19. The time line is:
0 1
…
?
–
$18,450
$500 $500 $500 $500 $500 $500 $500 $500 $500
Here, we need to find the length of an annuity. We know the interest rate, the PV, and the payments.
Using the PVA equation:
Now, we solve for t:
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:
The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks
in a year, so:
Intermediate
21. To find the FV of a lump sum with discrete compounding, we use:
FV = PV(1 + r)t
a.
0 6
$1,0 FV
00
b.
0 12
$1,0
00
FV
c.
0 72
$1,0
00
FV
d.
0 6
$1,0
00
FV
To find the future value with continuous compounding, we use the equation:
e. The future value increases when the compounding period is shorter because interest is earned
22. The total interest paid by First Simple Bank is the interest rate per period times the number of
periods. In other words, the interest by First Simple Bank paid over 10 years will be:
First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor
of $1, or:
(1 + r)10
Setting the two equal, we get:
23. Although the stock and bond accounts have different interest rates, we can draw one time line, but
we need to remember to apply different interest rates. The time line is:
0 1
…
360 361
…
660
Stock $750 $750 $750 $750 $750 C C C
Bond $250 $250 $250 $250 $250
We need to find the annuity payment in retirement. Our retirement savings end at the same time the
retirement withdrawals begin, so the PV of the retirement withdrawals will be the FV of the
retirement savings. So, we find the FV of the stock account and the FV of the bond account and add
the two FVs.
So, the total amount saved at retirement is:
$2,103,389.80 + 251,128.76 = $2,354,518.56
Solving for the withdrawal amount in retirement using the PVA equation gives us:
24. The time line is:
0 4
–$1 $4
Since we are looking to quadruple our money, the PV and FV are irrelevant as long as the FV is four
times as large as the PV. The number of periods is four, the number of quarters per year. So:
25. Here, we need to find the interest rate for two possible investments. Each investment is a lump sum,
so:
G:
0 6
–
$75,00
0
$125,0
00
H:
0 10
–
$75,00
0
$185,0
00
26. This is a growing perpetuity. The present value of a growing perpetuity is:
It is important to recognize that when dealing with annuities or perpetuities, the present value
equation calculates the present value one period before the first payment. In this case, since the first
payment is in two years, we have calculated the present value one year from now. To find the value
today, we discount this value as a lump sum. Doing so, we find the value of the cash flow stream
today is:
27. The dividend payments are made quarterly, so we must use the quarterly interest rate. The quarterly
interest rate is:
The time line is:
0 1
…
∞
PV $2.75 $2.75 $2.75 $2.75 $2.75 $2.75 $2.75 $2.75 $2.75
Using the present value equation for a perpetuity, we find the value today of the dividends paid must
be:
28. The time line is:
0 1234567…25
PV $5,50 $5,50 $5,50 $5,50 $5,50 $5,50 $5,50
00000 00
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:
This is the value of the annuity one period before the first payment, or Year 2. So, the value of the
cash flows today is:
29. The time line is:
0 1234567…20
PV $900 $900 $900 $900
We need to find the present value of an annuity. Using the PVA equation, and the 13 percent interest
rate, we get:
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:
30. The amount borrowed is the value of the home times one minus the down payment, or:
The time line is:
0 1
…
360
$520,00
0
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:
Now, at Year 8 (Month 96), we need to find the PV of the payments which have not been made. The
time line is:
96 97
…
360
PV $2,855.
38
$2,855.
38
$2,855.
38
$2,855.
38
$2,855.
38
$2,855.
38
$2,855.
38
$2,855.
38
$2,855.
