Fundamentals of corporate finance brealey chapter 04 the time value of money

Solutions to Chapter 4
The Time Value of Money
Note: Unless otherwise stated, assume that cash flows occur at the end of each year.
1.

a.
b.
c.
d.

100/(1.08)10
100/(1.08)20
100/(1.04)10
100/(1.04)20

2.

a.
b.
c.
d.

100  (1.08)10
100  (1.08)20
100  (1.04)10
100  (1.04)20

3.

With simple interest, you earn 4% of $1000, or $40 each year. There is no interest on
interest. After 10 years, you earn total interest of $400, and your account

accumulates to $1400. With compound interest, your account grows to 1000 
(1.04)10 = $1480. Therefore $80 is interest on interest.

4.

FV = 700

=
=
=
=

$46.32
$21.45
$67.56
$45.64
=
=
=
=

$215.89
$466.10
$148.02
$219.11

PV = 700/(1.05)5 = $548.47

5.
Present Value

Years

Future Value

Interest Rate*

a.

$400

11

$684

5% = ()1/11 – 1

b.

$183

4

$249

8% = ()1/4 – 1

c.

$300

7

$300

0% = ()1/7– 1

To find the interest rate, we rearrange the equation
FV = PV  (1 + r)n to conclude that r = ()1/n - 1
To use a financial calculator for (a) enter PV= (-)400, FV = 684, PMT = 0, n = 11
and compute the interest rate.

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6.

You should compare the present values of the two annuities.

a.
b.

Discount
Rate
5%
20%

Present Value of
10-year, $1000 annuity

7721.73
4192.47

Present Value of
15-year, $800 annuity
8303.73
3740.38

When the interest rate is low, as in part (a), the longer (i.e., 15-year) but
smaller annuity is more valuable because the impact of discounting on the
present value of future payments is less severe. When the interest rate is
high, as in part (b), the shorter but higher annuity is more valuable. In this
case, with the 20 percent interest rate, the present value of more distant
payments is substantially reduced, making it better to take the shorter but
higher annuity.
7.

PV = 200/1.05 + 400/1.052 + 300/1.053
= 190.48 + 362.81 + 259.15 = $812.44

8.

In these problems, you can either solve the equation provided directly, or you can
use your financial calculator setting PV = ()400, FV = 1000, PMT = 0, i as
specified by the problem. Then compute n on the calculator.
a.

400  (1 + .04)t = 1,000

t = 23.36 periods

b.

400  (1 + .08)t = 1,000

t = 11.91 periods

c.

400  (1 + .16)t = 1,000

t = 6.17 periods

Note: To solve directly, use the natural log function, ln. For example, for (a),
ln[ (1.04)t ] = ln[1000/400]
t × ln[1.04] = 0.91629
t = 0.91629/.03922 = 23.36 period.
Using the calculator: PV = (-)400, FV = 1000, i = 4, compute n to get n = 23.36.
9.

a.
b.
c.
d.

PV = 100 × PVIFA(.08,10) = 100 × 6.7101 = 671.01
PV = 100 × PVIFA(.08,20) = 100 × 9.8181 = 981.81
PV = 100 × PVIFA(.04,10) = 100 × 8.1109 = 811.09
PV = 100 × PVIFA(.04,20) = 100 × 13.5903 = 1,359.03

10.

a.
b.
c.
d.

FV = 100 × FVIFA(.08,10) = 100 × 14.4866 = 1,448.66
FV = 100 × FVIFA(.08,20) = 100 × 45.7620 = 4,576.20
FV = 100 × FVIFA(.04,10) = 100 × 12.0061 = 1,200.61
FV = 100 × FVIFA(.04,20) = 100 × 29.7781 = 2,977.81

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11.
APR

Compounding
Period

Per Period
Rate,
APR/m

Effective annual rate

12

a.

12%

1 month (m = 12/yr) .12/12 =.01

1.01  1 = .1268 = 12.68%

b.

8%

3 months (m = 4/yr) .08/4 = .02

1.02  1 = .0824 = 8.24%

c.

10%

6 months (m = 2/yr) .10/2 = .05

1.05  1 = .1025 = 10.25%

4

2

12.
Effective

Annual Rate,
EAR

Compounding
Period

Number of
Periods per
year, m

a.

10.0%

1 month

12

1.1 – 1
= .008

b.

6.09%

6 months

2

1.0609  1

= .03

c.

8.24%

3 months

4

1.0824  1
= .02

Per period rate,
(1+EAR)1/m -1

APR, m × per
period rate

1/12

12×.008 = .096
= 9.6%

1/2

1/4

2×.03 = .06
= 6%

4×.02 = .08
= 8%

13.

We need to find the value of n for which 1.08n = 2. You can solve to find that
n = 9.01 years. On a financial calculator you would enter PV = ()1, FV = 2,
PMT = 0, i = 8 and then compute n.

14.

Semiannual compounding means that the 8.5 percent loan really carries interest of
4.25 percent per half year. Similarly, the 8.4 percent loan has a monthly rate of
.7 percent.
APR

Period

m

Effective annual rate
= (1 + per period rate) m – 1

8.5%
8.4%

6 months
1 month

2

12

(1.0425)2  1 = .0868 = 8.68%
(1.007)12  1 = .0873 = 8.73%

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Choose the 8.5 percent loan for its slightly lower effective rate.

15.

APR = 1%  52 = 52%
52

EAR = (1 + .01)  1 = .6777 = 67.77%

16.

Our answer assumes that the investment was made at the beginning of 1900 and
now it is the end of 2008. Thus the investment was for 106 years (2008 – 1900 +
1).
a.
b.

17.

