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CHAPTER 6
DISCOUNTED CASH FLOW VALUATION
Learning Objectives
LO1 How to determine the future and present value of investments with multiple cash flows.
LO2 How loan payments are calculated and how to find the interest rate on a loan.
LO3 How loans are amortized or paid off.
LO4 How interest rates are quoted (and misquoted).
Answers to Concepts Review and Critical Thinking Questions
1. (LO1) The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and the
number of payments, or the life of the annuity, t.
2. (LO1) Assuming positive cash flows, both the present and the future values will rise.
3. (LO1) Assuming positive cash flows, the present value will fall and the future value will rise.
4. (LO1) It’s deceptive, but very common. The basic concept of time value of money is that a dollar today is not
worth the same as a dollar tomorrow. The deception is particularly irritating given that such lotteries are
usually government sponsored!
5. (LO1) If the total money is fixed, you want as much as possible as soon as possible. The team (or, more
accurately, the team owner) wants just the opposite.
6. (LO1) The better deal is the one with equal installments.
S6-1 Solutions to Questions and Problems
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.
Basic
1. (LO1) To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump
sum, we use:
t
PV = FV / (1 + r)
[email protected]% = $1,100 / 1.10 + $720 / 1.10 + $940 / 1.10 + $1,160 / 1.10 = $3,093.574
[email protected]% = $1,100 / 1.18 + $720 / 1.18 + $940 / 1.18 + $1,160 / 1.18 = $2,619.724
2 3 4
[email protected]% = $1,100 / 1.24 + $720 / 1.24 + $940 / 1.24 + $1,160 / 1.24 = $2,339.03
2. (LO1) To find the PVA, we use the equation:
PVA = C({1 – [1/(1 + r)] } / r )
At a 5 percent interest rate:
8
[email protected]%: PVA = $7,000{[1 – (1/1.05) ] / .05 } = $45,242.49
5
[email protected]%: PVA = $9,000{[1 – (1/1.05) ] / .05 } = $38,965.29
And at a 22 percent interest rate:
[email protected]%: PVA = $7,000{[1 – (1/1.22) ] / .22 } = $25,334.87
[email protected]%: PVA = $9,000{[1 – (1/1.22) ] / .22 } = $25,772.76
Notice that the PV of cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a 22 percent
interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the total cash flow is
more important since the cost of waiting (the interest rate) is not as great. At a higher interest rate, Y is more
valuable since it has larger cash flows. At the higher interest rate, these bigger cash flows early are more
important since the cost of waiting (the interest rate) is so much greater.
3. (LO1) To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a lump
sum, we use:
FV = PV(1 + r) t
[email protected]% = $700(1.08) + $950(1.08) + $1,200(1.08) + $1,300 = $4,585.88
3 2
[email protected]% = $700(1.11) + $950(1.11) + $1,200(1.11) + $1,300 = $4,759.84
3 2
[email protected]% = $700(1.24) + $950(1.24) + $1,200(1.24) + $1,300 = $5,583.36
Notice we are finding the value at Year 4, the cash flow at Year 4 is simply added to the FV of the other cash
flows. In other words, we do not need to compound this cash flow.
S6-2 4. (LO1) To find the PVA, we use the equation:
t
PVA = C({1 – [1/(1 + r)] } / r )
15
[email protected] yrs: PVA = $4,600{[1 – (1/1.08) ] / .08} = $39,373.60
[email protected] yrs: PVA = $4,600{[1 – (1/1.08) ] / .08} = $54,853.22
75
[email protected] yrs: PVA = $4,600{[1 – (1/1.08) ] / .08} = $57,320.99
To find the PV of a perpetuity, we use the equation:
PV = C / r
PV = $4,600 / .08 = $57,500.00
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 $179.01.
5. (LO1) Here we have the 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)] } / r )
15
PVA = $28,000 = $C{[1 – (1/1.0825) ] / .0825}
We can now solve this equation for the annuity payment. Doing so, we get:
C = $28,000 / 8.43035 = $3,321.33
6. (LO1) To find the PVA, we use the equation:
PVA = C({1 – [1/(1 + r)] } / r )
8
PVA = $65,000{[1 – (1/1.085) ] / .085} = $366,546.89
7. (LO1) Here we need to find the FVA. The equation to find the FVA is:
t
FVA = C{[(1 + r) – 1] / r}
FVA for 20 years = $3,000[(1.105 – 1) / .105] = $181,892.42
FVA for 40 years = $3,000[(1.105 – 1) / .105] = $1,521,754.74
Notice that because of exponential growth, doubling the number of periods does not merely double the FVA.
8. (LO1) Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity
payment. Using the FVA equation:
FVA = C{[(1 + r) – 1] / r}
10
$80,000 = $C[(1.065 – 1) / .065]
We can now solve this equation for the annuity payment. Doing so, we get:
C = $80,000 / 13.49442 = $5,928.38
S6-3 9. (LO2) Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity
payment. Using the PVA equation:
t
PVA = C({1 – [1/(1 + r)] } / r)
$30,000 = C{[1 – (1/1.08) ] / .08}
We can now solve this equation for the annuity payment. Doing so, we get:
C = $30,000 / 5.20637 = $5,762.17
10. (LO1) Here we have the quarterly annuity payment, the length of the annuity in years, and the interest rate
compounded monthly. We want to calculate the PVA for an annuity due.
