Page 1 of 32
Critical Thinking Questions
6.1 Identify the steps involved in computing the future value when you have multiple cash
flows.
First, prepare a time line to identify the size and timing of the cash flows. Second,
calculate the present value of each individual cash flow using an appropriate discount
rate. Finally, add up the present values of the individual cash flows to obtain the present
value of a cash flow stream. This approach is especially useful in the real world where
the cash flows for each period are not the same.
6.2 What is the key economic principle involved in calculating the present value and future
value of multiple cash flows?
Regardless of whether you are calculating the present value or the future value of a cash
flow stream, the key idea is to discount or compound the cash flows to the same point in
time.
6.3 What is the difference between a perpetuity and an annuity?
Discounted Cash Flows and Valuation
Page 2 of 32
A cash flow stream that consists of the same amount being received or paid on a periodic
basis is called an annuity. If the same payments are made periodically forever, the
contract is called a perpetuity.
6.4 Define annuity due. Would an investment be worth more if it was an ordinary annuity or
an annuity due? Explain.
When annuity cash flows occur at the beginning of each period, it is called an annuity
due. Annuity due will result in a bigger investment than an ordinary annuity because each
cash flow will accrue an extra interest payment.
6.5 Raymond Bartz is trying to choose between two equally risky annuities, each paying
$5,000 per year for five years. One is an ordinary annuity, and the other is an annuity
due. Which of the following statements is most correct?
a. The present value of the ordinary annuity must exceed the present value of the
annuity due, but the future value of an ordinary annuity may be less than the future
value of the annuity due.
b. The present value of the annuity due exceeds the present value of the ordinary
annuity, while the future value of the annuity due is less than the future value of the
ordinary annuity.
c. The present value of the annuity due exceeds the present value of the ordinary
annuity, and the future value of the annuity due also exceeds the future value of the
ordinary annuity.
Page 3 of 32
d. If interest rates increase, the difference between the present value of the ordinary
annuity and the present value of the annuity due remains the same.
c. The present value of the annuity due exceeds the present value of the ordinary
annuity, and the future value of the annuity due also exceeds the future value of the
ordinary annuity.
6.6 Which of the following investments will have the highest future value at the end of three
years? Assume that the effective annual rate for all investments is the same.
a. You earn $3,000 at the end of three years (a total of one payment).
b. You earn $1,000 at the end of every year for the next three years (a total of three
payments).
c. You earn $1,000 at the beginning of every year for the next three years (a total of
three payments).
c. Earning $1,000 at the beginning of each year for the next three years will have the
highest future value as it is an annuity due.
6.7 Explain whether or not each of the following statements is correct.
a. A 15-year mortgage will have larger monthly payments than a 30-year mortgage of
the same amount and same interest rate.
Page 4 of 32
This is a true statement. The 15-year mortgage will have higher monthly payments since
more of the principal will have to be paid each month than in the case of a 30-year
mortgage.
b. If an investment pays 10 percent interest compounded annually, its effective rate will
also be 10 percent.
This is true since the frequency of compounding is annual and hence the rate for a single
period is the same as the rate for a year.
6.8 When will the annual percentage rate (APR) be the same as the effective annual rate
(EAR)?
The annual percentage rate (APR) will be the same as the effective annual rate only if the
compounding period is annual, not otherwise.
6.9 Why is the EAR superior to the APR in measuring the true economic cost or return?
Unlike the APR, which reflects annual compounding, the EAR takes into account the
actual number of compounding periods. For example, suppose there are two investment
alternatives that both pay an APR of 10 percent. Assume that the first pays interest
annually and that the second pays interest quarterly. It would be a mistake to assume that
both investments will provide the same return. The real return on the first one is 10
Page 5 of 32
percent, but the second investment actually provides a return of 10.38 percent because of
the quarterly compounding. Thus, this is the superior investment!
6.10 Suppose two investments have equal lives and multiple cash flows. A high discount rate
tends to favor:
a. the investment with large cash flow early.
b. the investment with large cash flow late.
c. the investment with even cash flow.
d. neither investment since they have equal lives.
a. The investment with large cash flows early will be worth more compared to the one
with the large cash flows late. The cash flows that come in later will have a heavier
penalty when using a higher discount rate. Thus the investment with large cash flows
early will be favored.
