CHAPTER 2

TIME VALUE OF MONEY

True/False

Easy:
(2.2) Compounding Answer: a EASY
1
. One potential benefit from starting to invest early for retirement is
that the investor can expect greater benefits from the compounding of
interest.

a. True
b. False

(2.3) PV versus FV Answer: b EASY

2
. If the discount (or interest) rate is positive, the present value of an
expected series of payments will always exceed the future value of the
same series.

a. True
b. False

(2.3) PV versus FV Answer: a EASY

3
. Disregarding risk, if money has time value, it is impossible for the
present value of a given sum to exceed its future value.

a. True
b. False

(2.15) Effective annual rate Answer: b EASY

4
. If a bank compounds savings accounts quarterly, the nominal rate will
exceed the effective annual rate.

a. True
b. False

(2.17) Amortization Answer: a EASY

5
. The payment made each period on an amortized loan is constant, and it
consists of some interest and some principal. The closer we are to the
end of the loan's life, the greater the percentage of the payment that
will be a repayment of principal.

1. (2.2) Compounding Answer: a EASY

2. (2.3) PV versus FV Answer: b EASY

3. (2.3) PV versus FV Answer: a EASY

4. (2.15) Effective annual rate Answer: b EASY

Chapter 2: Time Value True/False Page 19

 a. True
b. False

Medium:
(2.2) Compounding Answer: b MEDIUM
6
. The greater the number of compounding periods within a year, then (1)
the greater the future value of a lump sum investment at Time 0 and (2)
the greater the present value of a given lump sum to be received at some
future date.

a. True
b. False

(2.2) Comparative compounding Answer: a MEDIUM

7
. Suppose an investor plans to invest a given sum of money. She can earn an
effective annual rate of 5% on Security A, while Security B will provide
an effective annual rate of 12%. Within 11 years' time, the compounded
value of Security B will be more than twice the compounded value of
Security A. (Ignore risk, and assume that compounding occurs daily.)

a. True
b. False

(2.3) PV of a sum Answer: a MEDIUM

8
. The present value of a future sum decreases as either the discount rate
or the number of periods per year increases.

a. True
b. False

(2.9) PV of an annuity Answer: a MEDIUM

9
. All other factors held constant, the present value of a given annual
annuity decreases as the number of discounting periods per year increases.

6. (2.2) Compounding Answer: b MEDIUM

7. (2.2) Comparative compounding Answer: a MEDIUM

Work out the numbers with a calculator:

PV 1000 FVA = $1,710.34
Rate on A 5% 2*FVA = $3,420.68
Rate on B 12% FVB = $3,478.55
Years 11 FVB > 2*FVA, so TRUE

8. (2.3) PV of a sum Answer: a MEDIUM

9. (2.9) PV of an annuity Answer: a MEDIUM

Page 20 True/False Chapter 2: Time Value

 a. True
b. False

(2.15) Effective and nominal rates Answer: a MEDIUM

10
. As a result of compounding, the effective annual rate on a bank deposit
(or a loan) is always equal to or greater than the nominal rate on the
deposit (or loan).

a. True
b. False

(2.15) Periodic and nominal rates Answer: a MEDIUM

11
. If we are given a periodic interest rate, say a monthly rate, we can
find the nominal annual rate by multiplying the periodic rate by the
number of periods per year.

a. True
b. False

(2.17) Amortization Answer: b MEDIUM

12
. When a loan is amortized, a relatively high percentage of the payment
goes to reduce the outstanding principal in the early years, and the
principal repayment's percentage declines in the loan's later years.

a. True
b. False

(2.17) Amortization Answer: b MEDIUM

13
. Midway through the life of an amortized loan, the percentage of the
payment that represents interest is equal to the percentage that
represents principal repayment. This is true regardless of the original
life of the loan.

a. True
b. False

Multiple Choice: Conceptual

Easy:
(2.1) Time lines Answer: a EASY
14
. Which of the following statements is NOT CORRECT?

a. A time line is meaningful only if all cash flows occur annually.

One could make up an example and see that the statement is true. Alternatively, one could simply
recognize that the PV of an annuity declines as the discount rate increases and recognize that more frequent
compounding increases the effective rate.

5. (2.17) Amortization Answer: a EASY

Chapter 2: Time Value True/False Page 21

 b. Time lines are useful for visualizing complex problems prior to
doing actual calculations.
c. Time lines can be constructed even in situations where some of the
cash flows occur annually but others occur quarterly.
d. Time lines can be constructed for annuities where the payments
occur at either the beginning or the end of periods.
e. The cash flows shown on a time line can be in the form of annuity
payments, but they can also be uneven amounts.

(2.1) Time lines Answer: b EASY

15
. Which of the following statements is CORRECT?

a. A time line is not meaningful unless all cash flows occur annually.
b. Time lines are useful for visualizing complex problems prior to
doing actual calculations.
c. Time lines cannot be constructed to deal with situations where some
of the cash flows occur annually but others occur quarterly.
d. Time lines can only be constructed for annuities where the payments
occur at the ends of the periods, i.e., for ordinary annuities.
e. Time lines cannot be constructed where some of the payments
constitute an annuity but others are unequal and thus are not part
of the annuity.

(2.3) Effects of factors on PVs Answer: b EASY

16
. You are analyzing the value of a potential investment by calculating the

14. (2.1) Time lines Answer: a EASY

10. (2.15) Effective and nominal rates Answer: a MEDIUM

11. (2.15) Periodic and nominal rates Answer: a MEDIUM

12. (2.17) Amortization Answer: b MEDIUM

13. (2.17) Amortization Answer: b MEDIUM

There is no reason to think that this statement would be true. Each portion of the payment representing interest
declines, while each portion representing principal repayment increases. Therefore, the statement is clearly false.
We could also work out some numbers to prove this point. Here's an example for a 3-year loan at a 10% annual
interest rate. The interest component is never equal to the principal repayment component.

Original loan 1000

Rate 10%
Life 3
Payment $402.11

Beg. Balance Interest Principal Ending Bal.

1 $1,000.00 $100.00 $302.11 $697.89
2 $697.89 $69.79 $332.33 $365.56
3 $365.56 $36.56 $365.56 $0.00
15. (2.1) Time lines Answer: b EASY

Page 22 True/False Chapter 2: Time Value

 sum of the present values of its expected cash flows. Which of the
following would lower the calculated value of the investment?

a. The cash flows are in the form of a deferred annuity, and they
total to $100,000. You learn that the annuity lasts for only 5
rather than 10 years, hence that each payment is for $20,000 rather
than for $10,000.
b. The discount rate increases.
c. The riskiness of the investment’s cash flows decreases.
d. The total amount of cash flows remains the same, but more of the
cash flows are received in the earlier years and less are received
in the later years.
e. The discount rate decreases.

(2.6) Annuities Answer: d EASY

17
. Which of the following statements is CORRECT?

a. The cash flows for an ordinary (or deferred) annuity all occur at
the beginning of the periods.
b. If a series of unequal cash flows occurs at regular intervals, such
as once a year, then the series is by definition an annuity.
c. The cash flows for an annuity due must all occur at the ends of the
periods.
d. The cash flows for an annuity must all be equal, and they must
occur at regular intervals, such as once a year or once a month.
e. If some cash flows occur at the beginning of the periods while
others occur at the ends, then we have what the textbook defines as
a variable annuity.

Medium:
(2.14) Solving for I with uneven cash flows Answer: c MEDIUM
18
. Which of the following statements is CORRECT?

a. If you have a series of cash flows, all of which are positive, you
can solve for I, where the solution value of I causes the PV of the
cash flows to equal the cash flow at Time 0.
b. If you have a series of cash flows, and CF0 is negative but all of
the other CFs are positive, you can solve for I, but only if the
sum of the undiscounted cash flows exceeds the cost.
c. To solve for I, one must identify the value of I that causes the PV
of the positive CFs to equal the absolute value of the PV of the
negative CFs. This is, essentially, a trial-and-error procedure
that is easy with a computer or financial calculator but quite
difficult otherwise.
d. If you solve for I and get a negative number, then you must have
made a mistake.
e. If CF0 is positive and all the other CFs are negative, then you
16. (2.3) Effects of factors on PVs Answer: b EASY

17. (2.6) Annuities Answer: d EASY

18. (2.14) Solving for I with uneven cash flows Answer: c MEDIUM

Chapter 2: Time Value True/False Page 23

 cannot solve for I.

(2.15) Effective annual rate Answer: e MEDIUM

19
. Which of the following bank accounts has the highest effective annual
return?

a. An account that pays 8% nominal interest with monthly compounding.

b. An account that pays 8% nominal interest with annual compounding.
c. An account that pays 7% nominal interest with daily (365-day)
compounding.
d. An account that pays 7% nominal interest with monthly compounding.
e. An account that pays 8% nominal interest with daily (365-day)
compounding.

(2.15) Quarterly compounding Answer: c MEDIUM

20
. Your bank account pays a 6% nominal rate of interest. The interest is
compounded quarterly. Which of the following statements is CORRECT?

a. The periodic rate of interest is 1.5% and the effective rate of

interest is 3%.
b. The periodic rate of interest is 6% and the effective rate of
interest is greater than 6%.
c. The periodic rate of interest is 1.5% and the effective rate of
interest is greater than 6%.
d. The periodic rate of interest is 3% and the effective rate of
interest is 6%.
e. The periodic rate of interest is 6% and the effective rate of

Page 24 True/False Chapter 2: Time Value

 interest is also 6%.

(2.17) Amortization Answer: c MEDIUM

21
. A $50,000 loan is to be amortized over 7 years, with annual end-of-year
payments. Which of these statements is CORRECT?

a. The annual payments would be larger if the interest rate were

lower.
b. If the loan were amortized over 10 years rather than 7 years, and
if the interest rate were the same in either case, the first
payment would include more dollars of interest under the 7-year
amortization plan.
c. The proportion of each payment that represents interest as opposed
to repayment of principal would be lower if the interest rate were
lower.
d. The last payment would have a higher proportion of interest than
the first payment.
e. The proportion of interest versus principal repayment would be the
same for each of the 7 payments.

