CHAPTER 4

QUESTIONS 4-1 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 rate—the 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).

4-2 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.

4-3 True, because of compounding effects—that 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. Using a financial calculator input N = 10, PV = -1, PMT = 0, FV = 2, and I/Y= ?. Solving for I/Yyou obtain 7.18%.

2. 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:

1

Chapter 4

2

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 = (1 + 0.05/365)365 - 1 = 5.1267% 4-5 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 3

r = 7%

PV = ? 500

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 rates—that is, opportunity costs.

4-6 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. 4-7 The concept of a perpetuity implies that payments will be received forever. FV of Perpetuity =

PVP(1 + r)∞ = ∞. 4-8 To compare APRs, you must compute the rEAR for each alternative. APRs are not comparable when

different compounding periods exist. EARs—that is, rEAR for alternatives—are comparable because these rates are adjusted for (include the effects of) interest compounding.

)816298.0(500)07.1(

5003 ==

= 408.15

Chapter 4

3

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 1 2 4-1

6%

-500 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

6% 0 1 2 3 4 5 4-2 PV = ? 1,000

( )26.747)74726.0(000,1

06.11000,1 5 ==

⎥⎥⎦

⎤

⎢⎢⎣

⎡=PV

Using a financial calculator, enter N = 5, I/Y= 6, and FV = 1,000; compute PV = -747.26 0 1 2 3 4 5 6 7 8 9 10

4-3 (1) 12%

PV = ? 1,552.90

( )99.499)321973.0(90.552,1

12.1190.552,1PV 10 ==

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

Using a financial calculator, enter N = 10, I/Y= 12, and FV = 1,552.90; compute PV = -499.99 0 1 2 3 4 5 6 7 8 9 10

(2)

PV = ? 1,552.90

PV = ? 1,552.90

6%

( )13.867)558395.0(90.552,1

06.1190.552,1PV 10 ==

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

Chapter 4

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 value—that is, discount—to 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 = ? 4-4 PV = -200 400

… 7%

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 1 2 3 4 5 6 4-5 1,000 FV = ?

r=14

PV = ? 2,000

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 1 2 3 4 5 4-6 a. r=? 6 12

Chapter 4

5

%87.141487.016

12r

)r1(1612

)r1(1PVFV

51

5

n

==−⎥⎦⎤

⎢⎣⎡=

⎥⎦

⎤⎢⎣

⎡

+=

⎥⎦

⎤⎢⎣

⎡

+=

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:

r

1 - )r + (1 PMT = FVA

n

n⎥⎥⎦

⎤

⎢⎢⎣

⎡ 0 1 2 3 4 5 6 7 8 9 10

a. 10%

400 400 400 400 400 400 400 400 400 400

FVA10 = ?

97.374,6)93742.15(400 10.0

1 - )10(1. 400 = FVA

10

10 ==⎥⎥⎦

⎤

⎢⎢⎣

⎡

Using a financial calculator, enter N = 10, I/Y= 10, and PMT = -400; compute FV = 6,374.97 0 1 2 3 4 5

b. 5%

200 200 200 200 200

FVA5 = ?

13.105,1)52563.5(200 05.0

1 - )05(1. 200 = FVA

5

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: )k1(

r1 - )r + (1

PMT = )DUEFVA(n

n⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+×⎥⎥⎦

⎤

⎢⎢⎣

⎡

a.

Chapter 4

6

0 1 2 3 4 5 6 7 8 9 10

10%

400 400 400 400 400 400 400 400 400 400

FVA(DUE)10 = ?

47.012,7)53117.17(400 )10.1(10.0

1 - )10(1. 400 = )DUEFVA(

10

10 ==⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

×⎥⎥⎦

⎤

⎢⎢⎣

⎡

Using a financial calculator, switch to BEGIN, enter N = 10, I/Y= 10, and PMT = -400;

compute FV = 7,012.47 b. 0 1 2 3 4 5

5%

200 200 200 200 200

FVA(DUE)5 = ?

38.160,1)80191.5(200 )05.1(05.0

1 - )05(1. 200 = )DUEFVA(

5

5 ==⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

×⎥⎥⎦

⎤

⎢⎢⎣

⎡

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:

r)r + (1 - 1

PMT = PVAn-

n⎥⎥⎦

⎤

⎢⎢⎣

⎡ 0 1 2 3 4 5 6 7 8 9 10

a. 10%

PVA10 = ? 400 400 400 400 400 400 400 400 400 400

83.457,2)14457.6(400 10.0

)10(1.11

400 = VAP10

10 ==

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

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

Chapter 4

7

0 1 2 3 4 5

b. 5%

PVA5 = ? 200 200 200 200 200

90.865)32948.4(200 05.0

)05(1.11

200 = VAP5

5 ==

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

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:

)r1(r

)r + (111

PMT = )DUEVA(Pn

n

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

+×

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ − a. 0 1 2 3 4 5 6 7 8 9 10

10%

400 400 400 400 400 400 400 400 400 400

PVA(DUE)10 = ?

61.703,2)75902.6(400 )10.1(10.0

)10(1.11

400 = )DUEVA(P10

10 ==

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

×

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

Using a financial calculator, switch to BEGIN, enter N = 10, I/Y= 10, and PMT = -400;

compute PV = 2,703.61 b. 0 1 2 3 4 5

5%

200 200 200 200 200

PVA(DUE)5 = ?

19.909)54595.4(200 )05.1(05.0

)05(1.11

200 = )DUEVA(P5

5 ==

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

×

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

Chapter 4

8

Using a financial calculator, switch to BEGIN, enter N = 5, I/Y= 5, and PMT = -200; compute PV = 909.19

4-11 PVP = $100/0.07 = $1,428.57. PVP = $100/0.14 = $714.29.

When the interest rate is doubled, the PV of the perpetuity is halved. 4-12 Cash Stream A Cash Stream B

0 1 2 3 4 0 1 2 3 4

a. PV=? 100 400 400 300 PV=? 300 400 400 100

8% 8%

57.9735090.2205329.3179355.3425926.92)08.1(

1300)08.1(

1400)08.1(

1400)08.1(

1100PV4321A

=+++=

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

75.011,15030.735329.3179355.3427778.277)08.1(

1100)08.1(

1400)08.1(

1400)08.1(

1300PV4321B

=+++=⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

Using a financial calculator, simply enter the cash flows into the cash flow register (be sure to

enter CF0 = 0), enter I/Y= 8, and press the NPV key to find NPV = PV = $973.57 for the first problem. Override I/Y= 8 with I/Y= 0 to find the next PV for Cash Stream A ($1,200). Repeat for Cash Stream B to get NPV = PV = $1,011.75 when I/Y= 8% and $1,200 when I/Y= 0%.

b. PVA = $100 + $400 + $400 + $400 + $300 = $1,200.

PVB = $300 + $400 + $400 + $400 + $100 = $1,200.

Chapter 4

9

0 1 2 3 4 5 years 12%

4-13 a. -500 FV = ? FV = 500(1.12)5 = 500(1.76234) = 881.17 Using a financial calculator, enter N = 5, I/Y= 12, and PV = -500; compute FV = 881.17 1 2 3 4 5 years

0 2 4 6 8 10 six-month periods 6%

b. -500 FV = ? FV = 500(1.06)10 = 500(1.79085) = 895.42 Using a financial calculator, enter N = 10, I/Y= 6, and PV = -500; compute FV = 895.42 1 2 3 4 5 years

0 4 8 12 16 20 quarters 3% c. -500 FV = ? FV = 500(1.03)20 = 500(1.80611) = 903.06 Using a financial calculator, enter N = 20, I/Y= 3, and PV = -500; compute FV = 903.06 1 2 3 4 5 years

0 12 24 36 38 60 months 1% … … … … … d.

-500 FV = ?

35.908)81670.1(500)01.1(5001212.01500FV 60

125

===⎟⎠⎞

⎜⎝⎛ +=

×

Using a financial calculator, enter N = 60, I/Y= 1, and PV = -500; compute FV = 908.35

0 1 2 3 4 5 years 12%

4-14 a. PV = ? -500

Chapter 4

10

PV = 500/(1.12)5 = 500(0.56743) = 283.71 Using a financial calculator, enter N = 5, I/Y= 12, and FV = -500; compute PV = 283.71 1 2 3 4 5 years

0 2 4 6 8 10 six-month periods 6%

b. PV = ? -500 FV = 500/(1.06)10 = 500(0.55839) = 279.20 Using a financial calculator, enter N = 10, I/Y= 6, and FV = -500; compute PV = 279.20 1 2 3 4 5 years

0 4 8 12 16 20 quarters 3% c. PV = ? -500 FV = 500/(1.03)20 = 500(0.55368) = 276.84 Using a financial calculator, enter N = 20, I/Y= 3, and FV = -500; compute PV = 276.84 1 2 3 4 5 years

0 12 24 36 38 60 months … … … … … d.

PV = ? 500

22.275)55045.0(500 + 1

1 500 =PV 1212.0 512

==

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞⎜

⎝⎛ ×

Using a financial calculator, enter N = 60, I/Y= 1, and FV = -500; compute PV = 275.22

0 1 2 3 9 10 periods 4-15 a. -400 -400 -400 -400 -400

6% …

FV = ?

