SOLUTIONS MANUAL
, CHAPTER 2
Present Value and the Opportunity Cost of Capital
Answers to Practice Questions
1. Let INV = investment required at time t = 0 (i.e., INV = -C0) and let x = rate of
return. Then x is defined as:
x = (C1 – INV)/INV
Therefore:
C1 = INV(1 + x)
It follows that:
NPV = C0 + {C1/(1 + r)}
NPV = -INV + {[INV(1 + x)]/(1 + r)}
NPV = INV {[(1 + x)/(1 + r)] – 1}
a. When x equals r, then:
[(1 + x)/(1 +r)] – 1 = 0
and NPV is zero.
b. When x exceeds r, then:
[(1 + x)/(1 + r)] – 1 > 0
and NPV is positive.
2. The face value of the treasury security is $1,000. If this security earns 5%, then
in one year we will receive $1,050. Thus:
NPV = C0 + [C1/(1 + r)] = -1000 + (1050/1.05) = 0
This is not a surprising result, because 5 percent is the opportunity cost of
capital, i.e., 5 percent is the return available in the capital market. If any
investment earns a rate of return equal to the opportunity cost of capital, the NPV
of that investment is zero.
1
,3. NPV = -$1,300,000 + ($1,500,000/1.10) = +$63,636
Since the NPV is positive, you would construct the motel.
Alternatively, we can compute r as follows:
r = ($1,500,000/$1,300,000) – 1 = 0.1538 = 15.38%
Since the rate of return is greater than the cost of capital, you would construct the
motel.
4.
Investment NPV Return
1) 18,000 18,000 − 10,000
− 10,000 + = $5,000 = 0.80 = 80.0%
1.20 10,000
2) 9,000 9,000 − 5,000
− 5,000 + = $2,500 = 0.80 = 80.0%
1.20 5,000
3) 5,700 5,700 − 5,000
− 5,000 + = −$250 = 0.14 = 14.0%
1.20 5,000
4) 4,000 4,000 − 2,000
− 2,000 + = $1,333.33 = 1.00 = 100.0%
1.20 2,000
a. Investment 1, because it has the highest NPV.
b. Investment 1, because it maximizes shareholders’ wealth.
5. a. NPV = (-50,000 + 30,000) + (30,000/1.07) = $8,037.38
b. NPV = (-50,000 + 30,000) + (30,000/1.10) = $7,272.73
Since, in each case, the NPV is higher than the NPV of the office building
($7,143), accept E. Coli’s offer. You can also think of it another way. The true
opportunity cost of the land is what you could sell it for, i.e., $58,037 (or
$57,273). At that price, the office building has a negative NPV.
6. The opportunity cost of capital is the return earned by investing in the best
alternative investment. This return will not be realized if the investment under
consideration is undertaken. Thus, the two investments must earn at least the
same return. This return rate is the discount rate used in the net present value
calculation.
2
, 7. a. NPV = -$2,000,000 + [$2,000,000 × 1.05)]/(1.05) = $0
b. NPV = -$900,000 + [$900,000 × 1.07]/(1.10) = -$24,545.45
The correct discount rate is 10% because this is the appropriate rate for an
investment with the level of risk inherent in Norman’s nephew’s restaurant. The
NPV is negative because Norman will not earn enough to compensate for the
risk.
c. NPV = -$2,000,000 + [$2,000,000 × 1.12]/(1.12) = $0
d. NPV = -$1,000,000 + ($1,100,000/1.12) = -$17,857.14
Norman should invest in either the risk-free government securities or the risky
stock market, depending on his tolerance for risk. Correctly priced securities
always have an NPV = 0.
8. a. Expected rate of return on project =
$2,100,000 − $ 2,000,000
= 0.05 = 5.0%
$2,000,000
This is equal to the return on the government securities.
b. Expected rate of return on project =
$963,000 − $ 900,000
= 0.07 = 7.0%
$900,000
This is less than the correct 10% rate of return for restaurants with similar
risk.
c. Expected rate of return on project =
$2,240,000 − $2,000,000
= 0.12 = 12.0%
$2,000,000
This is equal to the rate of return in the stock market.
d. Expected rate of return on project =
$1,100,000 − $1,000,000
= 0.10 = 10.0%
$1,000,000
This is less than the return in the equally risky stock market.