(2.17) Amortization Answer: a MEDIUM

22
. Which of the following statements regarding a 15-year (180-month)
$125,000 fixed-rate mortgage is NOT CORRECT? (Ignore all taxes and
transactions costs.)

a. The remaining balance after three years will be $125,000 less the
total amount of interest paid during the first 36 months.
b. Because it is a fixed-rate mortgage, the monthly loan payments
(that include both interest and principal payments) are constant.
c. Interest payments on the mortgage will steadily decline over time.
d. The proportion of the monthly payment that goes towards repayment of
principal will be higher 10 years from now than it will be the first
year.
e. The outstanding balance gets paid off at a faster rate in the later
years of a loan’s life.

(2.17) Amortization Answer: b MEDIUM

23
. Which of the following statements regarding a 30-year monthly payment
amortized mortgage with a nominal interest rate of 10% is CORRECT?

a. The monthly payments will decline over time.

b. A smaller proportion of the last monthly payment will be interest,
and a larger proportion will be principal, than for the first
monthly payment.
c. The total dollar amount of principal being paid off each month gets

Chapter 2: Time Value True/False Page 25

 smaller as the loan approaches maturity.
d. The amount representing interest in the first payment would be
higher if the nominal interest rate were 7% rather than 10%.
e. Exactly 10% of the first monthly payment represents interest.

(Comp: 2.2,2.7,2.8) Time value concepts Answer: a MEDIUM

24
. Which of the following investments will have the highest future value at
the end of 10 years? Assume that the effective annual rate for all
investments is the same.

a. Investment A pays $250 at the beginning of every year for the next
10 years (a total of 10 payments).
b. Investment B pays $125 at the end of every 6-month period for the
next 10 years (a total of 20 payments).
c. Investment C pays $125 at the beginning of every 6-month period for
the next 10 years (a total of 20 payments).
d. Investment D pays $2,500 at the end of 10 years (a total of one
payment).
e. Investment E pays $250 at the end of every year for the next 10
years (a total of 10 payments).

(Comp: 2.3,2.9,2.15) Various concepts Answer: d MEDIUM

25
. A Treasury bond promises to pay a lump sum of $1,000 exactly 3 years
from today. The nominal interest rate is 6%, semiannual compounding.
Which of the following statements is CORRECT?

a. The periodic interest rate is greater than 3%.

b. The periodic rate is less than 3%.
c. The present value would be greater if the lump sum were discounted
back for more periods.
d. The present value of the $1,000 would be smaller if interest were
compounded monthly rather than semiannually.
e. The PV of the $1,000 lump sum has a higher present value than the
PV of a 3-year, $333.33 ordinary annuity.

(Comp: 2.2,2.9,2.15,2.17) Various concepts Answer: c MEDIUM

26
. Which of the following statements is CORRECT, assuming positive interest
rates and other things held constant?

a. A 5-year, $250 annuity due will have a lower present value than a
similar ordinary annuity.
b. A 30-year, $150,000 amortized mortgage will have larger monthly
payments than an otherwise similar 20-year mortgage.
c. A typical investment's nominal interest rate will always be equal
to or less than its effective annual rate.
d. If an investment pays 10% interest, compounded annually, its

Page 26 True/False Chapter 2: Time Value

 effective annual rate will be less than 10%.
e. Banks A and B offer the same nominal annual rate of interest, but A
pays interest quarterly and B pays semiannually. Deposits in Bank
B will have the higher future value if you leave the funds on
deposit.

(Comp: 2.9,2.15,2.17) Various concepts Answer: e MEDIUM

27
. Which of the following statements is NOT CORRECT?

a. The present value of a 3-year, $150 annuity due will exceed the
present value of a 3-year, $150 ordinary annuity.
b. If a loan has a nominal annual rate of 8%, then the effective rate
can never be less than 8%.
c. If a loan or investment has annual payments, then the effective,
periodic, and nominal rates of interest will all be the same.
d. The proportion of the payment that goes toward interest on a fully
amortized loan declines over time.
e. An investment that has a nominal rate of 6% with semiannual
payments will have an effective rate that is less than 6%.

Chapter 2: Time Value True/False Page 27

 (Comp: 2.7,2.8,2.9) Annuities Answer: d MEDIUM
28
. You are considering two equally risky annuities, each of which pays
$5,000 per year for 10 years. Investment ORD is an ordinary (or
deferred) annuity, while Investment DUE is an annuity due. Which of the
following statements is CORRECT?

a. The present value of ORD must exceed the present value of DUE, but
the future value of ORD may be less than the future value of DUE.
b. The present value of DUE exceeds the present value of ORD, while
the future value of DUE is less than the future value of ORD.
c. The present value of ORD exceeds the present value of DUE, and the

28. (Comp: 2.7,2.8,2.9) Annuities Answer: d MEDIUM

19. (2.15) Effective annual rate Answer: e MEDIUM

By inspection, we can see that e dominates a and b, and that c dominates d because, with the same interest
rate, the account with the most frequent compounding has the highest EFF%. Thus, the correct answer
must be either e or c. Moreover, we can see by inspection that since c and e have the same compounding
frequency yet e has the higher nominal rate, e must have the higher EFF%. You could also prove that e is
the correct choice by calculating the EFF%s:

a. 8.300% = (1+0.08/12)12 – 1
b. 8.000% = (1+0.08/1)1 – 1
c. 7.250% = (1+0.07/365)365 – 1
d. 7.229% = (1+0.07/12)12 – 1
e. 8.328% = (1+0.08/365)365 – 1

20. (2.15) Quarterly compounding Answer: c MEDIUM

21. (2.17) Amortization Answer: c MEDIUM

a, d, and e can be ruled out as incorrect by simple reasoning. b is incorrect because interest in the first year
would be Loan amount * interest rate regardless of the life of the loan. That makes c the "logical guess." It
is also logical that the percentage of interest in each payment would be higher if the interest rate were
higher. Think about the situation where r = 0%, so interest would be zero. One could also set up an
amortization schedule and change the numbers to confirm that only c is correct.

23. (2.17) Amortization Answer: b MEDIUM

b is correct. a is clearly wrong, as are c and d. It is not obvious whether e is correct or not, but we could
set up an example to see:

Loan 100000 Term 30

Rate 10% Periods/Year 12
Periodic rate 0.0083333 Total periods 360

Payment -$877.57 Interest month 1 $833.33

Interest as % of total payment: 95%, which is much larger than 10%.

22. (2.17) Amortization Answer: a MEDIUM

a is not correct because we would subtract principal repaid, not interest paid. Thus a is the correct response
to this question. b is correct by definition. c is correct because the outstanding loan balance is declining. d
is clearly correct, as is e. One could also set up an amortization schedule to prove that the above statements

Page 28 Problems Chapter 2: Time Value

 future value of ORD also exceeds the future value of DUE.
d. The present value of DUE exceeds the present value of ORD, and the
future value of DUE also exceeds the future value of ORD.
e. If the going rate of interest decreases, say from 10% to 0%, the
difference between the present value of ORD and the present value
of DUE would remain constant.

Hard:
(2.15) Effective annual rates Answer: e HARD
29
. You plan to invest some money in a bank account. Which of the following
banks provides you with the highest effective rate of interest?

a. Bank 1; 6.1% with annual compounding.

b. Bank 2; 6.0% with monthly compounding.

are correct.

24. (Comp: 2.2,2.7,2.8) Time value concepts Answer: a MEDIUM

You could just reason this out, or you could do calculations to manually see which one is largest, as we
show below:
A dominates B because it receives the same total amount, but gets it faster, hence it can earn more interest
over the 10 years. A also dominates C and E for the same reason, and it dominates D because with D no
interest whatever is earned. We could also do these calculations to answer the question:

A $4,382.79 Largest EFF% 10.00% 10 250

B $4,081.59 NOM% 9.76% 125
C $4,280.81 125
D $2,500.00 2500
E $3,984.36 250

25. (Comp: 2.3,2.9,2.15) Various concepts Answer: d MEDIUM

26. (Comp: 2.2,2.9,2.15,2.17) Various concepts Answer: c MEDIUM

27. (Comp: 2.9,2.15,2.17) Various concepts Answer: e MEDIUM

29. (2.15) Effective annual rates Answer: e HARD

By inspection, we can see that e dominates b, c, and d because, with the same interest rate, the account with
the most frequent compounding has the highest EFF%. Thus, the correct answer must be either a or e.
However, we can cannot tell by inspection whether a or e provides the higher EFF%. We know that with
one compounding period an EFF% is 6.1%, so we can calculate e's EFF%. It is 6.183%, so e is the correct
answer.

a. = (1+0.061/12)12 – 1 = 6.100%
e. = (1+0.06/365)365 – 1 = 6.183%

Chapter 2: Time Value Problems Page 29

 c. Bank 3; 6.0% with annual compounding.
d. Bank 4; 6.0% with quarterly compounding.
e. Bank 5; 6.0% with daily (365-day) compounding.

Multiple Choice: Problems

Easy:
(2.2) FV of a lump sum Answer: d EASY
30
. What would the future value of $125 be after 8 years at 8.5% compound
interest?

a. $205.83
b. $216.67
c. $228.07
d. $240.08
e. $252.08

(2.2) FV of a lump sum Answer: a EASY

31
. Suppose you have $1,500 and plan to purchase a 5-year certificate of
deposit (CD) that pays 3.5% interest, compounded annually. How much
will you have when the CD matures?

a. $1,781.53
b. $1,870.61
c. $1,964.14
d. $2,062.34
e. $2,165.46

(2.2) FV of a lump sum Answer: c EASY

32
. Last year Toto Corporation's sales were $225 million. If sales grow at
6% per year, how large (in millions) will they be 5 years later?

a. $271.74
b. $286.05
c. $301.10
d. $316.16
e. $331.96

(2.2) FV of a lump sum Answer: b EASY

33
. How much would $1, growing at 3.5% per year, be worth after 75 years?

a. $12.54
b. $13.20

Page 30 Problems Chapter 2: Time Value

 c. $13.86
d. $14.55
e. $15.28

(2.2) FV of a lump sum Answer: b EASY

34
. You deposit $1,000 today in a savings account that pays 3.5% interest,
compounded annually. How much will your account be worth at the end of
25 years?

a. $2,245.08
b. $2,363.24
c. $2,481.41
d. $2,605.48
e. $2,735.75

(2.3) PV of a lump sum Answer: a EASY

35
. Suppose a U.S. government bond promises to pay $1,000 five years from
now. If the going interest rate on 5-year government bonds is 5.5%, how
much is the bond worth today?

a. $765.13
b. $803.39
c. $843.56
d. $885.74
e. $930.03

(2.3) PV of a lump sum Answer: e EASY

36
. How much would $5,000 due in 50 years be worth today if the discount
rate were 7.5%?

a. $109.51
b. $115.27
c. $121.34
d. $127.72
e. $134.45

(2.3) PV of a lump sum Answer: b EASY

37
. Suppose a U.S. treasury bond will pay $2,500 five years from now. If
the going interest rate on 5-year treasury bonds is 4.25%, how much is
the bond worth today?

a. $1,928.78
b. $2,030.30
c. $2,131.81
d. $2,238.40
e. $2,350.32

(2.4) Interest rate on a lump sum Answer: d EASY

38
. Suppose the U.S. Treasury offers to sell you a bond for $747.25. No
payments will be made until the bond matures 5 years from now, at which
time it will be redeemed for $1,000. What interest rate would you earn
if you bought this bond at the offer price?

