Calculations. Many hand-held and electronic financial calculators are available. They range in price from about $25 to several hundred dollars. The more expensive hand-held machines are typically programmable for more exotic financial functions. Lawyers rarely need such things. The Hewlett Packard HP 10Bii Financial Calculator is widely available and will serve nearly all needs of an attorney. Sharp and Texas Instruments also produce readily available, inexpensive calculators. When choosing a handheld calculator, be sure it has function keys for PV, FV, PMT, and P/YR. These cover Present Value, Future Value, Payment, and Payments per Year. Ideally, these will be the primary keys on the machine. Some Scientific Calculators have a shift function under which the calculator converts to a financial calculator. Be wary of these. The keys likely serve at least three, if not four functions, depending on whether the key is in normal mode, upward shift, downward shift, or even some other mode. This can be very confusing. Steven J. Willis 2009. All Rights Reserved. 3

Because handheld calculators are fairly inexpensive, lawyers should at least two: a Financial Calculator for financial calculations and a separate Scientific Calculator for the rare occasions they need to use trigonometry. Almost certainly they will have several simple calculators as well. Lawyers should also be wary of Calculators which use Reverse Polish Notation (RPN). In reverse polish notation, the mathematical operation follows the operands. For example, to add the numbers 3 and 5 in normal notation, one computes: (3 + 5) = 8 But in reverse polish notation, one would perform: 3, 5 + The calculator would have no equal sign, so one would not have that extra function. Early computers and calculators used this format. The HP 12C calculator still uses it and continues to be popular. Because the format does not require parenthesis and similar symbols, it can calculate some very complex functions more efficiently than standard notation. Simple polish notation is the opposite: the operator appears first: + 3,5 The term Polish dates from the creation of Polish notation by a famous Polish mathematician, Jan Łukasiewicz. The reverse method first appeared commonly in the 1960's. Typically, lawyers have little use for RPN. They should be generally familiar with it because they will encounter many people economists, real estate agents, and others who use the HP 12C or similar calculators. As these people may be expert witnesses, a lawyer needs to understand their terminology and methodology. C. TYPES OF FINANCIAL CALCULATIONS Six types of Financial Calculations are fundamental to a lawyers practice: 1. Future Value of a Sum 2. Present Value of a Sum Steven J. Willis 2009. All Rights Reserved. 4

3. Present Value of an Annuity 4. Future Value of an Annuity 5. Sinking Fund 6. Amortization The Electronic Calculators included with this Course include separate calculators for each type. In contrast, a handheld calculator performs all six functions using a single screen. I designed the calculators separately so as to avoid some common mistakes people make in using handheld calculators. With a handheld, a single machine has to be able to compute all six functions, which are variations of the same formula. With a computer, however, one has the luxury of linking separate calculators for each primary calculation and thus making the operations easier to use. 1. THE FUTURE VALUE OF A SUM This calculation computes the future amount or value of a current deposit. The distinction between an effective interest rate and a nominal interest rate is critical to understanding this course. Lesson Five covers this in greater depth. You should also consult the GLOSSARY. For example, $1,000 deposited today, earning 10% interest compounded annually a.k.a., 10% effective interest (EFF) - 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. The year two amount includes not only the $100 interest earned each year on the principal amount of $1,000, but it also includes an additional $10 interest earned on the first year's interest of $100: ten percent of $100 is $10. This illustrates the effect of compounding: interest accelerates over time as it compounds... or, as it earns interest on interest. mind: Why would a lawyer want to perform this calculation? Several examples come to 1. If you deposit money into an account, you can compute what it will be worth in the future. This is useful if: You want to save for a child s education. You are saving for retirement. Steven J. Willis 2009. All Rights Reserved. 5

You are saving for a large purchase. 2. If population growth rates continue at a constant rate, you can compute the population of an area after a given length of time. This is useful if: You are dealing with growth management. You are dealing with environmental issues, increasing at a constant rate. 3. If your client was owed a specific amount such as a debt or a judgment - 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. 4. If a budget item or cost increases at a particular rate, you can compute the amount for a future period. This is what the FUTURE VALUE OF A SUM CALCULATOR looks like. LESSON NINE explains how to use it. Steven J. Willis 2009. All Rights Reserved. 6