3
, CHAPTER 2
Present Value and the Opportunity Cost of Capital
Answers to Practice Questions
1. Let INV = investment required at time t = 0 (i.e., INV = -C0) and let x = rate of
return. Then x is defined as:
x = (C1 – INV)/INV
Therefore:
C1 = INV(1 + x)
It follows that:
NPV = C0 + {C1/(1 + r)}
NPV = -INV + {[INV(1 + x)]/(1 + r)}
NPV = INV {[(1 + x)/(1 + r)] – 1}
a. When x equals r, then:
[(1 + x)/(1 +r)] – 1 = 0
and NPV is zero.
b. When x exceeds r, then:
[(1 + x)/(1 + r)] – 1 > 0
and NPV is positive.
2. The face value of the treasury security is $1,000. If this security earns 5%, then
in one year we will receive $1,050. Thus:
NPV = C0 + [C1/(1 + r)] = -1000 + (1050/1.05) = 0
This is not a surprising result, because 5 percent is the opportunity cost of
capital, i.e., 5 percent is the return available in the capital market. If any
investment earns a rate of return equal to the opportunity cost of capital, the NPV
of that investment is zero.
1
,3. NPV = -$1,300,000 + ($1,500,000/1.10) = +$63,636
Since the NPV is positive, you would construct the motel.
Alternatively, we can compute r as follows:
r = ($1,500,000/$1,300,000) – 1 = 0.1538 = 15.38%
Since the rate of return is greater than the cost of capital, you would construct the
motel.
4.
Investment NPV Return
1) 18,000 18,000 − 10,000
− 10,000 + = $5,000 = 0.80 = 80.0%
1.20 10,000
2) 9,000 9,000 − 5,000
− 5,000 + = $2,500 = 0.80 = 80.0%
1.20 5,000
3) 5,700 5,700 − 5,000
− 5,000 + = −$250 = 0.14 = 14.0%
1.20 5,000
4) 4,000 4,000 − 2,000
− 2,000 + = $1,333.33 = 1.00 = 100.0%
1.20 2,000
a. Investment 1, because it has the highest NPV.
b. Investment 1, because it maximizes shareholders’ wealth.
5. a. NPV = (-50,000 + 30,000) + (30,000/1.07) = $8,037.38
b. NPV = (-50,000 + 30,000) + (30,000/1.10) = $7,272.73
Since, in each case, the NPV is higher than the NPV of the office building
($7,143), accept E. Coli’s offer. You can also think of it another way. The true
opportunity cost of the land is what you could sell it for, i.e., $58,037 (or
$57,273). At that price, the office building has a negative NPV.
6. The opportunity cost of capital is the return earned by investing in the best
alternative investment. This return will not be realized if the investment under
consideration is undertaken. Thus, the two investments must earn at least the
same return. This return rate is the discount rate used in the net present value
calculation.
2
, 7. a. NPV = -$2,000,000 + [$2,000,000 × 1.05)]/(1.05) = $0
b. NPV = -$900,000 + [$900,000 × 1.07]/(1.10) = -$24,545.45
The correct discount rate is 10% because this is the appropriate rate for an
investment with the level of risk inherent in Norman’s nephew’s restaurant. The
NPV is negative because Norman will not earn enough to compensate for the
risk.
c. NPV = -$2,000,000 + [$2,000,000 × 1.12]/(1.12) = $0
d. NPV = -$1,000,000 + ($1,100,000/1.12) = -$17,857.14
Norman should invest in either the risk-free government securities or the risky
stock market, depending on his tolerance for risk. Correctly priced securities
always have an NPV = 0.
8. a. Expected rate of return on project =
$2,100,000 − $ 2,000,000
= 0.05 = 5.0%
$2,000,000
This is equal to the return on the government securities.
b. Expected rate of return on project =
$963,000 − $ 900,000
= 0.07 = 7.0%
$900,000
This is less than the correct 10% rate of return for restaurants with similar
risk.
c. Expected rate of return on project =
$2,240,000 − $2,000,000
= 0.12 = 12.0%
$2,000,000
This is equal to the rate of return in the stock market.
d. Expected rate of return on project =
$1,100,000 − $1,000,000
= 0.10 = 10.0%
$1,000,000
This is less than the return in the equally risky stock market.
3
