lwww.wiley.com/col APPEDIX C USIG FIACIAL CALCULATORS OBJECTIVE 1 Use a financial calculator to solve time value of money problems. Illustration C-1 Financial Calculator Keys Business professionals, once they have mastered the underlying concepts in Appendix A, will often use a financial (business) calculator to solve time value of money problems. In many cases, they must use calculators if interest rates or time periods do not correspond with the information provided in the compound interest tables. To use financial calculators, you enter the time value of money variables into the calculator. Illustration C-1 shows the five most common keys used to solve time value of money problems. 1 where ege/warfield Use of Spreadsheets to Solve Time Value of Money Problems number of periods I interest rate per period (some calculators use I/YR or i) PV present value (occurs at the beginning of the first period) PMT payment (all payments are equal, and none are skipped) FV future value (occurs at the end of the last period) In solving time value of money problems in this appendix, you will generally be given three of four variables and will have to solve for the remaining variable. The fifth key (the key not used) is given a value of zero to ensure that this variable is not used in the computation. FUTURE VALUE OF A SIGLE SUM To illustrate the use of a financial calculator, let s assume that you want to know the future value of $50,000 invested to earn 11%, compounded annually for five years, as pictured in Illustration C-2. Illustration C-2 Future Value of a Single Sum Inputs: 5 11 50,000 0? Answer: 84,253 1 On many calculators, these keys are actual buttons on the face of the calculator; on others, they appear on the display after the user accesses a present value menu. 1080
The diagram shows you the information (inputs) to enter into the calculator: 5, I 11, PV 50,000, and PMT 0. You then press FV for the answer: $84,253. This is the same answer as shown on page 1008, when we used compound interest tables to compute the future value of a single sum. As indicated, the PMT key was given a value of zero because a series of payments did not occur in this problem. Plus and Minus The use of plus and minus signs in time value of money problems with a financial calculator can be confusing. Most financial calculators are programmed so that the positive and negative cash flows in any problem offset each other. In the future value problem above, we identified the 50,000 initial investment as a negative (outflow); the answer 84,253 was shown as a positive, reflecting a cash inflow. If the 50,000 were entered as a positive, then the final answer would have been reported as a negative ( 84,253). Hopefully, the sign convention will not cause confusion. If you understand what is required in a problem, you should be able to interpret a positive or negative amount in determining the solution to a problem. Compounding Periods In the problem above, we assumed that compounding occurs once a year. Some financial calculators have a default setting, which assumes that compounding occurs 12 times a year. You must determine what default period has been programmed into your calculator and change it as necessary to arrive at the proper compounding period. Rounding Most financial calculators store and calculate using 12 decimal places. As a result, because compound interest tables generally have factors only up to five decimal places, a slight difference in the final answer can result. In most time value of money problems, the final answer will not include more than two decimal points. Future Value of an Ordinary Annuity 1081 PRESET VALUE OF A SIGLE SUM To illustrate how a present value problem is solved using a financial calculator, assume that you want to know the present value of $84,253 to be received in five years, discounted at 11% compounded annually. Illustration C-3 pictures this problem. Inputs: 5 11? 0 84,253 Illustration C-3 Present Value of a Single Sum Answer: 50,000 In this case, you enter 5, I 11, PMT 0, FV 84,253, and then press the PV key to find the present value of $50,000. FUTURE VALUE OF A ORDIARY AUITY To illustrate the future value of an ordinary annuity, assume that you are asked to determine the future value of five $5,000 deposits made at the end of each of the next five years, each of which earns interest at 12%, compounded annually, as pictured in Illustration C-4.