2. THE PRESENT VALUE OF A SUM This calculation computes the present amount of a future deposit or debt. For example, $1,100 to be received in one year, discounted at 10% effective [ annual interest (EFF), has a present value of T $1,000 today. In comparison, $1,100 in two years The distinction between y an has a present value of $909.09. Discounting effective interest rate and p a $1,100 for five years produces a value of $683.01. nominal interest rate e is Discounting $1,100 for 100 years at 10% produces critical to understanding this a value of only $ 0.0798. a course. LESSON FIVE covers this in greater depth. qyou Thus, if you owe $1,100 100 years from should also consult u the now, you should be able to pay off the obligation GLOSSARY. o today with only 8, assuming the appropriate t effective interest rate is 10%. e Why would a lawyer want to know how to compute the Present Value f of a Sum? Several reasons come to mind: r o 1. If you owe money in the future, you can compute what it equals in current value. m 2. If you want a discount for an advance payment for goods or services, you t would h compute the present value of the future obligation. e 3. If you know the amount you need at a future time - such as for retirement d or college entrance - you can compute the present value needed to produce o that future amount. c u 4. For tax or accounting purposes, a lawyer may suggest his client deduct m the present value of a future cost if possible. While I.R.C. section 461(h) eof the Internal Revenue Code may not permit this in many situations, alternative n contractual relationships may make such a present value current deduction t possible. For a lawyer to compare the after-tax consequences of a present deduction versus a future deduction of a larger amount, he must first be capable o r of computing the present value. t This is what the PRESENT VALUE OF A SUM CALCULATOR looks like. LESSON h TEN explains how to use it. e s u m Steven J. Willis 2009. All Rights Reserved. m 7 a r y

3. THE PRESENT VALUE OF AN ANNUITY This calculation computes the present value of a series of equal payments made at regular intervals, earning a constant interest rate. For example, $1,000 deposited at the end of each year for ten years, earning 10% effective annual interest, has a present value today of $6,144.57. Similarly, $6,144.57 deposited today, earning 10% effective annual interest will produce a fund from which $1000 This is the amortization of the present value: it creates the annuity. This is an example of an annuity in arrears : one in which the payments occur at the end of each period. could be withdrawn for ten consecutive years, beginning one year from today. Each of these examples the annuity in arrears and the amortization - use the end mode function. In another example, $1,000 deposited at the beginning of each year for ten years, earning 10% effective annual interest, has a present value today of $6,759.02. Steven J. Willis 2009. All Rights Reserved. 8

This is an example of an annuity due : one in which the payments occur at the beginning of each period. The 10-year, 10% annuity due has a present value of $6,759.02. The annuity in arrears has a present LESSON FOUR will define the term annuity due. You may also consult the GLOSSARY. value of $6,144.57. The annuity due has a greater present value because it starts earlier: today (for the annuity due) and one year from now (for the annuity in arrears). Why would a lawyer want to know how to compute the Present Value of a Sum? Several reasons come to mind: 1. If you owe money at regular intervals in the future, you can compute what it equals in current value. This is useful if: 1. You owe money on a loan. The present value would be the payoff amount. 2. In a family law case, you owe alimony. The present value would be the lump sum amount to replace the periodic payments. 2. 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 is useful if: You need to compute the discount amount on an insurance or rental contract. 3. 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 tortfeasor s obligation. 4. If you have won a state lottery. The present value of the future payments would be the alternative amount that might be elected. 5. 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 would be the current value of the bond. This is what the PRESENT VALUE OF AN ANNUITY CALCULATOR looks like. LESSON ELEVEN explains how to use it. Steven J. Willis 2009. All Rights Reserved. 9

