Chapter 09: Time Value of Money
9-11
Appendix B
PV = FV × PVIF
Discount rate = 11%
24. Present value (LO3) Mr. Flint retired as president of Color Title Company but is currently
on a consulting contract for $45,000 per year for the next 10 years.
a. If Mr. Flint’s opportunity cost (potential return) is 10 percent, what is the present
value of his consulting contract?
b. Assuming that Mr. Flint will not retire for two more years and will not start to receive
his 10 payments until the end of the third year, what would be the value of his
deferred annuity?
9-24. Solution:
Using a Two Step Procedure
Appendix D
a. PVA = A × PVIFA (i = 10%, 10 periods)
Alternative Solution
Appendix D
Chapter 09: Time Value of Money
9-12
b. Deferred annuity-Appendix D
25 Quarterly compounding (LO5) Cousin Bertha invested $100,000 10 years ago at 12
percent, compounded quarterly. How much has she accumulated?
9-25. Solution:
Appendix A
26. Special compounding (LO5) Determine the amount of money in a savings account at the
end of five years, given an initial deposit of $3,000 and a 8 percent annual interest rate
when interest is compounded (a) annually, (b) semiannually, and (c) quarterly.
9-26. Solution:
Appendix A
FV = PV × FVIF
Chapter 09: Time Value of Money
9-13
27. Annuity due (LO4) As stated in the chapter, annuity payments are assumed to come at the
end of each payment period (termed an ordinary annuity). However, an exception occurs
when the annuity payments come at the beginning of each period (termed an annuity due).
To find the present value of an annuity due, subtract 1 from n and add 1 to the tabular
value. To find the future value of an annuity, add 1 to n and subtract 1 from the tabular
value. For example, to find the future value of a $100 payment at the beginning of each
period for five periods at 10 percent, go to Appendix C for n = 6 and i = 10 percent. Look
up the value of 7.716 and subtract 1 from it for an answer of 6.716 or $671.60 ($100 ×
6.716).
What is the future value of a 10-year annuity of $2,000 per period where payments
come at the beginning of each period? The interest rate is 8 percent.
9-27. Solution:
Appendix C
FVA = A × FVIFA
28. Annuity due (LO4) Related to the discussion in problem 27, what is the present value of
a 10-year annuity of $3,000 per period in which payments come at the beginning of each
period? The interest rate is 12 percent.
9-28. Solution:
Appendix D
PVA = A × PVIFA
29. Present value alternative (LO3) Your grandfather has offered you a choice of one of the
three following alternatives: $5,000 now; $1,000 a year for eight years; or $12,000 at the
end of eight years. Assuming you could earn 11 percent annually, which alternative should
you choose? If you could earn 12 percent annually, would you still choose the same
alternative?
9-29. Solution:
Chapter 09: Time Value of Money
9-14
(first alternative) Present value of $5,000 received now:
$5,000
(second alternative) Present value of annuity of $1,000 for eight
years: Appendix D
A IFA
IFA
PV A×PV
$1,000 PV (11%, 8years)
$1,000 5.146
$5,146
=
=
=
=
(third alternative) Present value of $12,000 received in eight
years: Appendix B
IF
IF
PV FV×PV
$12,000×PV (11%, 8years)
$12,000×.434
$5,208
=
=
=
=
Select $12,000 to be received in eight years.
9-29. (Continued)
Revised answers based on 12%.
Chapter 09: Time Value of Money
A IFA
IFA
PV A PV
$1,000 PV (12%, 8years)
$1,000 4.968
$4,968
=
=
=
=
12%: Appendix B
IF
IF
PV = FV×PV
= $12,000×PV (12%, 8years)
= $12,000×.404
= $4,848
the present value declines.
30. Payment required (LO4) You need $23,956 at the end of nine years, and your only
investment outlet is an 7 percent long-term certificate of deposit (compounded annually).
With the certificate of deposit, you make an initial investment at the beginning of the first
year.
a. What single payment could be made at the beginning of the first year to achieve this
objective?
b. What amount could you pay at the end of each year annually for nine years to achieve
this same objective?
9-30. Solution:
a. Appendix B
PV = FV × PVIF (7%, 9 periods)
Chapter 09: Time Value of Money
b. Appendix C
A = FVA/FVIFA
31. Quarterly compounding (LO5) Beverly Hills started a paper route on January 1, 2004.
Every three months, she deposits $300 in her bank account, which earns 8 percent annually
but is compounded quarterly. On December 31, 2007, she used the entire balance in her
bank account to invest in an investment at 12 percent annually. How much will she have on
December 31, 2010?
