# Chapter 4 Questions

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CHAPTER 4 QUESTIONS4-1 The opportunity cost rate is the rate of interest one could earn on an alternative investment with a risk equal to the risk of the investment in question. This is the value of r in the TVM equations, and it is shown on the top of a cash flow time line, between the first and second tick marks. It is not a single ratethe opportunity cost rate varies depending on the riskiness and maturity of an investment, and it also varies from year to year depending on inflationary expectations (see Chapter 5). True. The second series is an uneven payment stream, but it contains an annuity of $400 for 8 years. The series could also be thought of as a $100 annuity for 10 years plus an additional payment of $100 in Year 2, plus additional payments of $300 in Years 3 through 10. True, because of compounding effectsthat is, growth on growth. The following example demonstrates the point. The annual growth rate is r in the following equation: $1(1 + r)10 = $2. The term (1 + r)10 is the interest multiple for 10 years. We can find r in one of three ways: 1. 2. Using a financial calculator input N = 10, PV = -1, PMT = 0, FV = 2, and I/Y= ?. Solving for I/Yyou obtain 7.18%. Solve directly for r using the following method: FVn = PV(1 + r)n $2 = $1(1 + r)10 (1 + r)10 = $2/$1 = 2.0 r = (2.0)1/10 - 1 = 1.07177 - 1 = 0.07177 = 7.18% 3. Use the Rate function on a spreadsheet, which can be set up as follows, the solution is:

4-2

4-3

1

Chapter 4

Viewed another way, if earnings had grown at the rate of 10 percent per year for 10 years, then EPS would have increased from $1.00 to $2.59, found as follows: FV10 = PV(1.10)10 = $1(2.5937) = $2.59. Using a financial calculator, input N = 10, I/Y= 10, PV = -1, PMT = 0, and FV = ?. Solving for FV you obtain $2.59. This formulation recognizes the "interest on interest" phenomenon. The annual growth actually would be 10 percent per year only if the interest earned each year was not reinvested, thus compounded growth would not be possible. If the investor invested $1 at the beginning of each year at 10 percent, he or she would earn $0.10 each year. If the $.010 interest earned each year was taken out of the investment at the end of the year and deposited in a coffee can, then, at the end of 10 years, the investor would have $1 in the coffee can. The total value of the original $1 investment then would be $2 ($1 principal plus $1 interest in the coffee can). 4-4 For the same stated rate, more compounding is better. You would earn more interest on interest. Computing the effective annual rate for each alternative shows this to be true: EARsemiannual = (1 + 0.05/2)2 - 1 = 5.0625% EARdaily 4-5 = (1 + 0.05/365)365 - 1 = 5.1267%

To find the present value of an amount to be received in the future, we must take out the interest that the future amount can earn during the time period in question. The result of de-interesting the future amount is the present value, which represents the amount that must be invested today to grow to the future value at the given opportunity cost. For example, if you want to invest an amount today so that you have $500 in three years and your opportunity cost is 7 percent, the the following cash flow time line shows that the present value of the $500 is $408.15: 0 r = 7%= 500 (1.07)3 = 500(0.816298)

3 500

PV = ? = 408.15

If the opportunity cost is greater than 7 percent, then the present value will be lower because the amount invested today earns greater interest during the three-year period. In our example, PV = $375.66 when the opportunity cost is 10 percent. As you an see, then, the PV is dependent on interest ratesthat is, opportunity costs. 4-6 4-7 4-8 False. One can find the present value of an embedded annuity and add this PV to the PVs of the other individual cash flows to determine the present value of the stream of cash flows. The concept of a perpetuity implies that payments will be received forever. FV of Perpetuity = PVP(1 + r) = . To compare APRs, you must compute the rEAR for each alternative. APRs are not comparable when different compounding periods exist. EARsthat is, rEAR for alternativesare comparable because these rates are adjusted for (include the effects of) interest compounding.

2

Chapter 4

4-9

rEAR = APR is compounding occurs once per year; otherwise, rEAR > APR. This can be seen by computing rEAR when compounding occurs once per year: EAR = (1 + r/1)1 1.0 = r = APR = r x 1

4-10

An amortized loan is a loan for which a portion of the periodic payment includes interest that is charged for using the money and the remaining portion of the payment goes to repay the principal amount of the loan. An amortization schedule shows what portion of the periodic payment is the payment of interest and what portion is the repayment of the amount borrowed. ____________________________________________________________

PROBLEMS(Most solutions are rounded in the final answers, not in the intermediate computations.)0 -5006%

4-1

1

2 FV = ? FV2 = $500(1.06)2 = $500(1.1236) = $561.80

Using a financial calculator, enter N = 2, I/Y= 6, and PV = -500; compute FV = 561.80 0 4-2 1 2 3 4 5 1,000

6%

PV = ? 1 PV = 1,000 = 1,000(0.74726) = 747.26 5 (1.06 )

Using a financial calculator, enter N = 5, I/Y= 6, and FV = 1,000; compute PV = -747.260 1 2 3 4 5 6 7 8 9 10

4-3

(1)

12%

PV = ?

