Practice Problem Solutions – Chapter 6 (Part 2)
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps.
Due to space and readability constraints, when these intermediate steps are included in this solutions
manual, rounding may appear to have occurred. However, the final answer for each problem is found
without rounding during any step in the problem.
4.
To find the present value of annuity (PVA), we use the equation:
PVA = C({1 – [1/(1 + r) t]}/r)
0
PV
1
$4,350
15
$4,350
$4,350
$4,350
…
$4,350
$4,350
$4,350
$4,350
$4,350
PVA = $4,350{[1 – (1/1.06)15]/.06} = $42,248.28
PVA@15 yrs:
Calculator: Clear, 15 N, 6 I/Y, 4350 PMT, (CPT) PV
0
1
PV
$4,350
40
$4,350
$4,350
$4,350
…
$4,350
$4,350
$4,350
$4,350
$4,350
PVA = $4,350{[1 – (1/1.06)40]/.06} = $65,451.39
PVA@40 yrs:
0
1
PV
$4,350
75
$4,350
$4,350
$4,350
…
$4,350
$4,350
$4,350
$4,350
$4,350
PVA = $4,350{[1 – (1/1.06)75]/.06} = $71,582.94
PVA@75 yrs:
To find the PV of a perpetuity, we use the equation:
PV = C/r
0
1
…
∞
PV
$4,350
$4,350
$4,350
$4,350
$4,350
$4,350
$4,350
$4,350
$4,350
PV = $4,350/.06 = $72,500.00
Notice that as the length of the annuity payments increases, the present value of the annuity
approaches the present value of the perpetuity. The present value of the 75‐year annuity and the
present value of the perpetuity imply that the value today of all perpetuity payments beyond 75
years is only $917.06.
6.
The time line is:
0
1
2
3
4
5
6
7
$47,000
C
C
C
C
C
C
C
To find the present value of annuity (PVA), we use the equation:
PVA = C({1 – [1/(1 + r) t]}/r)
PVA = $47,000{[1 – (1/1.0717)]/.071}
PVA = $252,415.91
Calculator: Clear, 7 N, 47,000 PMT, 7.1 I/Y, (CPT) PV
7.
Here we need to find the future value of annuity (FVA). The equation to find the FVA is:
FVA = C{[(1 + r)t – 1]/r}
0
1
$4,500
20
$4,500
$4,500
$4,500
…
FVA for 20 years = $4,500[(1.09720 – 1)/.097]
FVA for 20 years = $249,119.03
Calculator: Clear, 20 N, 9.7 I/Y, 4500 PMT, (CPT) FV
$4,500
$4,500
$4,500
$4,500 $4,500
0
1
$4,500
40
$4,500
$4,500
…
$4,500
$4,500
$4,500
$4,500
$4,500
$4,500
FVA for 40 years = $4,500[(1.09740 – 1)/.097]
FVA for 40 years = $1,835,982.10
Notice that because of exponential growth, doubling the number of periods does not merely double
the FVA.
9.
The time line is:
0
1
2
3
4
5
$75,000
C
C
C
C
C
Here we have the present value of annuity (PVA), the length of the annuity, and the interest rate. We
want to calculate the annuity payment. Using the PVA equation:
PVA = C({1 – [1/(1 + r)t]}/r)
$75,000 = C{[1 – (1/1.0685)]/.068}
We can now solve this equation for the annuity payment. Doing so, we get:
C = $75,000/4.1222
C = $18,193.96
Calculator: Clear, 5 N, 6.8 I/Y, 75,000 PV, (CPT) PMT
14. As a borrower, you want the loan with the lowest interest rate. Since the quoted rates have
different compounding periods, we need to first calculate effective rates. To find the EAR, we use the
equation:
EAR = [1 + (APR/m)]m – 1
So, for each bank, the EAR is:
First National: EAR = [1 + (.1310/12)]12 – 1 = .1392, or 13.92%
EAR = [1 + (.1340/2)]2 – 1 = .1385, or 13.85%
First United:
Since First United has the lower effective rate, you would go to it for a loan.