1000  (1.05)109 = $204,001.61
PV  (1.05)109 = 1,000,000 implies that PV = $4,901.92

$1000  1.05 = $1050.00
$1050  1.05 = $1102.50
$52.50

First-year interest = $50
Second-year interest = $1102.50  $1050 =

After 9 years, your account has grown to 1000  (1.05)9 = $1551.33
After 10 years, your account has grown to 1000  (1.05)10 = $1628.89
Interest earned in tenth year = $1628.89  $1551.33 = $77.56
18.

Method 1: If you earned simple interest (without compounding) then the total
growth in your account after 25 years would be 4% per year  25 years = 100%,
and your money would double. With compound interest, your money would
grow faster, and therefore would require less than 25 years to double.
Method 2: Another quick way to answer the question is with the Rule of 72.
Dividing 72 by 4 gives 18 years, which is less than 25.
The exact answer is 17.673 years, found by solving 2000 = 1000  (1.04)n. [On
your calculator, input PV = (-) 1000, FV = 2000, i = 4, PMT = 0, and compute
the number of periods.]

19.

We solve 422.21  (1 + r)10 = 1000. This implies that r = 9%. [On your
calculator, input PV = (-)422.21, FV = 1000, n = 10, PMT = 0, and compute the
interest rate.]

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20.

The number of payment periods: n = 12 × 4 = 48.
If the payment is denoted PMT, then
PMT  annuity factor( %, 48 periods) = 8,000
PMT = $202.90.
The monthly interest rate is 10/12 = .8333 percent. Therefore, the effective annual
interest rate on the loan is (1.008333) 12  1 = .1047 = 10.47 percent.

21.

a.

PV = 100  annuity factor(6%, 3 periods)
= 100  = $267.30

b.

22.

a.

If the payment stream is deferred by an extra year, each payment will be
discounted by an additional factor of 1.06. Therefore, the present value is
reduced by a factor of 1.06 to 267.30/1.06 = $252.17.
This is an annuity problem with PV = (-)80,000, PMT = 600, FV = 0,
n = 20  12 = 240 months. Use a financial calculator to solve for i, the

monthly rate on this annuity: i = .5479%.
12

EAR = (1 +.005479)  1 = .06776 = 6.776%
APR = 12 × monthly interest rate = 12 × .5479%
= 6.5748%, compounded monthly

23.

b.

Again use a financial calculator and enter n = 240, i = .5%, FV = 0,
PV = ()80,000 and compute PMT = $573.14

a.

With PV = 9,000 and FV = 10,000, the annual interest rate is defined by
9,000  (1 + r) = 10,000,
which implies that r = 11.11%.

b.

Your present value is 10,000 (1  d), and the future value you pay back is 10,000.
Therefore, the annual interest rate is determined by:
PV  (1 + r) = FV

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[10,000 (1 – d)]  (1 + r) = 10,000
1+r=  r= 1=>d
Since 0 < d < 1, then 1 – d < 1 and d/(1 – d) > d. So r must be greater than d.
c.

With a discount interest loan, the discount is calculated as a fraction of the future
value of the loan. In fact, the proper way to compute the interest rate is as a
fraction of the funds borrowed. Since PV is less than FV, the interest payment is a
smaller fraction of the future value of the loan than it is of the present value.
Thus, the true interest rate exceeds the stated discount factor of the loan.

24.

If we assume cash flows come at the end of each period (ordinary annuity) when
in fact they actually come at the beginning (annuity due), we discount each cash
flow by one period too many. Therefore we can obtain the PV of an annuity due
by multiplying the PV of an ordinary annuity by (1 + r). Similarly, the FV of an
annuity due also equals the FV of an ordinary annuity times (1 + r). Because
each cash flow comes at the beginning of the period, it has an extra period to earn
interest compared to an ordinary annuity.

25.

a.

Solve for i in the following equation:
10,000 = 275 × PVIFA(i, 48)
Using the calculator, set PV = -10,000, PMT = 275, FV = 0, n = 48 and solve for i
i= 1.19544% per month
APR = 12 × 1.19544% = 14.3453%

EAR = (1 + .0119544)12 – 1 = .153271, or 15.3271%

b.

Annual payment = 12 × 275 = 3,300
Repeat the steps in (a) to find the EAR of this car loan to see which loan is
charging the lower interest rate:
Solve for i in the following equation:
10,000 = 3,330 × PVIFA(i, 4)
Using the calculator, set PV = -10,000, PMT = 3,300, FV = 0, n = 4 and solve for i
i= 12.11% per month
Little Bank's loan interest rate of 12.11% is less than the EAR of 15.53% on Big
Bank's loan. With a lower interest rate, Little Bank's loan is better.

c.

Find the annual loan payment, P, such that
10,000 = X × PVIFA(15.3271%, 4)
Using the calculator, set PV = -10,000, FV = 0, n = 4, i = 15.3271 and solve for
PMT = $3,525.86. By comparison, 12 times $275 per month is $3,300. The
annual payment on a 4-year loan equivalent to $275 per month for 48 months is
greater than 12 times the monthly payment of $275 because of the benefit of
delaying payment to the end of each year. The borrower gets to delay payment
and therefore is better off. If Little Bank doesn't charge at least $3,525.86
annually, it earns less on its loan than Big Bank earns on its loan.

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26.

27.

a.

Compare the present value of the lease to cost of buying the truck.
PV lease = 8,000 × PVIFA(7%, 6) = -$38,132.32
It is cheaper to lease than buy because by leasing the truck will cost only
$38,132.32, rather than $40,000. Of course, the crucial assumption here is that
the truck is worthless after 6 years. If you buy the truck, you can still operate it
after 6 years. If you lease it, you must return the truck and replace it.

b.

If the lease payments are payable at the start of each year, then the present value
of the lease payments are:
PV annuity due lease = 8,000 + 8,000 × PVIFA(7%, 5) = 8,000 + 32,801.58 =
$40,801.58. Note too that PV of an annuity due = PV of ordinary annuity  (1 +
r). Therefore, with immediate payment, the value of the lease payments increases
from its value in the previous problem to $38,132  1.07 = $40,801 which is
greater than $40,000 (the cost of buying a truck). Therefore, if the first payment
on the lease is due immediately, it is cheaper to buy the truck than to lease it.

a.

Compare the PV of the payments. Assume the product sells for $100.
Installment plan:
Down payment = .25 × 100 = 25
Three installments of .25 × 100 = 25

PV = 25 + 25  annuity factor(6%, 3 years) = $91.83.
Pay in full: Payment net of discount = $90. Choose this payment plan for its
lower present value of payments.
Note: The pay-in-full payment plan will have the lowest present value of
payments, regardless of the chosen product price.

b.

28.

Installment plan: PV = 25  annuity factor(6%, 4 years) = $86.63. Now the
installment plan offers the lower present value of payments.

a.

PMT  annuity factor(12%, 5 years) = 1000
PMT  3.6048 = 1000
PMT = $277.41

b.

If the first payment is made immediately instead of in a year, the annuity
factor will be greater by a factor of 1.12. Therefore
PMT  (3.6048  1.12) = 1000.
PMT = $247.69.

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29.

This problem can be approached in two steps. First, find the PV of the $10,000,
10-year annuity as of year 3, when the first payment is exactly one year away
(and is therefore an ordinary annuity). Then discount the value back to today.
Using a financial calculator,
1) PMT = 10,000; FV = 0; n = 10; i = 6%.
Compute PV3 = $73,600.87
2) PV0 = = = $61,796.71
A second way to solve the problem is the take the difference between a 13-year
annuity and a 3-year annuity, valued as of the end of year 0:
PV of delayed annuity = 10,000 × PVIFA(6%,13) – 10,000 × PVIFA(6%,3)
= 10,000 × (8.852683 – 2.673012) = 10,000 × 6.179671 = $61,796.71

30.

Note: Assume that this is a Canadian mortgage.
The monthly payment is based on a $175,000 loan with a 300-month (12 × 25
years) amortization. The posted interest rate of 6 percent has a 6-month
compounding period. Its EAR is (1 + .06/2) 2 – 1 = .0609, or 6.09%. The monthly
interest rate equivalent to 6.09% annual is (1.0609) 1/12 – 1 = .004939, or 0.4939%.
PMT  annuity factor(.4939%, 300) = 175,000
PMT = $1,119.71.
When the mortgage expires in 5 years, there will be 20 years remaining in the
amortization period, or 240 months. The loan balance in five years will be the
present value of the 240 payments:
Loan balance in 5 years = $1,119.71  Annuity factor (.4939%, 240 periods)
= $157,208.

31.

The EAR of the posted 7% rate is (1 + .07/2)2 – 1 = .071225. The monthly interest
rate equivalent is (1.071225)1/12 – 1 = .00575, or 0.575%. The payment on the
mortgage is computed as follows:
PMT  annuity factor (.575%, 300 periods) = 350,000
PMT = $2,451.44 per month.
If you pay the monthly mortgage payment in two equal installments, you will pay
$2,451.44/2, or $1,225.72 every two weeks. Thus each year you make 26 payments.
The bi-weekly equivalent of the 7% posted interest rate is (1.071225) 1/26 – 1 = .
002649, or .2649% every two weeks. Now calculate the number of periods it will
take to pay off the mortgage:

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$1,225.72  Annuity factor (.2649%, n periods) = $350,000
Using the calculator: PMT = 1,225.49, PV = (-)350,000, i = .2649 and compute n =
533.84. This is the number of bi-weekly periods. Divide by 26 to get the number of
years: 533.84/26 = 20.5 years. If you pay bi-weekly, the mortgage is paid off 5.5 years
sooner than if you pay monthly.
32.

a.

Input PV = (-)1,000, FV = 0, i = 8%, n = 4, compute PMT which equals
$301.92

b.
Time

0
1
2
3
4
c.

33.

Loan
Balance
$1,000.00
$778.08
$538.41
$279.56
0

Year End Interest
Due on Balance
$ 80.00
$62.25
$43.07
$22.37
0

Year End
Payment
$301.92
$301.92
$301.92

$301.92
—

Amortization
of Loan
$221.92
$239.67
$258.85
$279.56
—

301.92  annuity factor (8%, 3 years) = 301.92 × 2.5771 = $778.08, which equals
the loan balance after one year.

The loan repayment is an annuity with present value $4248.68. Payments are
made monthly, and the monthly interest rate is 1%. We need to equate this
expression to the amount borrowed, $4248.68, and solve for the number of
months, n. [On your calculator, input PV = () 4248.68, FV = 0, i = 1%, PMT =
200, and compute n.] The solution is n = 24 months, or 2 years.
The effective annual rate on the loan is (1.01) 12  1 = .1268 = 12.68%

34.

The present value of the $2 million, 20-year annuity, discounted at 8%, is
$19,636,295.
If the payment comes one year earlier, the PV increases by a factor of 1.08 to
$21,207,198.

35.

The real rate is zero. With a zero real rate, we simply divide her savings by the
years of retirement: $450,000/30 = $15,000 per year.

36.

Per month interest = 6%/12 = .5% per month
FV in 1 year (12 months) = 1000  (1.005)12 = $1,061.68

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FV in 1.5 years (18 months) = 1000  (1.005)18 = $1,093.93
37.

You are repaying the loan with an annuity of payments. The PV of those
payments must equal $100,000. Therefore,
804.62  annuity factor(r, 360 months) = 100,000
which implies that the interest rate is .750% per month.
[On your calculator, input PV = ()100,000, FV = 0, n = 360, PMT = 804.62, and
compute the interest rate.]
The effective annual rate is (1.00750)12  1 = .0938 = 9.38%.
If the lender is a Canadian financial institution, the quoted rate will be the APR
for a 6-month compounding period:
(1 + )2 – 1 = .0938
= (1.0938)1/2 -1 = .04585
APR = 2 × [(1.0938)1/2 -1] = .0917 or 9.17%, which is lower than the effective
annual rate.
Note: A simpler APR calculation is .750%  12 = 9%. However, this is not how
Canadian mortgage lenders calculate their APRs.

38.
39.

EAR = e.04 -1 = 1.0408 -1 = .0408 = 4.08%
The PV of the payments under option (a) is 11,000, assuming the $1,000 rebate is paid
immediately. The PV of the payments under option (b) is
$250  annuity factor(1%, 48 months) = $9,493.49
Option (b) is the better deal.

40.

100  e.10×6 = $182.21
100  e.06×10 = $182.21

41.

Your savings goal is 30,000 = FV. You currently have in the bank 20,000 = PV. The
PMT = (-) 100 and r = .5%. Solve for n to find n = 44.74 months.

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Note: You may have to solve this by trial-and-error if your calculator cannot handle
these numbers.
42.

The present value of your payments to the bank equals:
$100  annuity factor(8%, 10 years) = $671.01

The present value of your receipts is the value of a $100 perpetuity deferred for 10
years:
 = $578.99
This is a bad deal if you can earn 8% on your other investments.

43.

If you live forever, you will receive a $100 perpetuity which has present value 100/r.
Therefore, 100/r = 2500, which implies that r = 4 percent

44.

r = 10,000/125,000 = .08 = 8 percent.

45.

Suppose the purchase price is $1. If you pay today, you get the discount and pay
only $.97. If you wait a month, you must pay $1. Thus, you can view the deferred
payment as saving a cash flow of $.97 today, but paying $1 in a month. The
monthly rate is therefore .03/.97 = .0309, or 3.09%. The effective annual rate is
(1.0309)12  1 = .4408 = 44.08%.

46.

You borrow $1000 and repay the loan by making 12 monthly payments of $100.
We find that r = 2.923% by solving:
100  annuity factor(r, 12 months) = 1000
[On your calculator, input PV = ()1,000, FV = 0, n = 12, PMT = 100, and
compute the interest rate.]
The APR is therefore 2.923%  12 = 35.08%

and the effective annual rate is (1.02923)12  1 = .4130 = 41.30%
How do we know that the true rates must be greater than 20%? If you borrow
$1000 and repay $1200 in one year, the rate of interest on the loan is 20%. Here,
with add-on interest, you make the $1200 repayment sooner. Because of the time
value of money, the effective interest rate must be higher than 20%.

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47.

You will have to pay back the original $1000 plus 3  20% = 60% of the loan
amount, or $1600 over the three years. This implies monthly payments of
$1600/36 = $44.44
The monthly interest rate is obtained by solving:
44.44  annuity factor(r, 36) = 1000
On your calculator, input PV = ()1,000, FV = 0, n = 36, PMT = 44.44, and
compute the interest rate as 2.799% per month.
The APR is therefore 2.799%  12 = 33.59%,
and the effective annual rate is (1.02799)12  1 = .3927 = 39.27%

48.

For every $1000 you borrow, your present value is 1000 (1  d), and the future
value you pay back is 1000. Therefore, the annual interest rate is determined by:
PV  (1 + r) = FV
[10,000 (1 – d)]  (1 + r) = 10,000
1+r=  r= 1=>d
If d = 20%, then the effective annual interest rate is .2/.8 = .25 = 25%.

49.

The semi-annual interest rate paid at First National is 0.062/2 = .031, or 3.1%
every six months. After one year, each dollar invested will grow to:
$1  (1.031)2 = $1.06296 and the EAR is 6.296%
The monthly interest rate paid at Second National is .06/12 = .005, or 0.5% every
month. After one year, each dollar invested will grow to:
$1  (1.005)12 = $1.06168 and the EAR is 6.168%.
First National pays the higher effective annual rate.

50.

Since the $20 origination fee is taken out of the initial proceeds of the loan, the
amount actually borrowed is $1000  $20 = $980. The monthly rate is found by
solving the following equation for r:
90  annuity factor(r, 12) = 980

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r = 1.527% per month.
The effective rate is (1.01527)12 -1 = .1994 = 19.94%.
51.

Monthly interest rate = (1.08)1/12 – 1 = .006434 or .6434%
Football Quarterback:
Total salary paid over contract = 5 years × 3 million/year = $15 million
Monthly salary = 3 million/12 months per year = $250,000 at the end of each month

for 12 ×5, or 60 months
PV = 250,000 PVIFA(.6434%,60 months) = $12,411,236.45
Hockey Player
Total salary paid over contract = $4 million + 5 × $2.1 million = $14.5 million
Monthly salary = $2.1 million/12 months per year = $175,000 at the end of each
month
PV = 4 million + 175,000 PVIFA(.6434%,60 months) = $12,687,865.52
The quarterback is wrong. The hockey player’s contract has a higher present value.

52.

a.

Per month interest rate = 7%/12 = .005833333
48-month loan: PV = 400 × PVIFA(7%/12,48) = $16,704.08
60-month loan: PV = 400 × PVIFA(7%/12,60) = $20,200.80
Bill will buy a car for $16,704.08 if he arranges a 48-month loan and will buy
a car for $20,200.80 if he arranges a 60-month loan.

b.