Chapter 2: Time Value Problems Page 31

 a. 4.37%
b. 4.86%
c. 5.40%
d. 6.00%
e. 6.60%

(2.4) Growth rate Answer: b EASY

39
. Ten years ago, Levin Inc. earned $0.50 per share. Its earnings this
year were $2.20. What was the growth rate in Levin's earnings per share
(EPS) over the 10-year period?

a. 15.17%
b. 15.97%
c. 16.77%
d. 17.61%
e. 18.49%

(2.5) Number of periods Answer: e EASY

40
. How many years would it take $50 to triple if it were invested in a bank
that pays 3.8% per year?

a. 23.99
b. 25.26
c. 26.58
d. 27.98
e. 29.46

(2.5) Number of periods Answer: d EASY

41
. Last year Mason Corp's earnings per share were $2.50, and its growth
rate during the prior 5 years was 9.0% per year. If that growth rate
were maintained, how many years would it take for Mason’s EPS to double?

a. 5.86
b. 6.52
c. 7.24
d. 8.04
e. 8.85

(2.5) Number of periods Answer: e EASY

42
. You plan to invest in securities that pay 9.0%, compounded annually. If
you invest $5,000 today, how many years will it take for your investment
account to grow to $9,140.20?

Page 32 Problems Chapter 2: Time Value

 a. 4.59
b. 5.10
c. 5.67
d. 6.30
e. 7.00

(2.7) FV of an ordinary annuity Answer: c EASY

43
. You want to buy a new sports car 3 years from now, and you plan to save
$4,200 per year, beginning one year from today. You will deposit your
savings in an account that pays 5.2% interest. How much will you have
just after you make the 3rd deposit, 3 years from now?

a. $11,973.07
b. $12,603.23
c. $13,266.56
d. $13,929.88
e. $14,626.38

(2.7) FV of an ordinary annuity Answer: a EASY

44
. You want to go to Europe 5 years from now, and you can save $3,100 per
year, beginning one year from today. You plan to deposit the funds in a
mutual fund which you expect to return 8.5% per year. Under these
conditions, how much will you have just after you make the 5th deposit,
5 years from now?

a. $18,368.66
b. $19,287.09
c. $20,251.44
d. $21,264.02
e. $22,327.22

(2.8) FV of an annuity due Answer: a EASY

45
. You want to buy a new sports car 3 years from now, and you plan to save
$4,200 per year, beginning immediately. You will make 3 deposits in an
account that pays 5.2% interest. Under these assumptions, how much will
you have 3 years from today?

a. $13,956.42
b. $14,654.24
c. $15,386.95
d. $16,156.30
e. $16,964.11

(2.8) FV of an annuity due Answer: c EASY

Chapter 2: Time Value Problems Page 33

46
. You want to go to Europe 5 years from now, and you can save $3,100 per
year, beginning immediately. You plan to deposit the funds in a mutual
fund which you expect to return 8.5% per year. Under these conditions, how
much will you have just after you make the 5th deposit, 5 years from now?

a. $17,986.82
b. $18,933.49
c. $19,929.99
d. $20,926.49
e. $21,972.82

(2.9) PV of an ordinary annuity Answer: e EASY

47
. What is the PV of an ordinary annuity with 10 payments of $2,700 if the
appropriate interest rate is 6.5%?

a. $15,809.44
b. $16,641.51
c. $17,517.38
d. $18,439.35
e. $19,409.84

(2.9) PV of an ordinary annuity Answer: e EASY

48
. You have a chance to buy an annuity that pays $1,200 at the end of each
year for 3 years. You could earn 5.5% on your money in other
investments with equal risk. What is the most you should pay for the
annuity?

a. $2,636.98
b. $2,775.77
c. $2,921.86
d. $3,075.64
e. $3,237.52

(2.9) PV of an ordinary annuity Answer: b EASY

49
. Your aunt is about to retire, and she wants to buy an annuity that will
supplement her income by $65,000 per year for 25 years, beginning a year
from today. The going rate on such annuities is 6.25%. How much would
it cost her to buy such an annuity today?

a. $770,963.15
b. $811,540.16
c. $852,117.17
d. $894,723.02
e. $939,459.18

(2.9) PV of an annuity due Answer: a EASY

50
. What is the PV of an annuity due with 10 payments of $2,700 at an

Page 34 Problems Chapter 2: Time Value

 interest rate of 6.5%?

a. $20,671.48
b. $21,705.06
c. $22,790.31
d. $23,929.82
e. $25,126.31

(2.9) PV of an annuity due Answer: c EASY

51
. You have a chance to buy an annuity that pays $550 at the beginning of
each year for 3 years. You could earn 5.5% on your money in other
investments with equal risk. What is the most you should pay for the
annuity?

a. $1,412.84
b. $1,487.20
c. $1,565.48
d. $1,643.75
e. $1,725.94

(2.9) PV of an annuity due Answer: d EASY

52
. Your aunt is about to retire, and she wants to buy an annuity that will
provide her with $65,000 of income a year for 25 years, with the first
payment coming immediately. The going rate on such annuities is 6.25%.
How much would it cost her to buy the annuity today?

a. $739,281.38
b. $778,190.93
c. $819,148.35
d. $862,261.42
e. $905,374.49

(2.9) PV of an annuity due Answer: b EASY

53
. You own an oil well that will pay you $30,000 per year for 10 years,
with the first payment being made today. If you think a fair return on
the well is 8.5%, how much should you ask for if you decide to sell it?

a. $202,893
b. $213,572
c. $224,250
d. $235,463
e. $247,236

(2.9) PV of an ordinary annuity plus an ending payment Answer: e EASY

54
. What’s the present value of a 4-year ordinary annuity of $2,250 per year
plus an additional $3,000 at the end of Year 4 if the interest rate is 5%?

Chapter 2: Time Value Problems Page 35

 a. $8,508.74
b. $8,956.56
c. $9,427.96
d. $9,924.17
e. $10,446.50

(2.10) Payments on an ordinary annuity Answer: a EASY

55
. Suppose you inherited $275,000 and invested it at 8.25% per year. How
much could you withdraw at the end of each of the next 20 years?

a. $28,532.45
b. $29,959.08
c. $31,457.03
d. $33,029.88
e. $34,681.37

(2.10) Payments on an ordinary annuity Answer: d EASY

56
. Your uncle has $375,000 and wants to retire. He expects to live for
another 25 years, and to be able to earn 7.5% on his invested funds.
How much could he withdraw at the end of each of the next 25 years and
end up with zero in the account?

a. $28,843.38
b. $30,361.46
c. $31,959.43
d. $33,641.50
e. $35,323.58

(2.10) Payments on an annuity due Answer: c EASY

57
. Your uncle has $375,000 and wants to retire. He expects to live for
another 25 years, and he also expects to earn 7.5% on his invested
funds. How much could he withdraw at the beginning of each of the next
25 years and end up with zero in the account?

a. $28,243.21
b. $29,729.70
c. $31,294.42
d. $32,859.14
e. $34,502.10

(2.10) Payments on an annuity due Answer: d EASY

58
. Suppose you inherited $275,000 and invested it at 8.25% per year. How

Page 36 Problems Chapter 2: Time Value

 much could you withdraw at the beginning of each of the next 20 years?

a. $22,598.63
b. $23,788.03
c. $25,040.03
d. $26,357.92
e. $27,675.82

(2.10) Years to deplete an ordinary annuity Answer: a EASY

59
. Your uncle has $375,000 invested at 7.5%, and he now wants to retire.
He wants to withdraw $35,000 at the end of each year, beginning at the
end of this year. How many years will it take to exhaust his funds,
i.e., run the account down to zero?

a. 22.50
b. 23.63
c. 24.81
d. 26.05
e. 27.35

(2.10) Years to deplete an annuity due Answer: e EASY

60
. Your uncle has $500,000 invested at 7.5%, and he now wants to retire.
He wants to withdraw $40,000 at the beginning of each year, beginning
immediately. How many years will it take to exhaust his funds, i.e.,
run the account down to zero?

a. 23.16
b. 24.38
c. 25.66
d. 27.01
e. 28.44

(2.10) Interest rate implicit in an annuity Answer: b EASY

61
. You just won the state lottery, and you have a choice between receiving
$3,500,000 today or a 10-year annuity of $500,000, with the first
payment coming one year from today. What rate of return is built into
the annuity?

a. 6.72%
b. 7.07%
c. 7.43%
d. 7.80%
e. 8.19%

(2.10) Interest rate implicit in an annuity Answer: c EASY

Chapter 2: Time Value Problems Page 37

62
. Your girlfriend just won the Florida lottery. She has the choice of
$15,000,000 today or a 20-year annuity of $1,050,000, with the first
payment coming one year from today. What rate of return is built into
the annuity?

