Chapter Five
First Principles of Valuation:
The Time Value of Money
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PPTs t/a Fundamentals of Corporate Finance 3e
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5-1
Chapter Organisation
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Future Value and Compounding
Present Value and Discounting
More on Present and Future Values
Present and Future Values of Multiple Cash Flows
Valuing Equal Cash Flows: Annuities and Perpetuities
Comparing Rates: The Effect of Compounding Periods
Loan Types and Loan Amortisation
Summary and Conclusions
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PPTs t/a Fundamentals of Corporate Finance 3e
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5-2
Chapter Objectives
•
Distinguish between simple and compound interest.
•
Calculate the present value and future value of a single amount
for both one period and multiple periods.
•
Calculate the present value and future value of multiple cash
flows.
•
Calculate the present value and future value of annuities.
•
Compare nominal interest rates (NIR) and effective annual
interest rates (EAR).
•
Distinguish between the different types of loans and calculate the
present value of each type of loan.
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5-3
Time Value Terminology
0
1
2
PV
3
4
FV
•
Future value (FV) is the amount an investment is worth after
one or more periods.
•
Present value (PV) is the current value of one or more future
cash flows from an investment.
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5-4
Time Value Terminology
• The number of time periods between the present
value and the future value is represented by ‘t’.
• The rate of interest for discounting or
compounding is called ‘r’.
• All time value questions involve four values: PV,
FV, r and t. Given three of them, it is always
possible to calculate the fourth.
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5-5
Interest Rate Terminology
• Simple interest refers to interest earned only on the
original capital investment amount.
• Compound interest refers to interest earned on
both the initial capital investment and on the
interest reinvested from prior periods.
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5-6
Future Value of a Lump Sum
You invest $100 in a savings account that earns 10 per cent
interest per annum (compounded) for three years.
After one year:
After two years:
After three years:
$100  (1 + 0.10) = $110
$110  (1 + 0.10) = $121
$121  (1 + 0.10) = $133.10
Copyright  2004 McGraw-Hill Australia Pty Ltd
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5-7
Future Value of a Lump Sum
• The accumulated value of this investment at the
end of three years can be split into two
components:
– original principal
$100
– interest earned
$33.10
• Using simple interest, the total interest earned
would only have been $30. The other $3.10 is from
compounding.
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5-8
Future Value of a Lump Sum
• In general, the future value, FVt, of $1 invested
today at r per cent for t periods is:
FVt  $1 1  r 
t
• The expression (1 + r)t is the future value interest
factor (FVIF). Refer to Table A.1.
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5-9
Example—Future Value of a Lump
Sum
•
What will $1000 amount to in five years time if interest is 12
per cent per annum, compounded annually?
FV  $10001  0.12
 $1000  1.7623
 $1762.30
5
•
•
From the example, now assume interest is 12 per cent per
annum, compounded monthly.
Always remember that t is the number of compounding
periods, not the number of years.
FV  $1000 1  0.01
60
 $1000  1.8167
 $1816.70
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5-10
Interpretation
• The difference in values is due to the larger
number of periods in which interest can compound.
• Future values also depend critically on the
assumed interest rate—the higher the interest rate,
the greater the future value.
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5-11
Future Values at Different Interest
Rates
Number of
periods
Future value of $100 at various interest rates
5%
10%
15%
20%
1
$105.00
$110.00
$115.00
$120.00
2
$110.25
$121.00
$132.25
$144.00
3
$115.76
$133.10
$152.09
$172.80
4
$121.55
$146.41
$174.90
$207.36
5
$127.63
$161.05
$201.14
$248.83
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5-12
Future Value of $1 for Different
Periods and Rates
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5-13
Present Value of a Lump Sum
You need $1000 in three years time. If you can earn 10 per
cent per annum, how much do you need to invest now?
Discount one year:
Discount two years:
Discount three years:
$1000 (1 + 0.10) –1
$909.09 (1 + 0.10) –1
$826.45 (1 + 0.10) –1
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= $909.09
= $826.45
= $751.32
5-14
Interpretation
•
In general, the present value of $1 received in t periods of
time, earning r per cent interest is:
PV  $1 1  r 
$1

