rsm332 chapter 5 solutions
DESCRIPTION
Textbook solutionsTRANSCRIPT
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
28. Section: 5.4 Annuities and Perpetuities
Learning Objective: 5.4
Level of difficulty: Difficult
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
29. Section: 5.4 Annuities and Perpetuities
Learning Objective: 5.4
Level of difficulty: Difficult
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
Level of difficulty: Difficult
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 $57,257.
3
38. Section: 5.5 Nominal versus Effective Rates
Learning Objective: 5.5
Level of difficulty: Easy
Solution:
a. 000,53$)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
39. Section: 5.5 Nominal versus Effective Rates
Learning Objective: 5.5
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.
40. Section: 5.4 Annuities and Perpetuities
Learning Objective: 5.4
Level of difficulty: Easy
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 $16.67.
41. Section: 5.4 Annuities and Perpetuities
Learning Objective: 5.4
4
Level of difficulty: Easy
Solution:
Because the fees are paid at the start of the year, this is an annuity due.
47.303,21$)06.01(06.0
)06.01(
11
800,5$4
0
PV
42. Section: 5.5 Nominal versus Effective Rates
Learning Objective: 5.5
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
Level of difficulty: Difficult
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
Learning Objective: 5.4, 5.5
Level of difficulty: Difficult
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.
45. Section: 5.4 Annuities and Perpetuities
Learning Objective: 5.4
6
Level of difficulty: Easy
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
46. Section: 5.4 Annuities and Perpetuities
Learning Objective: 5.4
Level of difficulty: Medium
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
47. Section: 5.4 Annuities and Perpetuities
Learning Objective: 5.4
Level of difficulty: Medium
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
Learning Objective: 5.3, 5.4
7
Level of difficulty: Difficult
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.
Or using a financial calculator (TI BAII Plus),
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
Learning Objective: 5.3, 5.5
Level of difficulty: Difficult
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.
51. Section: 5.4 Annuities and Perpetuities
Learning Objectives: 5.4
Level of difficulty: Difficult
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
Learning Objective: 5.5, 5.6
Level of difficulty: Difficult
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
Level of difficulty: Difficult
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
54. Section: 5.5 Nominal versus Effective Rates
Learning Objective: 5.4, 5.5
Level of difficulty: Medium
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.
56. Section: 5.5 Nominal versus Effective Rates
Learning Objective: 5.5
Level of difficulty: Medium
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
Learning Objective: 5.6
Level of difficulty: Medium
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
Learning Objective: 5.6
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
Learning Objective: 5.7
Level of difficulty: Medium
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.
61. Section: 5.7 Comprehensive Examples
Learning Objective: 5.7
Level of difficulty: Easy
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.
62. Section: 5.7 Comprehensive Examples
Learning Objective: 5.7
Level of difficulty: Easy
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
Or using a financial calculator (TI BAII Plus),
Hit [2nd
] [BGN] [2nd
] [Set]
N=35, I/Y=12.6, PV=0, PMT= -9,000, CPT FV=5,039,384
63. Section: 5.7 Comprehensive Examples
Learning Objective: 5.7
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
Or using a financial calculator (TI BAII Plus),
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
Or using a financial calculator (TI BAII Plus),
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
Or using a financial calculator (TI BAII Plus),
N=120, I/Y=.9759, PV=-200,000, FV=0, CPT PMT=2,836
64. Section: 5.7 Comprehensive Examples
Learning Objective: 5.7
Level of difficulty: Difficult
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
Or using a financial calculator (TI BAII Plus),
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
Or using a financial calculator (TI BAII Plus),
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
Level of difficulty: Medium
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
69. Section: Appendix 5A: Growing Annuities and Perpetuities
Learning Objective: Appendix
Level of difficulty: Medium
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.
70. Section: Appendix 5A: Growing Annuities and Perpetuities
Learning Objective: Appendix
Level of difficulty: Medium
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.
71. Section: Appendix 5A: Growing Annuities and Perpetuities
Learning Objective: Appendix
Level of difficulty: Difficult
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.