( )( ) 32.272,5)18079.13(400

06.01)06.1(400

mr

1mr1

PMTFVA10

n

n ==⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −+=

Using a financial calculator, enter N = 5 x 2 = 10, I/Y= 12/2 = 6, PV = 0, and PMT = -400; compute FV = 5,272.32.

Chapter 4

11

0 1 2 3 4 18 19 20 periods b. -200 -200 -200 -200 -200 -200 -200

3% …

FV = ?

07.374,5)87037.26(20003.0

1)03.1(200FVA20

n ==⎥⎦

⎤⎢⎣

⎡ −=

Using a financial calculator, enter N = 5 x 4 = 20, I/Y= 12/4 = 3, PV = 0, and PMT = -200; compute FV = 5,374.07

Note that both solutions assume that the simple interest rate is compounded at the annuity period.

c. The annuity in part b earns more because some of the money is on deposit for a longer period

of time and thus earns more interest—the first payment is made in three months rather than in six months. Also, because compounding is more frequent, more interest is earned on interest.

0 1 2 3 9 10 periods 4-16 a. -400 -400 -400 -400 -400

6% …

PV = ?

( )( ) 03.944,2)36009.7(400

06.0)06.1(1400

mr

mr11

PMTPVA10

n

n ==⎥⎦

⎤⎢⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ +−=

−

Using a financial calculator, enter N = 5 x 2 = 10, I/Y= 12/2 = 6, FV = 0, and PMT = -400; compute PV = 2,944.03.

0 1 2 3 4 18 19 20 periods

b. -200 -200 -200 -200 -200 -200 -200

3% …

PV = ?

49.975,2)87747.14(20003.0

)03.1(1200FVA20

n ==⎥⎦

⎤⎢⎣

⎡ −=

−

Using a financial calculator, enter N = 5 x 4 = 20, I/Y= 12/4 = 3, FV = 0, and PMT = -200; compute PV = 2,975.49

Note that both solutions assume that the simple interest rate is compounded at the annuity period.

c. The annuity in part b requires the first payment to occur in three months, whereas the annuity

in part a requires a payment in six months. Thus, an amount invested today to create the annuity in part a would earn interest for a longer time—that is, six months—than would an amount invested today to create the annuity in part b—that is, three months. As a result, less would have to be invested today to create the annuity in part a than in part b.

Chapter 4

12

0 1 2 3 4 4-17 a. 10,000 10,000 10,000 10,000

r = 7%

11.872,33)38721.3(000,1007.0

1000,10PVA

4)07.1(1

==⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

Using a calculator, enter N = 4, I/Y= 7, PMT = 10,000, and FV = 0; compute PV = -33,872.11.

b. (1) At this point, we have a three-year $10,000 annuity at 7 percent. Input N = 3 to override the number of years from part a in your calculator’s TVM register, and you will find PV = 26,243.16.

You can also think of the problem as follows:

$33,872.11(1.07) ─ $10,000 = $26,243.16 Or,

16.243,26)624316.2(000,1007.0

1000,10PVA

3)07.1(1

==⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

(2) Zero after the last withdrawal.

0 1 2 10 11 12 Quarters … 4-18 12,000 PMT PMT PMT PMT PMT

r=12%/4=3%

55.205,195400.9

000,12PMT

)95400.9(PMT03.0

1PMT000,12

r

1PMTPVA

12)03.1(1

n)r1(1

==

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= +

Using a calculator, enter N = 12, I/Y= 3, PV = 12,000, and FV = 0; compute PV = -1,205.55 0 1 2 n-1 n Years … 4-19 12,000 -1,500 -1,500 -1,500 -1,500

r = 9%

Chapter 4

13

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= +

09.0

1500,1000,12

r

1PMTPVA

n)09.1(1

n)r1(1

Using a calculator, enter I/Y= 9, PV = 12,000, FV = 0, and PMT = -1,500; compute N = 14.77 ≈ 15 years

0 1 2 n-1 n … r = 6% 4-20 -1,750 -1,750 -1,750 PMT = ? FVA = 10,000

⎥⎦

⎤⎢⎣

⎡ −=

⎥⎦

⎤⎢⎣

⎡ −+=

06.01)06.1(750,1000,10

r1)r1(PMTFVA

n

n

Financial calculator: I/Y= 6, PV = 0, FV = 10,000, and PMT = -1,750; compute N = 5.06. This answer assumes that a payment of $1,750 will be made 6/100 of the way through Year 6.

Now find the FV of $1,750 for 5 years at 6 percent; it is $9,864.91. 91.864,9)63709.5(750,1

06.01)06.1(750,1FVA

5

==⎥⎦

⎤⎢⎣

⎡ −=

Using a calculator, enter N = 5, I/Y= 6, PV = 0, and PMT = -1,750; compute FV = 9,864.91 So the payment at the end of Year 5 will include an additional $135.09 = $10,000 - $9,864.91, which means the last investment will total $1,885.09 = $1,750 + $135.09. It will take 5 years to accumulate the $10,000 if, beginning one year from today, $1,750 is invested each year for the next four years at 6 percent, and a $1,885.09 investment is made at the end of Year 5.

4-21 The $2.9 million 30-year payment represents an annuity due. Therefore, compute the present value of

the annuity due.

( ) 3$46,809,116.141074)million)(1 9.2($05.105.0

)05.1(11

)million 9.2($)DUE(PVA30

==

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

=

Financial calculator: Switch to begin mode, n= 30, I/Y= 5, PMT = 2,900,000, and FV = 0; compute PV = -46,809,113. Because PVA(DUE) = $46,809,113, which is greater than the lump-sum payment of $44 million, the annuity option should be chosen.

Chapter 4

14

4-22 a. The $3.5 million 30-year payment represents an annuity due. Therefore, compute the present

value of the annuity due.

( ) 4$51,067,524.59072)million)(1 5.3($06.106.0

)06.1(11

)million 5.3($)DUE(PVA30

==

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −=

Financial calculator: Switch to begin mode, n = 30, I/Y= 6, PMT = 3,500,000, and FV = 0; compute PV = -51,067,524.

Because PVA(DUE) = $51,067,524, which is less than the lump-sum payment of $54 million,

the lump-sum payment should be chosen. b. Financial calculator: Switch to begin mode, N = 30, PV = -54,000,000, PMT = 3,500,000, and

FV = 0; compute I/Y= 5.44%. 4-23 These problems can all be solved using a financial calculator by entering the known values shown on

the time lines and then pressing the I/Y button. 0 1

r = ? a. +700 -749

%0.707.00.1700749r

)r1(1700749

)r1(1PVFV

1/1

1

n

==−⎟⎠⎞

⎜⎝⎛=

⎥⎦

⎤⎢⎣

⎡

+=

⎥⎦

⎤⎢⎣

⎡

+=

Using a financial calculator, enter N = 1, PV = 700, and FV = -749; compute I/Y= 7.0%

0 1 b.

r = ?

-700 +749

%0.707.00.1700749r

)r1(17007 49

)r1(1PVFV

1/1

1

n

==−⎟⎠⎞

⎜⎝⎛=

⎥⎦

⎤⎢⎣

⎡

+=

⎥⎦

⎤⎢⎣

⎡

+=

Using a financial calculator, enter N = 1, PV = -700, and FV = 749; compute I/Y= 7.0%

Chapter 4

15

0 1 2 3 4 5 6 7 8 9 10 c. +85,000 -201,229 Using a financial calculator, enter N = 10, PV = 85,000, and FV = -201,229; compute I/Y= 9.0% 0 1 2 3 4 5 d. +9,000 -2,684.80 -2,684.80 -2,684.80 -2,684.80 -2,684.80

Using a financial calculator, enter N = 5, PV = 9,000, and PMT = -2,684.80; compute I/Y=

15.0% 4-24 a. First City Bank: Effective rate = 7%. Second City Bank:

%66.60666.01.0 - 4

0.065 + 1 = rate Effective4

==⎟⎠⎞

⎜⎝⎛

You would choose the First City Bank.

b. If funds must be left on deposit until the end of the compounding period (one year for First City and one quarter for Second City), and you think there is a high probability that you will make a withdrawal during the year, the Second City account might be preferable. For example, if the withdrawal is made after 10 months, you would earn nothing on the First City account but (1.01625)3 - 1.0 = 4.95% on the Second City account.

r=?

%0.909.00.1000,85229,201r

)r1(100,85229,201

)r1(1PVFV

10/1

10

n

==−⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎥⎦

⎤⎢⎣

⎡

+=

⎥⎦

⎤⎢⎣

⎡

+=

r=?

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

+

+

r

18000.684,2000,9

r

1PMTPVA

5)r1(1

n)r1(1

Chapter 4

16

Twenty to 30 years ago, most banks and S&Ls were set up as described above, but now financial intermediaries pay interest from the day of deposit to the day of withdrawal, provided at least $1 is in the account at the end of the period.

0 1 2 3 4 17 18 months

r=? 4-25 -150,000 168,925

…

periodper %0.202.01)12617.1(1000,150925,168r

)r1(1000,150925,168

)r1(1PVFV

16667.06

1

6

n

==−=−⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡

+=

⎥⎦

⎤⎢⎣

⎡

+=

Using a calculator, enter N = 6 quarters, PV = -150,000, PMT = 0, and FV = 168,925; compute I/Y= 2% per quarter.