a. 2.79%
b. 3.10%
c. 3.44%
d. 3.79%
e. 4.17%

62. (2.10) Interest rate implicit in an annuity Answer: c EASY

N 20
PV $15,000,000
PMT $1,050,000
FV $0.00
I/YR 3.44%

31. (2.2) FV of a lump sum Answer: a EASY

N 5
I/YR 3.5%
PV $1,500
PMT $0
FV $1,781.53

32. (2.2) FV of a lump sum Answer: c EASY

N 5
I/YR 6.0%
PV $225.00
PMT $0.00
FV $301.10

33. (2.2) FV of a lump sum Answer: b EASY

N 75
I/YR 3.5%
PV $1.00
PMT $0.00
FV $13.20

34. (2.2) FV of a lump sum Answer: b EASY

N 25
I/YR 3.5%
PV $1,000

Page 38 Problems Chapter 2: Time Value

 (2.10) Interest rate implicit in an annuity due Answer: e EASY
63
. Assume that you own an annuity that will pay you $15,000 per year for 12
years, with the first payment being made today. Your uncle offers to
give you $120,000 for the annuity. If you sell it, what rate of return
would your uncle earn on his investment?

a. 6.85%
b. 7.21%
c. 7.59%
d. 7.99%
e. 8.41%

PMT $0
FV $2,363.24

35. (2.3) PV of a lump sum Answer: a EASY

N 5
I/YR 5.5%
PMT $0
FV $1,000.00
PV $765.13

36. (2.3) PV of a lump sum Answer: e EASY

N 50
I/YR 7.5%
PMT $0
FV $5,000
PV $134.45

37. (2.3) PV of a lump sum Answer: b EASY

N 5
I/YR 4.25%
PMT $0
FV $2,500.00
PV $2,030.30

30. (2.2) FV of a lump sum Answer: d EASY

N 8
I/YR 8.5%
PV $125
PMT $0
FV $240.08

38. (2.4) Interest rate on a lump sum Answer: d EASY

Chapter 2: Time Value Problems Page 39

 (2.11) PV of a perpetuity Answer: b EASY
64
. What’s the present value of a perpetuity that pays $250 per year if the
appropriate interest rate is 5%?

a. $4,750.00
b. $5,000.00
c. $5,250.00

N 5
PV $747.25
PMT $0
FV $1,000.00
I/YR 6.00%

39. (2.4) Growth rate Answer: b EASY

N 10
PV $0.50
PMT $0
FV $2.20
I/YR 15.97%

40. (2.5) Number of periods Answer: e EASY

I/YR 3.8%
PV $50.00
PMT $0
FV $150.00
N 29.46

41. (2.5) Number of periods Answer: d EASY

I/YR 9.0%
PV $2.50
PMT $0
FV $5.00
N 8.04

42. (2.5) Number of periods Answer: e EASY

I/YR 9.0%
PV $5,000.00
PMT $0
FV $9,140.20
N 7.00

43. (2.7) FV of an ordinary annuity Answer: c EASY

N 3

Page 40 Problems Chapter 2: Time Value

 d. $5,512.50
e. $5,788.13

(2.11) Rate of return on a perpetuity Answer: d EASY

65
. What’s the rate of return you would earn if you paid $950 for a
perpetuity that pays $85 per year?

a. 6.52%
b. 7.25%
c. 8.05%
d. 8.95%

I/YR 5.2%
PV $0.00
PMT $4,200
FV $13,266.56

44. (2.7) FV of an ordinary annuity Answer: a EASY

N 5
I/YR 8.5%
PV $0.00
PMT $3,100
FV $18,368.66

45. (2.8) FV of an annuity due Answer: a EASY

N 3
I/YR 5.2%
PV $0.00
PMT $4,200
FV $13,956.42

46. (2.8) FV of an annuity due Answer: c EASY

N 5
I/YR 8.5%
PV $0.00
PMT $3,100
FV $19,929.99

47. (2.9) PV of an ordinary annuity Answer: e EASY

N 10
I/YR 6.5%
PMT $2,700
FV $0.00
PV $19,409.84

48. (2.9) PV of an ordinary annuity Answer: e EASY

Chapter 2: Time Value Problems Page 41

 e. 9.84%

(2.12) PV of an uneven cash flow stream Answer: e EASY

66
. At a rate of 6.25%, what is the present value of the following cash flow
stream? $0 at Time 0; $75 at the end of Year 1; $225 at the end of Year
2; $0 at the end of Year 3; and $300 at the end of Year 4?

a. $411.57
b. $433.23
c. $456.03
d. $480.03

N 3
I/YR 5.5%
PMT $1,200
FV $0.00
PV $3,237.52

49. (2.9) PV of an ordinary annuity Answer: b EASY

N 25
I/YR 6.25%
PMT $65,000
FV $0.00
PV $811,540.16

50. (2.9) PV of an annuity due Answer: a EASY

N 10
I/YR 6.5%
PMT $2,700
FV $0.00
PV $20,671.48

51. (2.9) PV of an annuity due Answer: c EASY

N 3
I/YR 5.5%
PMT $550
FV $0.00
PV $1,565.48

52. (2.9) PV of an annuity due Answer: d EASY

N 25
I/YR 6.25%
PMT $65,000
FV $0.00

Page 42 Problems Chapter 2: Time Value

 e. $505.30

(2.12) PV of an uneven cash flow stream Answer: c EASY

67
. What is the present value of the following cash flow stream at an
interest rate of 12.0% per year? $0 at Time 0; $1,500 at the end of
Year 1; $3,000 at the end of Year 2; $4,500 at the end of Year 3; and
$6,000 at the end of Year 4.

a. $9,699.16
b. $10,209.64
c. $10,746.99

PV $862,261.42

53. (2.9) PV of an annuity due Answer: b EASY

N 10
I/YR 8.5%
PMT $30,000
FV $0.00
PV $213,572

54. (2.9) PV of an ordinary annuity plus an ending payment Answer: e EASY

N 4
I/YR 5.0%
PMT $2,250
FV $3,000
PV $10,446.50

56. (2.10) Payments on an ordinary annuity Answer: d EASY

N 25
I/YR 7.5%
PV $375,000
FV $0.00
PMT $33,641.50

55. (2.10) Payments on an ordinary annuity Answer: a EASY

N 20
I/YR 8.25%
PV $275,000
FV $0.00
PMT $28,532.45

57. (2.10) Payments on an annuity due Answer: c EASY

N 25
I/YR 7.5%

Chapter 2: Time Value Problems Page 43

 d. $11,284.34
e. $11,848.55

PV $375,000
FV $0.00
PMT $31,294.42

58. (2.10) Payments on an annuity due Answer: d EASY

N 20
I/YR 8.25%
PV $275,000
FV $0.00
PMT $26,357.92

59. (2.10) Years to deplete an ordinary annuity Answer: a EASY

I/YR 7.5%
PV $375,000
PMT $35,000
FV $0.00
N 22.50

60. (2.10) Years to deplete an annuity due Answer: e EASY

I/YR 7.5%
PV $500,000
PMT $40,000
FV $0.00
N 28.44

61. (2.10) Interest rate implicit in an annuity Answer: b EASY

N 10
PV $3,500,000
PMT $500,000
FV $0.00
I/YR 7.07%

63. (2.10) Interest rate implicit in an annuity due Answer: e EASY

Page 44 Problems Chapter 2: Time Value

Easy/Medium:
(2.12) PV of an uneven cash flow stream Answer: d EASY/MEDIUM
68
. An investment promises the following cash flow stream: $750 at Time 0;
$2,450 at the end of Year 1 (or at t = 1); $3,175 at the end of Year 2;
and $4,400 at the end of Year 3. At a discount rate of 8.0%, what is
the present value of the cash flow stream?

a. $7,916.51
b. $8,333.17

N 12
PV $120,000
PMT $15,000
FV $0.00
I/YR 8.41%

64. (2.11) PV of a perpetuity Answer: b EASY

I/YR 5.0%
PMT $250
PV $5,000.00

65. (2.11) Rate of return on a perpetuity Answer: d EASY

Cost (PV) $950

PMT $85
I/YR 8.95%

66. (2.12) PV of an uneven cash flow stream Answer: e EASY

I/YR = 6.25%
0 1 2 3 4
CFs: $0 $75 $225 $0 $300
PV of CFs: $0 $71 $199 $0 $235

PV = $505.30 Find the individual PVs and sum them. Automate the
PV = $505.30 process using Excel or a calculator, by inputting the
data into the cash flow register and pressing the NPV key.

67. (2.12) PV of an uneven cash flow stream Answer: c EASY

I/YR = 12.0%
0 1 2 3 4
CFs: $0 $1,500 $3,000 $4,500 $6,000
PV of CFs: $0 $1,339 $2,392 $3,203 $3,813

PV = $10,746.99 Found using the Excel NPV function

PV = $10,746.99 Found by summing individual PVs.
PV = $10,746.99 Found using the calculator NPV key.

Chapter 2: Time Value Problems Page 45

 c. $8,771.76
d. $9,233.43
e. $9,695.10

(2.12) PV of an uneven cash flow stream Answer: a EASY/MEDIUM

69
. What is the present value of the following cash flow stream if the
interest rate is 6.0% per year? 0 at Time 0; $1,000 at the end of Year
1; and $2,000 at the end of Years 2, 3, and 4.

a. $5,986.81
b. $6,286.16
c. $6,600.46
d. $6,930.49
e. $7,277.01

(2.15) FV of a lump sum, semiannual compounding Answer: c EASY/MEDIUM

70
. What’s the future value of $1,500 after 5 years if the appropriate
interest rate is 6%, compounded semiannually?

a. $1,819.33
b. $1,915.08
c. $2,015.87
d. $2,116.67
e. $2,222.50

(2.15) PV of a lump sum, semiannual compounding Answer: d EASY/MEDIUM

71
. What’s the present value of $1,500 discounted back 5 years if the

68. (2.12) PV of an uneven cash flow stream Answer: d EASY/MEDIUM

I/YR = 8.0%
0 1 2 3
CFs: $750 $2,450 $3,175 $4,400
PV of CFs: $750 $2,269 $2,722 $3,493

PV = $9,233.43 Found by summing individual PVs.

PV = $9,233.43 Found with a calculator or Excel to automate the
process. With a calculator, input the cash flows and I
into the cash flow register, then press the NPV key.

