1. Time Value of Money Problems
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
90.53
29.46
3.52
240.18
0
0
0
100
100
100
100
0
Cpt. PV =
Pmt =
FV =
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=
I=
PV =
Pmt =
Cpt. FV =
4
6
10%
12.25%
1,000
2,500
0
0
1,464.10
Future Value
5,001.01
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2. 3.
For each of the following, compute the interest rate:
Present Value
Years
$5,500
8
$12,000
$7,500
15
$60,000
N=
Cpt. I =
PV =
Pmt =
FV =
4.
8
14.8698%
-5,500
-7,500
0
0
$12,000
Future Value
15
10.2433%
Interest Rate
$60,000
For each of the following, compute the number of years:
Present Value
Interest Rate
Future Value
$300
5%
450
$27,500
10.125%
$60,000
Cpt. N =
I=
PV =
Pmt =
FV =
Years
8.3104
8.0891
5
10.125
-300
-27,500
0
0
450
60,000
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3. 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
PV = $170,000 × 
−
= $583,623.7 6
5 
0.14 × (1.14) 
 0.14
N=
5
I=
14
Cpt. PV =
Pmt =
FV =
583,623.76
170,000
0
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4. 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 = ∑
t =0
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
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5. 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 St = salary in year t
30
St
40,000 (1.05) t −1
=∑
(1.08) t
(1.08) t
t =1
30
PV = ∑
t =1
30
=∑
t =1
30
(40,000/1. 05)
38,095.24
=∑
t
(1.08 / 1.05)
(1.0286) t
t =1
 1

1
= 38,095.24 × 
−
= $760,379.2 1
30 
0.0286 × (1.0286) 
0.0286
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 × (1 + r) 
r
 1

1
$382,571.7 5 = C × 
−

0.08
0.08 × (1.08) 20 

 1

1

 = $38,965.78
−
0.08
0.08 × (1.08) 20 


C = $382,571.7 5
Or
a
N=
I=
b
c
30
30
20
2.86
8
8
Cpt. PV = 760,379.2
1
Pmt =
FV =
(0.05)(760,379.21) = 38,018.96 382,571.75
38,095.24
0
0
Cpt FV = 382,571.75
tvmreviewlecture-131226185711-phpapp02.doc
Cpt Pmt = 38,965.78
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6. 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
I=
CF1
100,000
Cpt NPV =
CF2
200,000
CF3
300,000
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12%
$62,258.56
Page 6

7. 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
PV = $1 million × 
−
= $8.559 million
15 
0.08 × (1.08) 
0.08

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
Cpt NPV =1.2552
F5= 4
CF6 = 1 + 1.5 = 2.5 F6 = 1
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8. 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 × (1.12) 
0.12
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
I=
12
12
12
Cpt. PV = 100,00
0
Pmt =
FV =
102,13
7
c
d
11,400/0.12 = 95,000 107,354
0
0
19,000
100,00
0
180,00
0
0
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9. 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 × (1 + r) 
r
 1

1
$20,000 = C × 
−
12 
0.08 × (1.08) 
0.08
 1

1

 = $2,653.90
−
0.08
0.08 × (1.08)12 


C = $20,000
N=
12
I=
8
PV =
20,000
Cpt. Pmt = 2,653.90
FV =
0
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10. 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
−
PV(savings) = Annual savings × 

0.10 × (1.10) 5 
0.10
Because PV(savings) must equal PV(boat):
 1

1
−
= $12,418
5 
0.10 × (1.10) 
0.10
Annual savings × 
 1
1

−
= $3,276
Annual savings = $12,418 
5 
 0.10 0.10 × (1.10) 
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
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11. 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:
rmonthly = rannual /12 = 0.10/12 = 0.0083 = 0.83%
The present value of the payments to Kangaroo Autos is:
 1

1
$1,000 + $300 × 
−
= $8,938
30 
0.0083 × (1.0083) 
 0.0083
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=
I=
30
10/12 = 0.8333
Cpt. PV = 7,934.11 → 7,934.11 + 1,000 = 8,934.11
Pmt =
FV =
300
0
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12. 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? Draw a NPV profile. At what rate would the NPV be
zero? Check your answer.
Time
T=0
Land
-50,000
Construction
120,000
T=1
T=2
- -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%
10%
15%
CF1
-100,000
Cpt NPV =
$25,011.3
4
$3,553.72
-$14,990.55
CF2
320,000
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13. NPV
30
20
10
0
-10
-20
0.05
0.10
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0.15 Rate of Interest
Page 13

14. 15.
You have just read an advertisement stating, “Pay us $100 a year for ten years and we will
pay you $100 a year thereafter in perpetuity.” If this is a fair deal, what is the rate of
interest?
One way to approach this problem is to solve for the present value of:
(1)
$100 per year for 10 years, and
(2)
$100 per year in perpetuity, with the first cash flow at year 11.
If this is a fair deal, these present values must be equal, and thus we can solve for the
interest rate (r).
The present value of $100 per year for 10 years is:
1

1
PV = $100 × −
10 
(r) × (1 + r) 
r
The present value, as of year 10, of $100 per year forever, with the first payment in
year 11, is: PV10 = $100/r
At t = 0, the present value of PV10 is:

