Time Value of Money Practice Problems - Solutions
1.
What is the PV of $100 received in: a. b. c. d.
Year 10 at a discount rate of 1 percent. Year 10 at a discount rate of 13 percent. Year 15 at a discount rate of 25 percent. Each of years 1 through 3 at a discount rate of 12 percent. a. PV = $100/1.0110 = $90.53 b. PV = $100/1.1310 = $29.46 c. PV = $100/1.2515 = $ 3.52 d. PV = $100/1.12 + $100/1.122 + $100/1.123 = $240.18
a
b
c
d
N = 10
10
15
3
I= 1
13
25
12
29.46
3.52
240.18
0
0
100
100
100
0
Cpt. PV = 90.53 Pmt = 0 FV = 100 2.
For each of the following, compute the future value: Present Value
Years
Interest Rate
$1,000
4
10%
$2,500
6
12.25%
N = 4
6
I = 10% PV = 1,000 Pmt = 0 Cpt. FV = 1,464.10
Time Value of Money Practice Problems
Future Value
12.25% 2,500 0 5,001.01
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3.
For each of the following, compute the interest rate: Present Value
Years
$5,500
8
$12,000
$7,500
15
$60,000
N = 8
15
Cpt. I = 10.2433% PV = -5,500 Pmt = 0 FV = $12,000 4.
Interest Rate
Future Value
14.8698% -7,500 0 $60,000
For each of the following, compute the number of years: Present Value
Years
Interest Rate
Future Value
$300
5%
450
$27,500
10.125%
$60,000
Cpt. N = 8.3104
8.0891
I= 5
10.125
PV = -300
-27,500
Pmt = 0 FV = 450
Time Value of Money Practice Problems
0 60,000
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5.
A factory costs $800,000. You believe that it will produce a cash flow of $170,000 a year for 10 years. If the opportunity cost of capital is 14 percent, what is the NPV of the factory? What will the factory be worth at the end of five years? The present value of the 10-year stream of cash inflows is:
1 1 PV $170,000 $886,739.6 6 10 0.14 0.14 (1.14) Thus: NPV = –$800,000 + $886,739.66 = +$86,739.66 CF0 = -800,000 CF1 = 170,000
I = 14 F1= 10
Cpt NPV = 86,739.66
At the end of five years, the factory’s value will be the present value of the five remaining $170,000 cash flows:
1 1 $583,623.7 6 PV $170,000 5 0.14 0.14 (1.14) N = 5 I = 14 Cpt. PV = 583,623.76 Pmt = 170,000 FV = 0
Time Value of Money Practice Problems
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6.
A machine costs $380,000 and it is expected to produce the following cash flows.
Year
1
2
3
4
5
6
7
8
9
10
CF ($000S)
50
57
75
80
85
92
92
80
68
50
If the cost of capital is 12 percent, what is the machine’s NPV? 10
NPV t0
Ct $50,000 $57,000 $75,000 $80,000 $85,000 $380,000 t 1.12 (1.12) 1.12 2 1.12 3 1.12 4 1.12 5
$92,000 $92,000 $80,000 $68,000 $50,000 $23,696.15 1.12 6 1.12 7 1.12 8 1.12 9 1.1210
Or CF0
-380
I=
12%
CF1
50
Cpt NPV =
23.69615
CF2
57
CF3
75
CF4
80
CF5
85
CF6
92
CF7
92
CF8
80
CF9
68
CF10
50
Time Value of Money Practice Problems
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7.
Mike Polanski is 30 years of age and his salary next year will be $40,000. Mike forecasts that his salary will increase at a steady rate of 5 percent per year until his retirement at age 60. a. b. c.
