# Integrated Case First National Bank Time Value of Money Analysis

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1 Integrated Case 5-42 First National Bank Time Value of Money Analysis You have applied for a job with a local bank. As part of its evaluation process, you must take an examination on time value of money analysis covering the following questions. A. Draw time lines for (1) a \$100 lump sum cash flow at the end of Year 2, (2) an ordinary annuity of \$100 per year for 3 years, and (3) an uneven cash flow stream of -\$50, \$100, \$75, and \$50 at the end of Years 0 through 3. ANSWER: [Show S5-1 through S5-4 here.] A time line is a graphical representation that is used to show the timing of cash flows. The tick marks represent end of periods (often years), so time 0 is today; Time 1 is the end of the first year, or 1 year from today; and so on Year I/YR% Lump sum 100 Cash flow I/YR% Annuity I/YR% Uneven cash flow stream A lump sum is a single flow; for example, a \$100 inflow in Year 2, as shown in the top time line. Chapter 5: Time Value of Money Integrated Case 1

2 An annuity is a series of equal cash flows occurring over equal intervals, as illustrated in the middle time line. An uneven cash flow stream is an irregular series of cash flows that do not constitute an annuity, as in the lower time line. -50 represents a cash outflow rather than a receipt or inflow. B. (1) What s the future value of \$100 after 3 years if it earns 10%, annual compounding? ANSWER: [Show S5-5 through S5-7 here.] Show dollars corresponding to question mark, calculated as follows: % 100 FV =? After 1 year: FV 1 = PV + I 1 = PV + PV(I) = PV(1 + I) = \$100(1.10) = \$ Similarly: FV 2 = FV 1 + I 2 = FV 1 + FV 1 (I) = FV 1 (1 + I) = \$110(1.10) = \$ = PV(1 + I)(1 + I) = PV(1 + I) 2. FV 3 = FV 2 + I 3 = FV 2 + FV 2 (I) = FV 2 (1 + I) = \$121(1.10) = \$ = PV(1 + I) 2 (1 + I) = PV(1 + I) 3. In general, we see that: FV N = PV(1 + I) N, So FV 3 = \$100(1.10) 3 = \$100(1.3310) = \$ Integrated Case Chapter 5: Time Value of Money

3 Note that this equation has 4 variables: FV N, PV, I/YR, and N. Here, we know all except FV N, so we solve for FV N. We will, however, often solve for one of the other three variables. By far, the easiest way to work all time value problems is with a financial calculator. Just plug in any three of the four values and find the fourth. Finding future values (moving to the right along the time line) is called compounding. Note that there are 3 ways of finding FV 3 : Regular calculator: 1. \$100(1.10)(1.10)(1.10) = \$ \$100(1.10) 3 = \$ Financial calculator: This is especially efficient for more complex problems, including exam problems. Input the following values: N = 3, I/YR = 10, PV = -100, PMT = 0, and solve for FV = \$ Spreadsheet: Spreadsheet programs are ideally suited for solving time value of money problems. The spreadsheet can be set up using the specific FV spreadsheet function or by entering a FV formula/equation. B. (2) What s the present value of \$100 to be received in 3 years if the interest rate is 10%, annual compounding? Answer: [Show S5-8 through S5-10 here.] Finding present values, or discounting (moving to the left along the time line), is the reverse of compounding, and the basic present value equation is the reciprocal of the compounding equation: Chapter 5: Time Value of Money Integrated Case 3

4 % PV =? 100 FV N = PV(1 + I) N transforms to: FVN PV = (1 + I) N =FV N I N =FV N (1 + I ) N Thus: PV = \$ = \$100(0.7513) = \$ The same methods (regular calculator, financial calculator, and spreadsheet program) used for finding future values are also used to find present values, which is called discounting. Using a financial calculator input: N = 3, I/YR = 10, PMT = 0, FV = 100, and then solve for PV = \$ C. What annual interest rate would cause \$100 to grow to \$ in 3 years? ANSWER: [Show S5-11 here.] \$100(1 + I) \$100(1 + I) 2 \$100(1 + I) 3 FV = \$100(1 + I) 3 = \$ Using a financial calculator; enter N = 3, PV = -100, PMT = 0, FV = , then press the I/YR button to find I/YR = 8%. Calculators can find interest rates quite easily, even when periods and/or interest rates are not whole numbers, and when 4 Integrated Case Chapter 5: Time Value of Money