4. THE FUTURE VALUE OF AN ANNUITY This calculation computes the future value of a series of equal payments made at regular intervals, earning a constant interest rate. We will deal with a level annuity: e.g., $1000 per month for ten years. An unlevel annuity e.g., $1000/month sometimes and $1500/month other times is beyond this course. We also deal with regular payments: e.g., every month or every quarter. An irregular annuity e.g., one that skips some otherwise regular payments is beyond this course. Further, although an annuity can have a varying interest rate, we will deal with one involving level payments. Hence, by definition, it will have a constant internal interest rate. When we compute the present value of the annuity, we will use a current interest rate (which we project to be valid for the annuity period). For example, $1,000 deposited annually beginning one year from now, earning 10% effective annual interest, will increase to $15,937.42 in ten years. This is an example of an annuity in arrears : one in which the payments occur at the end of each period. LESSON FOUR will define the term annuity in arrears. You may also consult the GLOSSARY. Steven J. Willis 2009. All Rights Reserved. 10

Similarly, if you need $15,937.42, in ten years, and you can earn 10% effective annual interest, you must deposit $1000 annually beginning one year from today. This is the sinking fund of the future value: it creates the annuity. The Future Value of an Annuity and a Sinking Fund are related. The Sinking Fund is the series of payments (the annuity) that equals the Future Value. For example, the annual deposit of $1000 (the Sinking Fund/Annuity), at 10% effective annual interest, beginning in one year, has a Future Value of $15,937.42. This is an example of an annuity in arrears : one in which the payments occur at the end of each period. The distinction between an effective interest rate and a nominal interest rate is critical to understanding this course. LESSON FIVE covers this in greater depth. You should also consult the GLOSSARY. In an alternate example, $1,000 deposited at the beginning of each year for ten years, earning 10% effective annual interest has a future value in ten years of $17,531.17. The 10-year, 10% annuity due has a future LESSON FOUR will define the value of $17,531.17. In contrast, the annuity in term annuity due. You may arrears has a future value of $ 15,937.42. The also consult the GLOSSARY. annuity due has a greater future value because it starts earlier: today (for the annuity due) and one year from now (for the annuity in arrears). 1. If you save money for a retirement plan at regular intervals, you can compute what it will be worth in the future. 2. If you save money for a child s education at regular intervals, 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 comparable to current values. Thus the answer might not be useful; however, two methods can be used to convert the answer to a useful number: Convert the amount to a present value. Modify the interest rate to reflect a real rate of return rather than the actual predicted rate. LESSONS SIX, SEVEN, and ELEVEN will explain these methods. This is what the FUTURE VALUE OF AN ANNUITY CALCULATOR looks like. LESSON TWELVE explains how to use it. Steven J. Willis 2009. All Rights Reserved. 11

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. LESSON FIVE defines nominal annual interest." You may For example, if you were to borrow $100,000 today and agreed to make 360 equal monthly also want to consult the GLOSSARY. payments at an interest rate of eight percent nominal annual interest, each payment would need to be $733.76, beginning one Never interchange interest rate terminology: nominal, annual, and effective percentage rates are terms of art: use them correctly. LESSON FIVE deals with the terminology distinctions in depth. You may also want to consult the GLOSSARY. month from now. Steven J. Willis 2009. All Rights Reserved. 12

Why would a lawyer want to know how to compute an Amortization? Several reasons come to mind: 1. If you want to purchase a home, this function will determine your monthly loan payments. 2. If you have student loans outstanding, this function will determine your monthly payments. 3. If you need to re-finance a loan or to combine various credit card obligations, this function will compute the monthly payments. This is what the AMORTIZATION CALCULATOR looks like. LESSON THIRTEEN explains how to use it. 6. SINKING FUND This calculation solves for the amount of the regular deposit needed, at a stated interest rate and period, to accumulate a future value. This is the opposite of the Steven J. Willis 2009. All Rights Reserved. 13