Page 6 of 32
Questions and Problems
BASIC
6.1 Future value with multiple cash flows: Konerko, Inc., expects to earn cash flows of
$13,227, $15,611, $18,970, and $19,114 over the next four years. If the company uses an
8 percent discount rate, what is the future value of these cash flows at the end of year 4?
Solution:
0 8% 1 2 3 4
├───────┼────────┼───────┼────────┤
$13,227 $15,611 $18,970 $19,114
6.2 Future value with multiple cash flows: Ben Woolmer has an investment that will pay
him the following cash flows over the next five years: $2,350, $2,725, $3,128, $3,366,
and $3,695. If his investments typically earn 7.65 percent, what is the future value of the
investment’s cash flows at the end of five years?
Solution:
0 7.65% 1 2 3 4 5
$74,472.48=
+++=
+++=
114,19$60.487,20$67.208,18$21.662,16$
114,19$)08.1(970,18$)08.1(611,15$)08.1(227,13$FV
123
4
Page 7 of 32
├───────┼────────┼───────┼────────┼───────┤
$2,350 $2,725 $3,128 $3,366 $3,695
$17,498.75=
++++=
++++=
695,3$50.623,3$89.624,3$45.399,3$91.155,3$
695,3$)0765.1(366,3$)0765.1(128,3$)0765.1(725,2$)0765.1(350,2$FV 1234
5
6.3 Future value with multiple cash flows: You are a freshman in college and are planning
a trip to Europe when you graduate from college at the end of four years. You plan to
save the following amounts starting today: $625, $700, $700, and $750. If the account
pays 5.75 percent annually, how much will you have at the end of four years?
Solution:
0 5.75% 1 2 3 4
├───────┼────────┼───────┼────────┤
$625 $700 $700 $750
$3,185.40=
+++=
+++=
13.79381.782$83.827
$63.781$
)0575.1(750$
)0575.1(700$)0575.
1(700$)0575.1(625$
FV
234
4
6.4 Present value with multiple cash flows: Saul Cervantes has just purchased some
equipment for his landscaping business. He plans to pay the following amounts at the end
of the next five years: $10,450, $8,500, $9,675, $12,500, and $11,635. If he uses a
discount rate of 10.875 percent, what is the cost of the equipment he purchased today?
Page 8 of 32
Solution:
0 10.875% 1 2 3 4 5
├───────┼────────┼───────┼────────┼───────┤
$10,450 $8,500 $9,675 $12,500 $11,635
$38,652.76=
++++=
++++=
82.943,6$33.271,823.098,7$35.914,6$03.425,9$
)10875.1(
635,11$
)10875.1(
500,12$
)10875.1(
675,9$
)10875.1(
500,8$
)10875.1(
450,10$
PV 5432
6.5 Present value with multiple cash flows: Jeremy Fenloch borrowed from his friend a
certain amount and promised to repay him the amounts of $1,225, $1,350, $1,500,
$1,600, and $1,600 over the next five years. If the friend normally discounts investments
at 8 percent annually, how much did Jeremy borrow?
Solution:
0 8% 1 2 3 4 5
├───────┼────────┼───────┼────────┼───────┤
$1,225 $1,350 $1,500 $1,600 $1,600
$5,747.40=
++++=
++++=
93.088,1$05.176,1$75.190,1$41.157,1$26.134,1$
)08.1(
600,1$
)08.1(
600,1$
)08.1(
500,1$
)08.1(
350,1$
)08.1(
225,1$
PV
5432
Page 9 of 32
6.6 Present value with multiple cash flows: Biogenesis, Inc., expects the following cash
flow stream over the next five years. The company discounts all cash flows at a 23
percent discount rate. What is the present value of this cash flow stream?
Solution:
0 23% 1
2
3 4 5
├───────┼────────┼───────┼────────┼───────┤
-$1,133,676 -$978,452 $275,455 $878,326 $1,835,444
2$384,711.7−=
+++−−=
+++
−
+
−
=
94.951,651$.43.738,383$09.025,148$37.739,646$80.687,921$
)23.1(
444,835,1$
)23.1(
326,878$
)23.1(
455,275$
)23.1(
452,978$
)23.1(
676,133,1$
PV 5432
6.7 Present value of an ordinary annuity: An investment opportunity requires a payment of
$750 for 12 years, starting a year from today. If your required rate of return is 8 percent,
what is the value of the investment today?
