rsm332 chapter 5 solutions

1

Chapter 5: Time Value of Money

Practice Problems

22. Section: 5.2 Simple Interest and 5.3 Compound Interest

Learning outcome: 5.2 and 5.3

Level of difficulty: Easy

Solution:

A. Value = P + (n x P x k) = $24 + (387 x $24 x 5%) = $488

B. 497,010,806,3$)05.01(24$ 387

387 yearsFV

27. Section: 5.4 Annuities and Perpetuities

Learning Objective: 5.4

Level of difficulty: Difficult

Solution:

The dividends for the first five years form an ordinary annuity. Starting in year 6, the reduced

dividend stream can be thought of as a perpetuity. However, the value of this perpetuity, as

determined by our formula, occurs at year 5 (one year before the first $2 dividend), and must be

discounted to the present:

27.20$5674.067.16$81.10$)12.01(

1

12.0

00.2$

12.0

)12.01(

11

00.3$5

5

0

PV




Solution:

%3125.012

0375.0monthlyk

Rent payments are typically made at the start of each month (so this is an annuity due). Over

three years, we would expect 36 monthly rent payments. However, the last month’s rent must be

paid up front, so the annuity includes only 35 payments; the present value of the last month’s rent

is $450 because it will be paid today.

77.393,15$450$)003125.01(003125.0

)003125.01(

11

450$35

0

PV

2




Solution:

It is tempting to view the first option as a perpetuity, but this would be incorrect as the man will

die at some time, and the payment will then cease. Thus, option one is an ordinary annuity, with

an uncertain number of payments. Option two is much easier to value; it includes exactly 240

monthly payments.

%5.012

06.0monthlyk

Using a financial calculator (TI BAII Plus),

N = 240, PMT = 3,500, I/Y = 0.5, FV=0, CPT PV = –488,532.70

For the first option to be a better deal, it must include enough payments so that its present value

is at least as great as for option two. Again using the calculator,

PV = –488,532.70, PMT = 2,785, I/Y = 0.5, CPT N = 420.29

So option one must continue for over 420 monthly payments to equal the value of option two.

This is just over 35 years. Hence, the man must live to be at least 100 years old for option one to

be a better deal.

35. Section: 5.1 Opportunity Cost, 5.2 Simple Interest, 5.3 Compound Interest, and 5.4 Annuities

and Perpetuities

Learning outcome: 5.1, 5.2, 5.3, 5.4


Solution:

The manager is confused. To make the choice between the two options you should consider the

present value of each set of payments, not the sum of the payments. Summing the payments

assumes that the opportunity cost is zero.

For example, if your opportunity cost is 10%, then the PV of Long is $161,009. The value of the

house if $250,000 but the cost of the loan (to you) is only $161,009 – a net benefit of $88,991.

The PV of the Short option is $216,289 – in this case, with an opportunity cost of 10%, the short

option costs me $55,280 more.

If instead, your opportunity cost is 1%, then the PV of the Long option is $390,647 while the PV

of the Short option is only $333,390. By taking the Short option, you will save $57,257.

3

38. Section: 5.5 Nominal versus Effective Rates



Solution:

a. 000,53$)06.1(000,50$)1(%6 01 kPVFVRateQuotedk year

b. 89.083,53$)0616778.1(000,50$%16778.6112

1 1

12

yearFV

QRk

c. 57.091,53$)0618313.1(000,50$%18313.61365

1 1

365

yearFV

QRk



Level of difficulty: Medium

Solution:

Step 1: determine monthly effective rate

= 0.66227%

Step 2: given the monthly effective rate, determine the quoted rate compounded monthly.

QR monthly = 12 x 0.66227

= 7.94724%

Therefore, 8% compounded quarterly is equivalent to 7.94724% compounded monthly.




Solution:

The value of any perpetual stream of payments can be valued as a perpetuity:

67.16$12.0

2$0

k

PMTPV

Each share is worth $16.67.



4


Solution:

Because the fees are paid at the start of the year, this is an annuity due.

47.303,21$)06.01(06.0

)06.01(

11

800,5$4

0

PV



Level of difficulty: Medium.

Solution:

a. m = 365: %.11.271)365

24.1( 365 k

b. m = 4: %.25.261)4

24.1( 4 k

c. m = 3: %.97.251)3

24.1( 3 k

d. m = 2: %.44.251)2

24.1( 2 k

e. Continuous compounding: %.12.27124. ek

f. The effective monthly rate for a. to d. is:

a. m=365, f=12 1)1( f

m

m

QRk = 1)

365

24.1( 12

365

=2.02%

b. m=4, f=12. 1)1( f

m

m

QRk = 1)

4

24.1( 12

4

=1.96%

c. m=3, f=12. 1)1( f

m

m

QRk = 1)

3

24.1( 12

3

=1.94%

d. m=2, f=12. 1)1( f

m

m

QRk = 1)

2

24.1( 12

2

=1.91%

43. Section: 5.4 Annuities and Perpetuities and 5.5 Nominal versus Effective Rates

Learning Objective: 5.4, 5.5


Solution:

5

Step 1: make the payment frequency match the compounding frequency. We need to convert the

6 percent compounded monthly to a quarterly effective rate.