calculation involving the Future Value of an Annuity. You should remember: the term Sinking Fund is another term for an Annuity. The sinking fund is the series of deposits (the annuity) that accumulates to a desired future value. For example, if you want to accumulate $25,000 in ten years and are willing to make ten equal annual deposits, beginning today, The distinction between an effective interest rate and a at an effective annual interest rate of ten percent, each deposit would need to be $1,426.03. nominal interest rate is Similarly, if you annually deposit $1,426.03, critical to understanding this beginning today, and you can earn 10% effective course. LESSON FIVE covers annual interest, in ten years, you will have $25,000. this in greater depth. You This is the Future Value of the Sinking Fund. The should also consult the fund is itself an Annuity. This is also an example of GLOSSARY. an annuity due : one in which the payments occur at the end of each period. LESSON FOUR will define the term annuity due. You may also consult the GLOSSARY. You should notice: The Future Value of an Annuity and a Sinking Fund are related. The Sinking Fund is the series of payments (the annuity) that equals the Future Value. For example, the annual deposit of $1000 (the Sinking Fund/Annuity), at 10% effective annual interest, beginning in one year, has a Future Value of $15,937.42. This is an example of an annuity in arrears : one in which the payments occur at the end of each period. LESSON FOUR will define the term annuity in arrears. You may also consult the GLOSSARY. The term Sinking Fund is another term for an Annuity. In an alternative example, If you want to accumulate $25,000 in ten years and are willing to make ten equal annual deposits, beginning one year from now, at an effective annual interest rate of ten percent, each deposit would need to be $1,568.63. The 10-year, 10% annuity due Sinking Fund requires $1,426.03 to produce $25,000. The annuity in arrears Sinking Fund requires $1,568.63 to produce the same $25,000. The annuity due has a lesser current value because it starts earlier: today (for the annuity due) and one year from now (for the annuity in arrears). Why would a lawyer want to know how to compute a Sinking Fund? Several reasons come to mind: 1. If you need to save for a child s education and know the future amount needed. Steven J. Willis 2009. All Rights Reserved. 14

2. If you need to save for retirement and know the future amount needed. 3. You need to pay off a bond or debenture in the future and want to save the principal necessary for the "balloon payment." This is what the SINKING FUND CALCULATOR looks like. explains how to use it. LESSON FOURTEEN D. CALCULATOR TERMINOLOGY Each calculation relies on the same basic formula, involving six factors, with the typical key label. The Calculators included with this Course use the words for each function. In contrast, handheld calculators use keys with abbreviations. Financial calculators have seven basic function keys: 1. Present Value (PV) Steven J. Willis 2009. All Rights Reserved. 15

2. Future Value (FV) 3. Payment (Pmt) 4. Interest (I) or (I/yr) 5. Number of Periods (N) 6. Payments per Year (P/Yr) 7. Mode For purposes of this Course, you will almost always know the mode. You must then know five of the remaining six functions. The calculator will solve for the remaining function. That is the point of a Financial Calculator: to solve for the missing function. If you only know four of the six functions, you must nevertheless estimate, logically deduce, or make up one of the two missing functions. No calculator can solve for two missing functions. For example, if you know: how much you borrowed for your home loan (PV), the interest rate (I/yr), that you intend to pay it off to zero (FV) and that you want monthly payments (P/yr), you cannot solve for both the time period and the payment amount. You may tell the calculator the time period perhaps 30 or 15 years and it will tell you the needed payment. Or, you may tell the calculator that you want to pay $1,500 monthly (Pmt), and it will tell you how long that will take (N). On a handheld calculator such as an HP10Bii the six functions are found along the top row of keys. All keys serve two functions with the orange (sometimes green) key serving as the shift key to change a key from one function to another. Steven J. Willis 2009. All Rights Reserved. 16

FV for Future Value. N for number of periods. I/YR for nominal interest per year. PV for present value. PMT for payment. P/YR for payments per year (or periods per year). Hewlett-Packard, Inc. Used with permission. On the Electronic Calculators used in this Course, all function keys are always displayed clearly. The Six critical function keys always appear. Also, several other important, but less critical function keys always appear. Also, the values for each function always appear. Steven J. Willis 2009. All Rights Reserved. 17