So the APR for this investment equals 2% x 4 = 8%. The effective annual rate of interest is:

rEAR = (1.02)4 - 1 = 8.24%

0 1 2 8 9 10 … r = ? 4-26 -13,250 2,345.05 2,345.05 2,345.05 2,345.05 2,345.05

Using a calculator, enter N = 10, PV = -13,250, PMT = 2,345.05, and FV = 0; compute I/Y= 12%. 0 1 2 28 29 30 … r = ? 4-27 85,000 8,273.59 8,273.59 8,273.59 8,273.59 8,273.59

Using a calculator, enter N = 30, PV = 85,000, PMT = -8,273.59, and FV = 0; compute I/Y= 9.0%. 4-28 a. Cost of points = ($250,000 - $40,000)(0.03) = $6,300 b. Bank of Middle Texas: n = 30 x 12 = 360, r = 8%/12 = 0.6667%, mortgage = $250,000 - $40,000 + $6,300 = $216,300

⎥⎥⎥

⎦⎢⎢⎢

⎣

=00666667.0

PMT300,216360)00666667.1(

⎤⎡ −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −= +

1

r

1PMTPVA

1

n)r1(1

Chapter 4

17

PMT = $216,300/136.28344 = $1,587.13 Calculator solution: N = 360, I/Y= 0.666667, PV = 216,300, and FV = 0; PMT = ? = -1,587.13 Bank of South Alaska: n = 30 x 12 = 360, r = 8.4%/12 = 0.7%, mortgage = $250,000 - $40,000 = $210,000

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= +

007.0

1PMT000,210

r

1PMTPVA

360)007.1(1

n)r1(1

PMT = $210,000/131.26156 = $1,599.86 Calculator solution: N = 360, I/Y= 0.7, PV = 210,000, and FV = 0; PMT = ? = -1,599.86 The mortgage from Bank of Middle Texas has monthly lower payments. c. First, determine the present value of the payment of the mortgage from Bank of South

Alaska—that is, $1,599.86—using the interest rate of the Bank of Middle Texas—that is, 8 percent.

43.034,218)28344.136(86.599,100666667.0

186.599,1PVA

360)00666667.1(1

==⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

Calculator solution: N = 360, I/Y= 0.6667, PMT = 1,599.86, and FV = 0; PV = ? = 218,034.43 Because the amount needed to purchase the house is $210,000, the cost of the points must be

$8,034.43. As a percent, the points would be: Points = $8,034.43/$210,000 = 0.0382 = 3.83% 4-29 With a calculator, enter I/Y= 10/2 = 5, N = 10 x 2 = 20, and PV = -10,000, and press PMT to get

PMT = $802.43. Or

Chapter 4

18

43.80246221.12000,10PMT

)46221.12(PMT05.0

1PMT000,10

r

1PMTPVA

20)05.1(1

n)r1(1

==

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= +

Set up an amortization table for Year 1:

Pmt of Period Beg Bal Payment Interest* Principal End Bal 1 $10,000.00 $802.43 $500.00 $302.43 $9,697.57 2 9,697.57 802.43 484.88 $984.88

Interest is computed by multiplying the “Beg Bal” by 5 percent. For example, the interest for the second six-month period is $9,697.57 x 0.05 = $484.88.

You can also work the problem with a calculator that has an amortization function. Find the interest in each six-month period, sum them, and you have the answer. Even simpler, with some calculators, you just need to input 2 for periods and press INT to get the interest during the first year, $984.88.

4-30 a. Using a financial calculator, enter N = 5, I/Y= 10, PV = 25,000, and FV = 0; compute PMT =

–6,594.94.

94.594,679079.3

000,25PMT

)79079.3(PMT10.0

1PMT000,25

r

1PMTPVA

5)10.1(1

n)r1(1

==

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= +

Set up an amortization schedule as described in the appendix to Chapter 4. Beginning Repayment Remaining Year Balance Payment Interest* of Principal Balance 1 $25,000.00 $ 6,594.94 $2,500.00 $ 4,094.94 $20,905.06 2 20,905.06 6,594.94 2,090.51 4,504.43 16,400.63 3 16,400.63 6,594.94 1,640.06 4,954.88 11,445.75 4 11,445.75 6,594.94 1,144.58 5,450.36 5,995.39 5 5,995.39 6,594.94 599.54 5,995.40 0 $32,974.69 $7,974.69 $25,000.00

Chapter 4

19

* Interest equals 10 percent of the outstanding balance at the beginning of the year. b. Here the loan size is doubled, so the payments also double in size to $13,189.87. Using a financial calculator, enter N = 5, I/Y= 10, PV = 50,000, and FV = 0; compute

PMT = –13,189.87

Notice the only difference from the computation of PMT in part a is that FV is twice as large ($50,000), so the payment (PMT) is double.

c. Using a financial calculator, enter N = 10, I/Y= 10, PV = 50,000, and FV = 0; compute PMT =

–8,137.27. Because the payments are spread out over a longer time period, more interest must be paid on

the loan. The total interest paid on the 10-year loan is $31,372.70 versus interest of $15,949.37 on the five-year loan; but the same $50,000 principal is repaid over a longer period, so the total payment per year is not doubled.

0 1 2 3 4 5 6 7 8 9 10 r=? 4-31 Z: -422.41 0 0 0 0 0 0 0 0 0 1,000.00 B: -500.00 74.50 74.50 74.50 74.50 74.50 74.50 74.50 74.50 74.50 74.50

a. Security Z:

%0.909.00.141.422

000,1r

)r1(1000,141.422

101

10

==−⎟⎠⎞

⎜⎝⎛=

⎥⎦

⎤⎢⎣

⎡

+=

Security B:

50.74500

r

1

r

150.74500

10)r1(1

10)r1(1

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

+

+

Solve for r using trial and error. Using a financial calculator, for Security Z, enter N = 10, PV = -422.41, PMT = 0, and FV =

1,000; compute I/Y= 9.00%. For Security B, enter N = 10, PV = -500, PMT = 74.50, and FV = 0; compute I/Y= 8%. (Alternatively, enter the values exactly as shown on the time line in the CF register, and use the IRR key to obtain the same answer.)

Chapter 4

20

b. Using a calculator, for Security Z, enter N = 10, I/Y= 6, PMT = 0, and FV = 1,000; compute PV = -558.39. The profit would be $558.39 - $422.41 = $135.98, and the percentage profit would be $135.98/$422.41 = 32.2%.

For Security B, enter N = 10, I/Y= 6, PMT = 74.50, and FV = 0; compute PV = -548.33. The profit is $48.33, and the percentage profit is 9.67%.

c. The value of Security Z would fall from $422.41 to $321.97, so a loss of $100.44, or 23.8

percent, would be incurred. The value of Security B would fall to $420.94, so the loss here would be $79.06, or 15.8 percent of the $500 original investment. The percentage loss for Security Z is 1.5 times greater than for Security B because the only cash flow for Security Z is 10 years from now.

The “actual” or “true” return on Security Z would remain at 9 percent, but the “actual” return on Security B would rise to 10.09 percent due to reinvestment of the $74.50 cash receipts at 12 percent.

4-32 a. If Jason makes his first withdrawal today, this is an annuity due: 1 4 Year 0 1 2 3 4 5 45 46 47 48 Quarters r=1%

… 10,000 PMT PMT PMT PMT PMT PMT PMT PMT PMT

Using a calculator, switch the END key to BGN, then enter I/Y= 12/12 = 1, N = 4 x 12 = 48, and PV = 10,000; compute PMT = -260.73.

73.26035370.38000,10PMT

)35370.38(PMT01.101.0

1PMT000,10

)r1(r

1PMTPVA

48)01.1(1

n)r1(1

==

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

×⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+×⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= +

b. If Jason makes his first withdrawal in one month, then this is an ordinary annuity: 1 4 Year 0 1 2 3 4 5 45 46 47 48 Quarters r=1%

… 10,000 PMT PMT PMT PMT PMT PMT PMT PMT PMT

Using a calculator, switch back to BGN, then enter I/Y= 12/12 = 1, N = 4 x 12 = 48, and PV = 10,000; compute PMT = -263.34.

Chapter 4

21

34.26397396.37000,10PMT

)97396.37(PMT01.0

1PMT000,10

r

1PMTPVA

48)01.1(1

n)r1(1

==

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= +

4-33 Here we want to have the same effective annual rate (rEAR) on the credit extended as on the bank

loan that will be used to finance the credit extension.