Solution:
0 8% 1 2 3 11 12
├───────┼────────┼───────┼………………┼───────┤
$750 $750 $750 $750 $750
1
2
3
4
5
-$1,133,676
-$978,452
$275,455
$878,326
$1,835,444
Page 10 of 32
Annual payment = PMT = $750
No. of payments = n = 12
Required rate of return = 8%
Present value of investment = PVA12
$5,652.06=
×=
−
×=
+
−
×=
5361.7750$
08.0
)08.1(
1
1
750$
)1(
1
1
PMTPVA
12
n
n
i
i
6.8 Present value of an ordinary annuity: Dynamics Telecommunications Corp. has made
an investment in another company that will guarantee it a cash flow of $22,500 each year
for the next five years. If the company uses a discount rate of 15 percent on its
investments, what is the present value of this investment?
Solution:
0 15% 1 2 3 4 5
├───────┼────────┼───────┼────────┼───────┤
$22,500 $22,500 $22,500 $22,500 $22,500
Annual payment = PMT = $22,500
No. of payments = n = 5
Required rate of return = 15%
Page 11 of 32
Present value of investment = PVA5
$75,423.49=
×=
−
×=
+
−
×=
3522.3500,22$
15.0
)15.1(
1
1
500,22$
)1(
1
1
PMTPVA
5
n
ni
i
6.9 Future value of an ordinary annuity: Robert Hobbes plans to invest $25,000 a year for
the next seven years in an investment that will pay him a rate of return of 11.4 percent.
He will invest at the end of each year. What is the amount that Mr. Hobbes will have at
the end of seven years?
Solution:
0 11.4% 1 2 3 6 7
├───────┼────────┼───────┼………………┼───────┤
$25,000 $25,000 $25,000 $25,000 $25,000
Annual investment = PMT = $25,000
No. of payments = n = 7
Investment rate of return = 11.4%
Future value of investment = FVA7
Page 12 of 32
5$247,609.9=
×=
−
×=
−+
×=
9044.9000,25$
114.0
1)114.1(
000,25$
1)1(
PMTFVA
7
n
n
i
i
6.10 Future value of an ordinary annuity: Cecelia Thomas is a sales executive at a
Baltimore firm. She is 25 years old and plans to invest $3,000 every year in an IRA
account, beginning at the end of this year until she turns 65 years old. If the IRA
investment will earn 9.75 percent annually, how much will she have in 40 years when she
turns 65 years old?
Solution:
0 9.75% 1 2 3 39 40
├───────┼────────┼───────┼………………┼───────┤
$3,000 $3,000 $3,000 $3,000 $3,000
Annual investment = PMT = $3,000
No. of payments = n = 40
Investment rate of return = 9.75%
Future value of investment = FVA40
.41$1,240,676=
×=
−
×=
−+
×=
5588.413000,3$
0975.0
1)0975.1(
000,
3$
1)1(
PMTFVA
40
n
n
i
i
Page 13 of 32
6.11 Future value of an annuity. Refer to Problem 6.10. If Cecelia Thomas starts saving at
the beginning of each year, how much will she have at age 65?
Solution:
0 9.75% 1 2 3 39 40
├───────┼────────┼───────┼………………┼───────┤
$3,000 $3,000 $3,000 $3,000 $3,000
Annual investment = PMT = $3,000
No. of payments = n = 40
Type of annuity = Annuity due
Investment rate of return = 9.75%
Future value of investment = FVA40
.36$1,361,642=
××=
−
×=
+
−+
×=
0975.15588.413000,3$)0975.1(
0975.0
1)0975.1(
000,3$
)1(
1)1(
PMTFVA
40
n
n
i
i
i
6.12 Computing annuity payment: Kevin Winthrop is saving for an Australian vacation in
three years. He estimates that he will need $5,000 to cover his airfare and all other
expenses for a week-long holiday in Australia. If he can invest his money in an S&P 500
Page 14 of 32
equity index fund that is expected to earn an average return of 10.3 percent over the next
three years, how much will he have to save every year, starting at the end of this year?
