9-38. Solution:
Payment #1
1
1
(1 )
1
$2,000 (1.09)
$1,834.86
n
PV FV
i
PV
PV
æ ö
= ´ ç ÷
+
è ø
= ´
=
Payment #2
2
1
(1 )
1
$3,500 (1.09)
$2,945.88
n
PV FV
i
PV
PV
æ ö
= ´ ç ÷
+
è ø
= ´
=
Payment #3
3
1
(1 )
1
$4,500 (1.09)
$3,474.83
n
PV FV
i
PV
PV
æ ö
= ´ ç ÷
+
è ø
= ´
=
7
1
1(1 )
1
1(1.09)
$5,000 .09
$25,164.76
n
A
A
A
i
PV A
i
PV
PV
æ ö
–
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
æ ö
–
ç ÷
= ´ ç ÷
ç ÷
ç ÷
è ø
=
Annuity Value at Time Zero
3
1
(1 )
1
$25,164.76 (1.09)
$19,431.81
n
PV FV
i
PV
PV
= ´ +
= ´
=
Total Present Value
Calculator Solution:
Answer: $1,834.86
N I/Y PV PMT FV
Answer: $2,945.88
N I/Y PV PMT FV
Answer: $3,474.83
Then find the present value of the deferred annuity.
N I/Y PV PMT FV
Answer: $25,164.76 as of the end of year 3.
Then, find its PV as of now:
N I/Y PV PMT FV
Answer: $19,431.81 as of now
Finally, find the total present value of all future payments.
8,255.57 + 19,431.81 = $27,687.38
First find the present value of the first three payments.
PV = FV × PVIF (Appendix B) i = 9%
Then find the present value of the deferred annuity.
Appendix D will give a factor for a seven period annuity (4th
year through the 10th year) at a discount rate of 9 percent. The
value of the annuity at the beginning of the fourth year is:
A IFA
PV A PV (9%,7periods)
$5,000 5.033 $25,165
= ´
= ´ =
This value at the beginning of year 4 (end of year 3) must now be
discounted back for three years to get the present value of the
deferred annuity. Use Appendix B.
IF
PV FV PV (9%,3periods)
$25,165 .772 $19.427.38
= ´
= ´ =
Finally, find the total present value of all future payments.
39. Bridget Jones has a contract in which she will receive the following payments for the next
five years: $1,000, $2,000, $3,000, $4,000, and $5,000. She will then receive an annuity
of $8,500 a year from the end of the 6th through the end of the 15th year. The appropriate
discount rate is 14 percent. If she is offered $30,000 to cancel the contract, should she do it?
9-39. Solution:
First Five Payments
1 :
2 :
2
1
$2,000 $1,538.94
(1.14)
PV = ´ =
3 :
3
1
$3,000 $2,024.91
(1.14)
PV = ´ =
4 :
4
1
$4,000 $2,368.32
(1.14)
PV = ´ =
5 :
5
1
$5,000 $2,596.84
(1.14)
PV = ´ =
Present Value of the Annuity
10
1
1(1 )
1
1(1.14)
$8,500 .14
$44,336.98
n
A
A
A
i
PV A
i
PV
PV
æ ö
–
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
æ ö
–
ç ÷
= ´ ç ÷
ç ÷
ç ÷
è ø
=
Discount five years for PV of deferred annuity
Since the present value of all future benefits under the contract is
Answer: $877.19
N I/Y PV PMT FV
Answer: $1,538.94
N I/Y PV PMT FV
Answer: $2,024.91
N I/Y PV PMT FV
Answer: $2,368.32
N I/Y PV PMT FV
Answer: $2,596.84
Then find the present value of the deferred annuity.
N I/Y PV PMT FV
Answer: $44,336.98 as of the end of year 5
Then find it PV as of now:
N I/Y PV PMT FV
Answer: $23,027.2 4 as of now
Finally, find the total present value of all future payments.
Since the present value of all future benefits under the contract is greater than $30,000, Bridget
Jones should not accept this amount to cancel the contract.
First find the present value of the first five payments.
PV = FV × PVIF (Appendix B) i = 14%
Then, find the present value of the deferred annuity.
Appendix D will give a factor for a 10-period annuity (6th year
through the 15th year) at a discount rate of 14 percent. The value
of the annuity at the beginning of the 6th year is:
A IFA
PV A PV (14%, 10 periods)
$8,500 5.216 $44,336
= ´
= ´ =
This value at the beginning of year 6 (end of year 5) must now be
discounted back for five years to get the present value of the
deferred annuity. Use Appendix B.
IF
PV = FV × PV (14%, 5 periods)
= $44,336 × .516 = $23,010.38
Next, find the total present value of all future payments.
40. Mark Ventura has just purchased an annuity to begin payment two years from today. The
annuity is for $8,000 per year and is designed to last 10 years. If the interest rate for this
problem calculation is 13 percent, what is the most he should have paid for the annuity?
9-40. Solution:
Annuity
10
1
1(1 )
1
1(1.13)
8,000 .13
$8,000 (5.426)
$43,409.95
n
A
A
A
A
i
PV A
i
PV
PV
PV
æ ö
–
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
æ ö
–
ç ÷
= ´ ç ÷
ç ÷
ç ÷
è ø
= ´
=
Discount off two years
2
1
(1 )
1
$43,409.95 (1.13)
$33,996.36
n
V FV
i
PV
PV
= ´ +
= ´
=
Calculator Solution:
N I/Y PV PMT FV
N I/Y PV PMT FV
Answer: $33,996.36 as of now
The maximum that should be paid for the annuity is $33,996.36.
Appendix D will give a factor for a 10-year annuity when the
appropriate discount rate is 13 percent (5.426). The value of the
annuity at the beginning of the year it starts (2011) is:
A IFA
PV A PV (13%, 10periods)
$8,000 5.426
$43,408
= ´
= ´
=
The present value at the beginning of 2014 is found using
Appendix B (two years at 13 percent). The factor is .783. Note
we are discounting from the beginning of 2016 to the beginning
of 2014.
IF
PV = FV × PV (13%, 2periods)
= $43,408 × .783
= $33,988
The maximum that should be paid for the annuity is $33,988.
41. Yield (LO9-4) If you borrow $9,441 and are required to pay back the loan in five equal
annual installments of $2,750, what is the interest rate associated with the loan?
9-41. Solution:
Calculator Solution:
N I/Y PV PMT FV
Answer: 14.00%
Appendix D
IFA A
PV PV / A (5 periods)
$9,441/$2,750
3.433
=
=
=
Go across period 5 until you find 3.433. Go up to the percentage
at the top of the column and find 14 percent.
42. Cal Lury owes $10,000 now. A lender will carry the debt for five more years at 10 percent
interest. That is, in this particular case, the amount owed will go up by 10 percent per year
for five years. The lender then will require that Cal pay off the loan over the next 12 years
at 11 percent interest. What will his annual payment be?