First, we must find the EAR of the bank loan:

rEAR = (1 + 0.15/12)12 - 1 = (1.0125)12 - 1 = 16.075%

The simple rate (APR) Sue should quote customers must be based on quarterly compounding:

0.1608 = (1 + rPER/4)4 - 1

1.1608 = (1 + rPER/4)4

(1.0380)1/4 = 1 + rPER/4

rPER = 0.0380 = 3.8% per quarter

rSIMPLE = 0.0380(4) = 15.19%. 4-34 Set the calculator to begin mode. a. Calculator solution: I/Y= 18/12 = 1.5, PV = 3,310, PMT = -150, and FV = 0; N = ? = 26.51

months, or 2.2 years. b. Calculator solution: I/Y= 18/12 = 1.5, PV = 3,310, PMT = -222, and FV = 0; N = ? = 16.72

months, or 1.4 years. c. Calculator solution: I/Y= 18/12 = 1.5, PV = 3,310, PMT = -360, and FV = 0; N = ? = 9.81

months, or 0.8 years. 4-35 a. n = 20 x 12 = 240, r = 9%/12 = 0.75, PVA = 95,000

Chapter 4

22

74.854144954.111

000,95PMT

)144954.111(PMT0075.0

1PMT000,95

r

1PMTPVA

240

n

)0075.1(1

)r1(1

==

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= +

Calculator solution: N = 240, I/Y= 0.75, PV = 95,000, and FV = 0; PMT = ? = -854.74 b. n = 10 x 12 = 120, r = 9%/12 = 0.75, PVA = 95,000

42.203,194169.78000,95PMT

)94169.78(PMT0075.0

1PMT000,95

r

1PMTPVA

120

n

)0075.1(1

)r1(1

==

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= +

Calculator solution: N = 120, I/Y= 0.75, PV = 95,000, and FV = 0; PMT = ? = -1,203.42 c. Calculator solution: I/Y= 0.75, PV = 95,000, PMT = -985, and FV = 0; N = ? = 171.98

months, or 14.3 years. 4-36 a. Car price, excluding rebates = $24,000 = price if 0 percent financing is taken PMT with 0% financing = $24,000/48 = $500 b. Car price with rebate = $24,000 - $3,000 = $21,000 = price if credit union loan is used

( )

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

+−

=

mr

111

PMTPVA mr

Chapter 4

23

( )

19.493$58032.42

000,21$PMT

1206.0

1

11

PMT000,21$48

1206.0

==

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

+−

=

c. Based on the monthly payment, the car should be purchased using the credit union loan; the

payment is lower ($493.19 ) than with the dealer’s 0 percent loan ($500). d. Three years remain on the loans: 0% financing balance = (3 x 12) x $500 = $18,000

Credit union loan: ( )

66.211,16)87102.32(19.493

1206.0

111

19.493PVA36

1206.0

==

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

+−

=

Calculator solution: N = 36, I/Y= 0.5, PMT = -493.19, and FV = 0; PV = 16,211.66. 4-37 a. First, determine the annual cost of college. The current cost is $12,500 per year, which will

escalate at a 5 percent inflation rate: College Current Years Inflation Cash Year Cost from Now Adjustment Required 1 $12,500 5 (1.05)5 $15,954 2 12,500 6 (1.05)6 16,751 3 12,500 7 (1.05)7 17,589 4 12,500 8 (1.05)8 18,468

Now put these costs on a cash flow time line and find the PV at the time the daughter starts college—that is, when she turns 18: 0 1 2 3 Year of College

18 19 20 21 Age 15,954 16,751 17,589 18,468 College cost

8%

15,510 15,080 14,660 61,204 General equation:

⎥⎦

⎤⎢⎣

⎡

+= n)r1(

1FVPV

Chapter 4

24

Thus, the college fund must accumulate $61,204 by the time the daughter reaches age 18.

b. The daughter has $7,500 now (age 13) to help achieve the educational goal. Five years hence

the $7,500, when invested at 8 percent, will be worth $11,020: 13 14 15 16 17 18

8%

7,500 FV = ? = $7,500(1.08)5 = $11,020.

c. The father needs to accumulate only $61,204 - $11,020 = $50,184. The key to completing the

problem at this point is to realize the series of deposits represent an ordinary annuity rather than an annuity due, despite the fact the first payment is made at the beginning of the first year. The reason it is not an annuity due is because there is no interest paid on the last payment, which occurs when the daughter is 18. Thus,

0 1 2 3 4 5 Year 13 14 15 16 17 18 Age 8%

PMT PMT PMT PMT PMT PMT FVA = 50,184

841,633593.7

184,50PMT

08.01)08.1(PMT184,50

r1)r1(PMTFVA

6

n

==

⎥⎦

⎤⎢⎣

⎡ −=

⎥⎦

⎤⎢⎣

⎡ −+=

In this case, if you want, you can assume that the father made plans for his daughter’s college education at the time she celebrated her 12th birthday. If you draw a cash flow time line using this assumption, you will see that the payments represent an ordinary annuity.

Another way to approach the problem is to treat the series of payments as a five-year annuity due with a lump sum deposit at the end of Year 5: $50,183 = FVA(DUE)5 + PMT =

841,633593.7

184,50PMT

PMT)08.1(08.0

1)08.1(PMT184,50

PMT)r1(r

1)r1(PMTFVA

5

n

==

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

×⎥⎦

⎤⎢⎣

⎡ −=

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+×⎥⎦

⎤⎢⎣

⎡ −+=

Chapter 4

25

0 1 2 29 30 Years 4-38 a. 0 1 2 3 4 58 59 60 Periods … -500 -500 -500 -500 -500 -500 -500

5%

( )( )

792,176)58372.353(500

05.01)05.1(500

mr

1mr1

PMTFVA

60

mn

==

⎥⎦

⎤⎢⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −+=

×

Using a calculator, enter N = 30 x 2 = 60, I/Y= 10/2 = 5, and PMT = -500; compute FV =

$176,792.

b. To solve this problem, we have to recognize that the answer given in part a is too high by the $10,000 withdrawal plus the interest the $10,000 would have earned for 10 years. The $10,000 withdrawal made 10 years before Kay's retirement would have been worth the following amount at retirement:

FV = $10,000(1.05)20 = $10,000(2.65330) = $26,533

So, considering the $10,000 withdrawal, the actual amount that Kay would have in her retirement fund would be:

Retirement fund balance = $176,792 - $26,533 = $150,259

0 1 4 5 Years 4-39 a. 0 1 2 3 4 16 17 18 19 20 Periods … -600 -600 -600 -600 -600 -600 -600 -600 -600 (1) Dealer's “special financing package”

33.827,10)04555.18(60001.0

1600PVA

20)01.1(1

==⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

Using a calculator, enter I/Y= 4/4 = 1, N = 5 x 4 = 20, and PMT = -600; compute PV =

10,827.33.

So Sarah must use $2,172.67 = $13,000 - $10,827.33 of the $3,000 in her checking account if the dealer's financing is used.

(2) Bank loan

Chapter 4

26

48.926,8)8875.14(60003.0

1600PVA

20)03.1(1

==⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

Using a calculator, enter I/Y= 12/4 = 3, N = 5 x 4 = 20, and PMT = -600; compute PV =

8,926.48.

The difference between this amount and the PVA of the dealer's “special financing package” is $1,900.86 = $10,827.36 - $8,926.50, so Sarah would have to negotiate a reduction in the sticker price equal to $1,901 to make the bank financing more attractive than the dealer’s financing.

4-40 Information given:

1. Janet will save for 40 years at 7 percent compounded annually, and then retire.

2. When she retires, Janet wants to take a trip around the world at a cost of $120,000.

3. After the trip, Janet wants to receive payments equal to $70,000 per year, and these payments are expected to last for 20 years.

4. Upon retirement, Janet's funds will earn 5 percent compounded annually.

The cash flow time line for Janet is:

25 26 27 28 65 66 67 84 85 Janet’s age 0 1 2 3 20 Payments 0 1 2 19 20 Withdrawals -PMT -PMT -PMT -PMT

5% 7%

120,000 Trip cost 70,000 70,000 70,000 70,000 Ret. inc.

To solve this problem, you must break it up into sub-problems. First, determine the amount Janet needs at retirement to be able to withdraw $70,000 per year for 20 years, beginning one year after retirement. This is an ordinary annuity. Using a calculator, enter I/Y= 5 (the rate after retirement is 5 percent), N = 20, and PMT = 70,000; compute PV to determine the value of the retirement annuity at retirement is equal to $872,355. The computation is:

355,87272.354,872)46331.12(000,7005.0

1000,70PVA

20)05.1(1

≈==⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

So, at retirement, including the amount needed for the trip around the world, Janet needs a total of $992,355 = $872,355 + $120,000. Thus, the cash flow time line for Janet today, when she is planning her retirement and wants to determine the amount she needs to contribute to the retirement fund, is as follows:

Chapter 4

27

The cash flow time line for Janet is: 25 26 27 28 63 64 65 Janet’s age 0 1 2 3 18 19 20 Payments to retirement fund -PMT -PMT -PMT -PMT -PMT -PMT

7%

992,355 = FVA

Using a calculator, enter N = 40, I/Y= 7%, and FV = 992,354; compute PMT = $4,971 each year. The computation is:

971,485.970,463544.199355,992PMT

)63511.199(PMT355,992

07.01)07.1(PMT355,992

r1)r1(PMTFVA

40

n

≈==

=⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+=

Janet needs to contribute $4,971 each year for 40 years to meet her retirement goals.

Chapter 4

28

INTEGRATIVE PROBLEM

4-41 ASSUME THAT YOU ARE NEARING GRADUATION AND THAT YOU HAVE APPLIED FOR A JOB WITH A LOCAL BANK. AS PART OF THE BANK'S EVALUATION PROCESS, YOU HAVE BEEN ASKED TO TAKE AN EXAMINATION THAT COVERS SEVERAL FINANCIAL ANALYSIS TECHNIQUES. THE FIRST SECTION OF THE TEST ADDRESSES TVM ANALYSIS. SEE HOW YOU WOULD DO BY ANSWERING THE FOLLOWING QUESTIONS:

A. DRAW CASH FLOW 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 THREE YEARS, AND (3) AN UNEVEN CASH FLOW STREAM OF -$50, $100, $75, AND $50 AT THE END OF YEARS 0 THROUGH 3.