Solution:
0 10.3% 1 2 3
├───────┼────────┼───────┤
PMT PMT PMT
FVAn = $5,000
Future value of annuity = FVA = $5,000
Return on investment = i = 10.3%
Payment required to meet target = PMT
Using the FVA equation:
$1,506.20=
=
−
=
−
×=
−+
×=
3196.3
000,5$
103.0
1)103.1(
000,5$
PMT
103.0
1)103.1(
PMT000,5$
1)1(
PMTFVA
3
3
n
n
i
i
Kevin has to save $1,506.20 every year for the next three years to reach his target of
$5,000.
6.13 Computing annuity payment: The Elkridge Bar & Grill has a seven-year loan of
$23,500 with Bank of America. It plans to repay the loan by paying in seven equal
cr
wry
ccc
Page 15 of 32
installments starting today. If the rate of interest is 8.4 percent, how much will each
payment be worth?
Page 16 of 32
0 1 2 3 6 7
├───────┼────────┼───────┼………………┼───────┤
PMT PMT PMT PMT PMT PMT
PVAn = $23,500 n = 7; i = 8.4%
Present value of annuity = PVA = $23,500
Return on investment = i = 8.4%
Payment required to meet target = PMT
Type of annuity = Annuity due
Using the PVA equation:
$4,221.07=
×
=
−
=
+
+
−
×=
084.11359
.5
500,23$
)084.1(
084.0
)084.1(
1
1
500
,23$
PMT
)1(
)1(
1
1
PMTPVA
7
n
n
i
i
i
Each payment made by Elkridge Bar & Grill will be $4,221.07, starting today.
6.14 Perpetuity: Your grandfather is retiring at the end of next year. Heould like to receive a
payment of $10,000 a year forever, starting when he retires. If he can invest at 6.5
percent, how much does need to invest to receive the desired cash flow?
Page 17 of 32
Solution:
Annual payment needed = PMT = $10,000
Investment rate of return = i = 6.5%
Term of payment = Perpetuity
Present value of investment needed = PV
5$153,846.1=
== 065.0
000,10$PMT
y Perpetuitof PV i
6.15 Perpetuity: Calculate the perpetuity payments for each of the following cases:
a. $250,000 invested at 6%
b. $50,000 invested at 12%
c. $100,000 invested at 10%
Solution:
a. Annual payment = PMT
Investment rate of return = i = 6%
Term of payment = Perpetuity
Present value of investment needed = PV = $250,000
$15,000=
×=×=
=
0.06$250,000PVPMT
PMT
y Perpetuitof PV
i
i
b. Annual payment = PMT
Page 18 of 32
Investment rate of return = i = 12%
Term of payment = Perpetuity
Present value of investment needed = PV = $50,000
$6,000=
×=×=
=
0.12$50,000PVPMT
PMT
y Perpetuitof PV
i
i
c. Annual payment = PMT
Investment rate of return = i = 10%
Term of payment = Perpetuity
Present value of investment needed = PV = $100,000
$10,000=
×=×=
=
0.10$100,000PVPMT
PMT
y Perpetuitof PV
i
i
6.16. Effective annual rate: Raj Krishnan bought a Honda Accord for a price of $17,345. He
put down $6,000 and financed the rest through the dealer at an APR of 4.9 percent for
four years. What is the effective annual rate (EAR) if payments are made monthly?
Solution:
Loan amount = PV = $11,345
Interest rate on loan = i = 4.9%
Frequency of compounding = m = 12
Effective annual rate = EAR
Page 19 of 32
5%=−=
−
+=−
+=
×
105.1
1
12
049.0
11
m
1EAR
121m
i
6.17 Effective annual rate: Cyclone Rentals borrowed $15,550 from a bank for three years. If
the quoted rate (APR) is 6.75 percent, and the compounding is daily, what is the effective
annual rate (EAR)?
Solution:
Loan amount = PV = $15,550
Interest rate on loan = i = 6.75%
Frequency of compounding = m = 365
Effective annual rate = EAR
7%=−=
−
+=−
+=
×
10698.1
1
365
0675.0
11
m
1EAR
3651m
i
6.18 Growing perpetuity: You are evaluating a growing perpetuity product from a large
financial services firm. The product promises an initial payment of $20,000 at the end of
this year and subsequent payments that will thereafter grow at a rate of 3.4 percent
annually. If you use a 9 percent discount rate for investment products, what is the present
value of this growing perpetuity?