 $100 
1
PV = 
×

10  
 (1 + r)   r 
Equating these two expressions for present value, we have:
1
 
 $100 
1
1
$100 × −
=
×

10 
10  
(r) × (1 + r)   (1 + r)   r 
r
Using trial and error or algebraic solution, we find that r = 7.18%.
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15. 16.
Which would you prefer?
a.
b.
c.
An investment paying interest of 12 percent compounded annually.
An investment paying interest of 11.7 percent compounded semiannually.
An investment paying 11.5 percent compounded continuously.
Assume the amount invested is one dollar.
Let A represent the investment at 12 percent, compounded annually.
Let B represent the investment at 11.7 percent, compounded semiannually.
Let C represent the investment at 11.5 percent, compounded continuously.
After one year:
FVA = $1 × (1 + 0.12)1
FVB = $1 × (1 + 0.0585)
FVC = $1 × e(0.115 × 1)
After five years:
= $1.1200
2
= $1.1204
= $1.1219
FVA = $1 × (1 + 0.12)5
FVB = $1 × (1 + 0.0585)
FVC = $1 × e(0.115 × 5)
After twenty years:
= $1.7623
10
= $1.7657
= $1.7771
FVA = $1 × (1 + 0.12)20
FVB = $1 × (1 + 0.0585)
= $9.6463
40
= $9.7193
FVC = $1 × e(0.115 × 20) = $9.9742
The preferred investment is C.
b
Nom
Cpt. Eff.
C/Y
c
11.7
Nom
c
11.5
Nom
11.5
12.0422
%
Cpt.
Eff.
12.185
%
Cpt.
Eff.
12.186%
2
C/Y
365
C/Y
730
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16. 17.
Fill in the blanks in the following table:
Nominal
Interest Rate
Inflation Rate
6.00%
1.00%
Real Interest Rate
10.00%
12.00%
9.00%
3.00%
1 + rnominal = (1 + rreal) × (1 + inflation rate)
Nominal
Interest Rate
6.00%
23.20%
9.00%
Inflation Rate
Real Interest Rate
1.00%
10.00%
5.83%
4.95%
12.00%
3.00%
1.06 = (1 + rreal)(1.01) → rreal = .0495
1 + rnominal = (1.1)(1.12) → rnominal = 0.2320
1.09 = (1.03) (1 + inflation rate) → inflation rate = 0.0583
18.
A leasing contract calls for an immediate payment of $100,000 and nine subsequent
$100,000 semiannual payments at six-month intervals. What is the PV of these payments if
the discount rate is 8 percent?
Because the cash flows occur every six months, we use a six-month discount rate, here
8%/2, or 4%. Thus:
 1

1
PV = $100,000 + $100,000 × 
−
= $843,533
9 
 0.04 0.04 × (1.04) 
N=
I=
9
8/2 = 4
Cpt. PV = 743,533 → 743,533 + 100,000 =843,533
Pmt =
FV =
100,000
0
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17. 19.
You estimate that by the time you retire in 35 years, you will have accumulated savings of
$2 million. If the interest rate is 8 percent and you live 15 years after retirement, what
annual level of expenditure will these savings support?
Unfortunately, inflation will eat into the value of your retirement income. Assume a 4
percent inflation rate and work out a spending program for your retirement that will allow
you to maintain a level real expenditure during retirement.
This is an annuity problem with the present value of the annuity equal to $2 million (as of
your retirement date), and the interest rate equal to 8 percent, with 15 time periods. Thus,
your annual level of expenditure (C) is determined as follows:
1

1
PV = C ×  −
t 
r × (1 + r) 
r
N=
I=
 1

1
$2,000,000 = C × 
−
15 
0.08
0.08 × (1.08) 

C = $2,000,000
or
15
8
PV =
2,000,000
Cpt. Pmt = 233,659
 1

1

 = $233,659
−
0.08
0.08 × (1.08)15 


FV =
0
With an inflation rate of 4 percent per year, we will still accumulate $2 million as of our
retirement date. However, because we want to spend a constant amount per year in real
terms (R, constant for all t), the nominal amount (C t ) must increase each year. For each
year t:
R = C t /(1 + inflation rate)t
Therefore:
PV [all C t ] = PV [all R × (1 + inflation rate)t] = $2,000,000
 (1 + 0 .04)1 (1 + 0.04)2
(1 + 0 .04)15 
R×
+
+. . . +
 = $2,000,000
1
(1 + 0 .08)2
(1+ 0.08)15 
 (1+ 0.08)
R × [0.9630 + 0.9273 + . . . + 0.5677] = $2,000,000
R × 11.2390 = $2,000,000
R = $177,952
Thus C1 = ($177,952 × 1.04) = $185,070, C2 = $192,473, etc.
N=
I=
PV =
15
Remember:
3.8462
1 + rnominal = (1 + rreal) × (1 + inflation rate)
2,000,000
1.08 = (1 + rreal)(1.04) → rreal = .038462
Cpt. Pmt = 177,952
20.
FV = 0
You are considering the purchase of an apartment complex that currently generates a net
cash flow of $400,000 per year. You normally demand a 10 percent rate of return on such
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Page 17