69. (2.12) PV of an uneven cash flow stream Answer: a EASY/MEDIUM

I/YR = 6.0%

0 1 2 3 4
CFs: $0 $1,000 $2,000 $2,000 $2,000
PV of CFs: $0 $943 $1,780 $1,679 $1,584

PV = $5,986.81 Found using the Excel NPV function

PV = $5,986.81 Found by summing individual PVs.
PV = $5,986.81 Found using the calculator NPV key.

Page 46 Problems Chapter 2: Time Value

 appropriate interest rate is 6%, compounded semiannually?

a. $956.95
b. $1,007.32
c. $1,060.33
d. $1,116.14
e. $1,171.95

Medium:
(2.10) Years to deplete an ordinary annuity Answer: b MEDIUM
72
. Your uncle has $300,000 invested at 7.5%, and he now wants to retire.
He wants to withdraw $35,000 at the end of each year, beginning at the
end of this year. He also wants to have $25,000 left to give you when
he ceases to withdraw funds from the account. For how many years can he
make the $35,000 withdrawals and still have $25,000 left in the end?

a. 14.21
b. 14.96
c. 15.71
d. 16.49
e. 17.32

(2.10) Years to deplete an annuity due Answer: c MEDIUM

73
. Your uncle has $300,000 invested at 7.5%, and he now wants to retire.
He wants to withdraw $35,000 at the beginning of each year, beginning
immediately. He also wants to have $25,000 left to give you when he
ceases to withdraw funds from the account. For how many years can he
make the $35,000 withdrawals and still have $25,000 left in the end?

a. 11.98
b. 12.61
c. 13.27
d. 13.94
e. 14.63

(2.10) Interest rate implicit in an annuity due Answer: a MEDIUM

74
. You agree to make 24 deposits of $500 at the beginning of each month
into a bank account. At the end of the 24th month, you will have $13,000
in your account. If the bank compounds interest monthly, what nominal
annual interest rate will you be earning?

a. 7.62%
b. 8.00%
c. 8.40%

Chapter 2: Time Value Problems Page 47

 d. 8.82%
e. 9.26%

(2.11) Payments on a perpetuity Answer: b MEDIUM

75
. What annual payment would you have to receive in order to earn a 7.5%
rate of return on a perpetuity that has a cost of $1,250?

a. $89.06
b. $93.75
c. $98.44
d. $103.36
e. $108.53

Page 48 Problems Chapter 2: Time Value

 (2.13) FV of an uneven cash flow stream Answer: e MEDIUM
76
. At a rate of 6.5%, what is the future value of the following cash flow
stream? $0 at Time 0; $75 at the end of Year 1; $225 at the end of Year
2; $0 at the end of Year 3; and $300 at the end of Year 4?

a. $526.01
b. $553.69
c. $582.83
d. $613.51

76. (2.13) FV of an uneven cash flow stream Answer: e MEDIUM

I/YR = 6.5%
0 1 2 3 4
CFs: $0 $75 $225 $0 $300
FV of CFs: $0 $91 $255 $0 $300

FV = $645.80 Found by summing individual FVs.

FV = $645.80 Found with the NFV key in some calculators.
FV = $645.80 Found with a calculator by first finding the PV of the
stream, then finding the FV of that PV.

PV of the stream: $501.99

FV of the PV: $645.80

70. (2.15) FV of a lump sum, semiannual compounding Answer: c EASY/MEDIUM

Years 5
Periods/Yr 2
Nom. I/YR 6.0%

N = Periods 10
PMT $0
I = I/Period 3.0%
PV $1,500 Could be found using a calculator, the equation, or Excel.
FV $2,015.87 Note that we must first convert to periods and rate per period.

71. (2.15) PV of a lump sum, semiannual compounding Answer: d EASY/MEDIUM

Years 5
Periods/Yr 2
Nom. I/YR 6.0%

FV $1,500

Chapter 2: Time Value Problems Page 49

 e. $645.80

(2.14) Interest rate built into uneven CF stream Answer: c MEDIUM

77
. An investment costs $1,000 (CF at t = 0) and is expected to produce cash
flows of $75 at the end of each of the next 5 years, then an additional
lump sum payment of $1,000 at the end of the 5th year. What is the
expected rate of return on this investment?

a. 6.77%
b. 7.13%
c. 7.50%
d. 7.88%

N = Periods 10
PMT $0
I = I/Period 3.0% Could be found using a calculator, the equation, or Excel.
PV $1,116.14 Note that we must first convert to periods and rate per period.

72. (2.10) Years to deplete an ordinary annuity Answer: b MEDIUM

I/YR 7.50%
PV $300,000
PMT $35,000
FV $25,000
N 14.96

73. (2.10) Years to deplete an annuity due Answer: c MEDIUM

I/YR 7.5%
PV $300,000
PMT $35,000
FV $25,000
N 13.27

74. (2.10) Interest rate implicit in an annuity due Answer: a MEDIUM

N 24
PV $0
PMT $500
FV $13,000
I/YR 7.62%

75. (2.11) Payments on a perpetuity Answer: b MEDIUM

Cost (PV) $1,250

I/YR 7.5%
PMT $93.75 Multiply cost by I.

77. (2.14) Interest rate built into uneven CF stream Answer: c MEDIUM

Page 50 Problems Chapter 2: Time Value

 e. 8.27%

(2.14) Interest rate built into uneven CF stream Answer: e MEDIUM

78
. An investment costs $725 and is expected to produce cash flows of $75 at
the end of Year 1, $100 at the end of Year 2, $85 at the end of Year 3,
and $625 at the end of Year 4. What rate of return would you earn if
you bought this investment?

a. 4.93%
b. 5.19%
c. 5.46%
d. 5.75%
e. 6.05%

(2.15) FV of a lump sum, monthly compounding Answer: b MEDIUM

79
. What’s the future value of $1,500 after 5 years if the appropriate
interest rate is 6%, compounded monthly?

a. $1,922.11
b. $2,023.28

0 1 2 3 4 5
CFs: -$1,000 $75 $75 $75 $75 $75
$1,000
-$1,000 $75 $75 $75 $75 $1,075

I/YR 7.50% I is the discount rate that causes the PV of the inflows
to equal the initial negative CF, and is found with
Excel's IRR function or by inputting the CFs into a
calculator and pressing` the IRR key.

78. (2.14) Interest rate built into uneven CF stream Answer: e MEDIUM

0 1 2 3 4
CFs: -$725 $75 $100 $85 $625
I/YR 6.05% I is the discount rate that causes the PV of the positive
inflows to equal the initial negative CF. I can be found
using Excel's IRR function or by inputting the CFs into a
calculator and pressing the IRR key.

79. (2.15) FV of a lump sum, monthly compounding Answer: b MEDIUM

Years 5
Periods/Yr 12
Nom. I/YR 6.0%

N = Periods 60
PMT $0
I/Period 0.5%
PV $1,500
FV $2,023.28 Could be found using a calculator, the equation, or Excel.
Note that we must first convert to periods and rate per period.

Chapter 2: Time Value Problems Page 51

 c. $2,124.44
d. $2,230.66
e. $2,342.19

(2.15) PV of a lump sum, monthly compounding Answer: d MEDIUM

80
. What’s the present value of $1,525 discounted back 5 years if the
appropriate interest rate is 6%, compounded monthly?

a. $969.34
b. $1,020.36
c. $1,074.06
d. $1,130.59
e. $1,187.12

(2.15) APR vs. EAR Answer: b MEDIUM

81
. Credit card issuers must by law print the Annual Percentage Rate (APR)
on their monthly statements. If the APR is stated to be 18.00%, with
interest paid monthly, what is the card's EFF%?

a. 18.58%
b. 19.56%
c. 20.54%
d. 21.57%
e. 22.65%

(2.15) Comparing the effective cost of two bank loans Answer: d MEDIUM

80. (2.15) PV of a lump sum, monthly compounding Answer: d MEDIUM

Years 5
Periods/Yr 12
Nom. I/YR 6.0%

N = Periods 60
PMT $0
I/Period 0.5%
FV $1,525
PV $1,130.59 Could be found using a calculator, the equation, or Excel.
Note that we must first convert to periods and rate per period.

81. (2.15) APR vs. EAR Answer: b MEDIUM

APR 18.00%
Periods/yr 12
EFF% 19.56%

Page 52 Problems Chapter 2: Time Value

82
. East Coast Bank offers to lend you $25,000 at a nominal rate of 7.5%,
compounded monthly. The loan (principal plus interest) must be repaid
at the end of the year. Midwest Bank also offers to lend you the
$25,000, but it will charge an annual rate of 8.3%, with no interest due
until the end of the year. What is the difference in the effective
annual rates charged by the two banks?

a. 0.93%
b. 0.77%
c. 0.64%
d. 0.54%
e. 0.43%

(2.15) Nominal rate vs. EFF% Answer: e MEDIUM

83
. Suppose a bank offers to lend you $10,000 for one year at a nominal
annual rate of 10.25%, but you must make interest payments at the end of
each quarter and then pay off the $10,000 principal amount at the end of
the year. What is the effective annual rate on the loan?

a. 6.99%
b. 7.76%

82. (2.15) Comparing the effective cost of two bank loans Answer: d MEDIUM

This problem can be worked most easily using the interest conversion feature of a calculator. It could also
be worked using the conversion formula. We used the conversion formula.

Nominal rate, East Coast Bank 7.5%

Nominal rate, Midwest Bank 8.3%
Periods/yr, East Coast 12
Periods/yr, Midwest 1
EFF% East Coast 7.76%
EFF% Midwest 8.30%
Difference 0.54%

83. (2.15) Nominal rate vs. EFF% Answer: e MEDIUM

Nominal I/YR 10.25%

Periods/yr 4
EFF% 10.65% Using conversion formula

You could also find the EFF% as follows:

Interest paid each quarter = Loan * rate/4 = quarterly PMT = $256.25

Then find the IRR as a quarterly rate and convert to an annual rate. This procedure is obviously longer.