1082 Appendix C Using Financial Calculators Illustration C-4 Future Value of an Ordinary Annuity Inputs: 5 12 0 5,000? Answer: 31,764.24 In this case, you enter 5, I 12, PV 0, PMT 5,000, and then press FV to arrive at the answer $31,764.24. 2 The $5,000 payments are shown as negatives because the deposits represent cash outflows that will accumulate with interest to the amount to be received (cash inflow) at the end of five years. FUTURE VALUE OF A AUITY DUE Recall from the discussion in Appendix A that in any annuity problem you must determine whether the periodic payments occur at the beginning or the end of the period. If the first payment occurs at the beginning of the period, most financial calculators have a key marked Begin (or Due ) that you press to switch from the end-of-period payment mode (for an ordinary annuity) to beginning-of-period payment mode (for an annuity due). For most calculators, the word BEGI is displayed to indicate that the calculator is set for an annuity due problem. (Some calculators use DUE.) To illustrate a future value of an annuity due problem, let s revisit a problem from Appendix A: Sue Lotadough plans to deposit $800 per year in a fund on each of her son s birthdays, starting today (his tenth birthday). All amounts on deposit in the fund will earn 6% compounded annually. Sue wants to know the amount she will have accumulated for college expenses on her son s eighteenth birthday. She will make eight deposits into the fund. (Assume no deposit will be made on the eighteenth birthday.) This problem is pictured in Illustration C-5. Illustration C-5 Calculator Solution for Future Value of an Annuity Due Inputs: 8 6 0 800? Answer: $8,393.05 In this case, you enter 8, I 6, PV 0, PMT 800, and then press FV to arrive at the answer of $8,393.05. You must be in the BEGI or DUE mode to solve this problem correctly. Before starting to solve any annuity problem, make sure that your calculator is switched to the proper mode. PRESET VALUE OF A ORDIARY AUITY To illustrate how to solve a present value of an ordinary annuity problem using a financial calculator, assume that you are asked to determine the present value of rental receipts of $6,000 each to be received at the end of each of the next five years, when discounted at 12%, as pictured in Illustration C-6. 2 ote that on page 1014 the answer using the compound interest tables is $31,764.25 a difference of 1 cent due to rounding.
Useful Applications of the Financial Calculator 1083 Inputs: 5 12? 6,000 0 Illustration C-6 Present Value of an Ordinary Annuity Answer: 21,628.66 In this case, you enter 5, I 12, PMT 6,000, FV 0, and then press PV to arrive at the answer of $21,628.66. 3 USEFUL APPLICATIOS OF THE FIACIAL CALCULATOR With a financial calculator you can solve for any interest rate or for any number of periods in a time value of money problem. Here are some examples of these applications. Auto Loan Assume you are financing a car with a three-year loan. The loan has a 9.5% nominal annual interest rate, compounded monthly. The price of the car is $6,000, and you want to determine the monthly payments, assuming that the payments start one month after the purchase. This problem is pictured in Illustration C-7. Inputs: 36 9.5 6,000? 0 Illustration C-7 Auto Loan Payments Answer: 192.20 By entering 36 (12 3), I 9.5, PV 6,000, FV 0, and then pressing PMT, you can determine that the monthly payments will be $192.20. ote that the payment key is usually programmed for 12 payments per year. Thus, you must change the default (compounding period) if the payments are different than monthly. Mortgage Loan Amount Let s say you are evaluating financing options for a loan on your house. You decide that the maximum mortgage payment you can afford is $700 per month. The annual interest rate is 8.4%. If you get a mortgage that requires you to make monthly payments over a 15-year period, what is the maximum purchase price you can afford? Illustration C-8 depicts this problem. Inputs: 180 8.4? 700 0 Illustration C-8 Mortgage Amount Answer: 71,509.81 3 If the rental payments were received at the beginning of the year, then it would be necessary to switch to the BEGI or DUE mode. In this case, the present value of the payments would be $24,224.10.