We must first calculate the quarterly interest rate. At 5% compounded monthly, the quarterly rate is:
r = (1 + .05/ 12) – 1 = .01255 = 1.26%
Six years is 24 quarters. Using the PVA equation for an ordinary annuity:
t
PVA = C({1 – [1/(1 + r)] } / r)
PVA = $1,059{[1 – (1/1.0126) ] / .0126}
We can now solve this equation for the annuity payment. Doing so, we get the PVA:
PVA = $21,827.68
Annuity due value = Ordinary annuity value x (1+r):
PVA due = $21,827.68 x (1 + .0126) = $22,101.61
11. (LO1) This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:
PV = C / r
PV = $20,000 / .08 = $250,000.00
12. (LO1) Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash
flows. Using the PV of a perpetuity equation:
PV = C / r
$280,000 = $20,000 / r
We can now solve for the interest rate as follows:
r = $20,000 / $280,000 = .0714 or 7.14%
13. (LO4) For discrete compounding, to find the EAR, we use the equation:
m
EAR = [1 + (APR / m)] – 1
EAR = [1 + (.07 / 4)] – 1 = .0719 or 7.19%
12
EAR = [1 + (.18 / 12)] – 1 = .1956 or 19.56%
S6-4 365
EAR = [1 + (.10 / 365)] – 1 = .1052 or 10.52%
To find the EAR with continuous compounding, we use the equation:
EAR = e – 1
.14
EAR = e – 1 = .1503 or 15.03%
14. (LO4) Here we are given the EAR and need to find the APR. Using the equation for discrete compounding:
m
EAR = [1 + (APR / m)] – 1
We can now solve for the APR. Doing so, we get:
APR = m[(1 + EAR) 1/m– 1]
2 1/2
EAR = .1220 = [1 + (APR / 2)] – 1 APR = 2[(1.1220) – 1] = .1185 or 11.85%
12 1/12
EAR = .0940 = [1 + (APR / 12)] – 1 APR = 12[(1.0940) – 1] = .0902 or 9.02%
EAR = .0860 = [1 + (APR / 52)] – 152 APR = 52[(1.0860) 1/52– 1] = .0826 or 8.26%
Solving the continuous compounding EAR equation:
q
EAR = e – 1
We get:
APR = ln(1 + EAR)
APR = ln(1 + .2380)
APR = .2135 or 21.35%
15. (LO4) For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR / m)] – 1m
So, for each bank, the EAR is:
First National: EAR = [1 + (.1310 / 12)] – 1 = .1392 or 13.92%
First United: EAR = [1 + (.1340 / 2)] – 1 = .1385 or 13.85%
Notice that the higher APR does not necessarily mean the higher EAR. The number of compounding periods
within a year will also affect the EAR.
16. (LO4) The reported rate is the APR, so we need to convert the EAR to an APR as follows:
m
EAR = [1 + (APR / m)] – 1
APR = m[(1 + EAR) 1/m– 1]
1/365
APR = 365[(1.14) – 1] = .1311 or 13.11%
This is deceptive because the borrower is actually paying annualized interest of 14% per year, not the 13.11%
reported on the loan contract.
S6-5 17. (LO1) For this problem, we simply need to find the FV of a lump sum using the equation:
t
FV = PV(1 + r)
It is important to note that compounding occurs semiannually. To account for this, we will divide the interest
rate by two (the number of compounding periods in a year), and multiply the number of periods by two. Doing
so, we get:
40
FV = $1,400[1 + (.096/2)] = $9,132.28
18. (LO1) For this problem, we simply need to find the FV of a lump sum using the equation:
FV = PV(1 + r) t
It is important to note that compounding occurs daily. To account for this, we will divide the interest rate by
365 (the number of days in a year, ignoring leap year), and multiply the number of periods by 365. Doing so,
we get:
5(365)
FV in 5 years = $6,000[1 + (.084/365)] = $9,131.33
FV in 10 years = $6,000[1 + (.084/365)] 10(36= $13,896.86
FV in 20 years = $6,000[1 + (.084/365)] 20(36= $32,187.11
19. (LO1) For this problem, we simply need to find the PV of a lump sum using the equation:
PV = FV / (1 + r) t
It is important to note that compounding occurs daily. To account for this, we will divide the interest rate by
365 (the number of days in a year, ignoring leap year), and multiply the number of periods by 365. Doing so,
we get:
PV = $45,000 / [(1 + .11/365) 6(36] = $23,260.62
20. (LO4) The APR is simply the interest rate per period times the number of periods in a year. In this case, the
interest rate is 25 percent per month, and there are 12 months in a year, so we get:
APR = 12(25%) = 300%
To find the EAR, we use the EAR formula:
EAR = [1 + (APR / m)] – 1 m
12
EAR = (1 + .25) – 1 = 1,355.19%
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. (LO2, 4) We first need to find the annuity payment. We have the PVA, the length of the annuity, and the
interest rate. Using the PVA equation:
PVA = C({1 – [1/(1 + r)] } / r)