9-31. Solution:
Appendix C
FVA = A × FVIFA (2%, 16 periods)
Appendix A
FV = PV × FVIF (12%, 3 periods)
32. Yield (LO4) Franklin Templeton has just invested $8,760 for her son (age one). This
money will be used for his son’s education 17 years from now. He calculates that he will
need $60,000 by the time the boy goes to school. What rate of return will Mr. Templeton
need in order to achieve this goal?
9-32. Solution:
Appendix B
IF
IF
PV
PV = (17 periods)
FV
$8,760
PV = .146 Rate of return 12%
$60,000 ==
Or
Chapter 09: Time Value of Money
9-17
Alternative solution
Appendix A
33. Yield with interpolation (LO4) On January 1, 2008, Mr. Dow bought 100 shares of stock
at $12 per share. On December 31, 2010, he sold the stock for $18 per share. What is his
annual rate of return? Interpolate to find the answer.
9-33. Solution:
Appendix B
IF
PV
PV FV
=
IF
$12
PV .667
$18
==
Return is between 14%-15% for 3 years
IF
IF
PV at 14% .675
PV at 15% .658
.017
−
IF
IF
PF at 14% .675
PV computed .667
.008
−
14% + (.008/.017) (1%)
14% + .471 (1%)
14.47%
Chapter 09: Time Value of Money
9-18
34. Yield with interpolation (LO4) C. D. Rom has just given an insurance company $30,000.
In return, he will receive an annuity of $3,200 for 20 years.
At what rate of return must the insurance company invest this $30,000 in order to make
the annual payments? Interpolate.
9-34. Solution:
Appendix D
IFA A
PV PV / A (20periods)
$30,000 / $3,200
9.375 is between 8% and 9% for 20 periods
=
=
=
IFA
IFA
PV at 8% 9.818
PV at 9% 9.129
.689
−
IFA
IFA
PV at 8% 9.818
PV computed 9.375
.443
−
8% + (.443/.689) (1%)
8% + .643 (1%) = 8.64%
35. Solving for an annuity (LO4) Alex Bell has just retired from the telephone company. His
total pension funds have an accumulated value of $200,000, and his life expectancy is 16
more years. His pension fund manager assumes he can earn a 12 percent return on his
assets.
What will be his yearly annuity for the next 16 years?
9-35. Solution:
Appendix D
A IFA
A PV / PV (12%,16periods)
$200,000 / 6.974
$28,677.95
=
=
=
Chapter 09: Time Value of Money
9-19
36. Solving for an annuity (LO4) Dr. Oats, a nutrition professor, invests $80,000 in a piece of
land that is expected to increase in value by 14 percent per year for the next five years. She
will then take the proceeds and provide herself with a 10-year annuity. Assuming a 14
percent interest rate for the annuity, how much will this annuity be?
9-36. Solution:
Appendix A
FV = PV × FVIF (14%, 5 periods)
37. Solving for an annuity (LO4) You wish to retire in 20 years, at which time you want to
have accumulated enough money to receive an annual annuity of $12,000 for 25 years after
retirement. During the period before retirement you can earn 8 percent annually, while
after retirement you can earn 10 percent on your money.
What annual contributions to the retirement fund will allow you to receive the $12,000
annuity?
9-37. Solution:
Determine the present value of an annuity during retirement:
Appendix D
A IFA
PV A PV (10%, 25years)
$12,000 9.077 $108,924
=
= =
Chapter 09: Time Value of Money
9-20
A IFA
A FV /FV (8%, 20years)
$108,924 = $2,380.23 annual contribution
45.762
=
=
38. Deferred annuity (LO3) Rusty Steele will receive the following payments at the end of
the next three years: $4,000, $7,000, and $9,000. Then from the end of the fourth year
through the end of the tenth year, he will receive an annuity of $10,000. At a discount rate
of 10 percent, what is the present value of all future benefits?
9-38. Solution:
First find the present value of the first three payments.
PV = FV × PVIF (Appendix B) i = 10%
Then find the present value of the deferred annuity.
Appendix D will give a factor for a seven period annuity (fourth
year through the tenth year) at a discount rate of 10 percent. The
value of the annuity at the beginning of the fourth year is:
AIFA
PV A PV (10%,7periods)
$10,000 4.868 $48,680
=
= =
This value at the beginning of year four (end of year three) must
now be discounted back for three years to get the present value of
the deferred annuity. Use Appendix B.