CHAPTER 4—TIME VALUE OF MONEY
TRUE/FALSE
1. Starting to invest early for retirement increases the benefits of compound interest.
2. Starting to invest early for retirement reduces the benefits of compound interest.
3. A time line is meaningful even if all cash flows do not occur annually.
4. A time line is not meaningful unless all cash flows occur annually.
5. Time lines can be constructed in situations where some of the cash flows occur annually but others
occur quarterly.
6. Time lines cannot be constructed in situations where some of the cash flows occur annually but others
occur quarterly.
7. Time lines can be constructed for annuities where the payments occur at either the beginning or the
end of the periods.
8. Time lines cannot be constructed for annuities unless all the payments occur at the end of the periods.
9. Some of the cash flows shown on a time line can be in the form of annuity payments while others can
be uneven amounts.
10. Some of the cash flows shown on a time line can be in the form of annuity payments but none can be
uneven amounts.
11. 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.
12. If the discount (or interest) rate is positive, the future value of an expected series of payments will
always exceed the present value of the same series.
13. Disregarding risk, if money has time value, it is impossible for the present value of a given sum to
exceed its future value.
14. Disregarding risk, if money has time value, it is impossible for the future value of a given sum to
exceed its present value.
15. If a bank compounds savings accounts quarterly, the nominal rate will exceed the effective annual rate.
16. If a bank compounds savings accounts quarterly, the effective annual rate will exceed the nominal rate.
17. A “growing annuity” is a cash flow stream that grows at a constant rate for a specified number of
periods.
18. A “growing annuity” is any cash flow stream that grows over time.
19. 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.
20. 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 smaller the present value of a given lump sum to be
received at some future date.
21. Suppose Sally Smith plans to invest $1,000. She can earn an effective annual rate of 5% on Security A,
while Security B has an effective annual rate of 12%. After 11 years, the compounded value of
Security B should be more than twice the compounded value of Security A. (Ignore risk, and assume
that compounding occurs annually.)
22. Suppose Randy Jones plans to invest $1,000. He can earn an effective annual rate of 5% on Security
A, while Security B has an effective annual rate of 12%. After 11 years, the compounded value of
Security B should be somewhat less than twice the compounded value of Security A. (Ignore risk, and
assume that compounding occurs annually.)
23. The present value of a future sum decreases as either the discount rate or the number of periods per
year increases, other things held constant.
24. The present value of a future sum increases as either the discount rate or the number of periods per
year increases, other things held constant.
25. All other things held constant, the present value of a given annual annuity decreases as the number of
periods per year increases.
26. All other things held constant, the present value of a given annual annuity increases as the number of
periods per year increases.
27. 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.
28. If we are given a periodic interest rate, say a monthly rate, we can find the nominal annual rate by
dividing the periodic rate by the number of periods per year.
29. 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).
30. As a result of compounding, the effective annual rate on a bank deposit (or a loan) is always equal to
or less than the nominal rate on the deposit (or loan).
31. 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.
32. When a loan is amortized, a relatively low percentage of the payment goes to reduce the outstanding
principal in the early years, and the principal repayment’s percentage increases in the loan’s later years.
33. 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.
34. 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 smaller the percentage of the
payment that will be a repayment of principal.
35. Midway through the life of an amortized loan, the percentage of the payment that represents interest
must be equal to the percentage that represents repayment of principal. This is true regardless of the
original life of the loan or the interest rate on the loan.
36. Midway through the life of an amortized loan, the percentage of the payment that represents interest
could be equal to, less than, or greater than to the percentage that represents repayment of principal.
The proportions depend on the original life of the loan and the interest rate.
MULTIPLE CHOICE
37. Which of the following statements is CORRECT?
a.
Some of the cash flows shown on a time line can be in the form of annuity payments, but
none can be uneven amounts.
b.
A time line is not meaningful unless all cash flows occur annually.
c.
Time lines are useful for visualizing complex problems prior to doing actual calculations.
d.
Time lines cannot be constructed in situations where some of the cash flows occur
annually but others occur quarterly.
e.
Time lines cannot be constructed for annuities where the payments occur at the beginning
of the periods.
38. Which of the following statements is CORRECT?
a.
Some of the cash flows shown on a time line can be in the form of annuity payments, but
none can be uneven amounts.
b.
A time line is not meaningful unless all cash flows occur annually.
c.
Time lines are not useful for visualizing complex problems prior to doing actual
calculations.
d.
Time lines cannot be constructed in situations where some of the cash flows occur
annually but others occur quarterly.
e.
Time lines can be constructed for annuities where the payments occur at either the
beginning or the end of the periods.
39. Which of the following statements is CORRECT?
a.