60
$61,800 = $C[1 – {1 / [1 + (.074/12)] } / (.074/12)]
S6-6 Solving for the payment, we get:
C = $61,800 / 50.02385 = $1,235.41
To find the EAR, we use the EAR equation:
m
EAR = [1 + (APR / m)] – 1
EAR = [1 + (.074 / 12)] – 1 = .0766 or 7.66%
22. (LO3) 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)] } / r)
t
$17,000 = $300{[1 – (1/1.009) ] / .009}
Now we solve for t:
t
1/1.009 = 1 – {[($17,000)/($300)](.009)}
1/1.009 = 0.49
1.009 = 1/(0.49) = 2.0408
t = ln 2.0408 / ln 1.009 = 79.62 months
23. (LO4) Here we are trying to find the interest rate when we know the PV and FV. Using the FV equation:
FV = PV(1 + r)
$4 = $3(1 + r)
r = 4/3 – 1 = 33.33% per week
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:
APR = (52)33.33% = 1,733.33%
And using the equation to find the EAR:
EAR = [1 + (APR / m)] – 1m
EAR = [1 + .3333] – 1 = 3,139,166.15,%
24. (LO1) Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash
flows. Using the PV of a perpetuity equation:
PV = C / r
$63,000 = $1,200 / r
We can now solve for the interest rate as follows:
r = $1,200 / $63,000 = .0190 or 1.90% per month
The interest rate is 1.90% per month. To find the APR, we multiply this rate by the number of months in a
year, so:
APR = (12)1.90% = 22.86%
And using the equation to find an EAR:
m
EAR = [1 + (APR / m)] – 1
S6-7 12
EAR = [1 + .0190] – 1 = 25.41%
25. (LO1) This problem requires us to find the FVA. The equation to find the FVA is:
t
FVA = C{[(1 + r) – 1] / r}
FVA = $250[{[1 + (.10/12) ] 360– 1} / (.10/12)] = $565,121.98
26. (LO1) In the previous problem, the cash flows are monthly and the compounding period is monthly. This
assumption still holds. Since the cash flows are annual, we need to use the EAR to calculate the future value of
annual cash flows. It is important to remember that you have to make sure the compounding periods of the
interest rate times with the cash flows. In this case, we have annual cash flows, so we need the EAR since it is
the true annual interest rate you will earn. So, finding the EAR:
EAR = [1 + (APR / m)] – 1 m
12
EAR = [1 + (.10/12)] – 1 = .1047 or 10.47%
Using the FVA equation, we get:
FVA = C{[(1 + r) – 1] / r}
30
FVA = $3,000[(1.1047 – 1) / .1047] = $539,686.21
27. (LO1) The cash flows are simply an annuity with four payments per year for four years, or 16 payments. We
can use the PVA equation:
t
PVA = C({1 – [1/(1 + r)] } / r)
PVA = $1,500{[1 – (1/1.0075) ] / .0075} = $22,536.47
28. (LO1) 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:
m
EAR = [1 + (APR / m)] – 1
EAR = [1 + (.11/4)] – 1 = .1146 or 11.46%
And now we use the EAR to find the PV of each cash flow as a lump sum and add them together:
2 4
PV = $900 / 1.1146 + $850 / 1.1146 + $1,140 / 1.1146 = $2,230.20
29. (LO1) Here the cash flows are annual and the given interest rate is annual, so we can use the interest rate
given. We simply find the PV of each cash flow and add them together.
3 4
PV = $2,800 / 1.0845 + $5,600 / 1.0845 + $1,940 / 1.0845 = $8,374.62
S6-8 Intermediate
30. (LO4) 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:
.06(10) = .6
First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor of $1, or:
10
(1 + r)
Setting the two equal, we get:
10
(.06)(10) = (1 + r) – 1
1/10
r = 1.6 – 1 = .0481 or 4.81%
31. (LO4) Here we need to convert an EAR into interest rates for different compounding periods. Using the
equation for the EAR, we get:
EAR = [1 + (APR / m)] – 1 m
2 1/2
EAR = .18 = (1 + r) – 1; r = (1.18) – 1 = .0863 or 8.63% per six months
4 1/4
EAR = .18 = (1 + r) – 1; r = (1.18) – 1 = .0422 or 4.22% per quarter
EAR = .18 = (1 + r) – 1; r = (1.18) 1/1– 1 = .0139 or 1.39% per month
Notice that the effective six month rate is not twice the effective quarterly rate because of the effect of
compounding.
32. (LO2) 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:
6
FV = $5,000 [1 + (.025/12)] = $5,062.83
This is the balance in six months. The FV in another six months will be:
FV = $5,062.83 [1 + (.17/12)] = $5,508.70
The problem asks for the interest accrued, so, to find the interest, we subtract the beginning balance
from the FV. The interest accrued is:
Interest = $5,508.70 – 5,000.00 = $508.70
33. (LO1) We need to find the annuity payment in retirement. Our retirement savings ends and 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.
