CHAPTER 10
TIME-VALUE-OF-MONEY CONCEPT
SOLUTIONS TO SELF-TEST EXERCISES
SELF-TEST EXERCISE 10.1:
CASH FLOW FORECAST OF A PHOTO CENTRE
CompuTech’s five-year cash flow forecast is as follows:
Years
Revenue (20%)
Operating expenses
- Salaries (8%)
- Materials (15%)
- Depreciation
Total expenses
Profit before taxes
Income tax expense (35%)
1
2
3
4
5___
$125,000 $150,000 $180,000 $216,000 $259,200
(25,000)
(50,000)
(15,000)
(90,000)
35,000
(12,250)
(27,000) (29,160) (31,493) (34,012)
(57,500) (66,125) (76,044) (87,451)
(15,000) (15,000) (15,000) (15,000)
(99,500) (110,285) (122,537) (136,463)
50,500
69,715
93,463 122,737
(17,675) (24,400) (32,712) (42,958)
Profit for the year
22,750
32,825
45,315
60,751
79,779
Add back depreciation
15,000
15,000
15,000
15,000
15,000
$ 37,750
$47,825
$60,315
$75,751
$94,779
Cash flow
SELF-TEST EXERCISE 10.2:
FUTURE VALUE OF A SINGLE SUM
Joan Miller has just inherited $30,000 wants to invest it. She is trying to choose between
GICs, safe mutual funds, or in stock options that are more risky. Based on her analysis,
the historical performance for each type of investment is 5%, 9%, and 12%, respectively.
If there are no withdrawals, how much would Joan have at the end of 20 years for each
choice?
By using Table A in Appendix B at the end of the book, and using the 5%, 9% and 12%
column and 20-year row, and multiplying the appropriate factors by the $30,000 amount,
we get the following:
GIC (5%)
Mutual funds (9%)
Stock market (12%)
$ 30,000 × 2.653 =
$ 30,000 × 5.604 =
$ 30,000 × 9.646 =
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$ 79,590
$ 168,120
$ 289,380
10-2 Chapter 10 Time-Value-of-Money Concepts
SELF-TEST EXERCISE 10.3:
FUTURE VALUE OF AN ANNUITY
Joan Miller was given a choice on her $30,000 inheritance between receiving (1) the full
payment today or (2) a $3,000 annuity for the next 20 years and a lump sum of $10,000 at
the end of the 20th year. If Joan can earn 9%, which option is the most attractive?
The first option is the most attractive because Joan would earn $168,120 instead of
$163,480.
Option 1
By going to Table A in Appendix B at the end of the book at the 9% and 20-year row,
you will find factor 5.604. The future value of the lump-sum amount is $168,120
calculated in the following way:
This future value is worth $168,120 ($30,000 × 5.604).
Option 2
By going to Table C in Appendix B at the end of the book at the 9% and 20-year row,
you find factor 51.160.
The future value of the $3,000.00 annuity is $153,480 ($3,000 × 51.160). By adding
this amount to $10,000, the future value of the second option will be $163,480.
SELF-TEST EXERCISE 10.4:
PRESENT VALUE OF A SINGLE SUM
The Millers would like to have a $30,000 education fund for their son Vincent, who is
now three years old, and the same amount for their daughter, Takara, who has just turned
one. The Millers expect that their children will start university when they are 20 years of
age. How much will the Millers have to invest today (in one lump sum) if the registered
education savings plan guarantees a 7% annual interest rate free from any income tax?
This problem can be solved by finding the present value of each of the two investments
separately and then by adding them up.
By going to Table B in Appendix B at the end of the book, at the 7% column, you will
have the following:
Number of
Years
17
19
Future
Value
$30,000
$30,000
$60,000
Total payment to be made today
PV Value
0.31657
0.27651
Lump Sum
Payment
$ 9,497.10
$ 8,295.30
$17,792.40
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Vincent
Takara
Chapter 10 Time-Value-of-Money Concepts 10-3
The Millers will have to pay $17,792.40 now to have $30,000 for each child by the time
they reach the age of 20 if they expect to earn 7% annually on the RESP.
SELF-TEST EXERCISE 10.5:
PRESENT AND FUTURE VALUES OF AN ANNUITY
Len has just won $100,000 at a casino. If money is worth 10%, would it be better for Len
to receive the full amount now or $15,000 each year for the next 10 years?
Questions
1. What is the value of each amount 10 years from now?
The lump sum option is more attractive: $259,400 versus $239,055.