12

14

112 4

12 4

.061 1

12

1 1

.061

12

.061

12

1.5075%

annual

quarterly annual

quarterly

k

k k

k

Step 2: Now we have an annuity of 5*4 = 20 quarterly payments, a present value of $50,000, and

an effective quarterly rate of 1.5075%. Solving for the payments we get $2,914.44.

44. Section: 5.4 Annuities and Perpetuities and 5.5 Nominal versus Effective Rates



Solution:

Step 1: make the payment frequency match the compounding frequency. We need to convert the

6% compounded quarterly to a monthly effective rate.

4

112

14 12

4 12

.061 1

4

1 1

.061

4

.061

4

0.4975%

annual

monthly annual

monthly

k

k k

k

Step 2: Now we have an annuity of 5*12 = 60 monthly payments, a present value of $150,000

and an effective monthly rate of 0.4975%. Solving for the payments we get $2,897.83.



6


Solution:

The future value amount is $40,000. The amount to be saved each year is really the payment on

an ordinary annuity:

71.898,3$07.0

1)07.01(000,40$

8

PMTPMT

Or using a financial calculator (TI BAII Plus),

N=8, I/Y=7, PV=0, FV= -40,000, CPT PMT= 3,898.71




Solution:

A. The future value of Jane’s account will be:

88.212,28$06.0

1)06.01(000,1$

17

17

FV

B. The grant has the effect of increasing the amount saved from $1,000 to $1,200. The future

value of the account will now be:

46.855,33$06.0

1)06.01(200,1$

17

17

FV




Solution:

Find the present value of the four-year annuity due:

26.497,14$)07.01(07.0

)07.01(

11

000,4$)1()1(

11

4

0

kk

kPMTPV

n

Now, discount this amount back three years:

08.834,11$)07.1(

126.497,14$

)1(

1330

kFVPV

49. Section: 5.3 Compound Interest


7


Solution:

a. We know the future value and present value amounts, as well as the monthly interest rate.

Finding the number of time periods (months) is most easily done with a financial calculator (TI

BAII Plus),

PV = 15,000, FV = -20,000, I/Y = 0.5, CPT N = 57.68

It will take nearly 58 months, or close to 5 years before Roger can afford to buy the car.

b. Solving the following equation for “n” we get:

005.

1)005.1(250$

)005.1(

000,15$000,20$

n

n n= 14.86.


I/Y=0.5, PV=15,000, FV= -20,000, PMT = 250, CPT N = 14.86

50. Section: 5.3 Compound Interest and 5.5 Nominal versus Effective Rates



Solution:

Let’s assume the present value of the investment is $1. The future value, after doubling, is then

$2.

a. Annually: With annual compounding, the effective rate is the same as the quoted rate, 9%.

Using a financial calculator (TI BAII Plus),

PV = –1, FV = 2, I/Y = 9, CPT N = 8.04

So the investment will double in just over 8 years.

b. Quarterly: With quarterly compounding, the effective annual rate is,

%3083.91)4

09.01( 4 k , and a financial calculator allows us to find:

PV = -1, FV = 2, I/Y = 9.3083, CPT N = 7.79

The higher effective rate means that only 7.79 years are needed to double the value of the

investment.


Learning Objectives: 5.4


Solution:

a. The present value of the annual payments can be found with a financial calculator, (TI BAII

Plus), N=9, PMT = -6,000, I/Y = 5.0, FV=0, CPT PV = 42,646.93

As this is less than $50,000, the immediate payment alternative is better.

8

b. This problem can be solved by trial and error, but the task is much easier with a financial

calculator, (TI BAII Plus), N=9, PMT = –6,000, PV = 50,000, FV=0, CPT I/Y = 1.5675%. At an

interest rate below 1.5675% per year, the nine-year annuity would be preferable; above the rate

the immediate payment is better.