360
Stock account: FVA = $600[{[1 + (.12/12) ] – 1} / (.12/12)] = $2,096,978.48
Bond account: FVA = $300[{[1 + (.07/12) ] 360– 1} / (.07/12)] = $365,991.30
So, the total amount saved at retirement is:
S6-9 $2,096,978.48 + 365,991.30 = $2,462,969.78
Solving for the withdrawal amount in retirement using the PVA equation gives us:
PVA = $2,462,969.78 = $C[1 – {1 / [1 + (.09/12)] } / (.09/12)]
C = $2,462,969.78 / 119.1616 = $20,669.15 withdrawal per month
34. (LO1) We need to find the FV of a lump sum in one year and two years. It is important that we use the
number of months in compounding since interest is compounded monthly in this case. So:
12
FV in one year = $1(1.0108) = $1.14
FV in two years = $1(1.0108) = $1.29
There is also another common alternative solution. We could find the EAR, and use the number of years as our
compounding periods. So we will find the EAR first:
12
EAR = (1 + .0108) – 1 = .1376 or 13.76%
Using the EAR and the number of years to find the FV, we get:
1
FV in one year = $1(1.1376) = $1.14
FV in two years = $1(1.1376) = $1.29
Either method is correct and acceptable. We have simply made sure that the interest compounding period is the
same as the number of periods we use to calculate the FV.
35. (LO1) Here we are finding the annuity payment necessary to achieve the same FV. The interest rate given is a
10 percent APR, with monthly deposits. We must make sure to use the number of months in the equation. So,
using the FVA equation:
480
FVA in 40 years = C[{[1 + (.11/12) ] – 1} / (.11/12)]
C = $1,000,000 / 8,600.127 = $116.28
FVA in 30 years = C[{[1 + (.11/12) ] 36– 1} / (.11/12)]
C = $1,000,000 / 2,804.52 = $356.57
FVA in 20 years = C[{[1 + (.11/12) ] 24– 1} / (.11/12)]
C = $1,000,000 / 865.638 = $1,155.22
Notice that a deposit for half the length of time, i.e. 20 years versus 40 years, does not mean that the annuity
payment is doubled. In this example, by reducing the savings period by one-half, the deposit necessary to
achieve the same terminal value is about nine times as large.
36. (LO2) 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:
(12/3)
FV = $3 = $1(1 + r)
r = .3160 or 31.60%
S6-10 37. (LO1) Since we have an APR compounded monthly and an annual payment, we must first convert the interest
rate to an EAR so that the compounding period is the same as the cash flows.
EAR = [1 + (.10 / 12)] – 1 = .104713 or 10.4713%
PVA = 190,000 {[1 – (1 / 1.104713) ] / .104713} = $155,215.98
2
PVA = 245,000 + $65,000{[1 – (1/1.104713) ] / .104713} = $157,100.43
You would choose the second option since it has a higher PV.
38. (LO1) We can use the present value of a growing perpetuity equation to find the value of your deposits today.
Doing so, we find:
t
PV = {C /(r – g)}{[1 – [(1 + g)/(1 + r)]}
PV = {$1,000,000/(.09 – .05)}{[1 – [(1 + .05)/(1 + .09)] } 25
PV = $15,182,293.68
39. (LO1) Since your salary grows at 4 percent per year, your salary next year will be:
Next year’s salary = $50,000 (1 + .04)
Next year’s salary = $52,000
This means your deposit next year will be:
Next year’s deposit = $52,000(.02)
Next year’s deposit = $1,040
Since your salary grows at 4 percent, you deposit will also grow at 4 percent. We can use the present value of a
growing perpetuity equation to find the value of your deposits today. Doing so, we find:
PV = {C /(r – g)}{[1 – [(1 + g)/(1 + r)]} t
40
PV = {$1,040/(.10 – .04)}{[1 – [(1 + .04)/(1 + .10)] }
PV = $15,494.64
Now, we can find the future value of this lump sum in 40 years. We find:
FV = PV(1 + r) t
40
FV = $15,494.64(1 + .10)
FV = $701,276.07
This is the value of your savings in 40 years.
40. (LO1) 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 $7,000 per year for 10 years at the various interest rates given are:
10
[email protected]% = $7,000{[1 – (1/1.10) ] / .10} = $43,011.97
[email protected]% = $7,000{[1 – (1/1.05) ] / .05} 10 = $54,052.14
[email protected]% = $7,000{[1 – (1/1.15) ] / .15}10 = $35,131.38
S6-11 41. (LO2) 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:
t
FVA = $20,000 = $225[{[1 + (.09/12)] – 1 } / (.09/12)]
Solving for t, we get:
t
1.0075 = 1 + [($20,000)/($225)](.09/12)
t = ln 1.66667 / ln 1.0075 = 68.37 payments
42. (LO2) 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:
PVA = $55,000 = $1,120[{1 – [1 / (1 + r)] }/ r]60
To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trial
and error. If you use trial and error, remember that increasing the interest rate lowers the PVA, and increasing
the interest rate decreases the PVA. Using a spreadsheet, we find:
r = 0.682%
The APR is the periodic interest rate times the number of periods in the year, so:
APR = 12(0.682%) = 8.18%
43. (LO2) Here we have the monthly annuity payment, the length of the annuity, and the annual interest rate. We
want to calculate the PVA for an annuity due.
We must first calculate the monthly interest rate. At 8% compounded annually, the monthly rate is:
r = .08 / 12 = .00667 = 0.67%
There are 39 monthly payments including the current July 1 payment. st Using the PVA equation for an
ordinary annuity:
t
PVA = C({1 – [1/(1 + r)] } / r)
Annuity due value = Ordinary annuity value x (1+r):
PVA due = C({1 – [1/(1 + r)] } / r) x (1+r)
PVA due = $1,800{[1 – (1/1.0067) ] / .0067} (1 + .0067)
PVA due = $62,047.27
44. (LO2) The amount of principal paid on the loan is the PV of the monthly payments you make. So, the present
value of the $1,100 monthly payments is:
PVA = $1,100[(1 – {1 / [1 + (.068/12)]} ) / (.068/12)] = $168,731.02
The monthly payments of $1,100 will amount to a principal payment of $168,731.02. The amount of principal
you will still owe is:
$220,000 – 168,731.02 = $51,268.98
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:
S6-12 Balloon payment = $51,268.98 [1 + (.068/12)] 360= $392,025.82
45. (LO1) 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:
PV of Year 1 CF: $1,500 / 1.10 = $1,363.64
PV of Year 3 CF: $1,800 / 1.10 3 = $1,352.37
4
PV of Year 4 CF: $2,400 / 1.10 = $1,639.23
So, the PV of the missing CF is:
$6,785 – 1,363.64 – 1,352.37 – 1,639.23 = $2,429.76
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:
2
$2,429.76(1.10) = $2,940.02
46. (LO1) To solve this problem, we simply 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:
2 3 4
$1,000,000 + $1,400,00051.09 + $1,800,000/1.69 + $2,200,000/1.09 7 $2,600,000/1.09 + 8 9
$3,000,000/1.09 + $3,400,000/1.09 + $3,800,000/1.09 + $4,200,000/1.09 + $4,600,000/1.09 +
$5,000,000/1.09 = $19,733,830.26
47. (LO4) 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 number of payments. 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 = 0.80($2,400,000) = $1,920,000
Using the PVA equation:
360
PVA = $1,920,000 = $13,000[{1 – [1 / (1 + r)] }/ r]
Unfortunately this equation cannot be solved to find the interest rate using algebra. To find the interest rate, we
need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. If you use trial
and error, remember that increasing the interest rate lowers the PVA, and increasing the interest rate decreases
the PVA. Using a spreadsheet, we find:
r = 0.598%
The APR is the monthly interest rate times the number of months in the year, so:
APR = 12(0.598%) = 7.17%
And the EAR is:
12
EAR = (1 + .00598) – 1 = .0742 or 7.42%
S6-13 48. (LO1) The profit the firm earns is just the PV of the sales price minus the cost to produce the asset. We find
the PV of the sales price as the PV of a lump sum:
PV = $145,000 / 1.13 = $100,492.27
And the firm’s profit is:
Profit = $100,492.27 – 94,000.00 = $6,492.27
To find the interest rate at which the firm will break even, we need to find the interest rate using the PV (or
FV) of a lump sum. Using the PV equation for a lump sum, we get:
$94,000 = $145,000 / ( 1 + r) 3
r = ($145,000 / $94,000) – 1 = .1554 or 15.54%
49. (LO1) 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 17 payments, so the PV of the annuity is:
17
PVA = $2,000{[1 – (1/1.10) ] / .10} = $16,043.11
Since this is an ordinary annuity equation, this is the PV one period before the first payment, so it is the PV at t
= 8. To find the value today, we find the PV of this lump sum. The value today is:
PV = $16,043.11 / 1.10 = $7,484.23
50. (LO1) This question is asking for the present value of an annuity, but the interest rate changes during the life
of the annuity. We need to find the present value of the cash flows for the last eight years first. The PV of these
cash flows is:
PVA = 21,500 [{1 – 1 / [1 + (.10/12)] } / (.10/12)] = $98,852.23
Note that this is the PV of this annuity exactly seven years from today. Now we can discount this lump sum to
today. The value of this cash flow today is:
84
PV = $98,852.23 / [1 + (.13/12)] = $39,985.62
Now we need to find the PV of the annuity for the first seven years. The value of these cash flows today is:
PVA = 11,500 [{1 – 1 / [1 + (.13/12)] } / (.13/12)] = $82,453.99
The value of the cash flows today is the sum of these two cash flows, so:
PV = $39,985.62 + 82,453.99 = $122,439.62
51. (LO1) Here we are trying to find the dollar amount invested today that will equal the FVA with a known
interest rate, and payments. First we need to determine how much we would have in the annuity account.