t
1  r 
t
•
The expression (1 + r)–t is the present value interest factor
(PVIF). Refer to Table A.2.
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5-15
Example—Present Value of a Lump
Sum
Your rich grandmother promises to give you $10 000 in 10 years
time. If interest rates are 12 per cent per annum, how much is
that gift worth today?
PV  $10 000  1  0.12
 $10 000  0.3220
 $ 3220
10
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5-16
Present Values at Different Interest
Rates
Number of
periods
Present value of $100 at various interest rates
5%
10%
15%
20%
1
$95.24
$90.91
$86.96
$83.33
2
$90.70
$82.64
$75.61
$69.44
3
$86.38
$75.13
$65.75
$57.87
4
$82.27
$68.30
$57.18
$48.23
5
$78.35
$62.09
$49.72
$40.19
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5-17
Present Value of $1 for Different
Periods and Rates
Present
value
of $1 ($)
r = 0%
1.00
.90
.80
.70
r = 5%
.60
.50
r = 10%
.40
.30
r = 15%
.20
r = 20%
.10
1
2
3
4
5
6
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7
8
9
10
Time
(years)
5-18
Solving for the Discount Rate
•
You currently have $100 available for investment for a 21year period. At what interest rate must you invest this amount
in order for it to be worth $500 at maturity?
•
Given any three factors in the present value or future value
equation, the fourth factor can be solved.
r can be solved in one of three ways:
• Use a financial calculator
• Take the nth root of both sides of the equation
• Use the future value tables to find a corresponding value. In
this example, you need to find the r for which the FVIF after
21 years is 5 (500/100).
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5-19
The Rule of 72
•
The ‘Rule of 72’ is a handy rule of thumb that states:
If you earn r per cent per year, your money will double in
about 72/r per cent years.
•
For example, if you invest at 6 per cent, your money will
double in about 12 years.
•
This rule is only an approximate rule.
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5-20
Future Value of Multiple Cash Flows
•
You deposit $1000 now, $1500 in one year, $2000 in two
years and $2500 in three years in an account paying 10 per
cent interest per annum. How much do you have in the
account at the end of the third year?
•
You can solve by either:
–
compounding the accumulated balance forward one
year at a time
–
calculating the future value of each cash flow first and
then totalling them.
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5-21
Solutions
•
Solution 1
–
–
–
•
End of year 1:
End of year 2:
End of year 3:
($1000  1.10) + $1500 = $2600
($2600  1.10) + $2000 = $4860
($4860  1.10) + $2500 = $ 846
Solution 2
$1000  (1.10)3
$1500  (1.10)2
$2000  (1.10)1
$2500  1.00
Total
=
=
=
=
=
$1331
$1815
$2200
$2500
$7846
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5-22
Solutions
Future value calculated by compounding forward one period at a time
0
2
1
3
4
5
Time
(years)
$0
$
0
$0
0
$2200
2000
x 1.1
$2000
$4620
2000
x 1.1
$4200
$7282
2000
x 1.1
$6620
$10 210.20
2000
x 1.1
$9282
2000.00
x 1.1
$12 210.20
Future value calculated by compounding each cash flow separately
0
2
1
3
4
5
Time
(years)
$2000
$2000
$2000
$2000
x 1.1
$2000.0
2200.0
x 1.12
2420.0
x 1.13
2662.0
x 1.14
2928.2
Total future value
$12 210.20
Figures 5.6/5.7 — Calculation of FV for Multiple Cash Flow Stream
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5-23
Present Value of Multiple Cash Flows
•
You will deposit $1500 in one year’s time, $2000 in two years
time and $2500 in three years time in an account paying 10
per cent interest per annum. What is the present value of
these cash flows?
•
You can solve by either:
–
discounting back one year at a time
–
calculating the present value of each cash flow first and
then totalling them.
Copyright  2004 McGraw-Hill Australia Pty Ltd
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5-24
Solutions
•
Solution 1
–
End of year 2:
– End of year 1:
– Present value:
•
($2500  1.10–1) + $2000
($4273  1.10–1) + $1500
($5385  1.10–1)
=
=
=
$4273
$5385
$4895
Solution 2
$2500  (1.10) –3
$2000  (1.10) –2
$1500  (1.10) –1
Total
=
=
=
=
$1878
$1653
$1364
$4895
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5-25
Solutions
Present value
0
2
1
3
4
5
calculated by
discounting each
$1000
$1000
$1000
$1000
$1000
x 1/1.06
Time
cash flow separately
(years)
$ 943.40
x
1/1.062
890.00
x 1/1.063
839.62
x 1/1.064
792.09
x 1/1.065
747.26
$4212.37
Total present value
r = 6%
Present value
0
1
2
3
4
5
calculated by
discounting back one
$4212.37
$3465.11
$2673.01
$1833.40
$ 943.40
0.00
1000.00
1000.00
1000.00
1000.00
1000.00
$4212.37
$4465.11
$3673.01
$2833.40
$1943.40
$1000.00
$
0.00
Time
period at a time
(years)
Total present value = $4212.37
r = 6%
Figures 5.8/5.9 — Calculation of PV for Multiple Cash Flow Stream
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5-26
Annuities
• An ordinary annuity is a series of equal cash flows
that occur at the end of each period for some fixed
number of periods.
• Examples include consumer loans and home
mortgages.
• A perpetuity is an annuity in which the cash flows
continue forever.
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5-27
Present Value of an Annuity