1084 Appendix C Using Financial Calculators Entering 180 (12 15 years), I 8.4, PMT 700, FV 0, and pressing PV, you find a present value of $71,509.81 the maximum house price you can afford, given that you want to keep your mortgage payments at $700. ote that by changing any of the variables, you can quickly conduct what-if analyses for different factual situations. Individual Retirement Account (IRA) Assume you opened an IRA on April 15, 2006, with a deposit of $2,000. Since then you have deposited $100 in the account every two weeks (26 deposits per year, with the first $100 deposit made on April 29, 2006). The account pays 7.6% annual interest compounded semi-monthly (with each deposit). How much will be in the account on April 15, 2016? Illustration C-9 depicts this problem. Illustration C-9 IRA Balance Inputs: 260 7.6 2,000 100? Answer: 43,131.79 By entering 260 (26 10 years), I 7.6, PV 2,000, PMT 100, and pressing FV, you determine the future value of $43,131.79. This is the amount that the IRA will grow to over the 10-year period. ote that in this problem we use four of the keys and solve for the fifth. Thus, we combine the future value of a single sum and of an annuity. Other problems similar to this are illustrated in Chapters 8 and 12. Summary of Learning Objective for Appendix C 1 Use a financial calculator to solve time value of money problems. Financial calculators can be used to solve the same and additional problems as those solved with time value of money tables. One enters into the financial calculator the amounts for all but one of the Exercises unknown elements of a time value of money problem (periods, interest rate, payments, future or present value). Particularly useful situations involve interest rates and compounding periods not presented in the tables. EC-1 (Determine Interest Rate) Reba McEntire wishes to invest $19,000 on July 1, 2008, and have it accumulate to $49,000 by July 1, 2018. Instructions Use a financial calculator to determine at what exact annual rate of interest Reba must invest the $19,000. EC-2 (Determine Interest Rate) On July 17, 2007, Tim McGraw borrowed $42,000 from his grandfather to open a clothing store. Starting July 17, 2008, Tim has to make ten equal annual payments of $6,500 each to repay the loan. Instructions Use a financial calculator to determine what interest rate Tim is paying. EC-3 (Determine Interest Rate) As the purchaser of a new house, Patty Loveless has signed a mortgage note to pay the Memphis ational Bank and Trust Co. $14,000 every 6 months for 20 years, at the end of which time she will own the house. At the date the mortgage is signed the purchase price was $198,000, and a down payment of $20,000 was made. The first payment will be made 6 months after the date the mortgage is signed. Instructions Using a financial calculator, compute the exact rate of interest earned on the mortgage by the bank.
Problems Problems 1085 PC-1 (Various Time Value of Money Situations) Using a financial calculator, provide a solution to each of the following questions. (a) What is the amount of the payments that Karla Zehms must make at the end of each of 8 years to accumulate a fund of $70,000 by the end of the eighth year, if the fund earns 7.25% interest, compounded annually? (b) Bill Yawn is 40 years old today, and he wishes to accumulate $500,000 by his sixty-fifth birthday so he can retire to his summer place on Lake Winnebago. He wishes to accumulate this amount by making equal deposits on his fortieth through sixty-fourth birthdays. What annual deposit must Bill make if the fund will earn 9.65% interest compounded annually? (c) Jane Mayer has a $26,000 debt that she wishes to repay 4 years from today; she has $17,000 that she intends to invest for the 4 years. What rate of interest will she need to earn annually in order to accumulate enough to pay the debt? PC-2 (Various Time Value of Money Situations) Using a financial calculator, solve for the unknowns in each of the following situations. (a) Wayne Eski wishes to invest $150,000 today to ensure payments of $20,000 to his son at the end of each year for the next 15 years. At what interest rate must the $150,000 be invested? (Round the answer to two decimal points.) (b) On June 1, 2008, Shelley Long purchases lakefront property from her neighbor, Joey Brenner, and agrees to pay the purchase price in seven payments of $16,000 each, the first payment to be payable June 1, 2009. (Assume that interest compounded at an annual rate of 7.35% is implicit in the payments.) What is the purchase price of the property? (c) On January 1, 2008, Cooke Corporation purchased 200 of the $1,000 face value, 8% coupon, 10-year bonds of Howe Inc. The bonds mature on January 1, 2018, and pay interest annually beginning January 1, 2009. Cooke purchased the bonds to yield 10.65%. How much did Cooke pay for the bonds? PC-3 (Various Time Value of Money Situations) Using a financial calculator, provide a solution to each of the following situations. (a) On March 12, 2008, William Scott invests in a $180,000 insurance policy that earns 5.25% compounded annually. The annuity policy allows William to receive annual payments, the first of which is payable to William on March 12, 2009. What will be the amount of each of the 20 equal annual receipts? (b) Bill Schroeder owes a debt of $35,000 from the purchase of his new sport utility vehicle. The debt bears annual interest of 9.1% compounded monthly. Bill wishes to pay the debt and interest in equal monthly payments over 8 years, beginning one month hence. What equal monthly payments will pay off the debt and interest? (c) On January 1, 2008, Sammy Sosa offers to buy Mark Grace s used snowmobile for $8,000, payable in five equal installments, which are to include 8.25% interest on the unpaid balance and a portion of the principal. If the first payment is to be made on January 1, 2008, how much will each payment be? (d) Repeat the requirements in part (c), assuming Sosa makes the first payment on December 31, 2008.