5 uneven cash flow streams are involved. (With uneven cash flows, we must use the CFLO function, and the interest rate is called the IRR, or internal rate of return; we will use this feature in capital budgeting.) D. If a company s sales are growing at a rate of 20% annually, how long will it take sales to double? ANSWER: [Show S5-12 here.] We have this situation in time line format: % -1 2 Say we want to find out how long it will take us to double our money at an interest rate of 20%. We can use any numbers, say \$1 and \$2, with this equation: FV N = \$2 = \$1(1 + I) N = \$1(1.20) N. We would plug I/YR = 20, PV = -1, PMT = 0, and FV = 2 into our calculator, and then press the N button to find the number of years it would take 1 (or any other beginning amount) to double when growth occurs FV Year at a 20% rate. The answer is 3.8 years, but some calculators will round this value up to the next highest whole number. The graph also shows what is happening. 4 Chapter 5: Time Value of Money Integrated Case 5

6 Optional Question A farmer can spend \$60/acre to plant pine trees on some marginal land. The expected real rate of return is 4%, and the expected inflation rate is 6%. What is the expected value of the timber after 20 years? ANSWER: FV 20 = \$60( ) 20 = \$60(1.10) 20 = \$ per acre. We could have asked: How long would it take \$60 to grow to \$403.65, given the real rate of return of 4% and an inflation rate of 6%? Of course, the answer would be 20 years. E. What s the difference between an ordinary annuity and an annuity due? What type of annuity is shown here? How would you change it to the other type of annuity? ANSWER: [Show S5-13 here.] This is an ordinary annuity it has its payments at the end of each period; that is, the first payment is made 1 period from today. Conversely, an annuity due has its first payment today. In other words, an ordinary annuity has end-of-period payments, while an annuity due has beginning-of-period payments. The annuity shown above is an ordinary annuity. To convert it to an annuity due, shift each payment to the left, so you end up with a payment under the 0 but none under the 3. F. (1) What is the future value of a 3-year, \$100 ordinary annuity if the annual interest rate is 10%? 6 Integrated Case Chapter 5: Time Value of Money

7 ANSWER: [Show S5-14 here.] % \$331 Go through the following discussion. One approach would be to treat each annuity flow as a lump sum. Here we have FVA N = \$100(1) + \$100(1.10) + \$100(1.10) 2 = \$100[1 + (1.10) + (1.10) 2 ] = \$100(3.3100) = \$ Future values of annuities may be calculated in 3 ways: (1) Treat the payments as lump sums. (2) Use a financial calculator. (3) Use a spreadsheet. F. (2) What is its present value? ANSWER: [Show S5-15 here.] % The present value of the annuity is \$ Here we used the lump sum approach, but the same result could be obtained by using a calculator. Input N = 3, I/YR = 10, PMT = 100, FV = 0, and press the PV button. Chapter 5: Time Value of Money Integrated Case 7

8 F. (3) What would the future and present values be if it was an annuity due? ANSWER: [Show S5-16 and S5-17 here.] If the annuity were an annuity due, each payment would be shifted to the left, so each payment is compounded over an additional period or discounted back over one less period. In our situation, the future value of the annuity due is \$364.10: FVA 3 Due = \$331.00(1.10) 1 = \$ This same result could be obtained by using the time line: \$ \$ \$ = \$ The best way to work annuity due problems is to switch your calculator to beg or beginning or due mode, and go through the normal process. Note that it s critical to remember to change back to end mode after working an annuity due problem with your calculator. In our situation, the present value of the annuity due is \$273.55: PVA 3 Due = \$248.68(1.10) 1 = \$ This same result could be obtained by using the time line: \$100 + \$ \$82.64 = \$ G. A 5-year \$100 ordinary annuity has an annual interest rate of 10%. (1) What is its present value? 8 Integrated Case Chapter 5: Time Value of Money

9 ANSWER: [Show S5-18 here.] % The present value of the annuity is \$ Here we used the lump sum approach, but the same result could be obtained by using a calculator. Input N = 5, I/YR = 10, PMT = 100, FV = 0, and press the PV button. G. (2) What would the present value be if it was a 10-year annuity? ANSWER: [Show S5-19 here.] The present value of the 10-year annuity is \$ To solve with a financial calculator, input N = 10, I/YR = 10, PMT = 100, FV = 0, and press the PV button. G. (3) What would the present value be if it was a 25-year annuity? ANSWER: The present value of the 25-year annuity is \$ To solve with a financial calculator, input N = 25, I/YR = 10, PMT = 100, FV = 0, and press the PV button. G. (4) What would the present value be if this was a perpetuity? ANSWER: The present value of the \$100 perpetuity is \$1,000. The PV is solved by dividing the annual payment by the interest rate: \$100/0.10 = \$1,000. Chapter 5: Time Value of Money Integrated Case 9