1,552.90

1 PV = 1,552.90 = 1,552.90(0.321973) = 499.99 10 (1.12 )

Using a financial calculator, enter N = 10, I/Y= 12, and FV = 1,552.90; compute PV = -499.990 1 2 3 4 5 6 7 8 9 10

(2)

6%

PV = ? PV = ? 1 PV = 1,552.90 = 1,552.90(0.558395) = 867.13 10 (1.06 ) 3

1,552.90 1,552.90

Chapter 4

Using a financial calculator, enter N = 10, I/Y= 6, and FV = 1,552.90; compute PV = -867.13 The present value represents the amount that needs to be invested today at the opportunity cost rate to generate the future amount. In essence, we take the interest out of the future valuethat is, discountto determine the current, or present value. For this problem, then, if $867.13 is invested today at 6 percent compounded annually, it will grow to $1,552.90 in 10 years.0 1 2 3 4 n=? 400

4-4

7%

PV = -200

Using a financial calculator, enter I/Y= 7, PV = -200, and FV = 400; compute N = 10.24 10 years If I/Y= 18%, N = 4.19 4 years 0 4-5 1 2 3 4 5 6 FV = ? 2,000

r=14

1,000 PV = ?

Using a calculator, enter N = 6, I/Y= 14, PMT = 0, and PV = 1,000; compute FV = 2,194.97 PV = 1,000(1.14)6 = 1,000(2.19497) = 2,194.97 $1,000 today is worth more. The future value of $1,000 at 14 percent over six years is $2,194.97, which is greater than the future $2,000.00. Alternatively, using a calculator, enter N = 6, I/Y= 14, PMT = 0, and FV = 2,000; compute PV = 911.17 PV = 2,000(1/1.14)6 = 2,000(0.455689) = $911.17 $1,000 today is worth more. The present value of $2,000 at 14 percent over six years is $911.17, which is less than $1,000.00. 0 4-6 a. 6 1 2 3 4 5 12

r=?

4

Chapter 4

1 FV = PV n (1 + r ) 1 12 = 6 5 (1 + r ) 12 r= 61 5

1 = 0.1487 = 14.87%

Using a calculator, enter N = 5, PV = -6, PMT = 0, and FV = 12; compute I/Y= 14.87% 15%. 4-7 The general formula for computing the future value of an ordinary annuity is: (1 + r )n - 1 = PMT FVA n r

0 a.

10%

1

2 400

3 400

4 400

5 400

6 400

7 400

8 400

9 400

10 400 FVA10 = ?

400

(1.10 )10 - 1 FVA10 = 400 = 400(15.93742) = 6,374.97 0.10

Using a financial calculator, enter N = 10, I/Y= 10, and PMT = -400; compute FV = 6,374.97 0 b. 200 200 200 200 200 FVA5 = ? (1.05 )5 - 1 = 200 FVA5 = 200(5.52563) = 1,105.13 0.05

5%

1

2

3

4

5

Using a financial calculator, enter N = 5, I/Y= 5, and PMT = -200; compute FV = 1,105.13 4-8 The general formula for computing the future value of an annuity due is: (1 + r )n - 1 FVA(DUE )n = PMT (1 + k ) r

a.5

Chapter 4

0 400

10%

1

2 400

3 400

4 400

5 400

6 400

7 400

8 400

9 400

10

400

FVA(DUE)10 = ?10 (1.10 ) - 1 FVA(DUE )10 = 400 (1.10) = 400(17.53117 ) = 7,012.47 0.10

Using a financial calculator, switch to BEGIN, enter N = 10, I/Y= 10, and PMT = -400; compute FV = 7,012.47 b. 0 2005%

1

2 200

3 200

4 200

5

200

FVA(DUE)5 = ? (1.05 )5 - 1 FVA(DUE )5 = 200 (1.05) = 200(5.80191) = 1,160.38 0.05

Using a financial calculator, switch to BEGIN, enter N = 5, I/Y= 5, and PMT = -200; compute FV = 1,160.38 4-9 The general formula for computing the present value of an ordinary annuity is:1 - (1 + r )- n = PMT PVA n r

0 a. PVA10 = ?

10%

1

2 400

3 400

4 400

5 400

6 400

7 400

8 400

9 400

10 400

400

1 1 10 (1.10 ) = 400(6.14457 ) = 2,457.83 P VA10 = 400 0.10

Using a financial calculator, enter N = 10, I/Y= 10, and PMT = -400; compute PV = 2,457.83

6

Chapter 4

0 b. PVA5 = ?

5%

1

2 200

3 200

4 200

5 200

200

1 1 (1.05 )5 = 200( 4.32948) = 865.90 P VA5 = 200 0.05

Using a financial calculator, enter N = 5, I/Y= 5, and PMT = -200; compute PV = 865.90 4-10 The general formula for computing the future value of an annuity due is: 1 1 n (1 + r ) (1 + r ) P VA(DUE )n = PMT r

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