Chapter 4: Net Prent Value
4.1 a. $1,000 1.0510 = $1,628.89
b. $1,000 1.0710 = $1,967.15
c. $1,000 1.0520 = $2,653.30
d. Interest compounds on the Interest already earned. Therefore, the interest earned in part c, $1,653.30, is more than double the amount earned in part a, $628.89.
4.2 a. $1,000 / 1.17 = $513.16
b. $2,000 / 1.1 = $1,818.18
c. $500 / 1.18 = $233.25
4.3 You can make your decision by computing either the prent value of the $2,000 that
you can receive in ten years, or the future value of the $1,000 that you can receive now.
Prent value: $2,000 / 1.0810 = $926.39
Future value: $1,000 1.0810 = $2,158.93
Either calculation indicates you should take the $1,000 now.
4.4 Since this bond has no interim coupon payments, its prent value is simply the prent value of the $1,000 that will be received in 25 years. Note: As will be discusd in the next chapter, the prent value of the payments associated with a bond is the price of that bond.
PV = $1,000 /1.125 = $92.30
4.5 PV = $1,500,000 / 1.0827 = $187,780.23
4.6 a. At a discount rate of zero, the future value and prent value are always the sa
me. Remember, FV = PV <1 + r> t. If r = 0, then the formula reduces to FV = PV. Therefore, the values of the options are $10,000 and $20,000, respectively. You should choo the cond option.
b. Option one: $10,000 / 1.1 = $9,090.91
Option two: $20,000 / 1.15 = $12,418.43
Choo the cond option.
c. Option one: $10,000 / 1.2 = $8,333.33
Option two: $20,000 / 1.25 = $8,037.55
Choo the first option.
d. You are indifferent at the rate that equates the PVs of the two alternatives. You know that rate must fall between 10% and 20% becau the option you would choo differs at the rates. Let r be the discount rate that makes you indifferent between the options.
$10,000 / <1 + r> = $20,000 / <1 + r>5
<1 + r>4 = $20,000 / $10,000 = 2
1 + r = 1.18921
r = 0.18921 = 18.921%
4.7 PV of Jones’ offer = $150,000 / <1.1>3 = $112,697.22
Since the PV of Jones’ offer is less than Smiths’ offer, $115,000, you should choo Smiths’ offer.
4.8 a. P0 = $1,000 / 1.0820 = $214.55
b. P10 = P0 <1.08>10 = $463.20
c. P15 = P0 <1.08>15 = $680.59
4.9 The $1,000 that you place in the account at the end of the first year will earn interest for six years. The $1,000 that you place in the account at the end of the cond year will earn interest for five years, etc. Thus, the account will have a balance of
$1,000 <1.12>6 + $1,000 <1.12>5 + $1,000 <1.12>4 + $1,000 <1.12>3
= $6,714.61
4.10 PV = $5,000,000 / 1.1210 = $1,609,866.18
4.11 a. The cost of investment is $900,000.
PV of cash inflows = $120,000 / 1.12 + $250,000 / 1.122 + $800,000 / 1.123
= $875,865.52
Since the PV of cash inflows is less than the cost of investment, you should not make the investment.
b. NPV = -$900,000 + $875,865.52
= -$24,134.48
c. NPV = -$900,000 + $120,000 / 1.11 + $250,000 / 1.112 + $800,000 / 1.113
= $-4,033.18
Since the NPV is still negative, you should not make the investment.
4.12 NPV = -<$340,000 + $10,000> + <$100,000 - $10,000> / 1.1
+ $90,000 / 1.12 + $90,000 / 1.13 + $90,000 / 1.14 + $100,000 / 1.15
= -$2,619.98
Since the NPV is negative, you should not buy it.
If the relevant cost of capital is 9 percent,
NPV = -$350,000 + $90,000 / 1.09 + $90,000 / 1.092 + $90,000 / 1.093
+ $90,000 / 1.094 + $100,000 / 1.095
= $6,567.93
Since the NPV is positive, you should buy it.
4.13 a. Profit = PV of revenue - Cost = NPV
NPV = $90,000 / 1.15 - $60,000 = -$4,117.08
No, the firm will not make a profit.
b. Find r that makes zero NPV.
$90,000 / <1+r>5 - $60,000 = $0
<1+r>5 = 1.5
r = 0.08447 = 8.447%
4.14 The future value of the decision to own your car for one year is the sum of the trade-in value and the benefit from owning the car. Therefore, the PV of the decision to own the car for one year is
$3,000 / 1.12 + $1,000 / 1.12 = $3,571.43
Since the PV of the roommate’s offer, $3,500, is lower than the aunt’s offer, you should accept aunt’s offer.
4.15 a. $1.000 <1.08>3 = $1,259.71
b. $1,000 [1 + <0.08 / 2>]2 3 = $1,000 <1.04>6 = $1,265.32
c. $1,000 [1 + <0.08 / 12>]12 3 = $1,000 <1.00667>36 = $1,270.24
d. $1,000 e0.08 3 = $1,271.25
e. The future value increas becau of the compounding. The account is earning interest on interest. Esntially, the interest is added to the account balance at the end of every compounding period. During the next period, the account earns interest on the new balance. When the compounding period shortens, the balance that earns interest is rising faster.