52. Section: 5.5 Nominal versus Effective Rates and 5.6 Loan or Mortgage Arrangements



Solution:

a. First, find the effective interest corresponding to the frequency of Jimmie’s car payments

(f =12); with monthly compounding, set m=12,

%70833.0112

%5.8111

1212

fm

monthlym

QRk

The 60 car payments form an “annuity” whose present value is the amount of the loan (the price

of the car):

98.594$0070833.0

)0070833.01(

11

000,29$60

PMTPMT

b. Use the effective monthly interest rate from part A, k=0.70833%

Period (1) Principal

Outstanding

(2)

Payment

(3) Interest

=k*(1)

(4) Principal

Repayment = (2)-

(3)

Ending Principal

= (1)-(4)

1 29,000.00 594.98 205.42 389.56 28,610.44

2 28,610.44 594.98 202.66 392.32 28,218.12

3 28,218.12 594.98 199.88 395.10 27,823.01

4 27,823.01 594.98 197.08 397.90 27,425.11

5 27,425.11 594.98 194.26 400.72 27,024.40

6 27,024.40 594.98 191.42 403.56 26,620.84

7 26,620.84 594.98 188.56 406.42 26,214.42

8 26,214.42 594.98 185.69 409.29 25,805.13

9 25,805.13 594.98 182.79 412.19 25,392.94

10 25,392.94 594.98 179.87 415.11 24,977.82

11 24,977.82 594.98 176.93 418.05 24,559.77

12 24,559.77 594.98 173.97 421.01 24,138.76

13 24,138.76 594.98 170.98 424.00 23,714.76

...

35 14,083.18 594.98 99.76 495.22 13,587.95

9

36 13,587.95 594.98 96.25 498.73 13,089.22

37 13,089.22 594.98 92.72 502.26 12,586.96

...

59 1,177.43 594.98 8.34 586.64 590.79

60 590.79 594.98 4.18 590.79 0.00

The first monthly payment repays $389.56 of the principal amount of the loan and the last

payment repays $590.79.

c. After three years, or 36 monthly payments, the principal outstanding is $13,089.22 (from the

amortization table).The present value of this amount is:

19.152,10$)0070833.01(

122.089,13$

360

PV

53. Section: 5.5 Nominal versus Effective Rates and 5.6 Loan or Mortgage Arrangements

Learning Objective: 5.5 and 5.6


Solution:

The 60 monthly payments form an annuity whose present value is $29,000. Finding the interest

rate is most easily done with a financial calculator (TI BAII Plus):

N=60, PMT=588.02, PV= -29,000, CPT I/Y = 0.6667%

Note that we used N=60 months, so the solution is a monthly interest rate, however, the problem

asks for the effective annual rate.

%30.81)006667.01(1)1( 1212 monthlykk

The quoted rate would be:

%00.8]1)0830.01[(12]1)1[( 121

121

kmQR

Or simply: %00.8006667.012 monthlykmQR




Solution:

Solve the annuity equation to find k, the interest rate:

10

?)1(

11

24.935,6$00.000,25$5

kk

k

The calculations are most easily done with a financial calculator (TI BAII Plus),

PV = -25,000, PMT=6,935.24, N= 5, FV=0, CPT I/Y = 12%

The effective annual interest rate is 12 percent. With annual compounding, the nominal rate (or

quoted rate) will also be 12 percent per year.




Solution:

a. Scott will pay interest of ($800–$750) = $50 after one week. This implies a nominal interest

rate of $50/$750 = 6.67% per week. With 52 weeks in the year, the nominal rate per year is then

52 x 6.67% = 346.84%.

b. The effective annual interest rate is %10.772,27210.271)0667.01( 52 k

57. Section: 5.6 Loan or Mortgage Arrangements



Solution:

a. In Canada, fixed-rate mortgages use semi-annual compounding of interest, so m=2. The

effective annual rate is therefore:

%5024.612

064.0111

2

m

m

QRk

b. With monthly payments, f=12. We can find the effective monthly interest rate from the

effective annual rate, k:

%5264.01%5024.6111 1211

fmonthly kk

c. The amortization period is 20 years, or 20 x 12 = 240 months. Josephine’s monthly payments

can be computed as:

69.322,1$005264.0

)005264.01(

11

000,180$240

PMTPMT

d. With monthly compounding and payments, the effective monthly interest rate is:

11

%530.0112

0636.0111

1212

fm

monthlym

QRk

Even though the quoted rate is lower at the Credit Union than at the Bank, the effective rate is

higher. Josephine should take the mortgage loan from Providence Bank in this case. The monthly

payment for the credit union mortgage would be $1,327.24, which, as expected, is higher than

that at Providence Bank.

59. Section: 5.6 Loan or Mortgage Arrangements


Level of Difficulty: Difficult

Solution:

Part 1: determine the principal outstanding after the 60th

payment (i.e., How much will the next

mortgage be for?)

Step 1: determine effective monthly rate:

Step 2: determine the monthly payments:

300

11

(1 0.00493862)$300,000

0.00493862

$1,919.4194

PMT

PMT

Step 3: determine Present Value of remaining (300 – 60) payments of $1,919.4194

300 60

11

(1 0.00493862)$1,919.4194

0.00493862

$269,510.0994

PV

PV

Part 2: determine new payments

Step 1: determine new effective monthly rate 1

2 12.08

1 1 0.006558202

monthlyk

Step 2: determine the new monthly payment

12 12

.061 1 0.00493862

2monthlyk

12

300 60

11

(1 0.00655820)$269,510.0994

0.00655820

$2,232.507688

PMT

PV

Franklin’s new payment is $2,232.51, an increase of $313.09.