Lump sum option
By going to Table A in Appendix B at the end of the book at the 10% and 10-year
row, you find factor 2.594. The future value of the $100,000 lump sum amount is
$259,400 ($100,000 × 2.594).
Annuity option
By going to Table C in Appendix B at the end of the book at the 10% and 10-year
row, you find factor 15.937. The future value of the $15,000 annuity is $239,055
($15,000 × 15.937).
2. What is today’s present value of each amount?
Just like in question one, the lump sum option is better: $100,000 versus $92,169.
Lump sum option
The $100,000 amount received today is worth $100,000.
Annuity option
By going to Table D in Appendix B at the end of the book at the 10% and 10-year
row, you find factor 6.1446. The present value of the $15,000 annuity is $92,169
($15,000 × 6.1446).
3. How much would Len have to receive each year to be equivalent to receiving
$100,000 today?
The yearly amount is approximately $16,275. This answer can be obtained in two ways.
a) By going to Table C in Appendix B at the end of the book at the 10% and 10-year
row, you find factor 15.937. By dividing this factor into the future value of the
annuity $259,400 (see question 1), you get $16,277.
© 2014 by Nelson Education Ltd.
10-4 Chapter 10 Time-Value-of-Money Concepts
b) By going to Table D in Appendix B at the end of the book at the 10% and 10-year
row, you find factor 6.1446. By dividing this factor in the $100,000 amount, you
get $16,274.
4. If Len received $15,000 each year instead of the $100,000 amount, what would the
effective interest rate or the internal rate of return (IRR) be?
Without a financial calculator or a spreadsheet, you can only find the answer by trial-anderror.
By going to Table D at the 9% and 10-year row, you find factor 6.4176. By multiplying
this factor by $15,000, you get $96,264. With a $3,736 negative net present value, it
means that the effective interest rate or the internal rate of return is less than 9.0%. By
using 8% instead, the present value is $100,652 ($15,000 × 6.7101) and gives a positive
$652 net present value, which means that the effective interest rate or the internal rate of
return is slightly more than 8%.
By using a financial calculator, you get exactly 8.14%.
SELF-TEST EXERCISE 10.6:
PRESENT VALUE OF UNEVEN SUMS
Use the cash flow forecast from Self-Test Exercise 10.1 to calculate the photo centre’s
present value. Assume that the company’s cost of capital is 10%.
The present value is $229,747.
Years
Cash Flow
1
$ 37,750
2
47,825
3
60,315
4
75,751
5
94,779
Total net present value (NPV)
Discount
Factor
0.90909
0.82645
0.75131
0.68301
0.62092
Present
Values
$ 34,318
39,525
45,315
51,739
58,850
$229,747
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Chapter 10 Time-Value-of-Money Concepts 10-5
SELF-TEST EXERCISE 10.7:
NET PRESENT VALUE AND NET FUTURE VALUE
Use the cash flow forecast from Self-Test Exercise 10.1 to calculate the photo centre’s
NPV and NFV. Assume that the company’s cost of capital is 10%.The net present value
is $79,947.
The net future value is $128,357.
Net Present Value (NPV)
By using the present value amount calculated is the Self-Test Exercise 10.6, the net
present value is calculated as follows:
Cash outflow (year 0)
Present value
Net present value (NPV)
$ 150,000
229,747
$ 79,947
Net future value (NFV)
Years
0
Cash Flow
Compound Factor
Future Values
$150,000
1.611
$ 241,650
1
37,750
2
47,825
3
60,315
4
75,751
5
94,779
Future value (FV)
1.464
1.331
1.210
1.100
1.000
55,266
63,655
72,981
83,326
94,779
$ 370,007
Total net future value (NFV)
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$ 128,357
10-6 Chapter 10 Time-Value-of-Money Concepts
SELF-TEST EXERCISE 10.8:
INTERNAL RATE OF RETURN
Use the cash flow forecast from Self-Test Exercise 10.1 to calculate the photo centre’s
IRR.
Using a discount rate of 26%, the net present value is +$134. By using a financial
calculator, the project gives an internal rate of return of 26.04%.
Present value using a 26% discount rate
Years
Cash Flow
1
$ 37,750
2
47,825
3
60,315
4
75,751
5
94,779
Total present value (PV)
Discount Factor
Present Values
0.79365
0.62988
0.49991
0.39675
0.31488
$ 29,960
30,124
30,152
30,054
29,844
$150,134
Net present value (NPV)
Present value of the cash inflows
Cash outflow
$150,134
150,000
Net present value (NPV)
$
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