38
The balloon payment will be:
31. The time line is:
0 12
$10,8
00
FV
Here, we need to find the FV of a lump sum, with a changing interest rate. We must do this problem
in two parts. After the first six months, the balance will be:
This is the balance in six months. The FV in another six months will be:
The problem asks for the interest accrued, so, to find the interest, we subtract the beginning balance
from the FV. The interest accrued is:
32. The time line is:
0 1
…
∞
–
$3,800,00
0
$267,0
00
$267,0
00
$267,0
00
$267,0
00
$267,0
00
$267,0
00
$267,0
00
$267,0
00
$267,0
00
The company would be indifferent at the interest rate that makes the present value of the cash flows
equal to the cost today. Since the cash flows are a perpetuity, we can use the PV of a perpetuity
equation. Doing so, we find:
PV = C / r
33. The company will accept the project if the present value of the increased cash flows is greater than
the cost. The cash flows are a growing perpetuity, so the present value is:
34. Since your salary grows at 3.7 percent per year, your salary next year will be:
Since your salary grows at 3.7 percent, your deposit will also grow at 3.7 percent. We can use the
present value of a growing annuity equation to find the value of your deposits today. Doing so, we
find:
Now, we can find the future value of this lump sum in 40 years. We find:
This is the value of your savings in 40 years.
35. The time line is:
0 1
…
15
PV $4,30
0
$4,30
0
$4,30
0
$4,30
0
$4,30
0
$4,30
0
$4,30
0
$4,30
0
$4,300
The relationship between the PVA and the interest rate is:
PVA falls as r increases, and PVA rises as r decreases
FVA rises as r increases, and FVA falls as r decreases
The present values of $4,300 per year for 15 years at the various interest rates given are:
36. The time line is:
0 1
…
?
–
$35,00
0
$240 $240 $240 $240 $240 $240 $240 $240 $240
Here, we are given the FVA, the interest rate, and the amount of the annuity. We need to solve for the
number of payments. Using the FVA equation:
37. The time line is:
0 1
…
60
–
$96,000
$1,95
0
$1,95
0
$1,95
0
$1,95
0
$1,95
0
$1,95
0
$1,95
0
$1,95
0
$1,950
Here, we are given the PVA, number of periods, and the amount of the annuity. We need to solve for
the interest rate. Using the PVA equation:
38. The time line is:
0 1
…
360
PV $950 $950 $950 $950 $950 $950 $950 $950 $950
The amount of principal paid on the loan is the PV of the monthly payments you make. So, the
present value of the $950 monthly payments is:
0 1
…
360
$62,506.
90
FV
This remaining principal amount will increase at the interest rate on the loan until the end of the loan
period. So the balloon payment in 30 years, which is the FV of the remaining principal, will be:
39. The time line is:
0 1 2 3 4
–$7,300 $1,50
0
? $2,70
0
$2,90
0
We are given the total PV of all four cash flows. If we find the PV of the three cash flows we know, and
subtract them from the total PV, the amount left over must be the PV of the missing cash flow. So, the
PV of the cash flows we know are:
So, the PV of the missing CF is:
$7,300 – 1,400.56 – 2,197.84 – 2,204.14 = $1,497.46
The question asks for the value of the cash flow in Year 2, so we must find the future value of this
amount. The value of the missing CF is:
40. The time line is:
0 1 2 3 4 5 6 7 8 9 10
$1M $1.275
M
$1.55
M
$1.825
M
$2.1
M
$2.375
M
$2.65
M
$2.925
M
$3.2
M
$3.475
M
$3.75
M
To solve this problem, we need to find the PV of each lump sum and add them together. It is
important to note that the first cash flow of $1 million occurs today, so we do not need to discount
that cash flow. The PV of the lottery winnings is:
41. Here, we are finding interest rate for an annuity cash flow. We are given the PVA, number of periods,
and the amount of the annuity. We need to solve for the interest rate. We should also note that the PV
of the annuity is not the amount borrowed since we are making a down payment on the warehouse.
The amount borrowed is:
Amount borrowed = .80($5,200,000) = $4,160,000
The time line is:
0 1
…
360
–
$4,160,000
$27,50
0
$27,50
0
$27,50
0
$27,50
0
$27,50
0
$27,50
0
$27,50
0
$27,50
0
$27,500
Using the PVA equation:
r = .578%
The APR is the monthly interest rate times the number of months in the year, so: