Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences
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Continuous Compounding of Interest
One question that arises naturally in the study of compound interest is: What happens
to the accumulated amount over a fixed period of time if the interest is computed more
and more frequently?
Intuition suggests that the more often interest is compounded, the larger the accu-
mulated amount will be. This is confirmed by the results of Example 3, where we
found that the accumulated amounts did in fact increase when we increased the num-
ber of conversion periods per year.
This leads us to another question: Does the accumulated amount keep growing
without bound, or does it approach a fixed number when the interest is computed more
and more frequently over a fixed period of time?
To answer this question, let’s look again at the compound interest formula:
(4)
Recall that m is the number of conversion periods per year. So to find an answer to
our question, we should let m get larger and larger in (4). If we let so that
m ru, then (4) becomes
Since
a
xy
(
a
x
)
y
Now let’s see what happens to the expression
as u gets larger and larger. From Table 2 you can see that, as u increases,
seems to approach the number 2.71828 (we have rounded all our calculations to five
decimal places).
It can be shown—although we will not do so here—that as u gets larger and
larger, the value of the expression approaches the irrational number
2.71828..., which we denote by e. (See the Exploring with Technology exercise
that follows.)
11
1
u
2
u
a1
1
u
b
u
a1
1
u
b
u
P ca1
1
u
b
u
d
rt
r
m
1
u
A P a1
1
u
b
urt
u
m
r
A P11 i2
n
P a1
r
m
b
mt
262 5 MATHEMATICS OF FINANCE
Exploring with
TECHNOLOGY
Investments that are allowed to grow over time can increase in value surprisingly
fast. Consider the potential growth of $10,000 if earnings are reinvested. More
specifically, suppose A
1
(t), A
2
(t), A
3
(t), A
4
(t), and A
5
(t) denote the accumulated
values of an investment of $10,000 over a term of t years and earning interest at
the rate of 4%, 6%, 8%, 10%, and 12% per year compounded annually.
1. Find expressions for A
1
(t), A
2
(t), . . . , A
5
(t).
2. Use a graphing utility to plot the graphs of A
1
, A
2
, . . . , A
5
on the same set of
axes, using the viewing window [0, 20] [0, 100,000].
3. Use
TRACE
to find A
1
(20), A
2
(20), . . . , A
5
(20) and then interpret your results.
TABLE 2
u
10 2.59374
100 2.70481
1000 2.71692
10,000 2.71815
100,000 2.71827
1,000,000 2.71828
a1 ⴙ
1
u
b
u
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 262
Using this result, we can see that, as m gets larger and larger, A approaches
P(e)
rt
Pe
rt
. In this situation, we say that interest is compounded continuously. Let’s
summarize this important result.
EXAMPLE 4
Find the accumulated amount after 3 years if $1000 is invested at
8% per year compounded (a) daily (assume a 365-day year) and (b) continuously.
Solution
a. Use Formula (3) with P 1000, r 0.08, m 365, and t 3. Thus, i
and n (365)(3) 1095, so
or $1271.22.
b. Here we use Formula (5) with P 1000, r 0.08, and t 3, obtaining
or $1271.25.
Observe that the accumulated amounts corresponding to interest compounded daily
and interest compounded continuously differ by very little. The continuous compound
interest formula is a very important tool in theoretical work in financial analysis.
Effective Rate of Interest
Example 3 showed that the interest actually earned on an investment depends on the
frequency with which the interest is compounded. Thus the stated, or nominal, rate of
8% per year does not reflect the actual rate at which interest is earned. This suggests
that we need to find a common basis for comparing interest rates. One such way of
comparing interest rates is provided by the use of the effective rate. The effective rate
is the simple interest rate that would produce the same accumulated amount in 1 year
⬇ 1271.25
A 1000e
10.082132
A 1000 a1
0.08
365
b
13652132
⬇ 1271.22
0.08
365
Continuous Compound Interest Formula
A Pe
rt
(5)
where
P Principal
r Nominal interest rate compounded continuously
t Time in years
A Accumulated amount at the end of t years
5.1 COMPOUND INTEREST 263
Exploring with
TECHNOLOGY
To obtain a visual confirmation of the fact that the expression
approaches the number e 2.71828...as u gets larger and larger, plot the
graph of f (x) in a suitable viewing window and observe that f (x)
approaches 2.71828...as x gets larger and larger. Use
ZOOM
and
TRACE
to find
the value of f (x) for large values of x.
11
1
x
2
x
11
1
u
2
u
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 263
as the nominal rate compounded m times a year. The effective rate is also called the
annual percentage yield.
To derive a relationship between the nominal interest rate, r per year compounded
m times, and its corresponding effective rate, R per year, let’s assume an initial invest-
ment of P dollars. Then the accumulated amount after 1 year at a simple interest rate
of R per year is
A P(1 R)
Also, the accumulated amount after 1 year at an interest rate of r per year compounded
m times a year is
Since and
t
1
Equating the two expressions gives
Divide both sides by
P
.
If we solve the preceding equation for R, we obtain the following formula for com-
puting the effective rate of interest.
EXAMPLE 5
Find the effective rate of interest corresponding to a nominal rate of
8% per year compounded (a) annually, (b) semiannually, (c) quarterly, (d) monthly,
and (e) daily.
Solution
a. The effective rate of interest corresponding to a nominal rate of 8% per year
compounded annually is, of course, given by 8% per year. This result is also
confirmed by using Equation (6) with r 0.08 and m 1. Thus,
b. Let r 0.08 and m 2. Then Equation (6) yields
so the effective rate is 8.16% per year.
0.0816
11.042
2
1
r
eff
a1
0.08
2
b
2
1
r
eff
11 0.08 2 1 0.08
1 R a1
r
m
b
m
P11 R2 P a1
r
m
b
m
i
r
m
A P11 i2
n
P a1
r
m
b
m
264 5 MATHEMATICS OF FINANCE
Effective Rate of Interest Formula
(6)
where
r
eff
Effective rate of interest
r Nominal interest rate per year
m Number of conversion periods per year
r
eff
a1
r
m
b
m
1
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 264
c. Let r ⫽ 0.08 and m ⫽ 4. Then Equation (6) yields
so the corresponding effective rate in this case is 8.243% per year.
d. Let r ⫽ 0.08 and m ⫽ 12. Then Equation (6) yields
so the corresponding effective rate in this case is 8.3% per year.
e. Let r ⫽ 0.08 and m ⫽ 365. Then Equation (6) yields
so the corresponding effective rate in this case is 8.328% per year.
If the effective rate of interest r
eff
is known, then the accumulated amount after t years
on an investment of P dollars may be more readily computed by using the formula
A ⫽ P(1 ⫹ r
eff
)
t
The 1968 Truth in Lending Act passed by Congress requires that the effective rate
of interest be disclosed in all contracts involving interest charges. The passage of this
act has benefited consumers because they now have a common basis for comparing
the various nominal rates quoted by different financial institutions. Furthermore,
knowing the effective rate enables consumers to compute the actual charges involved
in a transaction. Thus, if the effective rates of interest found in Example 4 were
known, then the accumulated values of Example 3 could have been readily found (see
Table 3).
Present Value
Let’s return to the compound interest Formula (3), which expresses the accumulated
amount at the end of n periods when interest at the rate of r is compounded m times a
year. The principal P in (3) is often referred to as the present value, and the accumu-
lated value A is called the future value, since it is realized at a future date. In certain
instances an investor may wish to determine how much money he should invest now,
⬇ 0.08328
r
eff
⫽ a1 ⫹
0.08
365
b
365
⫺ 1
⬇ 0.08300
r
eff
⫽ a1 ⫹
0.08
12
b
12
⫺ 1
⬇ 0.08243
⫽ 11.022
4
⫺ 1
r
eff
⫽ a1 ⫹
0.08
4
b
4
⫺ 1
5.1 COMPOUND INTEREST 265
Explore & Discuss
Recall the effective rate of inter-
est formula:
1. Show that
r ⫽ m[(1 ⫹ r
eff
)
1/m
⫺ 1]
2. A certificate of deposit (CD)
is known to have an effective
rate of 5.3%. If interest is
compounded monthly, find
the nominal rate of interest
by using the result of part 1.
r
eff
⫽ a1 ⫹
r
m
b
m
⫺ 1
TABLE 3
Nominal Frequency of Effective Initial Accumulated Amount
Rate, r Interest Payment Rate Investment after 3 Years
8% Annually 8% $1000 1000(1 ⫹ 0.08)
3
⬇ $1259.71
8 Semiannually 8.16 1000 1000(1 ⫹ 0.0816)
3
⬇ 1265.32
8 Quarterly 8.243 1000 1000(1 ⫹ 0.08243)
3
⬇ 1268.23
8 Monthly 8.300 1000 1000(1 ⫹ 0.08300)
3
⬇ 1270.24
8 Daily 8.328 1000 1000(1 ⫹ 0.08328)
3
⬇ 1271.22
87533_05_ch5_p257-312 2/18/08 10:41 AM Page 265
at a fixed rate of interest, so that he will realize a certain sum at some future date. This
problem may be solved by expressing P in terms of A. Thus, from Equation (3) we find
P A(1 i)
n
Here, as before, i , where m is the number of conversion periods per year.
EXAMPLE 6
How much money should be deposited in a bank paying interest at
the rate of 6% per year compounded monthly so that, at the end of 3 years, the accu-
mulated amount will be $20,000?
Solution
Here, r 0.06 and m 12, so and n (3)(12) 36. Thus,
the problem is to determine P given that A 20,000. Using Equation (7), we obtain
or $16,713.
EXAMPLE 7
Find the present value of $49,158.60 due in 5 years at an interest
rate of 10% per year compounded quarterly.
Solution
Using (7) with r 0.1 and m 4, so that , n (4)(5) 20, and
A 49,158.6, we obtain
or approximately $30,000.
If we solve Formula (5) for P, we have
A Pe
rt
and
P Ae
rt
(8)
which gives the present value in terms of the future (accumulated) value for the case
of continuous compounding.
APPLIED EXAMPLE 8
Real Estate Investment Blakely Investment
Company owns an office building located in the commercial district of a
city. As a result of the continued success of an urban renewal program, local
business is enjoying a miniboom. The market value of Blakely’s property is
where V(t) is measured in dollars and t is the time in years from the present. If the
expected rate of appreciation is 9% compounded continuously for the next 10 years,
find an expression for the present value P(t) of the market price of the property
that will be valid for the next 10 years. Compute P(7), P(8), and P(9), and then
interpret your results.
V1t2 300,000e
1t/2
P 149,158.62a1
0.1
4
b
20
⬇ 30,000.07
i
0.1
4
⬇ 16,713
P 20,000 a1
0.06
12
b
36
i
0.06
12
Present Value Formula for Compound Interest
P A(1 i)
n
(7)
r
m
266 5 MATHEMATICS OF FINANCE
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 266
Solution
Using Formula (8) with A V(t) and r 0.09, we find that the pres-
ent value of the market price of the property t years from now is
Letting t 7, 8, and 9, respectively, we find that
From the results of these computations, we see that the present value of the prop-
erty’s market price seems to decrease after a certain period of growth. This suggests
that there is an optimal time for the owners to sell. You can show that the highest
present value of the property’s market value is approximately $600,779, and that it
occurs at time t ⬇ 7.72 years, by sketching the graph of the function P.
The returns on certain investments such as zero coupon certificates of deposit
(CDs) and zero coupon bonds are compared by quoting the time it takes for each
investment to triple, or even quadruple. These calculations make use of the compound
interest Formula (3).
APPLIED EXAMPLE 9
Investment Options Jane has narrowed her
investment options down to two:
1. Purchase a CD that matures in 12 years and pays interest upon maturity at the
rate of 10% per year compounded daily (assume 365 days in a year).
2. Purchase a zero coupon CD that will triple her investment in the same period.
Which option will optimize her investment?
Solution
Let’s compute the accumulated amount under option 1. Here,
r 0.10 m 365 t 12
so n 12(365) 4380 and i . The accumulated amount at the end of
12 years (after 4380 conversion periods) is
or $3.32P. If Jane chooses option 2, the accumulated amount of her investment
after 12 years will be $3P. Therefore, she should choose option 1.
APPLIED EXAMPLE 10
IRAs Moesha has an Individual Retirement
Account (IRA) with a brokerage firm. Her money is invested in a money
market mutual fund that pays interest on a daily basis. Over a 2-year period in
which no deposits or withdrawals were made, her account grew from $4500 to
$5268.24. Find the effective rate at which Moesha’s account was earning interest
over that period (assume 365 days in a year).
Solution
Let r
eff
denote the required effective rate of interest. We have
5268.24 4500(1 r
eff
)
2
(1 r
eff
)
2
1.17072
1 r
eff
⬇ 1.081998 Take the square root on both sides.
or r
eff
⬇ 0.081998. Therefore, the effective rate was 8.20% per year.
A P a1
0.10
365
b
4380
⬇ 3.32P
0.10
365
P192 300,000e
0.09192 19/2
⬇ 598,115, or $598,115
P182 300,000e
0.09182 18/2
⬇ 600,640, or $600,640
P172 300,000e
0.09172 17/2
⬇ 599,837, or $599,837
300,000e
0.09t 1t/2
10 t 102
P1t2 V1t2e
0.09t
5.1 COMPOUND INTEREST 267
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 267
Using Logarithms to Solve Problems in Finance*
The next two examples show how logarithms can be used to solve problems involv-
ing compound interest.
EXAMPLE 11
How long will it take $10,000 to grow to $15,000 if the investment
earns an interest rate of 12% per year compounded quarterly?
Solution
Using Formula (3) with A 15,000, P 10,000, r 0.12, and m 4,
we obtain
15,000 10,000
(1.03)
4t
1.5
Taking the logarithm on each side of the equation gives
ln(1.03)
4t
ln 1.5
4t ln 1.03 ln 1.5 log
b
m
n
n
log
b
m
4t
t ⬇ 3.43
So it will take approximately 3.4 years for the investment to grow from $10,000 to
$15,000.
EXAMPLE 12
Find the interest rate needed for an investment of $10,000 to grow
to an amount of $18,000 in 5 years if the interest is compounded monthly.
Solution
Use Formula (3) with A 18,000, P 10,000, m 12, and t 5.
Thus i and n (12)(5) 60, so
18,000 10,000
Dividing both sides of the equation by 10,000 gives
or, upon simplification,
1.8
Now, we take the logarithm on each side of the equation, obtaining
ln ln 1.8
60 ln ln 1.8a1
r
12
b
a1
r
12
b
60
a1
r
12
b
60
a1
r
12
b
60
18,000
10,000
a1
r
12
b
12152
r
12
ln 1.5
4 ln 1.03
ln 1.5
ln 1.03
15,000
10,000
a1
0.12
4
b
4t
268 5 MATHEMATICS OF FINANCE
*Coverage of the next two examples is optional, and the corresponding exercises in Exercise Set 5.1 are marked optional. A
review of logarithms and examples in which the properties of logarithms are used to solve equations is given in Appendix C.
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 268
1. Find the simple interest on a $500 investment made for
2 yr at an interest rate of 8%/year. What is the accumulated
amount?
2. Find the simple interest on a $1000 investment made for
3 yr at an interest rate of 5%/year. What is the accumulated
amount?
3. Find the accumulated amount at the end of 9 mo on an
$800 deposit in a bank paying simple interest at a rate of
6%/year.
4. Find the accumulated amount at the end of 8 mo on a
$1200 bank deposit paying simple interest at a rate of
7%/year.
5. If the accumulated amount is $1160 at the end of 2 yr
and the simple rate of interest is 8%/year, then what is the
principal?
6. A bank deposit paying simple interest at the rate of 5%/year
grew to a sum of $3100 in 10 mo. Find the principal.
7. How many days will it take for a sum of $1000 to earn $20
interest if it is deposited in a bank paying ordinary simple
interest at the rate of 5%/year? (Use a 365-day year.)
8. How many days will it take for a sum of $1500 to earn $25
interest if it is deposited in a bank paying 5%/year? (Use a
365-day year.)
9. A bank deposit paying simple interest grew from an initial
sum of $1000 to a sum of $1075 in 9 mo. Find the interest
rate.
10. Determine the simple interest rate at which $1200 will
grow to $1250 in 8 mo.
5.1 COMPOUND INTEREST 269
1. Find the present value of $20,000 due in 3 yr at an interest
rate of 12%/year compounded monthly.
2. Paul is a retiree living on Social Security and the income
from his investment. Currently, his $100,000 investment in
a 1-yr CD is yielding 10.6% interest compounded daily. If
he reinvests the principal ($100,000) on the due date of the
CD in another 1-yr CD paying 9.2% interest compounded
daily, find the net decrease in his yearly income from his
investment.
Solutions to Self-Check Exercises 5.1 can be found on
page 272.
1. Explain the difference between simple interest and com-
pound interest.
2. What is the difference between the accumulated amount
(future value) and the present value of an investment?
3. What is the effective rate of interest?
ln
ln
6
1
0
.8
⬇ 0.009796
⬇ e
0.009796
ln
e
x
x
⬇ 1.009844
and
⬇ 1.009844 1
r ⬇ 0.1181
or 11.81% per year.
r
12
a1
r
12
b
a1
r
12
b
5.1 Exercises
5.1 Self-Check Exercises
5.1 Concept Questions
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 269
In Exercises 11–20, find the accumulated amount A if
the principal P is invested at the interest rate of r/year
for t yr.
11. P $1000, r 7%, t 8, compounded annually
12. P $1000, r %, t 6, compounded annually
13. P $2500, r 7%, t 10, compounded semiannually
14. P $2500, r 9%, t compounded semiannually
15. P $12,000, r 8%, t compounded quarterly
16. P $42,000, r %, t 8, compounded quarterly
17. P $150,000, r 14%, t 4, compounded monthly
18. P $180,000, r 9%, t , compounded monthly
19. P $150,000, r 12%, t 3, compounded daily
20. P $200,000, r 8%, t 4, compounded daily
In Exercises 21–24, find the effective rate corresponding
to the given nominal rate.
21. 10%/year compounded semiannually
22. 9%/year compounded quarterly
23. 8%/year compounded monthly
24. 8%/year compounded daily
In Exercises 25–28, find the present value of $40,000 due
in 4 yr at the given rate of interest.
25. 8%/year compounded semiannually
26. 8%/year compounded quarterly
27. 7%/year compounded monthly
28. 9%/year compounded daily
29. Find the accumulated amount after 4 yr if $5000 is invested
at 8%/year compounded continuously.
30. Find the accumulated amount after 6 yr if $6500 is invested
at 7%/year compounded continuously.
31. C
ONSUMER
D
ECISIONS
Mitchell has been given the option
of either paying his $300 bill now or settling it for $306
after 1 mo (30 days). If he chooses to pay after 1 mo, find
the simple interest rate at which he would be charged.
32. C
OURT
J
UDGMENT
Jennifer was awarded damages of
$150,000 in a successful lawsuit she brought against her
employer 5 yr ago. Interest (simple) on the judgment accrues
at the rate of 12%/year from the date of filing. If the case
were settled today, how much would Jennifer receive in the
final judgment?
33. B
RIDGE
L
OANS
To help finance the purchase of a new
house, the Abdullahs have decided to apply for a short-
term loan (a bridge loan) in the amount of $120,000 for a
term of 3 mo. If the bank charges simple interest at the rate
of 12%/year, how much will the Abdullahs owe the bank
at the end of the term?
6
1
4
7
3
4
10
1
2
,
10
1
2
,
8
1
2
34. C
ORPORATE
B
ONDS
David owns $20,000 worth of 10-yr
bonds of Ace Corporation. These bonds pay interest every
6 mo at the rate of 7%/year (simple interest). How much
income will David receive from this investment every
6 mo? How much interest will David receive over the life
of the bonds?
35. M
UNICIPAL
B
ONDS
Maya paid $10,000 for a 7-yr bond
issued by a city. She received interest amounting to $3500
over the life of the bonds. What rate of (simple) interest did
the bond pay?
36. Write Equation (1b) in the slope-intercept form and inter-
pret the meaning of the slope and the A-intercept in terms
of r and P.
Hint: Refer to Figure 1.
37. H
OSPITAL
C
OSTS
If the cost of a semiprivate room in a hos-
pital was $580/day 5 yr ago and hospital costs have risen at
the rate of 8%/year since that time, what rate would you
expect to pay for a semiprivate room today?
38. F
AMILY
F
OOD
E
XPENDITURE
Today a typical family of four
spends $600/month for food. If inflation occurs at the rate
of 3%/year over the next 6 yr, how much should the typi-
cal family of four expect to spend for food 6 yr from now?
39. H
OUSING
A
PPRECIATION
The Kwans are planning to buy a
house 4 yr from now. Housing experts in their area have
estimated that the cost of a home will increase at a rate of
5%/year during that period. If this economic prediction
holds true, how much can the Kwans expect to pay for a
house that currently costs $210,000?
40. E
LECTRICITY
C
ONSUMPTION
A utility company in a western
city of the United States expects the consumption of electric-
ity to increase by 8%/year during the next decade, due mainly
to the expected increase in population. If consumption does
increase at this rate, find the amount by which the utility
company will have to increase its generating capacity in or-
der to meet the needs of the area at the end of the decade.
41. P
ENSION
F
UNDS
The managers of a pension fund have in-
vested $1.5 million in U.S. government certificates of de-
posit that pay interest at the rate of 5.5%/year compounded
semiannually over a period of 10 yr. At the end of this
period, how much will the investment be worth?
42. R
ETIREMENT
F
UNDS
Five and a half years ago, Chris
invested $10,000 in a retirement fund that grew at the
rate of 10.82%/year compounded quarterly. What is his
account worth today?
43. M
UTUAL
F
UNDS
Jodie invested $15,000 in a mutual fund 4 yr
ago. If the fund grew at the rate of 9.8%/year compounded
monthly, what would Jodie’s account be worth today?
44. T
RUST
F
UNDS
A young man is the beneficiary of a trust
fund established for him 21 yr ago at his birth. If the orig-
inal amount placed in trust was $10,000, how much will he
receive if the money has earned interest at the rate of
8%/year compounded annually? Compounded quarterly?
Compounded monthly?
270
5 MATHEMATICS OF FINANCE
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 270
45. I
NVESTMENT
P
LANNING
Find how much money should be
deposited in a bank paying interest at the rate of 8.5%/year
compounded quarterly so that, at the end of 5 yr, the ac-
cumulated amount will be $40,000.
46. P
ROMISSORY
N
OTES
An individual purchased a 4-yr,
$10,000 promissory note with an interest rate of 8.5%/year
compounded semiannually. How much did the note cost?
47. F
INANCING A
C
OLLEGE
E
DUCATION
The parents of a child
have just come into a large inheritance and wish to estab-
lish a trust fund for her college education. If they estimate
that they will need $100,000 in 13 yr, how much should
they set aside in the trust now if they can invest the money
at %/year compounded (a) annually, (b) semiannually,
and (c) quarterly?
48. I
NVESTMENTS
Anthony invested a sum of money 5 yr ago in
a savings account that has since paid interest at the rate
of 8%/year compounded quarterly. His investment is now
worth $22,289.22. How much did he originally invest?
49. R
ATE
C
OMPARISONS
In the last 5 yr, Bendix Mutual Fund
grew at the rate of 10.4%/year compounded quarterly.
Over the same period, Acme Mutual Fund grew at the rate
of 10.6%/year compounded semiannually. Which mutual
fund has a better rate of return?
50. R
ATE
C
OMPARISONS
Fleet Street Savings Bank pays inter-
est at the rate of 4.25%/year compounded weekly in a sav-
ings account, whereas Washington Bank pays interest at
the rate of 4.125%/year compounded daily (assume a 365-
day year). Which bank offers a better rate of interest?
51. L
OAN
C
ONSOLIDATION
The proprietors of The Coachmen
Inn secured two loans from Union Bank: one for $8000 due
in 3 yr and one for $15,000 due in 6 yr, both at an interest
rate of 10%/year compounded semiannually. The bank has
agreed to allow the two loans to be consolidated into one
loan payable in 5 yr at the same interest rate. What amount
will the proprietors of the inn be required to pay the bank
at the end of 5 yr?
Hint: Find the present value of the first two loans.
52. E
FFECTIVE
R
ATE OF
I
NTEREST
Find the effective rate of inter-
est corresponding to a nominal rate of 9%/year com-
pounded annually, semiannually, quarterly, and monthly.
53. Z
ERO
C
OUPON
B
ONDS
Juan is contemplating buying a zero
coupon bond that matures in 10 yr and has a face value of
$10,000. If the bond yields a return of 5.25%/year, how
much should Juan pay for the bond?
54. R
EVENUE
G
ROWTH OF A
H
OME
T
HEATER
B
USINESS
Maxwell
started a home theater business in 2005. The revenue of his
company for that year was $240,000. The revenue grew by
20% in 2006 and by 30% in 2007. Maxwell projected that
the revenue growth for his company in the next 3 yr will be
at least 25%/year. How much does Maxwell expect his
minimum revenue to be for 2010?
55. O
NLINE
R
ETAIL
S
ALES
Online retail sales stood at $23.5 bil-
lion for the year 2000. For the next 2 yr, they grew by
8
1
2
33.2% and 27.8% per year, respectively. For the next 6 yr,
online retail sales were projected to grow at 30.5%, 19.9%,
24.3%, 14.0%, 17.6%, and 10.5% per year, respectively.
What were the projected online sales for 2008?
Source: Jupiter Research
56. P
URCHASING
P
OWER
The inflation rates in the U.S. econ-
omy for 2003 through 2006 are 1.6%, 2.3%, 2.7%, and
3.4%, respectively. What was the purchasing power of a
dollar at the beginning of 2007 compared to that at the
beginning of 2003?
Source: U.S. Census Bureau
57. I
NVESTMENT
O
PTIONS
Investment A offers a 10% return
compounded semiannually, and investment B offers a
9.75% return compounded continuously. Which invest-
ment has a higher rate of return over a 4-yr period?
58. E
FFECT OF
I
NFLATION ON
S
ALARIES
Leonard’s current annual
salary is $45,000. Ten yr from now, how much will he
need to earn in order to retain his present purchasing power
if the rate of inflation over that period is 3%/year com-
pounded continuously?
59. S
AVING FOR
C
OLLEGE
Having received a large inheritance,
Jing-mei’s parents wish to establish a trust for her college
education. If 7 yr from now they need an estimated
$70,000, how much should they set aside in trust now, if
they invest the money at 10.5% compounded quarterly?
Continuously?
60. P
ENSIONS
Maria, who is now 50 yr old, is employed by a
firm that guarantees her a pension of $40,000/year at age
65. What is the present value of her first year’s pension if
the inflation rate over the next 15 yr is 6%/year com-
pounded continuously? 8%/year compounded continu-
ously? 12%/year compounded continuously?
61. R
EAL
E
STATE
I
NVESTMENTS
An investor purchased a piece
of waterfront property. Because of the development of a
marina in the vicinity, the market value of the property is
expected to increase according to the rule
where V(t) is measured in dollars and t is the time (in yr)
from the present. If the rate of appreciation is expected to
be 9% compounded continuously for the next 8 yr, find an
expression for the present value P(t) of the property’s mar-
ket price valid for the next 8 yr. What is P(t) expected to be
in 4 yr?
62. The simple interest formula A P(1 rt) [Formula (1b)]
can be written in the form A Prt P, which is the
slope-intercept form of a straight line with slope Pr and
A-intercept P.
a. Describe the family of straight lines obtained by keep-
ing the value of r fixed and allowing the value of P to
vary. Interpret your results.
b. Describe the family of straight lines obtained by keep-
ing the value of P fixed and allowing the value of r to
vary. Interpret your results.
V1t 2 80,000e
1t/2
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