Solution:
Cash flow at t = 1 = CF1 = $20,000
Page 20 of 32
Annual growth rate = g = 3.4%
Discount rate = i = 9%
Present value of growing perpetuity = PVA∞
6$357,142.8=
−
=
−
=
∞)034.009.0(
000,20$
)g(
CF
PVA 1
i
INTERMEDIATE
6.19 Future value with multiple cash flows: Trigen Corp. is expecting to invest cash flows
of $331,000, $616,450, $212,775, $818,400, $1,239,644, and $1,617,848 in research and
development over the next six years. If the appropriate interest rate is 6.75 percent, what
is the future value of these investment cash flows?
Solution:
0 6.75% 1 2 3 4 5 6
├───────┼────────┼───────┼────────┼───────┼────────┤
$331,000 $616,450 $212,775 $818,400 $1,239,644 $1,617,848
.89$5,391,977=
+++++=
++
+++=
848,617,1$97.319,323,1$84.612,932$74.835,258$85.514,800$49.846,458$
848,617,1$)0765.1(644,239,1$
)0675.1(400,818$)0675.1(775,212$)0675.1(450,616$)0675.1(000,331$FV
1
2345
6
Page 21 of 32
6.20 Future value with multiple cash flows: Stephanie Watson plans to adopt the following
investment pattern beginning next year. She will invest $3,125 in each of the next three
years and will then make investments of $3,650, $3,725, $3,875, and $4,000 over the
following four years. If the investments are expected to earn 11.5 percent annually, how
much will she have at the end of the seven years?
Solution:
Expected rate of return = i = 11.5%
Investment period = n = 7 years
Future value of investment = FV
$34,231.57=
++++++=
+++
+++=
000,4$63.320,4$01.631,
4$61.059,5$03.830,4$48.385,5$81.004,6$
000,4$)115.1(875,3$)115.1
(725,3$
)115.1(650,3$)115.1(450,
616$)115.1(125,3$)115.1(125,3$FV
12
345
6
7
6.21 Present value with multiple cash flows: Carol Jenkins, a lottery winner, will receive the
following payments over the next seven years. If she can invest her cash flows in a fund
that will earn 10.5 percent annually, what is the present value of her winnings?
1
2
3
4
5
6
7
$200,000
$250,000
$275,000
$300,000
$350000
$400,000
$550,000
Solution:
Expected rate of return = i = 10.5%
Page 22 of 32
Investment period = n = 7 years
Future value of investment = FV
.71$1,496,377=
++
++++=
++++++=
77.417,273$47.728,219$
96.449,212$46.220,201$56.819,203$01.746,204$48.995,180$
)105.1(
000,550$
)105.1(
000,400$
)105.1(
000,350$
)105.1(
000,300$
)105.1(
000,275$
)105.1(
000,250$
)105.1(
000,200$
FV 7654321
7
6.22 Computing annuity payment: Gary Whitmore is a high school sophomore. He currently
has $7,500 in a money market account paying 5.65 percent annually. He plans to use this
and his savings over the next four years to buy a car at the end of his sophomore year in
college. He estimates that the car will cost him $12,000 in four years. How much should
he invest in the money market account every year for the next four years if he wants to
achieve his target?
Page 23 of 32
Solution:
Cost of car in four years = $12,000
Amount invested in money market account now = PV = $7,500
Return earned by investment = i = 5.65%
Value of current investment in 4 years = FV4
14.344,9$
)0565.1(500,7$)1(PVFV 44
4
=
=+= i
Balance of money needed to buy car = $12,000 – $9,344.14 = $2,655.86 = FVA
Payment needed to reach target = PMT
$610.27=
=
−
=
+−
=
−+
×=
351949.4
86.655,2$
0565.0
1)0565.1(
86.655,2$
)1(1
FVA
PMT
1)1(
PMTFVA
4n
n
i
i
i
i
6.23 Growing annuity: Modern Energy Company owns several gas stations. Management is
looking to open a new station in the western suburbs of Baltimore. One possibility they
are evaluating is to take over a station located at a site that has been leased from the
county. The lease, originally for 99 years, currently has 73 years before expiration. The
gas station generated a net cash flow of $92,500 last year, and the current owners expect
an annual growth rate of 6.3 percent. If Modern Energy uses a discount rate of 14.5
percent to evaluate such businesses, what is the present value of this growing annuity?