To fairly compare the two loans, both time periods must be the same. We
assume that Bill will keep the car for 5 years, regardless of which loan he
takes.
Bill’s wealth at the end of 5 years will depend on the value of the car and the
balance in his bank account.
Wealth in five years if take the 48-month loan and buy the $16,704.08 car:
(1) Value of car at the end of 5 years: starting value × (1 – depreciation rate) 5
= $16,704.08 × (1 - .18)5 = $6,192.87
(2) Savings = $400 invested each month for one year:

FV = 400 × FVIFA(5%/12,12) = $4,911.54
(3) Total wealth = $6,192.87 + $4,911.54 = $11,104.41
Wealth in five years if take the 60-month loan and buy the $20,200.80 car:
(1) Value of car at the end of 5 years: starting value × (1 – depreciation rate) 5
= $20,200.80 × (1 - .18)5 = $7,489.24
(2) Savings = none, spent all spare cash on car payments
(3) Total wealth = $7,489.24
We haven’t compared Bill’s happiness from owning the more expensive car to
his happiness from owning the less expensive car. On the other hand, we’ve

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Copyright © 20096 McGraw-Hill Ryerson Limited


not compare the cost of fuel or maintenance either. Whether the more
expensive car is worth it has not been established.
53.

a.

The present value of the ultimate sales price is 4 million/(1.08) 5 = $2.722
million.

b.

The present value of the sales price is less than the cost of the property, so this
would not be an attractive opportunity.

c.

The value of the total cash flows from the property is now
PV = .2  annuity factor(8%, 5 years) + 4/(1.08) 5
= .80 + 2.72 = $3.52 million
To solve with a calculator, enter: PMT = .2, FV = 4, i = 8%, n = 5 and compute
PV.
Therefore, the property is an attractive investment if you can buy it for $3
million.

54.

PV of cash inflows
= [120,000/1.12 + 180,000/1.122 + 300,000/1.123]
= $464,172
This exceeds the cost of the factory, so the investment is attractive.

55.

a.

The present value of the future payoff is 2000/(1.05) 10 = $1227.83. This is a
good deal: PV exceeds the initial investment.
You can solve this also by looking at the future value of investing $1,000 at
the opportunity cost of 5% for 10 years: FV = 1.05 10 × 1000 = $1628.9.
This is less than the $2000 payoff expected from the investment. The
investment is a good deal.
Another way to answer this question is to figure out the interest rate that
the investment is offering: Find r such that (1 + r) 10 × 1000 = 2000. Either
using the calculator or solving directly, gives r = .07177, or 7.177%. Clearly
it is better to invest at 7.177% than at 5%.

b.

The PV is now only 2000/(1.10)10 = $771.09, which is less than the initial
investment. Therefore, this is a bad deal.
Another way to look at the investment: Since we know that the investment

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Copyright © 20096 McGraw-Hill Ryerson Limited


offers a 7.177% return, it makes no sense to undertake the investment if we
can invest elsewhere and earn 10%.
56.

The future value of the payments into your savings fund must accumulate to
$500,000. We assume that payments are made at the end of the year. We choose
the payment so that PMT  future value of an annuity = $500,000. On your
calculator, enter: n = 40; i = 5; PV = 0; FV = 500,000. Compute PMT to be
$4,139.08.

57.

If you invest the $100,000 received in year 10 until your retirement in year 40, it
will grow to $100,000  (1.05)30 = $432,194. Therefore, your savings plan would
need to generate a future value of only $500,000 – $432,194 = $67,806. This
would require a savings stream of only $561.31.

58.

By the time you retire you will need

$40,000  future value annuity factor(5%, 20 periods) = $498,488.41.
The future value of the payments into your savings fund must accumulate to
$498,488.41. We choose the payment so that PMT  future value of an annuity =
$498,488.41. On your calculator, enter: n = 40; i = 5; PV = 0; FV = 498,488.41.
Compute PMT to be $4,126.57.

59.

After 30 years the couple will have accumulated the future value of a $3,000
annuity, plus the present value of the $10,000 gift. The sum of the savings from
these sources is:
$3,000  future value annuity factor (30,8%) =
$10,000  1.0825 =

$339,849.63
68,484.75
$408,334.38

If they wish to accumulate $800,000 by retirement, they need to save an
additional amount per year to provide additional accumulations of $391,665.62.
This requires additional annual savings of $3,457.40. [On your calculator, input i
= 8; n = 30;
PV = 0; FV = 391,665.62 and compute PMT.]
60.

a.

The present value of the planned consumption stream as of the retirement
date will be $30,000  annuity factor(25,8%) = $320,243.29. Therefore,
they need to have accumulated this level of savings by the time they retire.

So their savings plan must provide a future value of $320,243.29. With 50
years to save at 8%, the savings annuity must be $558.14

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Copyright © 20096 McGraw-Hill Ryerson Limited


Another way to think about this is to recognize that the present value of the
savings stream must equal the present value of the consumption stream.
The PV of consumption as of today is = $320,243.29/(1.08) 50 = $6,827.98
Therefore, we set the present value of savings equal to this value, and solve
for the required savings stream. Using the calculator: enter PV =
(-)6,827.98
n = 50, i = 8, PV = 0 and solve for PMT.
b.

61.

The couple needs to accumulate additional savings with a present value of
$60,000/(1.08)20 = $12,872.89. The total PV of savings is now $12,872.89 +
$6,827.98 = $19,700.77. Now we solve for the required savings stream as
follows: n = 50; i = 8; PV = ()19,700.77; FV = 0; and solve for PMT as
$1,610.40. They need to save $1,610.40 each year for the next 50 years.