ANSWER: Discuss basic time value concepts, terminology, and solution methods. A cash flow 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. LUMP-SUM AMOUNT—a single flow; for example, a $100 inflow in Year 2: 0 1 2 3 Year r% 100 Cash flow ANNUITY—a series of equal cash flows occurring over equal intervals: 0 1 2 3 Year r% 100 100 100 Cash flow UNEVEN CASH FLOW STREAM—an irregular series of cash flows that do not constitute an annuity: 0 1 2 3 Year r% -50 100 75 50 Cash flow CF = -50 represents a cash outflow rather than a receipt or inflow.

Chapter 4

29

B. (1) WHAT IS THE FUTURE VALUE OF AN INITIAL $100 AFTER THREE YEARS IF IT IS INVESTED IN AN ACCOUNT PAYING 10 PERCENT ANNUAL INTEREST?

ANSWER: Show dollars corresponding to question mark, calculated as follows: 0 1 2 3 10%

100 FV = ? After 1 year:

FV1 = PV + INT1 = PV + PV(r) = PV(1 + r) = $100(1.10) = $110.00.

Similarly: FV2 = FV1 + INT2 = FV1 + FV1(r) = FV1(1 + r) = $110(1.10) = $121.00 = PV(1 + r)(1 + r) = PV(1 + r)2. FV3 = FV2 + INT3 = FV2 + FV2(r) = FV2(1 + r) = $121(1.10) = $133.10 = PV(1 + r)2(1 + r) = PV(1 + r)3

In general, we see that: FVn = PV(1 + r)n,

so FV3 = $100(1.10)3 = $100(1.3310) = $133.10. Note that this equation has four variables: FVn, PV, r, and n. Here we know all except FVn, so we solve for FVn. However, often, we will 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 we generally find FV using one of these methods: (1) Numerical approach—use a regular calculator: FV3 = $100(1.10)3 = $133.10. (2) Financial calculator: This is especially efficient for more complex problems, including exam

problems. Input the following values: N = 3, I/Y = 10, PV = -100, and PMT = 0; compute FV = $133.10.

(3) Spreadsheet: Set up your spreadsheet and use the FV financial function similar to the following:

Chapter 4

30

Step 1: Set up the spreadsheet: Step 2: Select FV in the financial function category:

Step 3: Input the cell locations of the data: Step 4: Press OK to display the solution:

B. (2) WHAT IS THE PRESENT VALUE OF $100 TO BE RECEIVED IN 3 YEARS IF THE APPROPRIATE INTEREST RATE IS 10 PERCENT?

ANSWER: 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: 0 1 2 3 10% PV = ? 100

Chapter 4

31

FVn = PV(1 + r)n transforms to:

)r + (1

1 FV = )r + (1

FV =PV nnn

n

⎥⎥⎦

⎤

⎢⎢⎣

⎡

thus:

( ) 13.75$75134.0$100 = (1.10)

1 $100 = PV 3 =⎥⎦

⎤⎢⎣

⎡

The same methods (regular calculator, financial calculator, and spreadsheet) used for finding future values also are used to find present values, which is called discounting. Numerical (regular calculator) solution: Given above.

Financial calculator solution: Input N = 3, I/Y = 10, PMT = 0, and FV = 100; compute PV = $75.13.

Spreadsheet solution: Use the PV financial function that is available on the spreadsheet.

NSWERA : We have this situation in time line format:

0 1 2 3 n = ?

- 3

we want to find out how long it will take us to triple our money at an interest rate of 20 percent, we can use

FVn = $3 = $1(1 + r)n = $1(1.20)n

Numerical (regular calculator) solution: Use a trial-and-error method, substituting values in for n until

C. WE SOMETIMES NEED TO FIND HOW LONG IT WILL TAKE A SUM OF MONEY (OR ANYTHING ELSE) TO GROW TO SOME SPECIFIED AMOUNT. FOR EXAMPLE, IF A COMPANY'S SALES ARE GROWING AT A RATE OF 20 PERCENT PER YEAR, APPROXIMATELY HOW LONG WILL IT TAKE SALES TO TRIPLE?

… 20% 1 Ifany numbers, say, $1 and $3, with this equation:

the right side of the equation equals 3. Or, using more complex mathematics, we can solve the above equation as follows:

13)20.1(

)20.1(13

n

n

=

=

years 026.618232.009861.1

)20.1ln()3ln(n ===

Financial calculator solution: Input I/Y = 20, PV = -1, PMT = 0, and FV = 3; compute N = 6.026.

Chapter 4

32

Spreadsheet solution: Use the NPER financial function that is available on the spreadsheet.

hus, it takes approximately 6 periods for an amount to triple at a 20 percent interest rate.

**************************************************************************************

FV20 = $60(1 + 0.04 + 0.06)20 = $60(1.10)20 = $403.65 per acre.

e could have asked: How long would it take $60 to grow to $403.65, given the real rate of return of 4

******************

NSWER

T *OPTIONAL QUESTION: A FARMER CAN SPEND $60 PER ACRE TO PLANT PINE TREES ON SOME MARGINAL LAND. THE EXPECTED REAL RATE OF RETURN IS 4 PERCENT, AND THE EXPECTED INFLATION RATE IS 6 PERCENT. WHAT IS THE EXPECTED VALUE OF THE TIMBER AFTER 20 YEARS? Wpercent and an inflation rate of 6 percent. Of course, the answer would be 20 years. *********************************************************************

D. WHAT IS THE DIFFERENCE BETWEEN AN ORDINARY ANNUITY AND AN

0 1 2 3

ANNUITY DUE? WHAT TYPE OF ANNUITY IS SHOWN IN THE FOLLOWING CASH FLOW TIME LINE? HOW WOULD YOU CHANGE IT TO THE OTHER TYPE OF ANNUITY?

100 100 100

A : This is an ordinary annuity—its payments are at the end of each period; that is, the first payment

he annuity shown above is an ordinary annuity. To convert it to an annuity due, shift each payment to the

NSWER

is made one period from today. Conversely, an annuity due has its first payment today. In other words, an ordinary annuity has end-of-period payments, whereas an annuity due has beginning-of-period payments. Tleft, so you end up with a payment under the 0 but none under the 3.

E. (1) WHAT IS THE FUTURE VALUE OF A THREE-YEAR ORDINARY ANNUITY OF $100 IF THE APPROPRIATE INTEREST RATE IS 10 PERCENT?

A :

0 1 2 3

100 100 100

10% 110 121 331 One approach would be to treat each annuity flow as a lump sum as in the time line. Here we have:

Chapter 4

33

FVAn = $100(1.10)0 + $100(1.10)1 + $100(1.10)2

3.3100) = $331.00

Numerical solution:

= $100[(1.10)0 + (1.10)1 + (1.10)2] = $100(1.00 + 1.10 + 1.21) = $100(

00.331$)31000.3(100$

10.01)10.1(100$

r1)r1(PMTFVA

3

n

n

==

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+=

Financial calculator solution: Input N = 3, I/Y = 10, PV = 0, and PMT = -100; compute FV = 331

Spreadsheet solution: Use the FV financial function that is available on the spreadsheet, inputting 100 for

ANSWER

Pmt.

:

0 1 2 3

90.91 100 100 100

E. (2) WHAT IS THE PRESENT VALUE OF THE ANNUITY?

10% 82.64 75.13 248.68 The present value of the annuity is $248.68. Here we used the lump sum approach, but the same result could

Numerical solution:

be obtained by using a regular or financial calculator.

69.248$)48685.2(100$

10.0

1100$

r

1PMTPVA

3)10.1(1

n)r1(1

n

==

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −= +

Financial calculator solution: Input N = 3, I/Y = 10, PMT = 100, and FV = 0; compute PV = -248.69

Spreadsheet solution: Use the PV financial function that is available on the spreadsheet, inputting 100 for

Pmt.

Chapter 4

34

E. (3) WHAT WOULD THE FUTURE AND PRESENT VALUES BE IF THE ANNUITY WERE AN ANNUITY DUE?

ANSWER: 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.

Future Value of the Annuity Numerical solution:

10.364$)64100.3(100$

)10.1(10.0

1)10.1(100$

)r1(r

1)r1(PMT)DUE(FVA

3

n

n

==

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

×⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+×⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+=

Financial calculator solution: Switch your calculator to “BEG” or beginning or “DUE” mode, input N =

3, I/Y = 10, PV = 0, and PMT = -100; compute FV = 364.10. Remember to change back to “END” mode after working an annuity due problem with your calculator.

Spreadsheet solution: Use the FV financial function that is available on the spreadsheet, inputting Pmt =

100 and Type =1.

Present Value of the Annuity

Numerical solution:

55.273$)73554.2(100$

)10.1(10.0

1100$

)r1(r

1PMT)DUE(PVA

3)10.1(1

n)r1(1

n

==

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧×

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+×

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −= +

Financial calculator solution: Switch your calculator to “BEB” or beginning or “DUE” mode, input N =

3, I/Y = 10, PMT = 100, and FV = 0; compute PV = -248.69. Remember to change back to “END” mode after working an annuity due problem with your calculator.