The six main function keys are: Present Value (PV). This function inputs or solves for the value today of either a sum in the future or a series of payments in the future. Thus it is useful in both the Present Value of a Sum calculation as well as the Present Value of an annuity calculation. For example, if you know the future value, the interest rate and the time period, then you solve for the present value of the future amount. If, instead, you know the present value, then you must be solving for another function, such as the future value or the series of withdrawal payments that the present value can generate an amortization. Future Value (FV). This function inputs or solves for the value in the future of either a present sum or a series of payments in the future. Thus it is useful in both the Future Value of a Sum calculation as well as the Future Value of an annuity calculation. For example, if you know the present value, the interest rate and the time period, then you solve for the future value of the present amount. If, instead, you know the future value, then you must be solving for another function, such as the present value or the series of payments needed to generate the future value a sinking fund. Payment (PMT). This function inputs or solves for the value of a regular deposit or withdrawal. It is useful in the present and future value of an annuity calculation (the payment is the regular annuity amount). It is the function solved for in both the amortization and sinking fund calculations. For example, if you know the present or future value and the interest rate and time period, then you solve for the payment amount that the present sum would generate or the needed deposit to result in the future value. Or, if you know the payment value, then you must be solving for another function, such as the present or future value of the series of deposits. Interest (i/yr). This function inputs or solves for the nominal annual interest rate. It is essential in all the various calculations. For example, if you know the present and future value, the payment amount and the time and frequency period, then you solve for the interest rate. Or, if you know the interest rate, then you must be solving for another function, such as the present or future value of the series of deposits. LESSON FIVE focuses on interest rate terminology. It distinguishes nominal rates Steven J. Willis 2009. All Rights Reserved. 18

from periodic rates, from effective rates, and from an annual percentage rate. Number of Periods (N). This function inputs or solves for the total number of periods. It is essential in all calculations. For example, if you know the present and future value, the payment amount and the frequency and interest rate, then you solve for the number of periods. Or, if you know the number of periods, then you must be solving for another function, such as the present or future value of the series of deposits or for the interest rate. Payments Per Year (or, Periods Per Year) (P/yr). This function inputs or solves for the total number of periods or payments per year. Another way of defining this is as the compounding period: the Nominal Annual Interest Rate compounds at this frequency. As explained in LESSON FIVE: The compounding period (periods per year) and the payment period (payments per year) must either be identical or be 1. For example, if you know the present and future value, the payment amount, the total time period and interest rate, then you solve for the number of periods per year. Or, if you know the number of periods per year, then you must be solving for another function, such as the present or future value of the series of deposits or for the interest rate. Mode. This function defines the timing of the initial payment in a sinking fund, an amortization, or in an annuity. Begin Mode occurs when the first payment is at the beginning of each period. End Mode occurs when the first payment is at the end of each period. When using the Electronic Calculations provided with this Course, you may always highlight one of the function words such as Present Value or Mode using your cursor. This will cause a definition to appear. Steven J. Willis 2009. All Rights Reserved. 19

E. COMMON MISTAKES IN USING A CALCULATOR 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! Usually, they have violated one of the following six rules. Users of all handheld calculators risk making these mistakes. SIX COMMON CALCULATOR RULES 1. Clear the machine's memory when starting a new calculation. 2. Set the cash flows with the proper sign. 3. Set the Mode correctly. 4. Set the interest rate to compound for each payment period. 5. Set the periods per year correctly. 6. Set the display to the appropriate number of decimal places. 1. FIRST, CLEAR THE MACHINE AND DO SO CORRECTLY. 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 calculators, such as the HP 10Bii: the display shows only one function at a time, creating the risk that the user will not remember to clear all other functions. The Electronic Calculators included with this Course minimize this risk because the display shows all function values at all times; hence, users are unlikely to forget to set a function correctly. In contrast, with a handheld calculator, users can only see one function display at a time; as a result, they often forget to set one or more functions which retain settings from prior calculations. 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 Steven J. Willis 2009. All Rights Reserved. 20