Finding the FV of the annuity, we get:
FVA = $1,000 [{[ 1 + (.095/12)] 180– 1} / (.095/12)] = $395,948.63
Now we need to find the PV of a lump sum that will give us the same FV. So, using the FV of a lump sum
with continuous compounding, we get:
.09(15)
FV = $395,948.63 = PVe
PV = $395,948.63 e –1.3= $102,645.83
S6-14 52. (LO1) To find the value of the perpetuity at t = 7, we first need to use the PV of a perpetuity equation. Using
this equation we find:
PV = $5,000 / .057 = $87,719.30
Remember that the PV of a perpetuity (and annuity) equations give the PV one period before the first payment,
so, this is the value of the perpetuity at t = 14. To find the value at t = 7, we find the PV of this lump sum as:
PV = $87,719.30 / 1.057 = $59,507.30
53. (LO4) To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the interest
rate quoted in the problem is only relevant to determine the total interest under the terms given. The interest
rate for the cash flows of the loan is:
12
PVA = $20,000 = $1,916.67{(1 – [1 / (1 + r)] ) / r }
Again, we cannot solve this equation for r, so we need to solve this equation on a financial calculator, using a
spreadsheet, or by trial and error. Using a spreadsheet, we find:
r = 2.219% per month
So the APR is:
APR = 12(2.219%) = 26.62%
And the EAR is:
EAR = (1.02219) – 1 = .3012 or 30.12%
54. (LO1) The cash flows in this problem are semiannual, so we need the effective semiannual rate. The interest
rate given is the APR, so the monthly interest rate is:
Monthly rate = .10 / 12 = .00833
To get the semiannual interest rate, we can use the EAR equation, but instead of using 12 months as the
exponent, we will use 6 months. The effective semiannual rate is:
6
Semiannual rate = (1.00833) – 1 = .0511 or 5.11%
We can now use this rate to find the PV of the annuity. The PV of the annuity is:
PVA @ t = 9: $6,000{[1 – (1 / 1.0511) ] / .0511} = $46,094.33
Note, this is the value one period (six months) before the first payment, so it is the value at t = 9. So, the value
at the various times the questions asked for uses this value 9 years from now.
PV @ t = 5: $46,094.33 / 1.0511 = $30,949.21
Note, you can also calculate this present value (as well as the remaining present values) using the number of
years. To do this, you need the EAR. The EAR is:
12
EAR = (1 + .0083) – 1 = .1047 or 10.47%
So, we can find the PV at t = 5 using the following method as well:
4
PV @ t = 5: $46,094.33 / 1.1047 = $30,949.21
S6-15 The value of the annuity at the other times in the problem is:
PV @ t = 3: $46,094.33 / 1.0511 12 = $25,360.08
6
PV @ t = 3: $46,094.33 / 1.1047 = $25,360.08
PV @ t = 0: $46,094.33 / 1.0511 18 = $18,810.58
PV @ t = 0: $46,094.33 / 1.1047 9 = $18,810.58
55. (LO1)
a. Calculating the PV of an ordinary annuity, we get:
8
PVA = $950{[1 – (1/1.095) ] / .095} = $5,161.76
b. To calculate the PVA due, we calculate the PV of an ordinary annuity for t – 1 payments, and add the payment
that occurs today. So, the PV of the annuity due is:
7
PVA = $950 + $950{[1 – (1/1.095) ] / .095} = $5,652.13
56. (LO1) We need to use the PVA due equation, that is:
PVA due= (1 + r) PVA
Using this equation:
60
PVA due= $61,000 = [1 + (.0815/12)] × C[{1 – 1 / [1 + (.0815/12)] } / (.0815/12)
$60,588.50 = $C{1 – [1 / (1 + .0815/12) ]} / (.0815/12)
C = $1,232.87
Notice, when we find the payment for the PVA due, we simply discount the PV of the annuity due back one
period. We then use this value as the PV of an ordinary annuity.
57. (LO3) The payment for a loan repaid with equal payments is the annuity payment with the loan value as the
PV of the annuity. So, the loan payment will be:
5
PVA = $36,000 = C {[1 – 1 / (1 + .09) ] / .09}
C = $9,255.33
The interest payment is the beginning balance times the interest rate for the period, and the principal payment
is the total payment minus the interest payment. The ending balance is the beginning balance minus the
principal payment. The ending balance for a period is the beginning balance for the next period. The
amortization table for an equal payment is:
Beginning Total Interest Principal Ending
Year Balance Payment Payment Payment Balance
1 $36,000.00 $9,255.33 $3,240.00 $6,015.33 $29,984.67
2 29,984.67 9,255.33 2,698.62 6,556.71 23,427.96
3 23,427.96 9,255.33 2,108.52 7,146.81 16,281.15
4 16,281.15 9,255.33 1,465.30 7,790.02 8,491.13
5 8,491.13 9,255.33 764.20 8,491.13 0.00
S6-16 In the third year, $2,108.52 of interest is paid.