0 1 2 3 4
CFs: 10,000.00 -256.25 -256.25 -256.25 -256.25
-10,000.00
10,000.00 -256.25 -256.25 -256.25 -10,256.25

IRR (quarterly) = 2.56%

Annual effective rate = 10.65% vs. nominal rate = 10.25%

Chapter 2: Time Value Problems Page 53

 c. 8.63%
d. 9.59%
e. 10.65%

(2.15) Nominal rate vs. EFF% Answer: e MEDIUM

84
. Suppose a bank offers to lend you $10,000 for 1 year on a loan contract
that calls for you to make interest payments of $250.00 at the end of
each quarter and then pay off the principal amount at the end of the
year. What is the effective annual rate on the loan?

a. 8.46%
b. 8.90%
c. 9.37%
d. 9.86%
e. 10.38%

(2.15) Nominal rate vs. EAR Answer: e MEDIUM

85
. If a bank pays a 4.50% nominal rate, with monthly compounding on
deposits, what effective annual rate (EFF%) does the bank pay?

a. 3.01%
b. 3.35%
c. 3.72%
d. 4.13%
e. 4.59%

84. (2.15) Nominal rate vs. EFF% Answer: e MEDIUM

Interest payment: $250.00

0 1 2 3 4
CFs: 10,000.00 -250.00 -250.00 -250.00 -250.00
-10,000.00
10,000.00 -250.00 -250.00 -250.00 -10,250.00

IRR (quarterly) = 2.50%

Annual effective rate = 10.38% vs. nominal rate = 10.00%

85. (2.15) Nominal rate vs. EAR Answer: e MEDIUM

Nominal I/YR 4.50%

Periods/yr 12
EFF% 4.59%

Page 54 Problems Chapter 2: Time Value

 (2.15) Nominal rate vs. EAR Answer: b MEDIUM
86
. Suppose your credit card issuer states that it charges a 15.00% nominal
annual rate. If you must make monthly payments, which amounts to
monthly compounding, what is the effective annual rate?

a. 15.27%
b. 16.08%
c. 16.88%
d. 17.72%
e. 18.61%

(2.16) Interest charges, simple interest Answer: c MEDIUM

87
. Pace Co. borrowed $25,000 at a rate of 7.25%, simple interest, with
interest paid at the end of each month. The bank uses a 360-day year.
How much interest would Pace have to pay in a 30-day month?

a. $136.32
b. $143.49
c. $151.04
d. $158.59
e. $166.52

(2.16) Fractional time periods Answer: a MEDIUM

88
. Suppose you deposited $5,000 in a bank account that pays 5.25% with
daily compounding and a 360-day year. How much could you withdraw after
8 months, assuming each month has 30 days?

86. (2.15) Nominal rate vs. EAR Answer: b MEDIUM

Nominal I/YR 15.00%

Periods/yr 12
EFF% 16.08%

87. (2.16) Interest charges, simple interest Answer: c MEDIUM

Nominal I/YR 7.25% Days in month 30

Days/yr 360 Daily rate 0.020139%
Amount borrowed $25,000 Interest per day $5.03472
Interest per month $151.04

Chapter 2: Time Value Problems Page 55

 a. $5,178.09
b. $5,436.99
c. $5,708.84
d. $5,994.28
e. $6,294.00

(2.17) Loan amortization: payment Answer: a MEDIUM

89
. Suppose you borrowed $12,000 at a rate of 9% and must repay it in 4
equal installments at the end of each of the next 4 years. How large
would your payments be?

a. $3,704.02
b. $3,889.23
c. $4,083.69
d. $4,287.87
e. $4,502.26

(2.17) Loan amortization: payment Answer: c MEDIUM

90
. Suppose you are buying your first house for $210,000, and are making a
$20,000 down payment. You have arranged to finance the remaining amount
with a 30-year, monthly payment, amortized mortgage at a 6.5% nominal
interest rate. What will your equal monthly payments be?

a. $1,083.84
b. $1,140.88
c. $1,200.93

89. (2.17) Loan amortization: payment Answer: a MEDIUM

I/YR 9.0%
Years 4
Amount borrowed $12,000
Payments $3,704.02 Found with a calculator, as the PMT.

90. (2.17) Loan amortization: payment Answer: c MEDIUM

Years 30 Payments/year 12
N 360 Nominal rate 6.50%
Periodic rate 0.54% Purchase price $210,000
PV $190,000 Down payment $20,000
FV $0.00
PMT $1,200.93

88. (2.16) Fractional time periods Answer: a MEDIUM

Nominal I/YR 5.25% Rate/day 0.0146%

Number of months 8 Days on deposit 240
Days in year 360
Days in month 30
Amount deposited $5,000
Ending amount $5,178.09

Page 56 Problems Chapter 2: Time Value

 d. $1,260.98
e. $1,324.02

(2.17) Loan amortization: interest Answer: d MEDIUM

91
. Suppose you borrowed $12,000 at a rate of 9% and must repay it in 4
equal installments at the end of each of the next 4 years. How much
interest would you have to pay in the first year?

a. $925.97
b. $974.70
c. $1,026.00
d. $1,080.00
e. $1,134.00

(2.17) Loan amortization: interest Answer: d MEDIUM

92
. You plan to borrow $30,000 at a 7% annual interest rate. The terms
require you to amortize the loan with 6 equal end-of-year payments. How
much interest would you be paying in Year 2?

91. (2.17) Loan amortization: interest Answer: d MEDIUM

I/YR 9.0%
Years 4
Amount borrowed $12,000
Interest in Year 1 $1,080.00 Simply multiply the rate times the amount borrowed.

Chapter 2: Time Value Problems Page 57

 a. $1,548.79
b. $1,630.30
c. $1,716.11
d. $1,806.43
e. $1,896.75

(2.17) Loan amortization: interest Answer: c MEDIUM

93
. You plan to borrow $75,000 at a 7% annual interest rate. The terms
require you to amortize the loan with 10 equal end-of-year payments.
How much interest would you be paying in Year 2?

a. $4,395.19
b. $4,626.52
c. $4,870.02

93. (2.17) Loan amortization: interest Answer: c MEDIUM

Find the required payment:

N 10
I 7.0%
PV $75,000
FV $0
PMT $10,678.31

Amortization schedule (first 2 years)

Year Beg. Balance Payment Interest Principal End. Balance

1 75,000.00 10,678.31 5,250.00 5,428.31 69,571.69
2 69,571.69 10,678.31 4,870.02 5,808.29 63,763.39

92. (2.17) Loan amortization: interest Answer: d MEDIUM

Find the required payment:

N 6
I 7.0%
PV $30,000
FV $0
PMT $6,293.87

Amortization schedule (first 2 years)

Year Beg. Balance Payment Interest Principal End. Balance

1 30,000.00 6,293.87 2,100.00 4,193.87 25,806.13
2 25,806.13 6,293.87 1,806.43 4,487.45 21,318.68

Page 58 Problems Chapter 2: Time Value

 d. $5,113.52
e. $5,369.19

(2.17) Loan amortization: payment Answer: e MEDIUM

94
. Suppose you take out a $10,000 loan at a 6% nominal annual rate. The
terms of the loan require you to make 12 equal end-of-month payments
each year for 4 years, and then an additional final (balloon) payment of
$4,000 at the end of the last month. What will your equal monthly
payments be?

a. $131.06
b. $137.96
c. $145.22
d. $152.86
e. $160.91

(2.18) Growing annuity: calculating the real rate Answer: c MEDIUM

95
. You plan to make annual deposits into a bank account that pays a 5.00%
nominal annual rate. You think inflation will amount to 2.50% per year.
What is the expected annual real rate at which your money will grow?

a. 1.98%
b. 2.20%
c. 2.44%
d. 2.68%
e. 2.95%

(2.18) Growing annuity due: withdraw constant real amt Answer: e MEDIUM
96
. Your father now has $1,000,000 invested in an account that pays 9.00%.

94. (2.17) Loan amortization: payment Answer: e MEDIUM

Years 4 Nominal rate 6.0%

N 48 Payments/year 12
I 0.5% Monthly annuity, so interest must be calculated on monthly basis
PV $10,000
FV $4,000
PMT $160.91

95. (2.18) Growing annuity: calculating the real rate Answer: c MEDIUM

rNOM 5.00%
Inflation 2.50%
rr = [(1 + rNOM)/(1 + Inflation)] – 1
rr = 2.44%

Chapter 2: Time Value Problems Page 59

 He expects inflation to average 3%, and he wants to make annual constant
dollar (real) beginning-of-year withdrawals over each of the next 20
years and end up with a zero balance after the 20th year. How large
will his initial withdrawal (and thus constant dollar (real)
withdrawals) be?

a. $66,154.58
b. $69,636.40
c. $73,301.47
d. $77,159.45
e. $81,220.47

(Comp: 2.10,2.15) Annuity due, N, monthly compounding Answer: d MEDIUM

97
. You are considering investing in a Third World bank account that pays a
nominal annual rate of 18%, compounded monthly. If you invest $5,000 at
the beginning of each month, how many months will it take for your
account to grow to $250,000? Round fractional years up.

a. 23
b. 27
c. 32
d. 38
e. 44

(Comp: 2.10,2.15) Annuity, N, monthly compounding Answer: b MEDIUM

98
. You are considering investing in a bank account that pays a nominal
annual rate of 6%, compounded monthly. If you invest $5,000 at the end
of each month, how many months will it take for your account to grow to
$200,000? Round fractional years up.

a. 33
b. 37
c. 41
d. 45
e. 49

96. (2.18) Growing annuity due: withdraw constant real amt Answer: e MEDIUM

rNOM 9.00% Initial sum 1,000,000

Inflation 3.00% Years 20
rr = [(1 + rNOM)/(1 + growth] – 1
rr = 5.825243%
PMT = $81,220.47
97. (Comp: 2.10,2.15) Annuity due, N, monthly compounding Answer: d MEDIUM

I/YR 18.0%
I/MO 1.5% Monthly annuity due, so interest must be calculated on monthly basis
PV $0
PMT $5,000
FV $250,000
N 37.16 Rounded up 38

Page 60 Problems Chapter 2: Time Value

 (Comp: 2.10,2.15) Int rate, annuity, mos compounding Answer: d MEDIUM
99
. Your child’s orthodontist offers you two alternative payment plans. The
first plan requires a $4,000 immediate up-front payment. The second
plan requires you to make monthly payments of $137.41, payable at the
end of each month for 3 years. What nominal annual interest rate is
built into the monthly payment plan?

