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Chapter 09: Time Value of Money
9-21
IFA
PV FV PV (10%,3periods)
$48,680 .751 $36.559
=
= =
$52,736
9-38. (Continued) OR
Take the PVIFA for 10 years at 10% and subtract the PVIFA for 3
years at 10% to end up with the 7 year deferred annuity.
39. Present value (LO3) Kelly Greene has a contract in which she will receive the following
payments for the next five years: $3,000, $4,000, $5,000, $6,000, and $7,000. She will then
receive an annuity of $9,000 a year from the end of the sixth through the end of the 15th
year. The appropriate discount rate is 13 percent. If she is offered $40,000 to cancel the
contract, should she do it?
9-39. Solution:
First find the present value of the first five payments.
PV = FV × PVIF (Appendix B) i = 13%
Chapter 09: Time Value of Money
Then find the present value of the deferred annuity.
Appendix D will give a factor for a ten period annuity (sixth year
Chapter 09: Time Value of Money
40. Deferred annuity (LO3) Kay Mart has purchased an annuity to begin payment at the end
of 2113 (the date of the first payment). Assume it is now the beginning of 2011. The
annuity is for $12,000 per year and is designed to last eight years.
If the discount rate for the calculation is 11 percent, what is the most she should have paid
for the annuity?
9-40. Solution:
Appendix D will give a factor for an 8 year annuity when the
appropriate discount rate is 11 percent (5.146). The value of the
annuity at the beginning of the year it starts (2113) is:
A IFA
PV A PV (11%, 8periods)
$12,000 5.146
$61,752
=
=
=
The present value at the beginning of 2011 is found using
Appendix B (2 years at 11%). The factor is .812. Note we are
discounting from the beginning of 2113 to the beginning of 2011.
IF
PV = PV × PV (11%, 2periods)
= $61,752 × .812
= $50,142.62
41. Yield (LO4) If you borrow $9,725 and are required to pay back the loan in five equal
annual installments of $2,500, what is the interest rate associated with the loan?
9-41. Solution:
Appendix D
Chapter 09: Time Value of Money
9-24
IFA A
PV PV / A (5 periods)
$9,725/$2,500
3.890
=
=
=
Interest rate = 9 percent
42. Loan repayment (LO4) Tom Busby owes $20,000 now. A lender will carry the debt for
four more years at 8 percent interest. That is, in this particular case, the amount owed will
go up by 8 percent per year for four years. The lender then will require Busby to pay off the
loan over 12 years at 11 percent interest. What will his annual payment be?
9-42. Solution:
Appendix A
IFA
FV = PV × FV (8%, 4periods)
FV = $20,000 ×1.360
= $27,200 Amount owed at end of 4 periods
Appendix D
A IFA
A = PV /PV (11%,12periods)
= $27,200/6.492
= $4,189.77 Annual payment required
43. Loan repayment (LO4) If your aunt borrows $50,000 from the bank at 10 percent interest
over the eight-year life of the loan, what equal annual payments must be made to discharge
the loan, plus pay the bank its required rate of interest (round to the nearest dollar)? How
much of his first payment will be applied to interest? To principal? How much of her
second payment will be applied to each?
9-43. Solution:
Chapter 09: Time Value of Money
9-25
Appendix D
A IFA
A = PV / PV (10%, 8periods)
= $50,000 / 5.335
= $9,372.07 Annual payments
First payment:
Second payment: First determine remaining principal
44. Loan repayment (LO4) Jim Thorpe borrows $70,000 toward the purchase of a home at 12
percent interest. His mortgage is for 30 years.
a. How much will his annual payments be? (Although home payments are usually on a
monthly basis, we shall do our analysis on an annual basis for ease of computation.
We will get a reasonably accurate answer.)
b. How much interest will he pay over the life of the loan?
c. How much should he be willing to pay to get out of a 12 percent mortgage and into a
10 percent mortgage with 30 years remaining on the mortgage? Suggestion: Find the
annual savings and then discount them back to the present at the current interest rate
(10 percent).
9-44. Solution:
Appendix D
A IFA
a. A = PV / PV (12%, 30periods)
= $70,000 / 8.055
= $8,690.25
Chapter 09: Time Value of Money
b. $ 8,690.25 annual payments
× 30 years
$260,707.50 total payment
70,000.00 repayment of principal
$190,707.50 interest paid over life of loan
−
Appendix D
c. New payments at 10%
A IFA
A = PV / PV (10%, 30 periods)
= $70,000 / 9.427
= $7,425.48
9-44. (Continued)
Difference between old and new payments
$8,690.25 old
7,425.48 new
$1,264.77 difference
P.V. of difference – Appendix D
A IFA
PV A PV (assumes 10% discount rate, 30 periods)
$ 1,264.77 9.427
$11,922.99 Amount that could be paid to refinance
=
=
=
Chapter 09: Time Value of Money
9-27
45. Annuity with changing interest rates (LO4) You are chairperson of the investment fund
for the Continental Soccer League. You are asked to set up a fund of semiannual payments
to be compounded semiannually to accumulate a sum of $200,000 after 10 years at an 8
percent annual rate (20 payments). The first payment into the fund is to take place six
months from today, and the last payment is to take place at the end of the 10th year.
a. Determine how much the semiannual payment should be. (Round to whole numbers.)
On the day after the sixth payment is made (the beginning of the fourth year) the interest
rate goes up to a 10 percent annual rate, and you can earn a 10 percent annual rate on funds
that have been accumulated as well as all future payments into the fund. Interest is to be
compounded semiannually on all funds.
b. Determine how much the revised semiannual payments should be after this rate
change (there are 14 payments and compounding dates). The next payment will be in
the middle of the fourth year. (Round all values to whole numbers.)
9-45. Solution:
Appendix C
A IFA
a. A FV / FV
$200,000 / 29.778(4%, 20 periods)
$6,716
=
=
=
b. First determine how much the old payments are equal to after
6 periods at 4%. Appendix C.
A IFA
FV = A × FV (4%, 6 periods)
= $6,716 × 6.633
= $44,547
Appendix A
IF
FV = PV × FV (5%, 14 periods)
= $44,547 ×1.980
= $88,203