If the discount rate is 8 percent, what is the PV of these future salary payments? If Mike saves 5 percent of his salary each year and invests these savings at an interest rate of 8 percent, how much will he have saved by age 60? If Mike plans to spend these savings in even amounts over the subsequent 20 years, how much can he spend each year? a. Let S = salary in year t t
30
PV t 1
St (1.08) t
30
t 1
40,000 (1.05) t 1 30 (40,000/1. 05) t (1.08) t t 1 (1.08 / 1.05)
30
t 1
38,095.24 (1.0286) t
1 1 38,095.24 $760,379.2 1 30 0.0286 0.0286 (1.0286)
PV(salary) x 0.05 = $38,018.96 Future value = $38,018.96 x (1.08)30 = $382,571.75
1 1 PV C t r r (1 r) 1 1 $382,571.7 5 C 20 0.08 0.08 (1.08) 1 1 $38,965.78 C $382,571.7 5 0.08 0.08 (1.08) 20 Or a N = 30 I = 2.86
b
c
30
20
8
8
Cpt. PV = 760,379.21 (0.05)(760,379.21) = 38,018.96 382,571.75 Pmt = 38,095.24 FV = 0
Time Value of Money Practice Problems
0
Cpt Pmt = 38,965.78
Cpt FV = 382,571.75
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8.
A factory costs $400,000. It will produce a cash inflow of $100,000 in year 1, $200,000 in year 2, and $300,000 in year 3. The opportunity cost of capital is 12 percent. Calculate the NPV Period
Present Value
400,000.00
0 1
+100,000/1.12 =
+ 89,285.71
2
+200,000/1.122 =
+159,438.78
3
+300,000/1.123 =
+213,534.07
Total = NPV = $62,258.56 CF0
-400,000
CF1
100,000
CF2
200,000
CF3
300,000
Time Value of Money Practice Problems
I = 12% Cpt NPV = $62,258.56
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9.
Halcyon Lines is considering the purchase of new bulk carrier for $8 million. The forecasted revenues are $5 million a year and operating costs are $4 million. A major refit costing $2 million will be required after both the fifth and tenth years. After 15 years, the ship is expected to be sold for scrap at $1.5 million. If the discount rate is 8 percent, what is the ship’s NPV? We can break this down into several different cash flows, such that the sum of these separate cash flows is the total cash flow. Then, the sum of the present values of the separate cash flows is the present value of the entire project. (All dollar figures are in millions.)
Cost of the ship is $8 million PV = $8 million Revenue is $5 million per year, operating expenses are $4 million. Thus, operating cash flow is $1 million per year for 15 years.
1 1 $8.559 million PV $1 million 15 0.08 0.08 (1.08)
Major refits cost $2 million each, and will occur at times t = 5 and t = 10. PV = ($2 million)/1.085 + ($2 million)/1.0810 = $2.288 million Sale for scrap brings in revenue of $1.5 million at t = 15. PV = $1.5 million/1.0815 = $0.473 million
Adding these present values gives the present value of the entire project: NPV = $8 million + $8.559 million $2.288 million + $0.473 million NPV = $1.256 million CF0 = -8
I=8
CF1 = 1
F1= 4
CF2 = 1 – 2 = -1
F2 = 1
CF3 = 1
F3= 4
CF4 = 1 – 2 = -1
F4 = 1
CF5 = 1
F5= 4
Cpt NPV =-1.2552
CF6 = 1 + 1.5 = 2.5 F6 = 1
Time Value of Money Practice Problems
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10.
As winner of a breakfast cereal competition, you can choose one of the following prizes: a. $100,000 now. b. $180,000 at the end of 5 years. c. $11,400 a year forever. d. $19,000 for each of 10 years. e. $6,500 next year and increasing thereafter by 5 percent a year forever. If the interest rate is 12 percent, which is the most valuable prize? a. PV = $100,000 b. PV = $180,000/1.125 = $102,137 c. PV = $11,400/0.12 = $95,000 d.
1 1 PV $19,000 $107,354 10 0.12 0.12 (1.12)
e. PV = $6,500/(0.12 0.05) = $92,857 Prize (d) is the most valuable because it has the highest present value. a
b
N = 0
5
10
12
12
I = 12
c
d
Cpt. PV = 100,000 102,137 11,400/0.12 = 95,000 107,354 Pmt = 0
0
FV = 100,000 180,000
Time Value of Money Practice Problems
19,000 0
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11.