Notice that the higher APR does not necessarily result in the higher EAR. The number of compounding
periods within a year will also affect the EAR.
19. The APR is the interest rate per period times the number of periods in a year. In this case, the interest
rate is 27 percent per month, and there are 12 months in a year, so we get:
APR = 12(27%) = 324%
To find the EAR, we use the EAR formula:
EAR = [1 + (APR/m)]m – 1
EAR = (1 + .27)12 – 1
EAR = 1,660.53%
Notice that we didn’t need to divide the APR by the number of compounding periods per year. We
do this division to get the interest rate per period, but in this problem we are already given the
interest rate per period.
21. The time line is:
0
–$18,000
1
$450
?
$450
$450
$450
…
$450
$450
$450
$450
$450
Here we need to find the length of an annuity. We know the interest rate, the present value of
annuity (PVA), and the payments. Using the PVA equation:
PVA = C({1 – [1/(1 + r)t] }/r)
$18,000 = $450{[1 – (1/1.013)t]/.013}
Now we solve for t:
1/1.013t = 1 – ($18,000/$450)(.013)
1/1.013t = .48
1.013t = 1/.48 = 2.083
t = ln 2.083/ln 1.013
t = 56.83 months
Calculator: Clear, ‐18,000 PV, 450 PMT, 1.3 I/Y, (CPT) N
24. The time line is:
0
1
$475
360
$475
$475
$475
…
$475
$475
$475
$475
$475
This problem requires us to find the future value of annuity (FVA). The equation to find the FVA is:
FVA = C{[(1 + r)t – 1]/r}
FVA = $475[{[1 + (.10/12)]360 – 1}/(.10/12)]
FVA = $1,073,731.76
Monthly interest rate = 10% / 12 = 8.33%
Calculator: Clear, 475 PMT, 360 N, 8.33 I/Y, (CPT) FV
26. The time line is:
0
1
PV
$2,500
16
$2,500
$2,500
$2,500
…
$2,500
$2,500
$2,500
$2,500
$2,500
The cash flows are an annuity with four payments per year for four years, or 16 payments. We can
use the present value of annuity (PVA) equation:
PVA = C({1 – [1/(1 + r)t]}/r)
PVA = $2,500{[1 – (1/1.0057)16]/.0057}
PVA = $38,126.53
Calculator: Clear, 16 N, 0.57 I/Y, 2500 PMT, (CPT) PV
27. The time line is:
0
1
2
3
4
PV
$815
$990
$0
$1,520
The cash flows are annual and the compounding period is quarterly, so we need to calculate the EAR
to make the interest rate comparable with the timing of the cash flows. Using the equation for the
EAR, we get:
EAR = [1 + (APR/m)]m – 1
EAR = [1 + (.09/4)]4 – 1
EAR = .0931, or 9.31%
And now we use the EAR to find the PV of each cash flow as a lump sum and add them together:
PV = $815/1.0931 + $990/1.09312 + $1,520/1.09314
PV = $2,638.87
47. The time line is:
0
1
2
3
4
5
PV
6
$4,400
20
$4,400
…
$4,400
$4,400
We want to find the value of the cash flows today, so we will find the PV of the annuity, and then
bring the lump sum PV back to today. The annuity has 15 payments, so the PV of the annuity is:
PVA = $4,400{[1 – (1/1.08315)]/.083}
PVA = $36,981.52
Calculator: Clear, 4400 PMT, 8.3 I/Y, 15 N, (CPT) PV
Since this is an ordinary annuity equation, this is the PV one period before the first payment, so this
is the PV at t = 5. To find the value today, we find the PV of this lump sum. The value today is:
PV = $36,981.52/1.0835
PV = $24,822.33
Calculator: Clear, 8.3 I/Y, 5 N, 36981.52 FV, (CPT) PV