Spreadsheet solution: Use the PV financial function that is available on the spreadsheet, inputting Pmt =

100 and Type =1.

F. WHAT IS THE PRESENT VALUE OF THE FOLLOWING UNEVEN CASH FLOW

STREAM? THE APPROPRIATE INTEREST RATE IS 10 PERCENT, COMPOUNDED ANNUALLY.

0 10% 1 2 3 4 YEARS

Chapter 4

35

ANSWER: Here we have an uneven cash flow stream. The most straightforward approach is to find the present value of each cash flow and then sum the PVs as shown below: 0 1 2 3 4 10% 90.91 100 300 300 -50 247.93 225.39 (34.15) 530.08 Numerical solution:

09.530)1507.34$(3944.225$9339.247$9091.90$)10.1(

1)50$()10.1(

1300$)10.1(

1300$)10.1(

1100$

)r1(1CF

)r1(1CF

)r1(1CFPV

4321

nn2211

=−+++=⎥⎥⎦

⎤

⎢⎢⎣

⎡−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+= L

Financial calculator solution: Financial calculators have cash flow (CF) functions in which you would

input the cash flows, so they are in the calculator’s memory, input the interest rate, I, and then compute the NPV, which is the present value. In this case, CF0 = 0, CF1 = 100, CF2 = 300, CF3 = 300, and CF4 = -50.

Spreadsheet solution: Use the NPV financial function that is available on the spreadsheet. In Excel, the

NPV function computes the PV of CF1, CF2, CF3, and so forth; CF0 is not included in the computation.

ANSWER:

G. WHAT ANNUAL INTEREST RATE WILL CAUSE $100 TO GROW TO $125.97 IN THREE YEARS?

0 1 2 3 r = ? -100 125.97 FV = $100(1 + r)3 = $125.97

Chapter 4

36

Numerical solution:

%0.8080.00.100.100$97.125$r

)r1(100$97.125$

)r1(PVFV

31

3

n

==−⎟⎠

⎞⎜⎝

⎛=

+=

+=

Financial calculator solution: Input N = 3, PV = -100, PMT = 0, and FV = 125.97; compute I/Y = 8.0%

= r. Spreadsheet solution: Use the RATE financial function that is available on the spreadsheet

ANSWER: Investments that pay interest more frequently than once per year, for example—semiannually, quarterly, or daily—have higher future values because interest is earned on interest more often. Banks pay interest daily on passbook and most money fund accounts, so they use daily compounding.

INTEREST RATE CONSTANT? WHY? H. (2) DEFINE (I) THE STATED, OR QUOTED, OR SIMPLE, RATE (rSIMPLE), THE

ANNUAL PERCENTAGE RATE (APR), THE PERIODIC RATE (rPER), AND THE EFFECTIVE ANNUAL RATE (rEAR).

H. (1) WILL THE FUTURE VALUE BE LARGER OR SMALLER IF WE COMPOUND AN INITIAL AMOUNT MORE OFTEN THAN ANNUALLY, FOR EXAMPLE, EVERY SIX MONTHS, OR SEMIANNUALLY, HOLDING THE STATED

ANSWER: The quoted, or simple, 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 (rPER = rSIMPLE/m), and the effective annual rate (rEAR) is that rate of interest that would provide an identical future dollar value under annual compounding (rEAR = {1 + [rSIMPLE/m]}m - 1.0).

H. (3) WHAT IS THE EFFECTIVE ANNUAL RATE FOR A SIMPLE RATE OF 10 PERCENT, COMPOUNDED SEMIANNUALLY? COMPOUNDED QUARTERLY? COMPOUNDED DAILY?

ANSWER: The effective annual rate for 10 percent semiannual compounding, is 10.25 percent:

%25.101025.00.1210.01

1.0 - m

r + 1 = EAR

2

SIMPLEm

==−⎟⎠

⎞⎜⎝

⎛ +=

⎟⎠

⎞⎜⎝

⎛

For quarterly compounding, the effective annual rate is 10.38 percent: EAR = (1.025)4 ─ 1.0 = 1.1038 ─ 1.0 = 0.1038 = 10.38%. Daily compounding would produce EAR = 10.52%.

H. (4) WHAT IS THE FUTURE VALUE OF $100 AFTER THREE YEARS UNDER 10 PERCENT SEMIANNUAL COMPOUNDING? QUARTERLY COMPOUNDING?

Chapter 4

37

ANSWER: With semiannual compounding, the $100 is compounded over six semiannual periods at a 5.0 percent periodic rate: 1 2 3 Years 0 1 2 3 4 5 6 Six-month periods

5% -100 FV=? Numerical Solution:

01.134$)34010.1(100$)05.1(100$

20.10 + 1 $100

m

r1PVFV

6

3 2

nmSIMPLE

n

===

⎟⎠

⎞⎜⎝

⎛=

⎥⎦

⎤⎢⎣

⎡+=

×

×

Quarterly: FVn = $100(1.025)12 = $134.49

Another approach here would be to use the effective annual rate and compound over annual periods: Semiannually: $100(1.1025)3 = $134.01 Quarterly: $100(1.1038)3 = $134.49 Clearly, the return is higher when using quarterly compounding. Financial calculator solution: Semiannual compounding: Input N = 6, I/Y = 5, PV = -100, and PMT = 0;

compute FV = 134.01.

Spreadsheet solution: Use the FV financial function that is available on the spreadsheet; adjust the interest rate so that it represents the interest paid per period and N so that it equals the number of interest periods.

NSWERA : If annual compounding is used, then the simple rate will be equal to the effective annual rate. If

I. WILL THE EFFECTIVE ANNUAL RATE EVER BE EQUAL TO THE SIMPLE (QUOTED) RATE?

more frequent compounding is used, the effective annual rate will be greater than the simple rate. That is, rSIMPLE = rPER = rEAR when interest is compounded annually, whereas rSIMPLE < rEAR when interest is compounded more than once per year.

Chapter 4

38

100 100 100

J. (1) WHAT IS THE VALUE AT THE END OF YEAR 3 OF THE FOLLOWING CASH FLOW STREAM IF THE QUOTED INTEREST RATE IS 10 PERCENT, COMPOUNDED SEMIANNUALLY?

0 1 2 3 YEARS

ANSWER: 0 1 2 3 100 100 100.00

5%

110.25 = $100(1.05)2 121.55 = $100(1.05)4 331.80 Here we have a different situation. The payments occur annually, but compounding occurs each six months. Thus, we cannot use normal annuity valuation techniques. There are two approaches that can be applied: (1) Treat the cash flows as lump sums, as was done above, or (2) Treat the cash flows as an ordinary annuity, but use the effective annual rate:

10.25%. = 1 - 2

0.10 + 1 = 1 - m

r + 1 = r2

SIMPLEm

EAR ⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛

Now we have this three-period annuity: FVA3 = $100(1.1025)2 + $100(1.1025)1 + $100 = $331.80 Numerical solution:

80.331$)31801.3(100$

1025.01)1025.1(100$

r1)r1(PMTFVA

3

n

n

==

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+=

Financial calculator solution: Input N = 3, I/Y = 10.25, PV = 0, and PMT = -100; compute FV = 331.80 Spreadsheet solution: Use the FV financial function that is available on the spreadsheet, inputting 100 for

Pmt and 0.1025 for Rate.

J. (2) WHAT IS THE PV OF THE SAME STREAM?

Chapter 4

ANSWER: 0 1 2 3

5% 90.70 100 100 100 82.27 74.62 247.59 Numerical solution:

59.247$)47595.2(100$

1025.0

1100$

r

1PMTPVA

3)1025.1(1

n)r1(1

n

==

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −= +

Financial calculator solution: Input N = 3, I/Y = 10.25, PMT = 100, and FV = 0; compute PV = -247.59 Spreadsheet solution: Use the PV financial function that is available on the spreadsheet, inputting 100 for

Pmt and 0.1025 for Rate.

J. (3) IS THE STREAM AN ANNUITY?

ANSWER: The payment stream is an annuity in the sense of constant amounts at regular intervals, but the intervals do not correspond with the compounding periods. This kind of situation occurs often. In this situation the interest is compounded semiannually, so with a quoted rate of 10 percent, the rEAR will be 10.25 percent. Here we could find the effective rate and then treat it as an annuity. Enter N = 3, I/Y = 10.25, PMT = 100, and FV = 0; compute PV = -247.59.

ANSWER: rSIMPLE can only be used in the calculations when annual compounding occurs. If the simple rate of 10 percent was used to discount the payment stream the present value would be overstated by $272.32 ─ $247.59 = $24.73.

J. (4) AN IMPORTANT RULE IS THAT YOU SHOULD NEVER SHOW A SIMPLE RATE ON A TIME LINE OR USE IT IN CALCULATIONS UNLESS WHAT CONDITION HOLDS? (HINT: THINK OF ANNUAL COMPOUNDING, WHEN rSIMPLE = rEAR = rPER) WHAT WOULD BE WRONG WITH YOUR ANSWER TO QUESTIONS PARTS (1) AND (2) IF YOU USED THE SIMPLE RATE 10 PERCENT RATHER THAN THE PERIODIC RATE rSIMPLE /2 = 10%/2 = 5%?

