APPENDIX C
USING FINANCIAL CALCULATORS
OBJECTIVE
1
Use a financial calculator to solve time value of money problems.
B
usiness 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 cal-culator. Illustration C-1 shows the five most common keys used to solve time value of money problems.1
where
N 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 com-putation.
FUTURE VALUE OF A SINGLE SUM
T
o illustrate the use of a financial calculator, let’s assume that you want to know the fu-ture value of $50,000 invested to earn 11%, compounded annually for five years, as picfu-tured in Illustration C-2.1On 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.
N I PV PMT FV N Inputs: 5 11 –50,000 0 ? Answer: 84,253 I PV PMT FV Illustration C-1
Financial Calculator Keys
Illustration C-2
Calculator Solution for Future Value of a Single Sum
Use of Spreadsheets to Solve Time Value of Money Problems w w w .w ile y.c om /c o ll eg e/w arfield
1080
N
Inputs: 5 11 ? 0 84,253
Answer: –50,000
I PV PMT FV
The diagram shows you the information (inputs) to enter into the calculator: N 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 com-pute 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 nega-tive cash flows in any problem offset each other. In the future value problem above, we iden-tified 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 re-quired in a problem, you should be able to interpret a positive or negative amount in de-termining 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 calcula-tor 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 dif-ference in the final answer can result. In most time value of money problems, the final an-swer will not include more than two decimal points.
PRESENT VALUE OF A SINGLE SUM
T
o 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.In this case, you enter N 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 AN ORDINARY ANNUITY
T
o illustrate the future value of an ordinary annuity, assume that you are asked to de-termine 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.Illustration C-3
Calculator Solution for Present Value of a Single Sum
1082
Appendix C Using Financial CalculatorsIn this case, you enter N 5, I 12, PV 0, PMT 5,000, and then press FV to ar-rive at the answer $31,764.24.2The $5,000 payments are shown as negatives because the deposits represent cash outflows that will accumulate with interest to the amount to be re-ceived (cash inflow) at the end of five years.
FUTURE VALUE OF AN ANNUITY DUE
R
ecall 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 BEGIN 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 pic-tured in Illustration C-5.
2Note that on page 1014 the answer using the compound interest tables is $31,764.25—a difference of 1 cent due to rounding.
In this case, you enter N 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 BEGIN 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.
PRESENT VALUE OF AN ORDINARY ANNUITY
T
o 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. N Inputs: 5 12 0 –5,000 ? Answer: 31,764.24 I PV PMT FV N Inputs: 8 6 0 –800 ? Answer: $8,393.05 I PV PMT FV Illustration C-4Calculator Solution for Future Value of an Ordinary Annuity
Illustration C-5
Calculator Solution for Future Value of an Annuity Due
3If the rental payments were received at the beginning of the year, then it would be necessary to switch to the BEGIN or DUE mode. In this case, the present value of the payments would be $24,224.10.
N Inputs: 5 12 ? 6,000 0 Answer: –21,628.66 I PV PMT FV N Inputs: 36 9.5 6,000 ? 0 Answer: –192.20 I PV PMT FV N Inputs: 180 8.4 ? –700 0 Answer: 71,509.81 I PV PMT FV
By entering N 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. Note that the payment key is usually programmed for 12 payments per year. Thus, you must change the default (com-pounding 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.
In this case, you enter N 5, I 12, PMT 6,000, FV 0, and then press PV to arrive at the answer of $21,628.66.3
USEFUL APPLICATIONS OF THE FINANCIAL
CALCULATOR
W
ith a financial calculator you can solve for any interest rate or for any number of pe-riods 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 an-nual 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 pur-chase. This problem is pictured in Illustration C-7.
Illustration C-6
Calculator Solution for Present Value of an Ordinary Annuity
Illustration C-7
Calculator Solution for Auto Loan Payments
Illustration C-8
Calculator Solution for Mortgage Amount
1084
Appendix C Using Financial CalculatorsBy entering N 260 (26 10 years), I 7.6, PV 2,000, PMT 100, and press-ing 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. Note 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.
Entering N 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. Note 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.
N Inputs: 260 7.6 –2,000 –100 ? Answer: I PV PMT FV 43,131.79 Illustration C-9
Calculator Solution for IRA Balance
Summary of Learning Objective for Appendix C
1 Use a financial calculator to solve time value ofmoney 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
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.
Exercises
EC-1 (Determine Interest Rate) Reba McEntire wishes to invest $19,000 on July 1, 2008, and have it ac-cumulate 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 re-pay 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 National 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
Problems
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 accumu-late 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 de-posits 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 in-tends 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 pur-chase 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 Jan-uary 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 an-nually. 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 an-nual 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?