18. investments. Future cash flows are expected to grow with inflation at 4 percent year from
today’s level. How much would you be willing to pay for the complex if it:
a.
b.
Will produce cash flows forever?
Will have to be torn down in 20 years? Assume that the site will be worth $5 million
at the time of demolition costs. (The $5 million includes 20 years’ inflation.)
Now calculate the real discount rate corresponding to the 10 percent nominal rate. Redo the
calculations for part (a) and (b) using real cash flows. (Your answers should not change.)
a.
First, with nominal cash flows:
The nominal cash flows form a growing perpetuity at the rate of inflation, 4%. Thus,
the cash flow in one year will be $416,000 and:
b.
PV = $416,000/(0.10 – 0.04) = $6,933,333
The nominal cash flows form a growing annuity for 20 years, with an additional
payment of $5 million at year 20:
 1
1
( 1 + g)t  $5,000 ,000
PV = C1 
−
×
+
( 1 + r)t 
( 1 + r)t
(r-g) (r − g)

1
1
( 1.04 ) 20  $5,000 ,000
= $ 416 ,000 
−
×
+
10
10
( 1. ) 20 
10
( 1. )20
10
( 0. -0.04 ) ( 0. -0.04 )
=$5,418,389
Second, with real cash flows:
a. Here, the real cash flows are $400,000 per year in perpetuity, and we can find the real
rate (r) by solving the following equation:
b.
(1 + 0.10) = (1 + r)(1 + 0.04) ⇒ r = 0.05769 = 5.769%
PV = $400,000/0.057692 = $6,933,333
Now, the real cash flows are $400,000 per year for 20 years and $5 million (nominal)
in 20 years. In real terms, the $5 million dollar payment is:
$5,000,000/(1.04)20 = $2,281,935
Thus, the present value of the project is:

 $ 2 ,281,935
1
1
PV = $ 400 ,000 × 
−
+
= $5,418,510
20 
20
 ( 0.05769 ) ( 0.05769 )( 1.05769 )  ( 1.05769 )
[As noted in the statement of the problem, the answers agree, to within rounding errors.]
N=
I=
20
5.7692
Cpt. PV = 5,418,389
Pmt =
21.
400,000
FV = 2,281,935
Vernal Pool, a self-employed herpetologist, wants to put aside a fixed fraction of her annual
income as savings for retirement. Ms. Pool is now 40 years old and makes $40,000 a year.
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19. She expects her income to increase by 2 percentage points over inflation (e. g., 4 percent
inflation means a 6 percent increase in income). She wants to accumulate $500,000 in real
terms to retire at age 70. What fraction of her income does she need to set aside? Assume
her retirement funds are conservatively invested at an expected real rate of return of 5
percent a year. Ignore taxes.
Let x be the fraction of Ms. Pool’s salary to be set aside each year. At any point in the
future, t, her real income will be:
($40,000)(1 + 0.02) t
The real amount saved each year will be:
(x)($40,000)(1 + 0.02) t
The present value of this amount is:
( x )( $40,000)(1 + 0.02 ) t
(1 + 0.05) t
Ms. Pool wants to have $500,000, in real terms, 30 years from now. The present value of
this amount (at a real rate of 5 percent) is:
$500,000/(1 + 0.05)30
Thus:
30
( x )( $40,000)(1.02)
$500,000
=∑
30
(1.05)
(1.05) t
t =1
30
$500,000
( $40,000)(1.02)
= ( x )∑
30
(1.05)
(1.05) t
t =1
t
t
$115,688.72 = (x)($790,012.82)
x = 0.146
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20. 22.
You own a pipeline which will generate a $2 million cash flow over the coming year. The
pipeline’s operating costs are negligible, and it is expected to last for a very long time.
Unfortunately, the volume of oil shipped is declining, and cash flows are expected to decline
by 4 percent per year. The discount rate is 10 percent.
a.
b.
What is the PV of the pipeline’s cash flows if its cash flows are assumed to last
forever?
What is the PV of the cash flows if the pipeline is scrapped after 20 years?
a.
PV =
b.
This calls for the growing perpetuity formula with a negative growth rate
(g = –0.04):
$ 2 million
$ 2 million
=
= $14.29 million
0. − ( −0.04 )
10
0.
14
The pipeline’s value at year 20 (i.e., at t = 20), assuming its cash flows last
forever, is:
PV20 =
C 21
C ( 1 + g) 20
= 1
r −g
r −g
With C1 = $2 million, g = –0.04, and r = 0.10:
PV20 =
($ 2 million) ×( 1 − 0.04 )20
$0.884 million
=
= $ 6.314 million
0.
14
0.
14
Next, we convert this amount to PV today, and subtract it from the answer to Part
(a):
PV = $14.29 million −
$ 6.314 million
= $13.35 million
( 1. )20
10
Most of these problems and part of the solutions are from Chapter 3 in Principles of
Corporate Finance by Brealey, Myers, and Allen 8th edition. Part of the solutions were
generated by Dan Ervin
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