39

K. (1) CONSTRUCT AN AMORTIZATION SCHEDULE FOR A $1,000, 10 PERCENT ANNUAL RATE LOAN WITH THREE EQUAL INSTALLMENTS.

(2) WHAT IS THE ANNUAL INTEREST EXPENSE FOR THE BORROWER, AND THE ANNUAL INTEREST INCOME FOR THE LENDER, DURING YEAR 2?

Chapter 4

40

ANSWER: To begin, note that the face amount of the loan, $1,000, is the present value of a three-year annuity at a 10 percent rate: 0 1 2 3 10% -1,000 PMT PMT PMT

)PMT(1.10 + )PMT(1.10 + )PMT(1.10 =

)r+ PMT(1 + )r + PMT(1 + )r + PMT(1 = $1,000

)r + (1

1 PMT + )r+ (1

1 PMT + )ir+ (1

1 PMT = PVA

3-2-1-

3-2-1-

3213⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

We have an equation with only one unknown, so we can solve it to find PMT. Numerical solution:

11.402$48685.2

000,1$PMT

)48685.2(PMT10.0

1PMT000,1$

r

1PMTPVA

3)10.1(1

n)r1(1

n

==

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −= +

Financial calculator solution: Input N = 3, I/Y = 10, PV = 1,000, and FV = 0; compute PMT = -402.11 Spreadsheet solution: Use the PV financial function that is available on the spreadsheet, solving for Pmt. Now make the following points regarding the amortization schedule: (1) The $402.11 annual payment includes both interest and principal. Interest in the first year is

calculated as follows: Year 1 interest = r x beginning balance = 0.10 x $1,000 = $100.

(2) The repayment of principal is the difference between the $402.11 annual payment and the interest payment: Year principal repayment = $402.11 ─ $100 = $302.11.

(3) The loan balance at the end of the first year is: Year 1 ending balance = beginning balance ─ principal repayment = $1,000 ─ $302.11 = $697.89.

(4) We would continue these steps in the following years. (5) Notice that the interest each year declines because the beginning loan balance is declining. Because

the payment is constant, but the interest component declines, the principal repayment portion increases each year.

(6) 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 30 x 12 = 360 monthly payments if you get a 30-year, fixed rate mortgage.

Chapter 4

(7) The payment might 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.

(8) The lender received a 10 percent 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.

(9) Most financial calculators have amortization functions built in. The amortization schedule would be: Beginning Interest Principal Ending Year Balance Payment @ 10% Repayment Balance 1 $1,000.00 $402.11 $100.00 $302.11 $697.89 2 697.89 402.11 69.79 332.32 365.57 3 365.57 402.11 36.56 365.55 0.02 (rounding difference)

L. SUPPOSE ON JANUARY 1 YOU DEPOSIT $100 IN AN ACCOUNT THAT PAYS A SIMPLE, OR QUOTED, INTEREST RATE OF 11.33463 PERCENT, WITH INTEREST ADDED (COMPOUNDED) DAILY. HOW MUCH WILL YOU HAVE IN YOUR ACCOUNT ON OCTOBER 1, OR AFTER NINE MONTHS?

ANSWER: First, determine the effective annual rate of interest, with daily compounding:

12.0%. = 0.12 = 1 365

0.1133463 + 1 = EAR365

−⎟⎠⎞

⎜⎝⎛

Thus, if you left your money on deposit for an entire year, you would earn $12 of interest, and you would end up with $112. The question, however, is: How much will be in your account on October 1, 2005? Here you will be leaving the money on deposit for 9/12 = 3/4 = 0.75 of a year. 0 0.75 1

41

-100 FV=? 112

12%

You would use the regular set-up, but with the fraction of the year: Numerical solution: FV0.75 = $100(1.12)0.75 = $100(1.08871) = $108.87 Financial calculator solution: Input N = 0.75, I/Y = 12, PV = -100, and PMT = 0; compute FV = 108.87 Spreadsheet solution: Use the FV financial function that is available on the spreadsheet, inputting 0.75

for Nper.

M. NOW SUPPOSE YOU LEAVE YOUR MONEY IN THE BANK FOR 21 MONTHS. THUS, ON JANUARY 1 YOU DEPOSIT $100 IN AN ACCOUNT THAT PAYS A 11.33463 PERCENT COMPOUONDED DAILY. HOW MUCH WILL BE IN YOUR ACCOUNT ON OCTOBER 1 OF THE FOLLOWING YEAR?

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42

ANSWER: In this case, the money will be left on deposit for 1 + 9/12 = 1 + 3/4 = 1.75 of a year. 0 1 1.75 2 Years

12% -100 112 FV=? 125.44

Numerical solution: FV1.75 = $100(1.12)1.75 = $100(1.21936) = $121.94 Financial calculator solution: Input N = 1.75, I/Y = 12, PV = -100, and PMT = 0; compute FV = 121.94 Spreadsheet solution: Use the FV financial function that is available on the spreadsheet, inputting 1.75

for Nper.

N. SUPPOSE SOMEONE OFFERED TO SELL YOU A NOTE THAT CALLS FOR A $1,000 PAYMENT 15 MONTHS FROM TODAY. THE PERSON OFFERS TO SELL THE NOTE FOR $850. YOU HAVE $850 IN A BANK TIME DEPOSIT (SAVINGS INSTRUMENT) THAT PAYS A 6.76649 PERCENT SIMPLE RATE WITH DAILY COMPOUNDING, WHICH IS A 7 PERCENT EFFECTIVE ANNUAL INTEREST RATE; AND YOU PLAN TO LEAVE THE MONEY IN THE BANK UNLESS YOU BUY THE NOTE. THE NOTE IS NOT RISKY—THAT IS, YOU ARE SURE IT WILL BE PAID ON SCHEDULE. SHOULD YOU BUY THE NOTE? CHECK THE DECISION IN THREE WAYS: (1) BY COMPARING YOUR FUTURE VALUE IF YOU BUY THE NOTE VERSUS LEAVING YOUR MONEY IN THE BANK, (2) BY COMPARING THE PV OF THE NOTE WITH YOUR CURRENT BANK INVESTMENT, AND (3) BY COMPARING THE rEAR ON THE NOTE WITH THAT OF THE BANK INVESTMENT.

ANSWER: you can solve this problem in three ways: (1) by compounding the $850 now in the bank for 15 months and comparing that FV with the $1,000 the note will pay; (2) by finding the PV of the note and then comparing it with the $850 cost; and (3) by finding the effective annual rate of return on the note and comparing that rate with the 7 percent you are now earning, which is your opportunity cost of capital. All three procedures lead to the same conclusion. Here is the cash flow time line: 0 1 1.25 2 Years

7% -850 1,000 (1) Future Value Numerical solution: FV1.25 = $850(1.07)1.25 = $850(1.08825) = $925.01 < $1,000 FV of investment

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43

Financial calculator solution: Input N = 1.25, I/Y = 7, PV = -850, and PMT = 0; compute FV = 925.01

Spreadsheet solution: Use the FV financial function that is available on the spreadsheet, inputting

1.25 for Nper. (2) Present Value Numerical solution:

90.918$)91890.0(000,1$)07.1(

1000,1$PV 25.1 ==⎥⎦

⎤⎢⎣

⎡= > $850 cost of investment

Financial calculator solution: Input N = 1.25, I/Y = 7, PMT = 0, and FV = 1,000; compute PV = -

918.90 Spreadsheet solution: Use the PV financial function that is available on the spreadsheet, inputting

1.25 for Nper. (3) Effective Annual Rate of the Investment

%88.131388.01

850000,1r

)r1(850000,1

25.11

25.1

==−⎟⎠

⎞⎜⎝

⎛=

+=

Each computation shows that the investment should be made. If the $850 is invested and grows to $1,000 in 1¼ years, the investor will earn 13.9 percent, which is better than the bank rate of 7 percent.

O. SUPPOSE THE NOTE DISCUSSED IN PART N HAD A COST OF $850, BUT CALLS FOR FIVE QUARTERLY PAYMENTS OF $190 EACH, WITH THE FIRST PAYMENT DUE IN THREE MONTHS RATHER THAN $1,000 AT THE END OF 15 MONTHS. WOULD IT BE A GOOD INVESTMENT?

ANSWER: Here is the cash flow time line: ¼ ½ ¾ 1 1¼ Years 0 1 2 3 4 5 Quarters

1.706% 850 190 190 190 190 190 Rate per period = (1.07)0.25 – 1.0 = 1.70585 (1) Future Value

Chapter 4

44

Numerical solution:

97.982$)17352.5(190$0170585.0

1)0170585.1(190$FVA5

==⎥⎦

⎤⎢⎣

⎡ −= > $850(1.0170585)5 = $925.01

Financial calculator solution: Input N = 5, I/Y = 1.70585, PV = 0, and PMT = 190; compute FV =

982.97. Spreadsheet solution: Use the FV financial function that is available on the spreadsheet. (2) Present Value Numerical solution:

25.903)75397.4(190$0170585.0

1190$PVA )0170585.1(

1

==⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

Financial calculator solution: Input N = 5, I/Y = 1.70585, PMT = 190, and FV = 0; compute PV = -

903.25 Spreadsheet solution: Use the PV financial function that is available on the spreadsheet. (3) Effective Annual Rate of the Investment

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −= +

r

1190$850$ )r1(

1

Numerical solution: Use a trial-and-error method to determine k. Financial calculator solution: Input N = 5, PV = -850, PMT = 190, and FV = 0; compute I/Y =

3.8259 per quarter. EAR = (1.038259)4 – 1 = 0.1620 = 16.20% > 7% on bank deposit. Each computation shows that the investment should be made.