Note: Ignore taxes.
Monthly interest rate = .08/12 = .00666666
Borrow and buy the copier
Monthly loan payments : 20,000 = PMT × PVIFA(8%/12, 60)
PMT = $405.53
Cash flows if borrow/buy:

Cash
flows

0

1

2

Month
3

Loan

+20,000

-405.53

-405.53

-405.53

…

58

59

60

-405.53

-405.53

-405.53

Salvag
e

+5,000

PV(borrow/buy cash flows) = +20,000 – 405.53 × PVIFA(.08/12,60) + 5,000 × PVIF(.08/12,60)
= + 3,356.05

Cash flows if lease:
Cash
flows

0

1

2

Month
3

Lease

+20,000
-X

-X

-X

-X

…

58

59

60

-X

-X

0

Salvag
e
PV(Lease cash flows) = +20,000 – X – X × PVIFA(.08/12, 59)

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0


Find X such that PV(borrow/buy cash flows) = PV(lease cash flows):
3,356.05 = +20,000 – X – X × PVIFA(.08/12, 59)
16,643.95 = X + X × PVIFA(.08/12, 59)
= X × annuity paid at the beginning of the period (.08/12, 60)
This can be solved using a financial calculator. Set the calculator for payments at the
beginning of the period. (With a BAII Plus: enter 2 nd PMT (which is BEG), followed by
2nd Enter (which is SET). You should see the word END on the screen change to the
word BEG.)
PV = (-) 16,643.95, i = .08/12, n = 60, FV = 0, solve for PMT = 335.24
The lease payment equivalent to borrowing and buying is $335.24 per month, paid at the
start of each month.
62.

63.

a.

$60,000/8.2 = $7,317.07. Her real income increased from $6,000 to
$7,317.07.

b.

Years to retirement = 2008 – 1950 = 58 years
Salary inflation rate, s: (1 + s)58 × $6,000 = $60,000
s = (60,000/6,000)1/58 – 1 = .0405, or 4.05%
Cost of goods inflation rate, c: (1 + c)58 × 1 = 8.2
c = 8.21/58 – 1 = .03694, or 3.694%

1 + nominal rate = (1 + real rate)  (1 + inflation rate)
a.
b.
c.

64.

1.04  1.0 = 1.04; nominal rate = 4%
1.04  1.04 = 1.0816; nominal rate = 8.16%
1.04  1.06 = 1.1024; nominal rate = 10.24%

real interest rate =  1
a. 1.08/1 – 1 = .080 = 8.0%
b. 1.08/1.03 – 1 = .0485 = 4.85%
c. 1.08/1.06 – 1 = .0189 = 1.89%

65.

a.
b.

100/(1.08)3
100/(1.03)3

c.

real interest rate

d.

= .04854, or 4.854%
91.51/(1.04854)3 = $79.38

= $79.38 present value
= $91.51 real value
=1

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Copyright © 20096 McGraw-Hill Ryerson Limited


66.

Standard & Poor’s
Expected results: Students gain experience working with real data to calculate
nominal rates of growth and converting them to real growth rates.
According to the company profile, Thomson Reuters Corporation provides
intelligent information for businesses and professionals in the financial, legal, tax
and accounting, scientific, healthcare, and media markets worldwide.
Based on the annual income statement, the compound annual growth rates
(CAGR) over the period of Dec. 03 to Dec. 07 will be calculated as follow:
CAGR(sales) = (7,296/7,606)1/5 – 1 = –0.0083 = –0.83%
CAGR(net income) = (3,998/879)1/5 – 1 = 0.354 = 35.4%
CAGR(dividends) = (0.980/1.153)1/5 – 1 = –0.032 = –3.2%
Inflation Calculator can be found at
/>Enter $100 for 2003, then calculate the value of the $100
for 2008. The result will be $115.45 for 2008.
Now, compound average annual inflation rate can
be calculated as (115.45/100)1/5 – 1 = 0.0292 =

2.92%.

Source: />
Real CAGR(sales) = (1 – 0.0083)/(1 + 0.0292) –1 = –0.0364 = –3.64%
Real CAGR(net income) = (1 + 0.354)/(1 + 0.0292) –1 = 0.316 = 31.6%
Real CAGR(dividends) = (1 – 0.032)/(1 + 0.0292) – 1 = –0.0595 = –5.95%
67.

Internet: Inflation and Investment Returns
a.
Expected results: Students gain some perspective on inflation rates over
the past century.
The inflation calculator provides a comparison between dollar amounts in
two different years considering the inflation rate. Enter $5,000 for 1914
and 2008 for comparison, click CALCULATE and you will get $93,166.67.

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Copyright © 20096 McGraw-Hill Ryerson Limited


Source: />
b.

The Inflation Calculator (above figure) shows the Average Annual Rate of
Inflation equal to 3.16%.

c.

Tip: As we write this, the button for the Investment calculator is at the top
right side of the inflation calculator screen.

Expected Results: As we write this, the calculator assumes annual
payments. The number of years is equal to the selected year minus the
current year. For the example below, the number of years assumed is 4 (=
2012 – 2008). For the numbers below, the real rate of interest is
1.04/1.0316 - 1 = .008142691, or .8142691%.
After-inflation value of investment (in 4 years) = $100,000/1.0316 4 =
$88,298.77249
Total interest earned (nominal):
= $100,000 ×FVIF(4.0%,4) – $100,000 = $16,985.86
Interest earned after inflation can also be calculated as: nominal
interest/inflation factor = $16,985.86/(1.0316) 4 = $14,998.31
Total future value:
= $100,000 ×FVIF(.8142691%,4) = $103,297.08

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Copyright © 20096 McGraw-Hill Ryerson Limited


Source: />
d.

