9-46. Solution:
PV of college costs five years from today (Part 1)
4
1
1(1 )
1
1(1.10)
$17,000 .10
$17,000 (3.170)
$53,887.71
n
A
A
A
A
i
PV A i
PV
PV
PV
–+
= ´
–
= ´
= ´
=
Accumulation of $2,000 per year for 10 years (Part 2)
10
(1 ) 1
(1.10) 1
$2,000 .10
$2,000 (15.937)
$31,874.85
n
A
A
A
A
i
FV A i
FV
FV
FV
+ –
= ´
–
= ´
= ´
=
Part 1 minus Part 2
$22,012.86 Additional funds required in five years
Additional contribution required for next five years
5
(1 ) 1
(1 ) 1
$22,012.86
(1.10) 1
.10
$22,012.86
6.105
$3,605.65
n
A
A
n
i
FV A i
FV
Ai
i
A
A
A
+ –
= ´
=+ –
=–
=
=
Calculator Solution:
Present value of college costs
N I/Y PV PMT FV
Accumulation based on investing $2,000 per year for 10 years.
N I/Y PV PMT FV
Additional funds required five years from now when Brittany starts college:
= 53,887.71 – 31,874.85 = $22,012.86
Added contribution for the next five years
N I/Y PV PMT FV
Present value of college costs
Appendix D
A IFA
PV = A × PV (10%, 4 periods)
= $17,000 × 3.170
= $53,890
Accumulation based on investing $2,000 per year for 10 years.
Appendix C
A IFA
FV = A × FV (10%, 10 periods)
= $2,000 × 15.937
= $31,874
Additional funds required five years from now.
$53,890 PV of college costs
31,874 Accumulation based on $2,000 per year investment
$22,016 Additional funds required
Added contribution for the next five years
Appendix C
A IFA
A = FV /FV (10%, 5 periods)
= $22,016/6.105
= $3,606.22
Connect-Only Problem
47. Special consideration of annuities and time periods (LO9-4) Your parents have
accumulated a $120,000 nest egg. They have been planning to use this money to pay college
costs to be incurred by you and your sister, Courtney. However, Courtney has decided to forgo
college and start a nail salon. Your parents are giving Courtney $15,000 to help her get started,
and they have decided to take year-end vacations costing $10,000 per year for the next four
years.
How much money will your parents have at the end of four years to help you with graduate
school, which you will start then? You plan to work on a master’s and perhaps a PhD. If graduate
school costs $26,353 per year, approximately how long will you be able to stay in school based
on these funds? Use 9 percent as the appropriate interest rate throughout this problem. Round all
values to whole numbers.
9.47. Solution:
Funds available after Nail Salon
PV of vacations
4
1
1(1 )
1
1(1.09)
$10,000 .09
$10,000 (3.2397)
$32,397
n
A
A
A
A
i
PV A i
PV
PV
PV
–+
= ´
–
= ´
= ´
=
Funds available after vacations
$105,000
$ 72,603
Funds four years later for graduate school
4
(1 )
$72,603 (1.09)
$102,485
n
FV PV i
FV
FV
= ´ +
= ´
=
Number of years of graduate school
1
1(1 )
1
1(1 )
1
1$102,485
(1.09)
.09 $26,353
1
1 .3500
(1.09)
1.6500
(1.09)
(1.09) 1.538
ln(1.09) ln(1.538)
ln(1.538)
ln(1.09)
5
n
A
n
A
n
n
n
n
i
PV A i
PV
i
i A
n
n
n
–+
= ´
–+=
–
=
– =
– =–
=
´ =
=
=
N I/Y PV PMT FV
Less present value of vacation
Funds available four years later for graduate school:
N I/Y PV PMT FV
Number of years of graduate education
N I/Y PV PMT FV
Funds available after the nail salon
$120,000 Funding available before the nail salon
Less present value of vacation
Appendix D
A IFA
PV A PV (9%, 4 periods)
$10,000 3.240 = $32,400
= ´
= ´
$105,000
– 32,400
Appendix A
IF
FV PV FV (9%, 4 periods)
$72,600 1.412
$102,511 Funds available for starting graduate school
= ´
= ´
=
Number of years of graduate education
Appendix D
A
IFA
PV
PV (9%)
A
$102,511 3.890 (rounded)
$26,353
=
= =
with i = 9%, n = 5 for 3.890, the answer is five years.
COMPREHENSIVE PROBLEM
Medical Research Corporation (Comprehensive time value of money) Dr. Harold Wolf of
Medical Research Corporation (MRC) was thrilled with the response he had received from drug
companies for his latest discovery, a unique electronic stimulator that reduces the pain from
arthritis. The process had yet to pass rigorous Federal Drug Administration (FDA) testing and
was still in the early stages of development, but the interest was intense. He received the three
offers described following this paragraph. (A 10 percent interest rate should be used throughout
this analysis unless otherwise specified.)
Offer I – $1,000,000 now plus $200,000 from year 6 through 15. Also, if the product did over
$100 million in cumulative sales by the end of year 15, he would receive an additional
$3,000,000. Dr. Wolf thought there was a 70 percent probability this would happen.
Offer II – Thirty percent of the buyer’s gross profit on the product for the next four years. The
buyer in this case was Zbay Pharmaceutical. Zbay’s gross profit margin was 60 percent. Sales in
year one were projected to be $2 million and then expected to grow by 40 percent per year.
Offer III – A trust fund would be set up for the next eight years. At the end of that period, Dr.
Wolf would receive the proceeds (and discount them back to the present at 10 percent). The trust
fund called for semiannual payments for the next eight years of $200,000 (a total of $400,000 per
year).
The payments would start immediately. Since the payments are coming at the beginning of
each period instead of the end, this is an annuity due. To look up the future value of an annuity
due in the tables, add 1 to n (16 + 1) and subtract 1 from the value in the table. Assume the
annual interest rate on this annuity is 10 percent annually (5 percent semiannually). Determine
the present value of the trust fund’s final value. Hint: See page 280for a discussion of calculating
an annuity due.
Required: Find the present value of each of the three offers and indicate which one has the
highest present value.
CP 9-1. Solution:
Medical Research Corporation
Offer I
$1,000,000 now plus:
+ $200,000 from year 6 through 15 (deferred annuity)
Appendix D
A IFA
PV = A × PV (10%, 10 years)
= $200,000 × 6.145
= $1,229,000 (percent value at the beginning of
year 6; i.e., the end of year 5)
Appendix B
IF
PV FV PV (10%, 5 years)
$1,229,000 .621
$763,209
+ .70 × $3,000,000 = $2,100,000
Appendix B
IF
PV FV PV (10%, 15 years)
$2,100,000 .239 $501,900
Total value of Offer I
$1,000,000 Payment today
Offer II
Gross Profit Payment 30%
Year Sales (60% of Sales) of Gross Profit
1 $2,000,000 $1,200,000 $360,000
2 2,800,000 1,680,000 504,000
Appendix B
Year Payment PV Factor PV
1 $360,000 .909 $327,240
Offer III
Future value of an annuity due (Appendix C)
8 years – semiannually
n = 16 + 1 = 17
A IFA
FV A FV
$200,000 24.840
$4,968,000 value of trust fund after eight years
= ´
= ´
=
Present value of trust fund (Appendix B)
IF
PV A PV (10%, 8 years)
$4,968,000 .467
$2,320,056 Total value of offer III
CP 9-1. (Continued)
Summary
Value of Offer I $2,265,109
Select Offer III