Siegfried Basset is 65 years of age and has a life expectancy of 12 more years. He wishes to invest $20,000 in an annuity that will make a level payment at the end of each year until his death. If the interest rate is 8 percent, what income can Mr. Basset expect to receive each year? Mr. Basset is buying a security worth $20,000 now. That is its present value. The unknown is the annual payment. Using the present value of an annuity formula, we have:
1 1 PV C t r r (1 r) 1 1 $20,000 C 12 0.08 0.08 (1.08)
C $20,000
1 1 $2,653.90 12 0.08 0.08 (1.08)
N = 12 I= 8 PV = 20,000 Cpt. Pmt = 2,653.90 FV = 0
Time Value of Money Practice Problems
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12.
David and Helen Zhang are saving to buy a boat at the end of five years. If the boat costs $20,000 and they can earn 10 percent a year on their savings, how much do they need to put aside at the end of years 1 through 5? Assume the Zhangs will put aside the same amount each year. One approach to solving this problem is to find the present value of the cost of the boat and then equate that to the present value of the money saved. From this equation, we can solve for the amount to be put aside each year. PV(boat) = $20,000/(1.10)5 = $12,418
1 1 5 0.10 0.10 (1.10)
PV(savings) = Annual savings
Because PV(savings) must equal PV(boat):
1 1 $12,418 5 0.10 0.10 (1.10)
Annual savings
1 1 $3,276 5 0.10 0.10 (1.10)
Annual savings $12,418
Another approach is to find the value of the savings at the time the boat is purchased. Because the amount in the savings account at the end of five years must be the price of the boat, or $20,000, we can solve for the amount to be put aside each year. If x is the amount to be put aside each year, then: x(1.10)4 + x(1.10)3 + x(1.10)2 + x(1.10)1 + x = $20,000 x(1.464 + 1.331 + 1.210 + 1.10 + 1) = $20,000 x(6.105) = $20,000 x = $ 3,276 Or N = 5 I = 10 PV = 0 Cpt. Pmt = 3,275.95 FV = 20,000
Time Value of Money Practice Problems
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13.
Kangaroo Autos is offering free credit on a new $10,000 car. You pay $1,000 down and then $300 a month for the next 30 months. Turtle Motors next door does not offer free credit but will give you $1,000 off the list price. If the rate of interest is 10 percent a year, which company is offering the better deal? The fact that Kangaroo Autos is offering “free credit” tells us what the cash payments are; it does not change the fact that money has time value. A 10 percent annual rate of interest is equivalent to a monthly rate of 0.83 percent: r monthly = r annual /12 = 0.10/12 = 0.0083 = 0.83% The present value of the payments to Kangaroo Autos is:
1 1 $8,938 $1,000 $300 30 0.0083 0.0083 (1.0083) A car from Turtle Motors costs $9,000 cash. Therefore, Kangaroo Autos offers the better deal, i.e., the lower present value of cost. Or N = 30 I = 10/12 = 0.8333 Cpt. PV = 7,934.11 7,934.11 + 1,000 = 8,934.11 Pmt = 300 FV = 0
Time Value of Money Practice Problems
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14.
You are building an office building and construction will require two years. The contractor requires a $120,000 down payment now and commitment of the land with a market value of $50,000. The contractor will be paid $100,000 in 1 year and a final payment of $100,000 at the completion of construction in 2 years. Your real estate advisor estimates the office building will be worth $420,000 when completed. What is the NPV if the cost of capital is 5 percent, 10 percent, and 15 percent?
Time
T=0
Land
-50,000
T=1
T=2
Construction -120,000 -100,000 -100,000 Payoff
420,000
Total CFs -170,000 -100,000
320,000
The NPVs are: at 5 percent
NPV $170,000
$100,000 $320,000 $25,011 1.05 (1.05) 2
at 10 percent
NPV $170,000
$100,000 320,000 $3,554 1.10 (1.10) 2
at 15 percent
NPV $170,000
$100,000 320,000 $14,991 1.15 (1.15) 2
The figure below shows that the project has zero NPV at about 11 percent. As a check, NPV at 11 percent is:
NPV $170,000
$100,000 320,000 $371 1.11 (1.11) 2
CF0
-170,000
I=
5%
CF1
-100,000
Cpt NPV =
$25,011.34 $3,553.72
CF2
320,000
Time Value of Money Practice Problems
10%
15% -$14,990.55
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