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.
b.
A time line is not meaningful unless all cash flows occur annually.
c.
Time lines are not useful for visualizing complex problems prior to doing actual
calculations.
d.
Time lines can be constructed to deal with situations where some of the cash flows occur
annually but others occur quarterly.
e.
Time lines can only be constructed for annuities where the payments occur at the end of
the periods, i.e., for ordinary annuities.
40. Which of the following statements is CORRECT?
a.
A time line is not meaningful unless all cash flows occur annually.
b.
Time lines are not 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 end of
the periods, i.e., for ordinary annuities.
e.
Time lines can be constructed where some of the payments constitute an annuity but
others are unequal and thus are not part of the annuity.
41. You plan to analyze the value of a potential investment by calculating the sum of the present values of
its expected cash flows. Which of the following would lower the calculated value of the investment?
a.
The discount rate decreases.
b.
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.
c.
The discount rate increases.
d.
The riskiness of the investment’s cash flows decreases.
e.
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.
42. You plan to analyze the value of a potential investment by calculating the sum of the present values of
its expected cash flows. Which of the following would increase the calculated value of the investment?
a.
The discount rate increases.
b.
The cash flows are in the form of a deferred annuity, and they total to $100,000. You learn
that the annuity lasts for 10 years rather than 5 years, hence that each payment is for
$10,000 rather than for $20,000.
c.
The discount rate decreases.
d.
The riskiness of the investment’s cash flows increases.
e.
The total amount of cash flows remains the same, but more of the cash flows are received
in the later years and less are received in the earlier years.
43. Which of the following statements is CORRECT?
a.
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.
b.
The cash flows for an ordinary (or deferred) annuity all occur at the beginning of the
periods.
c.
If a series of unequal cash flows occurs at regular intervals, such as once a year, then the
series is by definition an annuity.
d.
The cash flows for an annuity due must all occur at the ends of the periods.
e.
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.
44. Which of the following statements is CORRECT?
a.
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.
b.
The cash flows for an ordinary (or deferred) annuity all occur at the beginning of the
periods.
c.
If a series of unequal cash flows occurs at regular intervals, such as once a year, then the
series is by definition an annuity.
d.
The cash flows for an annuity due must all occur at the beginning of the periods.
e.
The cash flows for an annuity may vary from period to period, but they must occur at
regular intervals, such as once a year or once a month.
45. Your bank account pays a 5% nominal rate of interest. The interest is compounded quarterly. Which of
the following statements is CORRECT?
a.
The periodic rate of interest is 5% and the effective rate of interest is also 5%.
b.
The periodic rate of interest is 1.25% and the effective rate of interest is 2.5%.
c.
The periodic rate of interest is 5% and the effective rate of interest is greater than 5%.
d.
The periodic rate of interest is 1.25% and the effective rate of interest is greater than 5%.
e.
The periodic rate of interest is 2.5% and the effective rate of interest is 5%.
46. Your bank account pays an 8% nominal rate of interest. The interest is compounded quarterly. Which
of the following statements is CORRECT?
a.
The periodic rate of interest is 8% and the effective rate of interest is also 8%.
b.
The periodic rate of interest is 2% and the effective rate of interest is 4%.
c.
The periodic rate of interest is 8% and the effective rate of interest is greater than 8%.
d.
The periodic rate of interest is 4% and the effective rate of interest is less than 8%.
e.
The periodic rate of interest is 2% and the effective rate of interest is greater than 8%.
47. A $250,000 loan is to be amortized over 8 years, with annual end-of-year payments. Which of these
statements is CORRECT?
a.
The proportion of interest versus principal repayment would be the same for each of the 8
payments.
b.
The annual payments would be larger if the interest rate were lower.
c.
If the loan were amortized over 10 years rather than 8 years, and if the interest rate were
the same in either case, the first payment would include more dollars of interest under the
8-year amortization plan.
d.
The proportion of each payment that represents interest as opposed to repayment of
principal would be lower if the interest rate were lower.
e.
The last payment would have a higher proportion of interest than the first payment.
48. A $150,000 loan is to be amortized over 6 years, with annual end-of-year payments. Which of these
statements is CORRECT?
a.
The proportion of interest versus principal repayment would be the same for each of the 7
payments.
b.
The annual payments would be larger if the interest rate were lower.
c.
If the loan were amortized over 10 years rather than 6 years, and if the interest rate were
the same in either case, the first payment would include more dollars of interest under the
6-year amortization plan.
d.
The proportion of each payment that represents interest as opposed to repayment of
principal would be higher if the interest rate were lower.
e.
The proportion of each payment that represents interest versus repayment of principal
would be higher if the interest rate were higher.