a. 12.31%
b. 12.96%
c. 13.64%
d. 14.36%
e. 15.08%

Medium/Hard:
(2.10) N, lifetime vs. annual pmts Answer: e MEDIUM/HARD
100
. Your subscription to Investing Wisely Weekly is about to expire. You
plan to subscribe to the magazine for the rest of your life, and you can
renew it by paying $75 annually, beginning immediately, or you can get a
lifetime subscription for $750, also payable immediately. Assuming you
can earn 5.5% on your funds and the annual renewal rate will remain
constant, how many years must you live to make the lifetime subscription
the better buy? Round fractional years up. (Hint: Be sure to remember
that you are solving for how many years you must live, not for how many
payments must be made.)

a. 7
b. 8
c. 9
d. 11
e. 13

(2.15) Non-annual compounding Answer: b MEDIUM/HARD

99. (Comp: 2.10,2.15) Int rate, annuity, mos compounding Answer: d MEDIUM

N 36
PV $4,000
PMT $137.41
FV $0
I/MO 1.20% Monthly annuity, so interest must be calculated on monthly basis
I/YR 14.36%

98. (Comp: 2.10,2.15) Annuity, N, monthly compounding Answer: b MEDIUM

I/YR 6.0%
I/MO 0.5% Monthly annuity, so interest must be calculated on monthly basis
PV $0
PMT $5,000
FV $200,000
N 36.56 Rounded up: 37

Chapter 2: Time Value Problems Page 61

101
. You just deposited $2,500 in a bank account that pays a 12% nominal
interest rate, compounded quarterly. If you also add another $5,000 to
the account one year (12 months) from now and another $7,500 to the
account two years from now, how much will be in the account three years
(12 quarters) from now?

a. $17,422.59
b. $18,339.57
c. $19,256.55
d. $20,219.37
e. $21,230.34

(2.15) Compare effective cost of two bank loans Answer: d MEDIUM/HARD

102
. Merchants Bank offers to lend you $30,000 at a nominal rate of 6.0%,
simple interest, with interest paid quarterly. Gold Coast Bank offers
to lend you the $30,000, but it will charge 7.0%, simple interest, with
interest paid at the end of the year. What's the difference in the
effective annual rates charged by the two banks?

a. 1.49%
b. 1.24%
c. 1.04%
d. 0.86%
e. 0.69%

101. (2.15) Non-annual compounding Answer: b MEDIUM/HARD

Interest rate 12.0%

Periods/year 4 Years on Quarters Ending
Quarterly rate 3.0% Deposit on Deposit Amount
1st deposit $2,500 3 12 $3,564.40
2nd deposit $5,000 2 8 $6,333.85
3rd deposit $7,500 1 4 $8,441.32
$18,339.57

100. (2.10) N, lifetime vs. annual pmts Answer: e MEDIUM/HARD

Find N for an annuity due with the indicated terms to determine how long you must
live to make the lifetime subscription worthwhile.
Interest rate 5.5%
Annual cost $75
Lifetime subscription cost $750
Number of payments made 13.76 Rounded up: 14

Recall that we used BEGIN mode (because it is an annuity due), so it takes 14 payments to make the
lifetime subscription better. Since the 1st payment occurs today, the 14th payment occurs at t = 13, which
is 13 years from now.

So, you must live for: 14 − 1 = 13 years. 13

Page 62 Problems Chapter 2: Time Value

 (2.17) Loan amortization: principal repayment Answer: b MEDIUM/HARD
103
. Suppose you borrowed $12,000 at a rate of 9% and must repay it in 4
equal installments at the end of each of the next 4 years. By how much
would you reduce the amount you owe in the first year?

a. $2,492.82
b. $2,624.02
c. $2,755.23
d. $2,892.99
e. $3,037.64

(2.17) Loan amortization: ending balance Answer: e MEDIUM/HARD

104
. Suppose you borrowed $12,000 at a rate of 9% and must repay it in 4
equal installments at the end of each of the next 4 years. How much
would you still owe at the end of the first year, after you have made
the first payment?

102. (2.15) Compare effective cost of two bank loans Answer: d MEDIUM/HARD

Students must understand that "simple interest with interest paid quarterly" means that the bank gets the
interest at the end of each quarter, hence it can invest it, presumably at the same nominal rate. This results
in the same effective rate as if it were stated as "6%, quarterly compounding."

Nominal rate, Merchants Bank 6.0%

Periods/yr, Merchants 4
Nominal rate, Gold Coast Bank 7.0%
Periods/yr, Gold Coast 1
EFF% Merchants 6.14%
EFF% Gold coast 7.00%
Difference 0.86%

103. (2.17) Loan amortization: principal repayment Answer: b MEDIUM/HARD

Interest rate 9.0%

Years 4
Amount borrowed $12,000

Step 1: Find the PMT $3,704.02

Step 2: Find the 1st year's interest $1,080.00
Step 3: Subtract the interest from the payment; this is repayment of principal $2,624.02

Chapter 2: Time Value Problems Page 63

 a. $7,636.79
b. $8,038.73
c. $8,461.82
d. $8,907.18
e. $9,375.98

(Comp: 2.2,2.10) Retirement planning Answer: c MEDIUM/HARD

105
. Your sister turned 35 today, and she is planning to save $5,000 per year
for retirement, with the first deposit to be made one year from today.
She will invest in a mutual fund that will provide a return of 8% per
year. She plans to retire 30 years from today, when she turns 65, and she
expects to live for 25 years after retirement, to age 90. Under these
assumptions, how much can she spend in each year after she retires? Her
first withdrawal will be made at the end of her first retirement year.

a. $47,888
b. $50,408
c. $53,061
d. $55,714
e. $58,500

Hard:

105. (Comp: 2.2,2.10) Retirement planning Answer: c MEDIUM/HARD

Interest rate 8.0%

Years to retirement 30
Years in retirement 25
Amount saved per year $5,000

Step 1: Find the amount at age 65; use the FV function $566,416
Step 2: Find the PMT for a 25-year ordinary annuity using that FV as the PV $53,061

104. (2.17) Loan amortization: ending balance Answer: e MEDIUM/HARD

Interest rate 9.0%

Years 4
Amount borrowed $12,000

Step 1: Find the PMT $3,704.02

Step 2: Find the 1st year's interest $1,080.00
Step 3: Subtract the interest from the payment; this is repayment of principal $2,624.02
Step 4: Subtract the repayment of principal from the beginning amount owed $9,375.98

Page 64 Problems Chapter 2: Time Value

 (2.17) Loan amort: int rate, % of pmt toward principal Answer: e HARD
106
. Your company has just taken out a 1-year installment loan for $72,500. The
nominal rate is 12.0%, but with equal end-of-month payments. What percentage
of the 2nd monthly payment will go toward the repayment of principal?

a. 73.01%
b. 76.85%
c. 80.89%
d. 85.15%
e. 89.63%

(2.17) Loan amort: pmt and % of pmt toward interest Answer: b HARD
107
. A homeowner just obtained a 30-year amortized mortgage loan for $150,000
at a nominal annual rate of 6.5%, with 360 end-of-month payments. What
percentage of the total payments made during the first 3 months will go
toward payment of interest?

106. (2.17) Loan amort: int rate, % of pmt toward principal Answer: e HARD

N 12
rNOM 12.0%
Periodic r 1.0%
PV $72,500
PMT $6,442
FV $0 % paid toward prin. = 89.63%

Amortization schedule(first 4 years)

Month Beg. Balance Payment Interest Principal Ending Balance

1 72,500.00 6,441.54 725.00 5,716.54 66,783.46
2 66,783.46 6,441.54 667.83 5,773.70 61,009.76
3 61,009.76 6,441.54 610.10 5,831.44 55,178.32
4 55,178.32 6,441.54 551.78 5,889.75 49,288.57

107. (2.17) Loan amort: pmt and % of pmt toward interest Answer: b HARD

Years 30 Periods/yr 12
Nominal r 6.50% N (12 mo.) 360
PV $150,000 I 0.54%
FV $0 Total pmts $2,844.31
PMT $948.10 Interest $2,435.29
% interest 85.62%

Amortization schedule(first 3 months)

Year Beg. Balance Payment Interest Principal Ending Balance

1 150,000.00 948.10 812.50 135.60 149,864.40
2 149,864.40 948.10 811.77 136.34 149,728.06
3 149,728.06 948.10 811.03 137.08 149,590.99
Total payments: 2,844.31 2,435.29 409.01

Chapter 2: Time Value Problems Page 65

 a. 81.34%
b. 85.62%
c. 89.90%
d. 94.40%
e. 99.12%

(2.18) Growing annuity: withdrawing constant real amt Answer: e HARD

108
. Your father now has $1,000,000 invested in an account that pays 9.00%.
He expects inflation to average 3%, and he wants to make annual constant
dollar (real) end-of-year withdrawals over each of the next 20 years and
end up with a zero balance after the 20th year. How large will his
initial withdrawal (and thus constant dollar (real) withdrawals) be?

a. $68,139.22
b. $71,725.49
c. $75,500.52
d. $79,474.23
e. $83,657.08

(2.18) Growing annuity Answer: c HARD

109
. You anticipate that you will need $1,500,000 when you retire 30 years
from now. You plan to make 30 deposits, beginning today, in a bank
account that will pay 6% interest, compounded annually. You expect to
receive annual raises of 4%, so you will increase the amount you deposit
each year by 4%. (That is, your 2nd deposit will be 4% greater than
your first, the 3rd will be 4% greater than the 2nd, etc.) How much
must your 1st deposit be if you are to meet your goal?