60. Section: 5.7 Comprehensive Examples



Solution:

a. Timmy’s savings extend right to age 61 (end of each year), so this is an ordinary annuity.

67.777,327,1$10.0

1)10.01(000,3$

40

40

FV

Yes, Timmy will achieve his goal by a comfortable margin.

b. In the equation for part A set FV = $1,000,000, and solve for the number of years, n. This is

easiest done with a financial calculator (TI BAII Plus),

FV = –1,000,000, I/Y = 10, PMT = 3,000, CPT N = 37.1.

Timmy will hit the $1 million dollar mark in just over 37 years, or shortly after his 58th

birthday.




Solution:

This is an ordinary annuity.

45.174,953$10.0

1)10.01(000,30$

15

15

FV

No, Tommy will not quite achieve his goal before retirement.




Solution:

Annual investment = Annual income – Annual expenditure = $45,000 – $36,000 = $9,000.

This is an annuity due.

13

)1(1)1(

kk

kPMTFV

n

n

384,039,5$)126.1)(2749.497)(000,9()126.1(126.

1)126.1(000,9$

35


Hit [2nd

] [BGN] [2nd

] [Set]

N=35, I/Y=12.6, PV=0, PMT= -9,000, CPT FV=5,039,384



Level of Difficulty: Difficult

Solution:

a. PV=$200,000, monthly rate=12%/12=1%, N = (10)(12)=120 months

01.

)01.1(

11

000,200120

PMT

01.

)01.1(

11

/000,200120

PMT

So, PMT=$2,869


N=120, I/Y=1, PV=-200,000, FV=0, CPT PMT=2,869

b. Remaining months to pay=120 – 18=102 months

01.

)01.1(

11

869,2102

0PV =$182,920


N=102, I/Y=1, PMT=- 2,869, FV=0, CPT PV=182,920

14

c. kmonthly= 1)2

12.1( 12

2

=.9759%

009759.

)009759.1(

11

000,200120

PMT

009759.

)009759.1(

11

/000,200120

PMT

So, PMT=$2,836


N=120, I/Y=.9759, PV=-200,000, FV=0, CPT PMT=2,836




Solution:

Investor A:

k=e.15

– 1=16.183424%.

1st, consider an ordinary annuity and the present value of the investment when A turns 25 years

old is:

16183424.

)16183424.1(

11

500,5$8

25PV =$23,749.19


N=8, I/Y=16.183424, PMT=5,500, FV=0, CPT PV=- 23,749.19

2nd

, discount this amount for five years back to today when she is 20.

PV=FV/(1+k)5=$23,749.19/(1.16183424)

5=$11,218.3231

Or, N=5, I/Y=16.183424, PMT=0, FV=- 23,749.19, CPT PV=11,218.3231

Investor B:

k= 1)4

16.1( 4 =16.985856%

15

)16985856.1(16985856.

)16985856.1(

11

3231.218,11$10

PMT

PMT=$2,057.38


Hit [2nd

] [BGN] [2nd

] [Set]

N=10, I/Y=16.985856, PV=11,218.3231, FV=0, CPT PMT= - 2,057.38

Therefore, Investor B has to make a yearly payment of $2,057.38 so that the present value of the

two investments is the same.

68. Section: Appendix 5A: Growing Annuities and Perpetuities

Learning Objective: Appendix


Solution:

= $1,816.67

The most I’d pay is the present value of the investment. In this case the cash flows start

immediately ($100) and then grow by 3% per year. The present value, or the maximum I’d be

willing to pay, is $1,816.67




Solution:

To solve this we need to realize that the present value of a perpetuity (growing or otherwise)

occurs one period prior to the first cash flow. Hence, using the growing perpetuity formula will

give us the value of the cash flows in year 4. We need to discount those back to time 0.

16

= $1,180.71

The most I’d be willing to pay for this investment is $1,180.71.




Solution:

Present value of Grow: 100

$10,000.05 .04

GROWPV

Present value of Shrink:1000

$14,285.71.05 ( .02)

SHRINKPV

Grow exceeds the cost by $9,000 while Shrink exceeds the investment cost by $13,285.71 Shrink

is preferred, as it exceeds the investment cost by the most.




Solution:

= PMT1 *1.8946799

17

.02

The initial deposit is $10,555.87

If Xiang made constant deposits (i.e., no growth), he would have to deposit $15,051.44 per year

for the next 30 years.

rsm332 chapter 5 solutions

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