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14. The time line is:
0
1
…
∞
PV
$18,000
$18,000
$18,000
$18,000
$18,000
$18,000
$18,000
$18,000
$18,000
This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:
PV = C/r
PV = $18,000/.043
EAR = [1 + (APR/m)]m – 1
We can now solve for the APR. Doing so, we get:
APR = m[(1 + EAR)1/m – 1]
EAR = .104 = [1 + (APR/2)]2 – 1 APR = 2[(1.104)1/2 – 1] = .1014, or 10.14%
APR = .1432, or 14.32%
17. For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR/m)]m – 1
periods within a year will also affect the EAR.
18. The cost of a case of wine is 10 percent less than the cost of 12 individual bottles, so the cost of a case
will be:
Cost of case = (12)($10)(1 – .10)
Cost of case = $108
PVA = (1 + r) C({1 – [1/(1 + r)t] }/r)
$108 = (1 + r) $10({1 – [1/(1 + r)12]/r)
Solving for the interest rate, we get:
EAR = (1 + .0198)52 – 1
EAR = 1.7668, or 176.68%
The analysis appears to be correct. He really can earn about 177 percent buying wine by the case. The
only question left is this: Can you really find a fine bottle of Bordeaux for $10?
19. The time line is:
0
1
…
?
–$18,700
$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 PV, and the payments.
Using the PVA equation:
PVA = C({1 – [1/(1 + r)t]}/r)
$18,700 = $450{ [1 – (1/1.013)t ]/.013}
Now, we solve for t:
1/1.013t = 1 – [($18,700)(.013)/($450)]
1.013t = 1/(.4598) = 2.175
t = ln 2.175/ln 1.013
t = 60.16 months
20. The time line is:
0
1
$3
$4
Here, we are trying to find the interest rate when we know the PV and FV. Using the FV equation:
APR = (52)33.33% = 1,733.33%
And using the equation to find the EAR:
EAR = [1 + (APR/m)]m – 1
EAR = [1 + .3333]52 – 1 = 313,916,515.69%
Intermediate
21. To find the FV of a lump sum with discrete compounding, we use:
FV = PV(1 + r)t
a.
0
6
$1,500
FV
Bond account: FVA = $325[{[1 + (.061/12) ]360 – 1}/(.061/12)] = $332,782.27
So, the total amount saved at retirement is:
PVA = $2,220,083.01 = C[1 – {1/[1 + (.069/12)]300}/(.069/12)]
C = $2,220,083.01/142.773
C = $15,549.74 withdrawal per month
24. The time line is:
0
4
–$1
$3
Since we are looking to triple our money, the PV and FV are irrelevant as long as the FV is three times
as large as the PV. The number of periods is four, the number of quarters per year. So:
FV = $3 = $1(1 + r)(12/3)
r = .3161, or 31.61%
25. Here, we need to find the interest rate for two possible investments. Each investment is a lump sum,
so:
(1 + r)11 = $235,000/$55,000
r = (4.2727)1/11 – 1 = .1411, or 14.11%
26. This is a growing perpetuity. The present value of a growing perpetuity is:
PV = C/(r – g)
PV = $210,000/(.11 – .025)
PV = $2,470,588.24
Quarterly rate = Stated rate/4
Quarterly rate = .055/4
Quarterly rate = .0138
The time line is:
0
1
…
∞
PV
$1.75
$1.75
$1.75
$1.75
$1.75
$1.75
$1.75
$1.75
$1.75
Using the present value equation for a perpetuity, we find the value today of the dividends paid must
be:
PV = C/r
PV = $1.75/.0138
PV = $127.27
28. The time line is:
0
1
2
3
4
5
6
7
…
25
PV
$5,700
$5,700
$5,700
$5,700
$5,700
$5,700
$5,700
We can use the PVA annuity equation to answer this question. The annuity has 23 payments, not 22
payments. Since there is a payment made in Year 3, the annuity actually begins in Year 2. So, the
value of the annuity in Year 2 is:
PVA = C({1 – [1/(1 + r)t]}/r )
PVA = $5,700({1 – [1/(1 + .068)23]}/.068)
PV = $57,305.23
29. The time line is:
0
1
2
3
4
5
6
7
…
20
PV
$825
$825
$825
$825
We need to find the present value of an annuity. Using the PVA equation, and the 12 percent interest
rate, we get:
PVA = C({1 – [1/(1 + r)t]}/r )
PVA = $825({1 – [1/(1 + .12)15]}/.12)
PVA = $5,618.96
This is the value of the annuity in Year 5, one period before the first payment. Finding the value of
this amount today, we find:
PV = FV/(1 + r)t
PV = $5,618.96/(1 + .09)5
PV = $3,651.94
30. The amount borrowed is the value of the home times one minus the down payment, or:
Amount borrowed = $825,000(1 – .20)
Amount borrowed = $660,000
The time line is:
0
1
…
360
$660,000
C
C
C
C
C
C
C
C
C
The monthly payments with a balloon payment loan are calculated assuming a longer amortization
schedule, in this case, 30 years. The payments based on a 30-year repayment schedule would be:
PVA = $660,000 = C({1 – [1/(1 + .054/12)360]}/(.054/12))
C = $3,706.10
–$1,650,000
$185,000
$185,000
$185,000
$185,000
$185,000
$185,000
$185,000
$185,000
$185,000
The company should accept the project since the cost is less than the increased cash flows.
34. Since your salary grows at 3.4 percent per year, your salary next year will be:
Next year’s salary = $75,000(1 + .034)
Next year’s salary = $77,550
This means your deposit next year will be:
Next year’s deposit = $77,550(.10)
Next year’s deposit = $7,755
Since your salary grows at 3.4 percent, you deposit will also grow at 3.4 percent. We can use the
present value of a growing annuity equation to find the value of your deposits today. Doing so, we
find:
PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}
PV = $7,755{[1/(.095 – .034)] – [1/(.095 – .034)] × [(1 + .034)/(1 + .095)]35}
PV = $110,031.91
Now, we can find the future value of this lump sum in 35 years. We find:
FV = PV(1 + r)t
FV = $110,031.91(1 + .095)35
FV = $2,636,409.31
This is the value of your savings in 35 years.