To determine the interest rate used in the calculator, plug in a future target of
$100,000. The investment needed today is $96,808.16. The interest rate, r, must
solve the following equation:
$96,808.16 = $100,000 × PVIF(r, 4)
Using a hand-held calculator, with PV = (-) 96,808.16, FV=100,000, n = 4,
PMT=0, calculate r = 0.8142697%. With some rounding errors, we get almost the
same real rate of interest. Assuming that it is the correct real rate of interest, the
future target must also be a real amount, the amount in terms of today’s dollars.
For example, if you want $150,000 in 2012’s dollars, that equates to $150,000/

(1.0316)4 = $132,448.16 in 2008 dollars. If the real rate of interest is
0.8142697%, you need to invest $128,220.63 today, in 2008.

Source: />
68.

a.

Since the nominal cash flows are expected to grow with inflation, the
expected annual real cash flow is $100,000. The real rate is 1.08/1.03 – 1 =
4.85%. So the present value is:

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Copyright © 20096 McGraw-Hill Ryerson Limited


$100,000  annuity factor(4.85%, 5 years) = $434,749
b.

69.

If cash flow is level in nominal terms, use the 8% nominal interest rate to
discount. The annuity factor is now 3.99271 and the cash flow stream is
worth only $399,271.

a.

$1 million will have a real value of $1 million/(1.03) 45 = $264,439.

b.

At a real rate of 2%, this can support a real annuity of $228,107/annuity
factor(2%, 20 years).
= $264,439/16.3514 = $16,172
To solve this on a calculator, input n = 20, i = 2, PV = 264,439, FV = 0, and
compute PMT.

70.

According to the Rule of 72, at an interest rate of 8%, it will take 72/8 = 9 years
for your money to double. For it to quadruple, your money must double, and then
double again. This will take approximately 18 years.

71.

(1.23)12 – 1 = 10.99. Prices increased by 1,099 percent per year.

72.

Using the perpetuity formula, the 4% consol will sell for
£4/.06 = £66.67. The 2 1/2% consol will sell for £2.50/.06 = £41.67.

73.

The savings calculator can be reached directly from the following link:
/>Total value after 30 years without any savings = $1,000 x (1.06 30) = $5,743.49
In the savings calculator enter Years to Save = 30, Total Cost of Expense = $0,
Current Savings = $1,000, Deposit Amount = $0, Compounded Annual Rate of
Return = 6%, and click Calculate, you will get the same result.
Total value with after 30 years with $200 savings per month:

(1 + monthly rate) 12 –1 = .06  monthly rate = .004867551
Number of periods = 30 x 12 = 360 months
Assuming the payments are made at the start of each period:
Total Value = $5,743.49 + $200 x FVIFA(360, .4867551%) x (1.004867551)
= $5,743.49 + $195,851.31
= $201,594.80
In the calculator enter $200 as Deposit Amount and choose Monthly Deposit
Frequency, click calculate and you will get the same result with some
rounding errors.

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Copyright © 20096 McGraw-Hill Ryerson Limited


74.

75.

Expected Result: Using "To Buy or To Lease" calculator from
www.smartmoney.com, enter $20,000 as the Price of Car, $350 as Down
Payment, $350 as Monthly payment of lease, 36 months as Lease term, 10%
Rate of return, and $10,000 as the value of the car at the end of the lease. The
calculator will calculate the value of alternative investments at the end of
lease term of $11,510 which is more than the value of the car at the end of
the lease ($10,000). Thus, lease is a better option.
a.

$30,000  annuity factor(10%, 15 years) = $228,182

b.

Fin the annual payment, PMT, such that PMT × future value annuity
factor(10%, 30 years) = 228,182. Using the calculator, PV = 0, n=30, i=
10%, FV= (-) 228,182. Compute PMT = 1,387. You must save $1,387
annually.

c.

1.00  (1.04)30 = $3.24

d.

We repeat part (a) using the real rate of 1.10/1.04 – 1 = .0577 or 5.77%
The retirement goal in real terms is
$30,000  annuity factor(5.77%, 15 years) = $295,797

e.

The future value of your 30-year saving stream must equal this value. So we
solve for payment (PMT) in the following equation
PMT  future value annuity factor(5.77%, 30 years) = $295,797
PMT  75.930 = $295,797
PMT = 3,896
You must save $3896 per year in real terms. This value is much higher than
the answer to (b) because the rate at which purchasing power grows is less
than the nominal interest rate, 10%.

f.

76.

If the real amount saved is $3,896 and prices rise at 4 percent per year, then
the amount saved at the end of one year in nominal terms will be $3,896 
1.04 = $4,052. The thirtieth year will require nominal savings of 3,896 
(1.04)30 = $12,636.

In the 113 years since the capture of Ned Kelley, from 1880 to 1993, one dollar
invested in the bank would have grown to be $35.14 (= 1 × (1.032) 113 ). By
contrast, that same dollar invested in the Australian stock market would have
grown to $28, 431.22 (= $1 × (1.095)113). My clients are reasonable people but
believe that $1 is a ridiculously low amount. Given the 3% annual inflation, the

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Copyright © 20096 McGraw-Hill Ryerson Limited


real value in 1993 of $1 paid in 1880 is only $0.035 (= $1/(1.03) 113). Surely the
efforts of the trackers is worth more than 3 ½ cents! We think a reasonable
payment is $14,233 each, half way between the value of $1 invested in the bank
and the value of a $1 invested in the stock market.
77.

The interest rate per three months is 12%/4 = 3%. So the value of the perpetuity is
$100/.03 = $3,333.

78.

FV = PV  (1 + r1)  (1 + r2) = 1  (1.08)  (1.10) = $1.188
PV = = = $0.8418

79.