10 H. A 20-year-old student wants to save \$3 a day for her retirement. Every day she places \$3 in a drawer. At the end of each year, she invests the accumulated savings (\$1,095) in a brokerage account with an expected annual return of 12%. (1) If she keeps saving in this manner, how much will she have accumulated at age 65? ANSWER: [Show S5-20 and S5-21 here.] If she begins saving today, and sticks to her plan, she will have saved \$1,487, by the time she reaches 65. With a financial calculator, enter the following inputs: N = 45, I/YR = 12, PV = 0, PMT = -1095, then press the FV button to solve for \$1,487, H. (2) If a 40-year-old investor began saving in this manner, how much would he have at age 65? ANSWER: [Show S5-22 here.] This question demonstrates the power of compound interest and the importance of getting started on a regular savings program at an early age. The 40-year old investor will have saved only \$146, by the time he reaches 65. With a financial calculator, enter the following inputs: N = 25, I/YR = 12, PV = 0, PMT = -1095, then press the FV button to solve for \$146, H. (3) How much would the 40-year-old investor have to save each year to accumulate the same amount at 65 as the 20-year-old investor? ANSWER: [Show S5-23 here.] Again, this question demonstrates the power of compound interest and the importance of getting started on a regular savings program at an early age. The 40-year old investor will have to save \$11, every year, or \$30.56 per day, in order 10 Integrated Case Chapter 5: Time Value of Money

11 to have as much saved as the 20-year old investor by the time he reaches 65. With a financial calculator, enter the following inputs: N = 25, I/YR = 12, PV = 0, FV = , then press the PMT button to solve for \$11, I. What is the present value of the following uneven cash flow stream? The annual interest rate is 10% Years ANSWER: [Show S5-24 and S5-25 here.] Here we have an uneven cash flow stream. The most straightforward approach is to find the PVs of each cash flow and then sum them as shown below: Years 10% (34.15) Note that the \$50 Year 4 outflow remains an outflow even when discounted. There are numerous ways of finding the present value of an uneven cash flow stream. But by far the easiest way to deal with uneven cash flow streams is with a financial calculator. Calculators have a function that on the HP-17B is called CFLO, for cash flow. Other calculators could use other designations such as CF 0 and CF j, but they explain how to use them in the manual. Anyway, you would input the cash flows, so they are in the Chapter 5: Time Value of Money Integrated Case 11

12 calculator s memory, then input the interest rate, I/YR, and then press the NPV or PV button to find the present value. J. (1) Will the future value be larger or smaller if we compound an initial amount more often than annually (e.g., semiannually, holding the stated (nominal) rate constant)? Why? ANSWER: [Show S5-26 here.] Accounts that pay interest more frequently than once a year, for example, semiannually, quarterly, or daily, have future values that are higher because interest is earned on interest more often. Virtually all banks now pay interest daily on passbook and money fund accounts, so they use daily compounding. J. (2) Define (a) the stated (or quoted or nominal) rate, (b) the periodic rate, and (c) the effective annual rate (EAR or EFF%). ANSWER: [Show S5-27 and S5-28 here.] The quoted, or nominal, rate is merely the quoted percentage rate of return, the periodic rate is the rate charged by a lender or paid by a borrower each period (periodic rate = I NOM /M), and the effective annual rate (EAR) is the rate of interest that would provide an identical future dollar value under annual compounding. J. (3) What is the EAR corresponding to a nominal rate of 10% compounded semiannually? Compounded quarterly? Compounded daily? ANSWER: [Show S5-29 through S5-31 here.] The effective annual rate for 10% semiannual compounding, is 10.25%: EAR = Effective annual rate = 1 + I NOM M M Integrated Case Chapter 5: Time Value of Money