1  1/1  r t 
PV  C  

r


C = equal cash flow
• The discounting term is called the present value
interest factor for annuities (PVIFA). Refer to Table
A.3.
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5-28
• Example 1
You will receive $500 at the end of each of the
next five years. The current interest rate is 9
per cent per annum. What is the present value
of this series of cash flows?


1  1/1.095 
PV  $500  

0.09


 $500 3.8897
 $1944.85
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5-29
• Example 2
You borrow $7500 to buy a car and agree to
repay the loan by way of equal monthly
repayments over five years. The current
interest rate is 12 per cent per annum,
compounded monthly. What is the amount of
each monthly repayment?


1  1/1.0160 
$7 500  C  

0.01


C  $7 500  39.1961
 $191.35
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5-30
Future Value of an Annuity


1  r   1
FV  C 
t
r
• The compounding term is called the future value
interest factor for annuities (FVIFA). Refer to Table
A.4.
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5-31
Example—Future Value of an
Annuity
What is the future value of $200 deposited at the
end of every year for 10 years if the interest rate is
6 per cent per annum?
1.0610  1


FV  $200  
0.06
 $200  13.181
 $2636.20
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5-32
Perpetuities
• The future value of a perpetuity cannot be
calculated as the cash flows are infinite.
• The present value of a perpetuity is calculated as
follows:
C
PV 
r
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5-33
Comparing Rates
• The nominal interest rate (NIR) is the interest rate
expressed in terms of the interest payment made
each period.
• The effective annual interest rate (EAR) is the
interest rate expressed as if it was compounded
once per year.
• When interest is compounded more frequently than
annually, the EAR will be greater than the NIR.
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5-34
Calculation of EAR
m
 NIR 
EAR  1 

1

m 

m = number of times the interest is compounded
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5-35
Comparing EARS
•
Consider the following interest rates quoted by three banks:
–
Bank A: 15%, compounded daily
–
Bank B:15.5%, compounded quarterly
–
Bank C:16%, compounded annually
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5-36
Comparing EARS
EARBank A
 0.15
 1 

365


365
 1  16.18%
4
 0.155
EARBank B  1 
 1  16.42%

4 

1
 0.16
EARBank C  1 
 1  16%

1 

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5-37
Comparing EARS
• Which is the best rate? For a saver, Bank B offers
the best (highest) interest rate. For a borrower,
Bank C offers the best (lowest) interest rate.
• The highest NIR is not necessarily the best.
• Compounding during the year can lead to a
significant difference between the NIR and the
EAR.
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5-38
Types of Loans
• A pure discount loan is a loan where the borrower
receives money today and repays a single lump
sum in the future.
• An interest-only loan requires the borrower to only
pay interest each period and to repay the entire
principal at some point in the future.
• An amortised loan requires the borrower to repay
parts of both the principal and interest over time.
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5-39
Amortisation of a Loan
Year
Beginning
Balance
Total
Payment
Interest
Paid
Principal
Paid
Ending
Balance
1
$5000.00
$1285.46
$450.00
$835.46
$4164.54
2
$4164.54
$1285.46
$374.81
$910.65
$3253.89
3
$3253.89
$1285.46
$292.85
$992.61
$2261.28
4
$2261.28
$1285.46
$203.52
$1081.94
$1179.33
5
$1179.33
$1285.46
$106.13
$1179.33
$0.00
$6427.30
$1427.30
$5000.00
Totals
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5-40