Total interest over life of the loan = $3,240 + 2,698.62 + 2,108.52 + 1,465.30 + 764.20 = $10,276.64
58. (LO3) This amortization table calls for equal principal payments of $7,200 per year. The interest payment
is the beginning balance times the interest rate for the period, and the total payment is the principal payment
plus the interest payment. The ending balance for a period is the beginning balance for the next period. The
amortization table for an equal principal reduction is:
Beginning Total Interest Principal Ending
Year Balance Payment Payment Payment Balance
1 $36,000.00 $10,440.00 $3,240.00 $7,200.00 $28,800.00
2 28,800.00 9,792.00 2,592.00 7,200.00 21,600.00
3 21,600.00 9,144.00 1,944.00 7,200.00 14,400.00
4 14,400.00 8,496.00 1,296.00 7,200.00 7,200.00
5 7,200.00 7,848.00 648.00 7,200.00 0.00
In the third year, $1,944 of interest is paid.
Total interest over life of the loan = $3,240 + 2,592 + 1,944 + 1,296 + 648 = $9,720
Notice that the total payments for the equal principal reduction loan are lower. This is because more principal
is repaid early in the loan, which reduces the total interest expense over the life of the loan.
Challenge
59. (LO1) The cash flows for this problem occur monthly, and the interest rate given is the EAR. Since the cash
flows occur monthly, we must get the effective monthly rate. One way to do this is to find the APR based on
monthly compounding, and then divide by 12. So, the pre-retirement APR is:
EAR = .11 = [1 + (APR / 12)] – 1; APR = 12[(1.11) 1/12– 1] = 10.48%
And the post-retirement APR is:
EAR = .08 = [1 + (APR / 12)] – 1; APR = 12[(1.08) 1/12– 1] = 7.72%
First, we will calculate how much he needs at retirement. The amount needed at retirement is the PV of the
monthly spending plus the PV of the inheritance. The PV of these two cash flows is:
PVA = $20,000{1 – [1 / (1 + .0772/12) 12(]} / (.0772/12) = $2,441,554.61
PV = $750,000 / [1 + (.0772/12)] 24= $160,911.16
So, at retirement, he needs:
$2,441,544.61 + 160,911.16 = $2,602,465.76
He will be saving $2,100 per month for the next 10 years until he purchases the cabin. The value of his savings
after 10 years will be:
FVA = $2,000[{[ 1 + (.1048/12)] 12(1– 1} / (.1048/12)] = $421,180.66
S6-17 After he purchases the cabin, the amount he will have left is:
$421,180.66 – 325,000 = $96,180.66
He still has 20 years until retirement. When he is ready to retire, this amount will have grown to:
FV = $96,180.66[1 + (.1048/12)] 12(2= $775,438.43
So, when he is ready to retire, based on his current savings, he will be short:
$2,602,465.76 – 775,438.43 = $1,827,027.33
This amount is the FV of the monthly savings he must make between years 10 and 30. So, finding the annuity
payment using the FVA equation, we find his monthly savings will need to be:
FVA = $1,827,027.33 = C[{[ 1 + (.1048/12)] 12(20– 1} / (.1048/12)]
C = $2,259.65
60. (LO1) To answer this question, we should find the PV of both options, and compare them. Since we are
purchasing the car, the lowest PV is the best option. The PV of the leasing is simply the PV of the lease
payments, plus the $1. The interest rate we would use for the leasing option is the same as the interest rate of
the loan. The PV of leasing is:
12(3)
PV = $1 + $380{1 – [1 / (1 + .08/12) ]} / (.08/12) = $12,127.49
The PV of purchasing the car is the current price of the car minus the PV of the resale price. The PV of the
resale price is:
PV = $15,000 / [1 + (.08/12)] 12(3= $11,808.82
The PV of the decision to purchase is:
$28,000 – 11,808.82 = $16,191.18
In this case, it is cheaper to lease the car than buy it since the PV of the leasing cash flows is lower. To find the
breakeven resale price, we need to find the resale price that makes the PV of the two options the same. In other
words, the PV of the decision to buy should be:
$28,000 – PV of resale price = $12,127.49
PV of resale price = $15,872.51
The resale price that would make the PV of the lease versus buy decision is the FV of this value, so:
Breakeven resale price = $15,872.51[1 + (.08/12)] 12(3= $20,161.86
S6-18 61. (LO1) To find the quarterly salary for the player, we first need to find the PV of the current contract. The cash
flows for the contract are annual, and we are given a daily interest rate. We need to find the EAR so the
interest compounding is the same as the timing of the cash flows. The EAR is:
EAR = [1 + (.055/365)] 365– 1 = 5.65%
The PV of the current contract offer is the sum of the PV of the cash flows. So, the PV is:
2 3
PV = $8,000,000 + $4,000,000/1.4565 + $4,800,000/1.05655+ $5,700,000/1.0565 6
+ $6,400,000/1.0565 + $7,000,000/1.0565 + $7,500,000/1.0565
PV = $36,764,432.45
The player wants the contract increased in value by $750,000, so the PV of the new contract will be:
PV = $36,764,432.45 + 750,000 = $37,514,432.45
The player has also requested a signing bonus payable today in the amount of $9 million. We can simply
subtract this amount from the PV of the new contract. The remaining amount will be the PV of the future
quarterly paychecks.