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4-42 Computer-Related Problem

a. Payment = $3,523.79. INPUT DATA: KEY OUTPUT: Loan amount $30,000 Payment $3,523.79 Interest rate 10.00% Number of years 20 MODEL-GENERATED DATA: Amortization schedule: Principal Remaining PV of Year Payment Interest Repayment Balance Payments 1 $ 3,523.79 $ 3,000.00 $ 523.79 $29,476.21 $ 3,203.44 2 3,523.79 2,947.62 576.17 28,900.04 2,912.22 3 3,523.79 2,890.00 633.78 28,266.26 2,647.47 4 3,523.79 2,826.63 697.16 27,569.10 2,406.80 5 3,523.79 2,756.91 766.88 26,802.22 2,188.00 6 3,523.79 2,680.22 843.57 25,958.65 1,989.09 7 3,523.79 2,595.87 927.92 25,030.73 1,808.26 8 3,523.79 2,503.07 1,020.72 24,010.01 1,643.87 9 3,523.79 2,401.00 1,122.79 22,887.22 1,494.43 10 3,523.79 2,288.72 1,235.07 21,652.16 1,358.57 11 3,523.79 2,165.22 1,358.57 20,293.58 1,235.07 12 3,523.79 2,029.36 1,494.43 18,799.15 1,122.79 13 3,523.79 1,879.92 1,643.87 17,155.28 1,020.72 14 3,523.79 1,715.53 1,808.26 15,347.02 927.92 15 3,523.79 1,534.70 1,989.09 13,357.93 843.57 16 3,523.79 1,335.79 2,188.00 11,169.94 766.88 17 3,523.79 1,116.99 2,406.80 8,763.14 697.16 18 3,523.79 876.31 2,647.47 6,115.67 633.78 19 3,523.79 611.57 2,912.22 3,203.44 576.17 20 3,523.79 320.34 3,203.44 0.00 523.79 $70,475.77 $40,475.77 $30,000.00 $30,000.00

b. Payment = $7,047.58 INPUT DATA: KEY OUTPUT: Loan amount $60,000 Payment $7,047.58 Interest rate 10.00% Number of years 20

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46

MODEL-GENERATED DATA: Amortization schedule: Principal Remaining PV of Year Payment Interest Repayment Balance Payments 1 $ 7,047.58 $ 6,000.00 $ 1,047.58 $58,952.42 $ 6,406.89 2 7,047.58 5,895.24 1,152.34 57,800.09 5,824.44 3 7,047.58 5,780.01 1,267.57 56,532.52 5,294.95 4 7,047.58 5,653.25 1,394.33 55,138.19 4,813.59 5 7,047.58 5,513.82 1,533.76 53,604.43 4,375.99 6 7,047.58 5,360.44 1,687.13 51,917.30 3,978.17 7 7,047.58 5,191.73 1,855.85 50,061.45 3,616.52 8 7,047.58 5,006.15 2,041.43 48,020.02 3,287.75 9 7,047.58 4,802.00 2,245.58 45,774.45 2,988.86 10 7,047.58 4,577.44 2,470.13 43,304.31 2,717.15 11 7,047.58 4,330.43 2,717.15 40,587.17 2,470.13 12 7,047.58 4,058.72 2,988.86 37,598.31 2,245.58 13 7,047.58 3,759.83 3,287.75 34,310.56 2,041.43 14 7,047.58 3,431.06 3,616.52 30,694.04 1,855.85 15 7,047.58 3,069.40 3,978.17 26,715.86 1,687.13 16 7,047.58 2,671.59 4,375.99 22,339.87 1,533.76 17 7,047.58 2,233.99 4,813.59 17,526.28 1,394.33 18 7,047.58 1,752.63 5,294.95 12,231.33 1,267.57 19 7,047.58 1,223.13 5,824.44 6,406.89 1,152.34 20 7,047.58 640.69 6,406.89 0.00 1,047.58 $140,951.55 $80,951.55 $60,000.00 $60,000.00

c. Payment = $12,321.39. INPUT DATA: KEY OUTPUT: Loan amount $60,000 Payment $12,321.39 Interest rate 20.00% Number of years 20 MODEL-GENERATED DATA: Amortization schedule: Principal Remaining PV of Year Payment Interest Repayment Balance Payments 1 $ 12,321.39 $ 12,000.00 $ 321.39 $59,678.61 $10,267.83 2 12,321.39 11,935.72 385.67 59,292.94 8,556.52 3 12,321.39 11,858.59 462.80 58,830.13 7,130.44 4 12,321.39 11,766.03 555.37 58,274.77 5,942.03 5 12,321.39 11,654.95 666.44 57,608.33 4,951.69 6 12,321.39 11,521.67 799.73 56,808.60 4,126.41 7 12,321.39 11,361.72 959.67 55,848.93 3,438.67 8 12,321.39 11,169.79 1,151.61 54,697.33 2,865.56 9 12,321.39 10,939.47 1,381.93 53,315.40 2,387.97 10 12,321.39 10,663.08 1,658.31 51,657.09 1,989.97 11 12,321.39 10,331.42 1,989.97 49,667.12 1,658.31 12 12,321.39 9,933.42 2,387.97 47,279.15 1,381.93

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47

13 12,321.39 9,455.83 2,865.56 44,413.59 1,151.61 14 12,321.39 8,882.72 3,438.67 40,974.91 959.67 15 12,321.39 8,194.98 4,126.41 36,848.50 799.73 16 12,321.39 7,369.70 4,951.69 31,896.81 666.44 17 12,321.39 6,379.36 5,942.03 25,954.78 555.37 18 12,321.39 5,190.96 7,130.44 18,824.35 462.80 19 12,321.39 3,764.87 8,556.52 10,267.83 385.67 20 12,321.39 2,053.57 10,267.83 0.00 321.39 $246,427.84 $186,427.84 $60,000.00 $60,000.00

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ETHICAL DILEMMA

IT’S ALL CHINESE TO ME! Ethical dilemma: Terry Zupita must decide whether she should invest in Universal Autos, which is an American firm that is partnering with the Chinese government to manufacture U.S. automobiles in China. If she makes the investment, it appears that Ms. Zupita’s money will grow substantially during the next few years. However, friends and relatives have told her that UA might use child labor in its Chinese plants, and that employees often are abused in such plants. Should Terry purchase UA? If she does, she could be supporting exploitation of children and mistreatment of employees. Discussion questions: ● Is there an ethical problem? If so, what is it? Terry has no proof that child labor is used in Chinese manufacturing plants or workers are

abused. As a result, there is a question as to whether an ethical dilemma exists. If she is bothered by the information provided by her friends and relatives, Terry should investigate further to determine the validity of the comments she has heard about the treatment of Chinese employees in the UA plant in Shanghai.

● What are the implications if Terry invests in UA? If the information about child labor and employee abuse are correct and Terry invests in UA,

she would be supporting a company that is willing to violate the rules of ethics that we aspire to in the United States. In addition, her friends might consider her investment as a signal that she condones such unethical behavior. On the other hand, if Terry takes the information provided by her friends and relatives at face value and decides not to invest in UA, then she might be losing a return that certainly will be higher than can be earned by investing in Treasury bonds.

● Should Terry invest in UA?

You might get some interesting responses to this question. Some students might say that it would be immoral to invest in UA, given the rumors about its manufacturing plant in China. Others might say that investors should purchase the financial asset that is expected to generate the higher return, and that they should not get involved with the micromanagement of the firm. Using the information provided in the Ethical Dilemma, if students compute the expected annual returns on the UA investment and the T-bond, they will find that both

Chapter 4

49

investments are expected to provide a return equal to approximately 9 percent. Based on this information, the answer should be clear—that is, Terry should invest in the T-bond (assuming she is risk-averse), because it provides the same return for lower risk than UA. Of course, this situation generally does not exist because T-bonds normally earn a return that is lower than returns on risky securities. But this case is designed to see whether students look through the “smoke” to find the numbers.

References:

The following articles might be assigned for background material: Yochi J. Dreazen, “U.S. Investigates Firm Building Embassy in Iraq,” The Wall Street Journal, June 7, 2007, A1+.

Betsy Atkins, “Is Corporate Social Responsibility Responsible?,” November 28, 2006,

Forbes.com. Ruth David, “Indian Law Does Little for Littlest Laborers,” October 10, 2006, Forbes.com.

Yochi J. Dreazen, “Probe Targets Ex-Navy Official for Link to Disgraced Contractor,” The Wall Street Journal, May 5, 2006, A1+. Deborah Orr, “Slave Chocolate?,” April 24, 2006, Forbes.com. Yochi J. Dreazen, “Contractor Admits Bribing a U.S. Official in Iraq,” The Wall Street Journal, April 19, 2006, B1+. Laura Santini, “Lone Star Offers Korea a Donation as Probe Grows,” The Wall Street Journal, April 18, 2006, C1+.