13 If I NOM = 10% and interest is compounded semiannually, then: EAR = = (1.05) = = = 10.25%. For quarterly compounding, the effective annual rate is 10.38%: (1.025) = = = 10.38%. Daily compounding would produce an effective annual rate of 10.52%. J. (4) What is the future value of \$100 after 3 years under 10% semiannual compounding? Quarterly compounding? ANSWER: [Show S5-32 here.] Under semiannual compounding, the \$100 is compounded over 6 semiannual periods at a 5.0% periodic rate: I NOM = 10%. FV N = MN INOM 1 + M 0.10 = \$ = \$100(1.05) 6 = \$ Quarterly: FV N = \$100(1.025) 12 = \$ The return when using quarterly compounding is clearly higher. Another approach here would be to use the effective annual rate and compound over annual periods: Semiannually: \$100(1.1025) 3 = \$ Quarterly: \$100(1.1038) 3 = \$ (3) Chapter 5: Time Value of Money Integrated Case 13

14 K. When will the EAR equal the nominal (quoted) rate? ANSWER: [Show S5-33 here.] If annual compounding is used, then the nominal rate will be equal to the effective annual rate. If more frequent compounding is used, the effective annual rate will be above the nominal rate. L. (1) What is the value at the end of Year 3 of the following cash flow stream if interest is 10%, compounded semiannually? (Hint: You can use the EAR and treat the cash flows as an ordinary annuity or use the periodic rate and compound the cash flows individually.) Periods ANSWER: [Show S5-34 through S5-36 here.] Periods 5% = \$100(1.05) = \$100(1.05) 4 \$ Here we have a different situation. The payments occur annually, but compounding occurs each 6 months. Thus, we cannot use normal annuity valuation techniques. 14 Integrated Case Chapter 5: Time Value of Money

15 L. (2) What is the PV? ANSWER: [Show S5-37 here.] Periods 5% \$ PV = 100(1.05) \$ To use a financial calculator, input N = 3, I/YR = 10.25, PMT = 100, FV = 0, and then press the PV key to find PV = \$ L. (3) What would be wrong with your answer to Parts L(1) and L(2) if you used the nominal rate, 10%, rather than the EAR or the periodic rate, I NOM /2 = 10%/2 = 5% to solve the problems? ANSWER: I NOM can be used in the calculations only when annual compounding occurs. If the nominal rate of 10% were used to discount the payment stream, the present value would be overstated by \$ \$ = \$ M. (1) Construct an amortization schedule for a \$1,000, 10% annual interest loan with 3 equal installments. (2) What is the annual interest expense for the borrower and the annual interest income for the lender during Year 2? ANSWER: [Show S5-38 through S5-44 here.] To begin, note that the face amount of the loan, \$1,000, is the present value of a 3-year annuity at a 10% rate: Chapter 5: Time Value of Money Integrated Case 15

16 % -1,000 PMT PMT PMT PVA 3 = PMT + PMT + PMT 1 + I 1 + I 1 + I \$1,000 = PMT(1 + I) -1 + PMT(1 + I) -2 + PMT(1 + I) -3. We have an equation with only one unknown, so we can solve it to find PMT. The easy way is with a financial calculator. Input N = 3, I/YR = 10, PV = -1000, FV = 0, and then press the PMT button to get PMT = , rounded to \$ Amortization Schedule: Beginning Payment of Ending Period Balance Payment Interest Principal Balance 1 \$1, \$ \$ \$ \$ * *Due to rounding, the third payment was increased by \$0.02 to cause the ending balance after the third year to equal \$0. Now make the following points regarding the amortization schedule: The \$ annual payment includes both interest and principal. Interest in the first year is calculated as follows: 1st year interest = I Beginning balance = 0.1 \$1,000 = \$100. The repayment of principal is the difference between the \$ annual payment and the interest payment: 1st year principal repayment = \$ \$100 = \$ Integrated Case Chapter 5: Time Value of Money

17 The loan balance at the end of the first year is: 1st year ending balance = Beginning balance Principal repayment = \$1,000 \$ = \$ We would continue these steps in the following years. Notice that the interest each year declines because the beginning loan balance is declining. Since the payment is constant, but the interest component is declining, the principal repayment portion is increasing each year. The interest component is an expense that is deductible to a business or a homeowner, and it is taxable income to the lender. If you buy a house, you will get a schedule constructed like ours, but longer, with = 360 monthly payments if you get a 30- year, fixed-rate mortgage. The payment may have to be increased by a few cents in the final year to take care of rounding errors and make the final payment produce a zero ending balance. The lender received a 10% rate of interest on the average amount of money that was invested each year, and the \$1,000 loan was paid off. This is what amortization schedules are designed to do. Most financial calculators have amortization functions built in. Chapter 5: Time Value of Money Integrated Case 17

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