Solution:
Page 24 of 32
Time for lease to expire = n = 73 years
Last year’s net cash flow = CF0 = $92,500
Expected annual growth rate = g = 6.3%
Firm’s required rate of return = i = 14.5%
Expected cash flow next year = CF1 = $92,500(1 + g) = $92,500(1.063)
= $98,327.50
Present value of growing annuity = PVAn
.54$1,193,831=
×=
−×
−
=
+
+
−×
−
=
995593.085.115,199,1$
145.1
063.1
1
)063.0145.0(
50.327,98$
i1
g1
1
)g(
CF
PVA
73n
1
n
i
6.24 Future value of an annuity due: Jeremy Denham plans to save $5,000 every year for
the next eight years, starting today. At the end of eight years, Jeremy will turn 30 years
old and plans to use his savings toward the down payment on a house. If his investment
in a mutual fund will earn him 10.3 percent annually, how much will he have saved in
eight years when he will need the money to buy a house?
Solution:
0 10.3% 1 2 3 7 8
├───────┼────────┼───────┼………………┼───────┤
$5,000 $5,000 $5,000 $5,000 $5,000
Annual investment = PMT = $5,000
Page 25 of 32
No. of payments = n = 8
Type of annuity = Annuity due
Investment rate of return = 10.3%
Future value of investment = FVA8
$63,760.19=
××=
−
×=
+
−+
×=
103.15612.11000,5$)103.1(
103.0
1)103.1(
000,5$
)1(
1)1(
PMTFVA
8
n
n
i
i
i
6.25 Present value of an annuity due: Grant Productions has borrowed a huge sum from the
California Finance Company at a rate of 17.5 percent for a seven-year period. The loan
calls for a payment of $1,540,862.19 each year beginning today. What is the amount
borrowed by this company? Round to the nearest dollar.
Solution:
0 17.5% 1 2 3 6 7
├───────┼────────┼───────┼………………┼───────┤
PMT =$1,540,862.19 at the beginning of each year
Annual payment = PMT = $1,540,862.19
Type of annuity = Annuity due
No. of payments = n = 7
Required rate of return = 17.5%
Present value of investment = PVA8
Page 26 of 32
$7,000,000.98$6,999,999 ≅=
××=
−
×=
+
+
−
×=
175.18663.319.862,540,1$)175.1(
175.0
)175.1(
1
1
19.862,540,1$
)1(
)1(
1
1
PMTPVA
7
n
n
i
i
i
6.26 Present value of an annuity due: Sharon Kabana has won a state lottery and will
receive a payment of $89,729.45 every year, starting today for the next 20 years. If she
invests the proceeds at a rate of 7.25 percent, what is the present value of the cash flows
that she will receive? Round to the nearest dollar.
Solution:
0 7.25% 1 2 3 19 20
├───────┼────────┼───────┼………………┼───────┤
PMT = $89,729.45 at the beginning of each year
Annual payment = PMT = $89,729.45
Type of annuity = Annuity due
No. of payments = n = 20
Required rate of return = 7.25%
Present value of investment = PVA20
Page 27 of 32
$1,000,0005$999,999.9 ≅=
××=
−
×=
+
+
−
×=
0725.13912.1045.729,89$)0725.1(
0725.0
)0725.1(
1
1
45.729,89$
)1(
)1(
1
1
PMTPVA
20
n
n
i
i
i
6.27 Perpetuity: Calculate the present value of the following perpetuities:
a. $1,250 discounted back to the present at 7%
b. $7,250 discounted back to the present at 6.33%
c. $850 discounted back to the present at 20%
Solution:
a. Annual payment = PMT =$1,250
Investment rate of return = i = 7%
Term of payment = Perpetuity
Present value of investment needed = PV
$17,857.14=
== 07.0
250,1$PMT
y Perpetuitof PV i
b. Annual payment = PMT =$7,250
Investment rate of return = i = 6.33%
Term of payment = Perpetuity.