49. Which of the following statements regarding a 20-year (240-month) $225,000, fixed-rate mortgage is
CORRECT? (Ignore taxes and transactions costs.)
a.
The outstanding balance declines at a slower rate in the later years of the loan’s life.
b.
The remaining balance after three years will be $225,000 less one third of the interest paid
during the first three years.
c.
Because it is a fixed-rate mortgage, the monthly loan payments (which include both
interest and principal payments) are constant.
d.
Interest payments on the mortgage will increase steadily over time, but the total amount of
each payment will remain constant.
e.
The proportion of the monthly payment that goes towards repayment of principal will be
lower 10 years from now than it will be the first year.
50. Which of the following statements regarding a 15-year (180-month) $225,000, fixed-rate mortgage is
CORRECT? (Ignore taxes and transactions costs.)
a.
The outstanding balance declines at a faster rate in the later years of the loan’s life.
b.
The remaining balance after three years will be $125,000 less one third of the interest paid
during the first three years.
c.
Because the outstanding balance declines over time, the monthly payments will also
decline over time.
d.
Interest payments on the mortgage will increase steadily over time, but the total amount of
each payment will remain constant.
e.
The proportion of the monthly payment that goes towards repayment of principal will be
lower 10 years from now than it will be the first year.
51. Which of the following statements regarding a 30-year monthly payment amortized mortgage with a
nominal interest rate of 8% is CORRECT?
a.
Exactly 8% of the first monthly payment represents interest.
b.
The monthly payments will decline over time.
c.
A smaller proportion of the last monthly payment will be interest, and a larger proportion
will be principal, than for the first monthly payment.
d.
The total dollar amount of principal being paid off each month gets smaller as the loan
approaches maturity.
e.
The amount representing interest in the first payment would be higher if the nominal
interest rate were 6% rather than 8%.
52. Which of the following statements regarding a 20-year monthly payment amortized mortgage with a
nominal interest rate of 10% is CORRECT?
a.
Exactly 10% of the first monthly payment represents interest.
b.
The monthly payments will increase over time.
c.
A larger proportion of the first monthly payment will be interest, and a smaller proportion
will be principal, than for the last monthly payment.
d.
The total dollar amount of interest being paid off each month gets larger as the loan
approaches maturity.
e.
The amount representing interest in the first payment would be higher if the nominal
interest rate were 7% rather than 10%.
53. At the end of 10 years, which of the following investments would have the highest future value?
Assume that the effective annual rate for all investments is the same and is greater than zero.
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 (just one payment).
e.
Investment E pays $250 at the end of every year for the next 10 years (a total of 10
payments).
54. Of the following investments, which would have the lowest present value? Assume that the effective
annual rate for all investments is the same and is greater than zero.
a.
Investment A pays $250 at the end 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 (just one payment).
e.
Investment E pays $250 at the beginning of every year for the next 10 years (a total of 10
payments).
55. A U.S. Treasury bond will 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 PV of the $1,000 lump sum has a higher present value than the PV of a 3-year,
$333.33 ordinary annuity.
b.
The periodic interest rate is greater than 3%.
c.
The periodic rate is less than 3%.
d.
The present value would be greater if the lump sum were discounted back for more
periods.
e.
The present value of the $1,000 would be smaller if interest were compounded monthly
rather than semiannually.
56. A U.S. Treasury bond will 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 PV of the $1,000 lump sum has a smaller present value than the PV of a 3-year,
$333.33 ordinary annuity.
b.
The periodic interest rate is greater than 3%.
c.
The periodic rate is less than 3%.
d.
The present value would be greater if the lump sum were discounted back for more
periods.
e.
The present value of the $1,000 would be larger if interest were compounded monthly
rather than semiannually.
57. Which of the following statements is CORRECT, assuming positive interest rates and holding other
things constant?
a.
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 provide the higher future value if you
leave your funds on deposit.
b.
The present value of a 5-year, $250 annuity due will be lower than the PV of a similar
ordinary annuity.
c.
A 30-year, $150,000 amortized mortgage will have larger monthly payments than an
otherwise similar 20-year mortgage.
d.
A bank loan’s nominal interest rate will always be equal to or less than its effective annual
rate.
e.
If an investment pays 10% interest, compounded annually, its effective annual rate will be
less than 10%.
58. Which of the following statements is CORRECT, assuming positive interest rates and holding other
things constant?
a.
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 provide the higher future value if you
leave your funds on deposit.
b.
The present value of a 5-year, $250 annuity due will be lower than the PV of a similar
ordinary annuity.
c.
A 30-year, $150,000 amortized mortgage will have larger monthly payments than an
otherwise similar 20-year mortgage.
d.