entered can be cleared, and a different function key by which all information can be cleared. a. Electronic Calculators All Electronic Calculators included with this Course have a Clear All button. Pressing it sets all functions to the default value often zero or 1. The computer's Back Space key will erase individual digits and the delete key will erase an entire highlighted number. Because the included Electronic Calculators display 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 a number in the box below. Use the backspace key or delete key to erase it. Or, click on the Clear All Key to clear the box. b. HP 10Bii Calculator The HP 10Bii Calculator has three levels for the clear function. 1. The C key - when pressed in un-shifted 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. Steven J. Willis 2009. All Rights Reserved. 21

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. Press the following keys: 50 + 20 + 30 40 The display will read 110. EXAMPLE 2 (HP 10Bii ) If you input 523 but intended 524, you may use the backward arrow key to erase the 3. Then simply enter the number 4. The display will read 524. In contrast, the C key will clear the entire number 523. Press the following keys: 523 The display will read 524. 4 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 HP 10Bii calculator, 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: Steven J. Willis 2009. All Rights Reserved. 22

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 re-set it manually through the procedure described below or remove the battery. 2. SET THE CASH FLOWS WITH THE PROPER SIGN. Many, but not all, calculators require that cash flows be directional. 1 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 the Electronic Calculators. a. Electronic Calculators The Electronic Calculators eliminate 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 Electronic Calculators eliminate the second most common difficulty faced by users of many handheld calculators. b. HP 10Bii Calculator The HP 10Bii Calculator requires cash flows to be entered with opposite signs. As shown in EXAMPLE 3, failure to do so will prompt the display no SoLution. You may enter a negative number in two ways. For example, to input the number (1000), first enter the positive number 1000, and then press the plus/minus key: 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. 1 Many other calculators -such as those manufactured by Texas Instruments - do not require opposite signs between present value and future value. Steven J. Willis 2009. All Rights Reserved. 23

EXAMPLE 3 (HP 10Bii ) Suppose you want to compute the Nominal 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 negative number, press the following keys: The display will read -500. 500 If, instead, you entered both numbers the 500 Present Value and the 1000 Future Value as positive numbers, the display would read: no SoLution 3. SET THE MODE CORRECTLY. Calculations involving annuities, sinking funds and amortizations, require a mode setting: either Begin Mode or End Mode. This is true of all calculators, including the HP 10Bii as well as the Electronic Calculators. Begin Mode applies if payments (or deposits or withdrawals) 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 amortization - 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. Steven J. Willis 2009. All Rights Reserved. 24

a. Electronic Calculators The Electronic Calculators have keys labeled Begin and End to designate the mode. Click on Begin to set the Calculator in Begin Mode; or, click on End to set it in End Mode. Also, when in Begin Mode, the words begin mode 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. b. HP 10Bii 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, 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. The display on most hand-held calculators does not indicate End Mode. Instead, it only indicates mode when the Calculator is in Begin Mode. As a result, users often fail to notice the mode and thus obtain an incorrect answer. A common mistake among calculator users involves Begin Mode annuities, sinking funds, or amortizations. Because the HP10Bii display does not indicate End Mode, the user may forget to change the mode to Begin, thus producing significant (but not obvious) incorrect results. The default setting is for End Mode because that is consistent with most amortizations, a common calculation involving Mode. However, sinking funds and annuities commonly use Begin Mode, necessitating a different calculator setting. Steven J. Willis 2009. All Rights Reserved. 25

For example, if you were saving for a child s education 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 interest 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. 4. SET THE INTEREST RATE TO COMPOUND FOR EACH PAYMENT PERIOD. Steven J. Willis 2009. All Rights Reserved. 26

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 expect to earn 6% nominal annual interest compounded monthly (after tax). To determine how much you must deposit if you start one month from now, press: 12 216 6 100,000 The display will read 258.16. To determine how much you must deposit if you start today, press: The display will read 256.88. Steven J. Willis 2009. All Rights Reserved. 27