Page 28 of 32
Present value of perpetuity = PV
7$114,533.9=
== 0633.0
250,7$PMT
y Perpetuitof PV i
c. Annual payment = PMT =$850
Investment rate of return = i = 20%
Term of payment = Perpetuity.
Present value of investment needed = PV
$4,250=
== 20.0
850$PMT
y Perpetuitof PV i
6.28 Effective annual rate: Find the effective annual interest rate (EAR) on each of the
following:
a. 6% compounded quarterly.
b. 4.99% compounded monthly.
c. 7.25% compounded semi-annually.
d. 5.6% compounded daily.
Solution:
a. Interest rate = i = 6%
Frequency of compounding = m = 4
Effective annual rate = EAR
Page 29 of 32
%6 14.106136.1
1
4
06.0
11
m
1EAR
41m
=−=
−
+=−
+=
×
i
b. Interest rate = i = 4.99%
Frequency of compounding = m = 12
Effective annual rate = EAR
%11.510511.1
1
12
0499.0
11
m
1EAR
121m
=−=
−
+=−
+=
×
i
c. Interest rate = i = 7.25%
Frequency of compounding = m = 2
Effective annual rate = EAR
%38.710738.1
1
2
0725.0
11
m
1EAR
21m
=−=
−
+=−
+=
×
i
d. Interest rate = i = 5.6%
Frequency of compounding = m = 365
Effective annual rate = EAR
%76.510576.1
1
365
056.0
11
m
1EAR
3651m
=−=
−
+=−
+=
×
i
6.29 Effective annual rate: Which of the following investments has the highest effective
Page 30 of 32
annual rate (EAR)?
a. A bank CD that pays 8.25% interest quarterly.
b. A bank CD that pays 8.25% monthly.
c. A bank CD that pays 8.45% annually.
d. A bank CD that pays 8.25% semiannually.
e. A bank CD that pays 8% daily (on a 365-day basis).
Solution:
a. Interest rate on CD = i = 8.25%
Frequency of compounding = m = 4
Effective annual rate = EAR
%51.8108509.1
1
4
0825.0
11
m
1EAR
41m
=−=
−
+=−
+=
×
i
b. Interest rate on CD = i = 8.25%
Frequency of compounding = m = 1
Effective annual rate = EAR
%57.810857.1
1
12
0825.0
11
m
1EAR
121m
=−=
−
+=−
+=
×
i
c. Interest rate on CD = i = 4.99%
Frequency of compounding = m = 12
Page 31 of 32
Effective annual rate = EAR
%45.810845.1
1
1
0845.0
11
m
1EAR
11m
=−=
−
+=−
+=
×
i
d. Interest rate on CD = i = 8.25%
Frequency of compounding = m = 2
Effective annual rate = EAR
%42.810842.1
1
2
0825.0
11
m
1EAR
21m
=−=
−
+=−
+=
×
i
e. Interest rate on CD = i = 8%
Frequency of compounding = m = 365
Effective annual rate = EAR
%33.810833.1
1
365
08.0
11
m
1EAR
3651m
=−=
−
+=−
+=
×
i
The bank CD that pays 8.25 percent monthly has the highest yield.
6.30 Effective annual rate: You are considering three alternative investments: (1) a three-
year bank CD paying 7.5 percent interest compounded quarterly; (2) a three-year bank
CD paying 7.3 percent interest compounded monthly; and (3) a three-year bank CD
paying 7.75 percent interest compounded annually. Which investment has the highest
effective annual rate?
Page 32 of 32
Solution:
(1) Interest rate on CD = i = 75%
Frequency of compounding = m = 4
Effective annual rate = EAR
%71.710771.1
1
4
075.0
11
m
1EAR
41m
=−=
−
+=−
+=
×
i
(2) Interest rate on CD = i = 7.3%
Frequency of compounding = m = 12
Effective annual rate = EAR
%55.710755.1
1
12
073.0
11
m
1EAR
121m
=−=
−
+=−
+=
×
i
(3) Interest rate on CD = i = 7.75%
Frequency of compounding = m = 1
Effective annual rate = EAR
%75.710775.1
1
1
0775.0
11
m
1EAR
11m
=−=
−
+=−
+=
×
i
The three-year bank CD paying 7.75 percent interest compounded annually has the
highest effective yield.