A bank loan’s nominal interest rate will always be equal to or greater than its effective
annual rate.
e.
If an investment pays 10% interest, compounded quarterly, its effective annual rate will be
greater than 10%.
59. Which of the following statements is CORRECT?
a.
An investment that has a nominal rate of 6% with semiannual payments will have an
effective rate that is smaller than 6%.
b.
The present value of a 3-year, $150 annuity due will exceed the present value of a 3-year,
$150 ordinary annuity.
c.
If a loan has a nominal annual rate of 8%, then the effective rate can never be greater than
8%.
d.
If a loan or investment has annual payments, then the effective, periodic, and nominal
rates of interest will all be different.
e.
The proportion of the payment that goes toward interest on a fully amortized loan
increases over time.
60. Which of the following statements is CORRECT?
a.
An investment that has a nominal rate of 6% with semiannual payments will have an
effective rate that is smaller than 6%.
b.
The present value of a 3-year, $150 ordinary annuity will exceed the present value of a 3-
year, $150 annuity due.
c.
If a loan has a nominal annual rate of 7%, then the effective rate will never be less than
7%.
d.
If a loan or investment has annual payments, then the effective, periodic, and nominal
rates of interest will all be different.
e.
The proportion of the payment that goes toward interest on a fully amortized loan
increases over time.
61. You are considering two equally risky annuities, each of which pays $15,000 per year for 20 years.
Investment ORD is an ordinary (or deferred) annuity, while Investment DUE is an annuity due. Which
of the following statements is CORRECT?
a.
If the going rate of interest decreases from 10% to 0%, the difference between the present
value of ORD and the present value of DUE would remain constant.
b.
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.
c.
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.
d.
The present value of ORD exceeds the present value of DUE, and the future value of ORD
also exceeds the future value of DUE.
e.
The present value of DUE exceeds the present value of ORD, and the future value of DUE
also exceeds the future value of ORD.
62. You are considering two equally risky annuities, each of which pays $25,000 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.
If the going rate of interest decreases from 10% to 0%, the difference between the present
value of ORD and the present value of DUE would remain constant.
b.
A rational investor would be willing to pay more for DUE than for ORD, so their market
prices should differ.
c.
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.
d.
The present value of ORD exceeds the present value of DUE, and the future value of ORD
also exceeds the future value of DUE.
e.
The present value of ORD exceeds the present value of DUE, while the future value of
DUE exceeds the future value of ORD.
63. Which of the following statements is CORRECT?
a.
If CF0 is positive and all the other CFs are negative, then you cannot solve for I.
b.
If you have a series of cash flows, each of which is 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.
c.
If you have a series of cash flows, and CF0 is negative but each of the following CFs is
positive, you can solve for I, but only if the sum of the undiscounted cash flows exceeds
the cost.
d.
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.
e.
If you solve for I and get a negative number, then you must have made a mistake.
64. Which of the following statements is CORRECT?
a.
If CF0 is positive and all the other CFs are negative, then you can still solve for I.
b.
If you have a series of cash flows, each of which is 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.
c.
If you have a series of cash flows, and CF0 is negative but each of the following CFs is
positive, you can solve for I, but only if the sum of the undiscounted cash flows exceeds
the cost.
d.
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 FV of the negative CFs. It is impossible to find the value of
I without a computer or financial calculator.
e.
If you solve for I and get a negative number, then you must have made a mistake.
65. Which of the following bank accounts has the highest effective annual return?
a.
An account that pays 8% nominal interest with daily (365-day) compounding.
b.
An account that pays 8% nominal interest with monthly compounding.
c.
An account that pays 8% nominal interest with annual compounding.
d.
An account that pays 7% nominal interest with daily (365-day) compounding.
e.
An account that pays 7% nominal interest with monthly compounding.
66. Which of the following bank accounts has the lowest effective annual return?
a.
An account that pays 8% nominal interest with daily (365-day) compounding.
b.
An account that pays 8% nominal interest with monthly compounding.
c.
An account that pays 8% nominal interest with annual compounding.
d.
An account that pays 7% nominal interest with daily (365-day) compounding.
e.
An account that pays 7% nominal interest with monthly compounding.
67. 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.
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.
68. Ellen now has $125. How much would she have after 8 years if she leaves it invested at 8.5% with
annual compounding?
a.
$205.83
b.
$216.67
c.
$228.07
d.
$240.08
e.
$252.08
69. How much would Roderick have after 6 years if he has $500 now and leaves it invested at 5.5% with
annual compounding?
a.
$591.09
b.
$622.20
c.
$654.95
d.
$689.42
e.
$723.89