$37,514,432.45 – 9,000,000 = $28,514,432.45
To find the quarterly payments, first realize that the interest rate we need is the effective quarterly rate. Using
the daily interest rate, we can find the quarterly interest rate using the EAR equation, with the number of days
being 91.25, the number of days in a quarter (365 / 4). The effective quarterly rate is:
91.25
Effective quarterly rate = [1 + (.055/365)] – 1 = .01384 or 1.384%
Now we have the interest rate, the length of the annuity, and the PV. Using the PVA equation and solving for
the payment, we get:
PVA = $28,514,432.45 = C{[1 – (1/1.01384) ] / .01384}
C = $1,404,517.39
62. (LO1)
a. Calculating the monthly interest rate:
r = .109/12 = .00908 = 0.91%
Using the PVA equation to find the monthly payments:
PVA = $17,679 - $2,500 - $1,000 = C[{1 – [1 / (1 + r)] }/ r]8
48
PVA = $14,179 = C[{1 – [1 / (1 + .0091)] }/ .0091]
Solving for the monthly payment:
C = $365.78
Now calculate the PV of this option with an annual discount rate of 9%:
The monthly interest rate = .09 / 12 = .0075 = 0.75%
48
PV = $2,500 + $365.78[{1 – [1 / (1 + .0075)] }/ .0075]
PV = $2,500 + $14,698.79 = $17,198.79
S6-19 b. Calculating the monthly interest rate:
r = .015/12 = .00125 = 0.13%
Using the PVA equation to find the monthly payments:
PVA = $17,679 - $2,500 = C[{1 – [1 / (1 + r)] }/ r]
PVA = $14,179 = C[{1 – [1 / (1 + .0013)] }/ .0013]
Solving for the monthly payment:
C = $326.01
Now calculate the PV of this option with an annual discount rate of 9%:
The monthly interest rate = .09 / 12 = .0075 = 0.75%
48
PV = $2,500 + $326.01[{1 – [1 / (1 + .0075)] }/ .0075]
PV = $2,500 + $13,100.64 = $15,600.64
Option b is the better deal.
63. (LO4) To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the interest
rate quoted in the problem is only relevant to determine the total interest under the terms given. The cash flows
of the loan are the $20,000 you must repay in one year, and the $17,200 you borrow today. The interest rate of
the loan is:
$20,000 = $17,200(1 + r)
r = ($20,000 – 17,200) – 1 = .1628 or 16.28%
Because of the discount, you only get the use of $17,200, and the interest you pay on that amount is 16.28%,
not 14%.
64. (LO1) Here we have cash flows that would have occurred in the past and cash flows that would occur in the
future. We need to bring both cash flows to today. Before we calculate the value of the cash flows today, we
must adjust the interest rate so we have the effective monthly interest rate. Finding the APR with monthly
compounding and dividing by 12 will give us the effective monthly rate. The APR with monthly compounding
is:
1/12
APR = 12[(1.09) – 1] = 8.65%
To find the value today of the back pay from two years ago, we will find the FV of the annuity, and then find
the FV of the lump sum. Doing so gives us:
12
FVA = ($44,000/12) [{[ 1 + (.0865/12)] – 1} / (.0865/12)] = $45,786.76
FV = $45,786.76(1.09) = $49,907.57
Notice we found the FV of the annuity with the effective monthly rate, and then found the FV of the lump sum
with the EAR. Alternatively, we could have found the FV of the lump sum with the effective monthly rate as
long as we used 12 periods. The answer would be the same either way.
Now, we need to find the value today of last year’s back pay:
12
FVA = ($46,000/12) [{[ 1 + (.0865/12)] – 1} / (.0865/12)] = $47,867.98
S6-20 Next, we find the value today of the five year’s future salary:
12(5)
PVA = ($49,000/12){[{1 – {1 / [1 + (.0865/12)] }] / (.0865/12)}= $198,332.55
The value today of the jury award is the sum of salaries, plus the compensation for pain and suffering, and
court costs. The award should be for the amount of:
Award = $49,907.57 + 47,867.98 + 198,332.55 + 100,000 + 20,000 = $416,108.10
As the plaintiff, you would prefer a lower interest rate. In this problem, we are calculating both the PV and FV
of annuities. A lower interest rate will decrease the FVA, but increase the PVA. So, by a lower interest rate, we
are lowering the value of the back pay. But, we are also increasing the PV of the future salary. Since the future
salary is larger and has a longer time, this is the more important cash flow to the plaintiff.
65. (LO4) Again, to find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan is
in the form of a lump sum, the amount you will repay is the FV of the principal amount, which will be:
Loan repayment amount = $10,000(1.09) = $10,900
The amount you will receive today is the principal amount of the loan times one minus the points.
Amount received = $10,000(1 – .03) = $9,700
Now, we simply find the interest rate for this PV and FV.
$10,900 = $9,700(1 + r)
r = ($10,900 / $9,700) – 1 = .1237 or 12.37%
66. (LO4) This is the same question as before, with different values. So:
Loan repayment amount = $10,000(1.12) = $11,200
Amount received = $10,000

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