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F. (1) What is the future value of a 3-year, $100 ordinary annuity if the
annual interest rate is 4%?
ANSWER: [Show S5–14 here.]
0 1 2 3
| | | |
F. (2) What is its present value?
ANSWER: [Show S5–15 here.]
0 1 2 3
| | | |
4%
4%
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F. (3) What would the future and present values be if it was an annuity
due?
ANSWER: [Show S5–16 and S5-17 here.] If the annuity were an annuity due,
each payment would be shifted to the left, so each payment is
compounded over an additional period or discounted back over one
less period.
G. A 5-year $100 ordinary annuity has an annual interest rate of 4%.
(1) What is its present value?
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ANSWER: [Show S5–18 here.]
0 1 2 3 4 5
| | | | | |
100 100 100 100 100
96.15
G. (2) What would the present value be if it was a 10-year annuity?
G. (3) What would the present value be if it was a 25-year annuity?
G. (4) What would the present value be if this was a perpetuity?
4%
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H. A 20-year-old student wants to save $5 a day for her retirement.
Every day she places $5 in a drawer. At the end of each year, she
invests the accumulated savings ($1,825) in a brokerage account
with an expected annual return of 8%.
(1) If she keeps saving in this manner, how much will she have
accumulated at age 65?
ANSWER: [Show S5–20 and S5-21 here.] If she begins saving today, and sticks
H. (2) If a 40-year-old investor began saving in this manner, how much
would he have at age 65?
ANSWER: [Show S5–22 here.] This question demonstrates the power of
H. (3) How much would the 40-year-old investor have to save each year to
accumulate the same amount at 65 as the 20-year-old investor?
ANSWER: [Show S5–23 here.] Again, this question demonstrates the power of
I. What is the present value of the following uneven cash flow stream?
The annual interest rate is 4%.
0 1 2 3 4 Years
| | | | |
0 100 300 300 –50
ANSWER: [Show S5–24 and S5-25 here.] Here we have an uneven cash flow
stream. The most straightforward approach is to find the PVs of each
4%
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115
J. (1) Will the future value be larger or smaller if we compound an initial
amount more often than annually (e.g., semiannually, holding the
stated (nominal) rate constant)? Why?
ANSWER: [Show S5–26 here.] Accounts that pay interest more frequently than
J. (2) Define (a) the stated (or quoted or nominal) rate, (b) the periodic
rate, and (c) the effective annual rate (EAR or EFF%).
J. (3) What is the EAR corresponding to a nominal rate of 4% compounded
semiannually? Compounded quarterly? Compounded daily?
ANSWER: [Show S5–29 through S5-31 here.] The effective annual rate for 4%
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J. (4) What is the future value of $100 after 3 years under 4% semiannual
compounding? Quarterly compounding?
ANSWER: [Show S5–32 here.] Under semiannual compounding, the $100 is
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117
K. When will the EAR equal the nominal (quoted) rate?
ANSWER: [Show S5–33 here.] If annual compounding is used, then the
L. (1) What is the value at the end of Year 3 of the following cash flow
stream if interest is 4% compounded semiannually? (Hint: You can
use the EAR and treat the cash flows as an ordinary annuity or use
the periodic rate and compound the cash flows individually.)
0 2 4 6 Periods
| | | | | | |
0 100 100 100
ANSWER: [Show S5–34 through S5-36 here.]
0 2 4 6 Periods
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L. (2) What is the PV?
ANSWER: [Show S5–37 here.]
0 2 4 6 Periods
| | | | | | |
L. (3) What would be wrong with your answer to parts L(1) and L(2) if you
used the nominal rate, 4%, rather than the EAR or the periodic rate,
INOM/2 = 4%/2 = 2% to solve the problems?
ANSWER: INOM can be used in the calculations only when annual compounding
M. (1) Construct an amortization schedule for a $1,000, 4% annual interest
loan with three equal installments.
(2) What is the annual interest expense for the borrower and the annual
interest income for the lender during Year 2?
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Amortization Schedule:
Beginning Payment of Ending
Period Balance Payment Interest Principal Balance
Now make the following points regarding the amortization
schedule:
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• Notice that the interest each year declines because the beginning