108. (2.18) Growing annuity: withdrawing constant real amt Answer: e HARD

rNOM 9.00% Initial sum 1,000,000

Inflation 3.00% Years 20
rr = [(1 + rNOM)/(1 + growth] – 1
rr = 5.825243%
PMT = $81,220.47
Adj. PMT = $83,657.08

Page 66 Problems Chapter 2: Time Value

 a. $10,216.60
b. $10,754.31
c. $11,320.33
d. $11,886.35
e. $12,480.66

(2.18) Growing annuity Answer: a HARD

110
. You want to accumulate $2,500,000 in your 401(k) plan by your retirement
date, which is 35 years from now. You will make 30 deposits into your
plan, with the first deposit occurring today. The plan's rate of return
typically averages 9%. You expect to increase each deposit by 2% as your
income grows with inflation. (That is, your 2nd deposit will be 2% greater
than your first, the 3rd will be 2% greater than the 2nd, etc.) How much
must your 1st deposit at t = 0 be to enable you to meet your goal?

a. $8,718.90
b. $9,154.84
c. $9,612.58
d. $10,093.21
e. $10,597.87

(Comp: 2.7,2.10) Retirement planning Answer: a HARD

111
. Steve and Ed are cousins who were both born on the same day. Both turned
25 today. Their grandfather began putting $2,500 per year into a trust
fund for Steve on his 20th birthday, and he just made a 6th payment into
the fund. The grandfather (or his estate's trustee) will continue with
these $2,500 payments until a 46th and final payment is made on Steve's
65th birthday. The grandfather set things up this way because he wants
Steve to work, not to be a "trust fund baby," but he also wants to
ensure that Steve is provided for in his old age.

Until now, the grandfather has been disappointed with Ed, hence has not
given him anything. However, they recently reconciled, and the
grandfather decided to make an equivalent provision for Ed. He will
make the first payment to a trust for Ed later today, and he has
instructed his trustee to make additional equal annual payments each
year until Ed turns 65, when the 41st and final payment will be made.
If both trusts earn an annual return of 8%, how much must the
grandfather put into Ed's trust today and each subsequent year to enable
him to have the same retirement nest egg as Steve after the last payment

Chapter 2: Time Value Problems Page 67

 is made on their 65th birthday?

a. $3,726
b. $3,912
c. $4,107
d. $4,313
e. $4,528

Page 68 Problems Chapter 2: Time Value

 (Comp: 2.2,2.7) FV of uneven CF stream Answer: d HARD
112
. After graduation, you plan to work for Dynamo Corporation for 12 years and
then start your own business. You expect to save and deposit $7,500 a
year for the first 6 years and $15,000 annually for the following 6 years,
with the first deposit being made a year from today. In addition, your
grandfather just gave you a $25,000 graduation gift which you will deposit
immediately. If the account earns 9% compounded annually, how much will
you have when you start your business 12 years from now?
112. (Comp: 2.2,2.7) FV of uneven CF stream Answer: d HARD

There are 3 cash flow streams: the gift and the two annuities. The gift will grow for 12 years. Then there is
a 6-year annuity that will compound for an additional 6 years. Finally, there is a second 6-year annuity. The
sum of the compounded values of those three sets of cash flows is the final amount.

Amount Amount
at Year at Year
6 12
Interest rate 9.0%
1st annuity $7,500 $56,425 $94,630
2nd annuity $15,000 NA $112,850
Gift $25,000 NA $70,317
Total years 12
Annuity years 6 Final amt: $277,797

109. (2.18) Growing annuity Answer: c HARD

Chapter 2: Time Value Problems Page 69

 a. $238,176
b. $250,712
c. $263,907
d. $277,797
e. $291,687

(Comp: 2.2,2.3,2.10,2.12) Find CF for given return Answer: c HARD

113
. You are negotiating to make a 7-year loan of $25,000 to Breck Inc. To
repay you, Breck will pay $2,500 at the end of Year 1, $5,000 at the end
of Year 2, and $7,500 at the end of Year 3, plus a fixed but currently
unspecified cash flow, X, at the end of Years 4 through 7. Breck is

Step 1. Calculate the purchasing power of $1,500,000 in 30 years at an inflation rate of 4%:

N 30
I/YR 4.0%
PMT $0.00
FV $1,500,000
PV $462,478.00

Step 2. Calculate the real rate of return on the growing annuity:

rNOM 6.0%
Inflation 4.0%
rr = [(1 + rNOM)/(1 + Inflation)] – 1
rr = 1.92308%

Step 3. Calculate the required initial payment of the growing annuity by using inputs converted to "real"
terms:

N 30
I/YR 1.92308%
PV $0.00
FV 462,478.00
PMT $11,320.33

110. (2.18) Growing annuity Answer: a HARD

Step 1. Calculate the purchasing power of $2,500,000 in 35 years at an inflation rate of 2%:

N 35
I/YR 2.0%
PMT $0.00
FV $2,500,000
PV $1,250,069.03

Step 2. Calculate the real rate on the growing annuity:

rNOM 9.0%
Inflation 2.0%
rr = [(1 + rNOM)/(1 + Inflation)] – 1
rr = 6.86275%

Page 70 Problems Chapter 2: Time Value

 essentially riskless, so you are confident the payments will be made,
and you regard 8% as an appropriate rate of return on low risk 7-year
loans. What cash flow must the investment provide at the end of each of
the final 4 years, that is, what is X?

a. $4,271.67
b. $4,496.49
c. $4,733.15
d. $4,969.81
e. $5,218.30

(Comp: 2.2,2.3,2.10,2.12) Saving for college Answer: e HARD

Step 3. Calculate the required initial payment of the growing annuity by using inputs converted to "real"
terms:

N 35
I/YR 6.86275%
PV $0.00
FV 1,250,069.03
PMT $8,718.90

111. (Comp: 2.7,2.10) Retirement planning Answer: a HARD

Steve's retirement account Ed's retirement account

No. of payments thus far, at end of day 6 1
Number of remaining payments 40 40
N 46 N 41
I/YR 8.0% I/YR 8.0%
PV $0 PV $0
PMT $2,500 FV $1,046,065

FV Ed's FV should equal this: $1,046,065 PMT $3,726

113. (Comp: 2.2,2.3,2.10,2.12) Find CF for given return Answer: c HARD

This is a relatively easy problem to work with Excel, but it is quite difficult to work it with a calculator
because it is hard to conceptualize how to set it up for an efficient calculator solution. We would not use it
for a regular classroom exam, but it might be appropriate for a take-home or online exam.

I = 8%
0 1 2 3 4 5 6 7

-$25,000 $2,500 $5,000 $7,500 X X X X

Calculator solution:

Step 1. Use the CF register to find the NPV of the 4 known cash flows, CF0 to CF3: -$12,444.75
Step 2. Find the FV of this NPV at the end of period 3, i.e., compound the NPV for 3 years. -$15,676.80
Step 3. Now find the PMT for a 4-year annuity with this PV. $4,733.15

Excel solution:

Chapter 2: Time Value Problems Page 71

114
. John and Daphne are saving for their daughter Ellen's college education.
Ellen is now 10 years old and will be entering college 8 years from now
(t = 8). College tuition and expenses at State U. are currently $14,500
a year, but they are expected to increase at a rate of 3.5% a year.
They expect Ellen to graduate in 4 years. (If Ellen wants to go to
graduate school, she will be on her own.) Tuition and other costs will
be due at the beginning of each school year (at t = 8, 9, 10, and 11).
So far, John and Daphne have accumulated $15,000 in the college savings
account. Their long-run financial plan is to add an additional $5,000
at the beginning of each of the next 4 years (at t = 0, 1, 2, and 3).
Then they plan to make 4 equal annual contributions at the end of each
of the following 5 years (t = 4, 5, 6, 7, and 8). They expect their
investment account to earn 9%. How large must the annual payments be at
t = 4, 5, 6, 7, and 8 to meet Ellen's anticipated college costs?

a. $777.96
b. $818.91
c. $862.01

Set the problem up as shown below. Put a guess—we initially guessed $5,000—in the boxed cell under the
first X. The IRR initially is greater than 8%, so lower the guess, and keep iterating until IRR = 8%. This
value of X is the required payment for the investment to provide the 8% rate of return. The problem can be
worked faster if you use Goal Seek. Here you would highlight the cell with the IRR, then tell Excel to
change the Year 4 cell reference to the value that causes IRR = 8%. It turns out to be $4,733.15. If input
values are changed PMT does not change automatically—you must repeat this step again.

0 1 2 3 4 5 6 7

-$25,000 $2,500 $5,000 $7,500 $4,733.15 $4,733.15 $4,733.15 $4,733.15

IRR = 8.00%

Page 72 Problems Chapter 2: Time Value

 d. $907.38
e. $955.13

114. (Comp: 2.2,2.3,2.10,2.12) Saving for college Answer: e HARD

Current college costs $14,500

College cost inflation 3.5%
Account return 9.0%
First 4 payments $5,000
Current account balance $15,000

First, determine each year of college's costs.

Year 1 of college (t = 8) = 19,093.73

Year 2 of college (t = 9) = 19,762.01
Year 3 of college (t = 10) = 20,453.68
Year 4 of college (t = 11) = 21,169.56

The PV (at t = 8) of all college costs is: 70,786.26. This is what they need at t = 8.

After the first 4 payments, the college account will have (at t = 3): $42,291.08

5 more contributions are left in order to get the required funds for college costs.

N 5
I 9.0%
PV $42,291
FV $70,786.26
PMT $955.13

This problem can also be solved with Excel using Goal Seek:

College Need to Have FV of

Period = t Costs: at t = 8 Initial Balance Payments: FV of Pmts

now 0 14,500.00 15,000.00 5,000.00 9,962.81

1 15,007.50 5,000.00 9,140.20
2 15,532.76 5,000.00 8,385.50
3 16,076.41 5,000.00 7,693.12
4 16,639.08 955.13 1,348.25
5 17,221.45 955.13 1,236.92
6 17,824.20 955.13 1,134.79
7 18,448.05 955.13 1,041.09
8 19,093.73 70,786.26 29,888.44 955.13 955.13
9 19,762.01 40,897.82

Chapter 2: Time Value Problems Page 73

 CHAPTER 2
ANSWERS AND SOLUTIONS

10 20,453.68
11 21,169.56 Amt. needed – FV initial bal – FV of Pmts = 0.00

Use Goal Seek to set blue pmt such that we get zero for the pink sum. Note that the Goal Seek solution
step must be repeated again if input values change. It doesn't change automatically with input changes.

Page 74 Answers Chapter 2: Time Value