You earned compound interest of 8% for 8 years and 6% for 13 years. Your $1000
has grown to
1000  (1.08)8  (1.06)13 = $3,947.90.

80.

The answer to this question can be found in various ways. The key is to pick a
common point in time to measure all cash flows. Here we pick today as the common
reference point.
Monthly interest rate = 1.061/12 – 1 = .004868 = .4868%.
Present value today of the funds need for boat: 150,000/(1.06) 4 = 118,814
Funds need for monthly expenses (this is an annuity due)
= (2200 + 2200×annuity factor(.4868%, 23 months))/ (1.06) 5
= 37,333.29
Funds need for emergencies = 45,000/(1.06)5 = 33,626.62
Total funds needed = 118,814 + 37,333.29 + 33,626.62 = 189,774
The present value of the savings stream must equal the present value of the
expenditures:
PMT × annuity factor (.4868%,60 months) = 189,774
The monthly savings must be $3,654.9. (Expect slight variations due to rounding)

81.

Interest rate on parents’ car loan = .024/12 = .002 = .2%
Monthly car repayment: PMT × annuity factor (.2%, 48) = 4,000
Using the calculator to find PMT = $87.48
Monthly opportunity cost of funds = (1.06) 1/12 – 1 = .004868 = .4868%
Summary of Car Costs:

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Copyright © 20096 McGraw-Hill Ryerson Limited


Car
payment
Operating
cost
Total costs

0

1
87.48

2
87.48

200

200

200

200

287.48

287.48

Month
3
87.48

…
…

47
87.48

48
87.48

200

…

200

0

287.48

…

287.48

87.48

Present value of car costs
= 200 + 287.48 × annuity factor (.4868%, 47 months) + 87.48/(1.004868) 1/48
= $12,320.5
You have to decide whether to charge your friends at the beginning or the end of
each month. In this calculation, we assume that your friends will pay you at the
start of each month. The total amount
of money needed each month to cover the car costs, given a 6% EAR is:
PMT × annuity due factor (.4868%, 48) = 12,320.5
Switch your calculator to the annuity due setting, and solve for PMT =$287
[n=48, i=.4868, FV=0, PV= (-)12,320.5]
If your three friends are to cover the cost of the car, they each must pay $287/3 =
$95.67 a month. If you share in the cost, dividing it four ways, you each pay
$287/4 = $71.75.
Does it make financial sense to buy the car? Given that the cost of a monthly bus
pass is $80, it does not make sense to charge your friends much more than $80 per
month, unless the bus trip to school is extremely long relative to the time taken in
the car. Likewise, you too will not want to pay much more than $80 a month
either. Of course, you will have access to the car at times when your friends do
not. It depends on the value of the convenience of having access to the car. If you
charge yourself $80 a month, then you should ask your friends for $69 a month.
Perhaps that might be viewed as a fair trade-off.
We have not considered the impact of inflation on the costs of operating the car.
The car payments won’t change with inflation but the operating costs will. You
can redo the analysis assuming a 2% annual rate of inflation and see how much
higher must be the monthly charge.
One final note: You may want to consider the benefits for the air quality of
taking public transit to school. We have not factored into our analysis the
economic costs to society of air pollution from the car.
82.

a.

Assume there are 26 events per year (52 weekends/2). In 5 years, you
attend 26×5 = 130 events. The bi-weekly interest rate is (1.09) 1/26 – 1 = .

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Copyright © 20096 McGraw-Hill Ryerson Limited


00332 = .332% (assuming that 9% rate is an effective annual rate).
Cost of Renting
Cost of renting a van per weekend = $100
Mileage charge per weekend = 200 km × $.5 = $100
Fuel costs per weekend = 200 km × $.75 = $150
Total cost = $350 per weekend
PV of cost of renting = 350 × annuity factor(.332%, 130 events) =
$36,904.6
Cost of Owning
Weekend operating costs = 200 km × ($.25 + $.75) = $200
PV of weekend operating costs = 200 × annuity factor(.332%, 130 events)
= $21,088.3
Cost today of buying van = $20,000
Expected selling price in 5 years = (1 - .1) 5 × 20,000 = 11,809.8
PV of selling price in 5 years = 11,809.8/(1.09)5 = 7,675.6
PV of insurance (assume insurance is paid in advance)
= 1200 × annuity due factor(9%, 5 years) = 5,087.7
Total cost of owning = 21,088.3 + 20,000 - 7,675.6 + 5,087.7 = $38,500.4
Extra cost of purchasing the van rather than renting
= $38,500.4 - $36,904.6 = $1,595.8
Although the total cost is higher, the van is available to drive at other

times. If costs of another vehicle can be saved, then it makes sense to buy.
Otherwise, it is cheaper to rent, if inflation is not considered.
b.

We use the principle of discounting real cash flows at the real discount
rate. Assume all cash flows are in current year dollars, including the
expected resale value of the car. The real discount rate is (1.09/1.03) – 1 = .
05825 effective annual rate. The bi-weekly equivalent rate is (1.05825) 1/26 –
1 = .00218 = .218%. Recalculate the present value of the costs at the real
discount rate:
Cost of Renting
PV of cost of renting = 350 × annuity factor(.218%, 130 events) =
$39,583.6
Cost of Owning
PV of weekend operating costs = 200 × annuity factor(.218%, 130 events)
= $22,619.2
Cost today of buying van = $20,000
PV of selling price in 5 years = 11,809.8/(1.05825)5 = $8,898.2
PV of insurance (assume insurance is paid in advance)
= 1200 × annuity due factor(5.825%, 5 years) = $5,374.8
Total cost of owning = 22,619.2 + 20,000 - 8,898.2 + 5,374.8 = $39,095.8

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Copyright © 20096 McGraw-Hill Ryerson Limited