Chapter 3
The Time Value of
Money:
An Introduction to
Financial Mathematics
Copyright  2006 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and Pinder
Prepared by Dr Buly Cardak
3–1
Learning Objectives
•
Understand and solve problems involving simple
interest and compound interest, including
accumulating, discounting and making
comparisons using the effective interest rate.
•
Value, as at any date, contracts involving multiple
cash flows.
•
Distinguish between different types of annuity
and calculate their present and future values.
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PPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and Pinder
Prepared by Dr Buly Cardak
3–2
Learning Objectives (cont.)
•
Apply knowledge of annuities to solve a range of
problems, including problems involving principaland-interest loan contracts.
•
Distinguish between ordinary and general
annuities and make basic calculations involving
general annuities.
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PPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and Pinder
Prepared by Dr Buly Cardak
3–3
Fundamental Concepts
•
Cash flows — fundamental to finance, the
funds that flow between parties either now
or in the future as a consequence of a
financial contract.
•
Rate of return — relates cash inflows to cash
outflows.
C1  C0
r
C0
where:
C1 = cash inflow at time 1
C0 = cash inflow at time 0
r = rate of return per period
Copyright  2006 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and Pinder
Prepared by Dr Buly Cardak
3–4
Fundamental Concepts (cont.)
•
Interest rate — special case of rate of return
(used when the financial agreement is in the
form of debt).
•
Time value of money
–
Money received now can be invested to earn additional
cash (interest).
–
Relates to opportunity cost of giving up money or
resources for a period of time — either forgone
investments or consumption, whatever the next best
alternative is.
Copyright  2006 McGraw-Hill Australia Pty Ltd
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Prepared by Dr Buly Cardak
3–5
Time Value of Money
•
An investment decision will involve an outlay of cash
made in one period with the expectation of cash inflows
in future periods.
•
As a significant amount of time may elapse between the
outflow of cash and the subsequent inflows, the significance
of this time should be considered.
•
To ignore differences in the timing of cash flows is to ignore
the importance of the time value of money.
•
Cash flows that occur at different points in time cannot simply
be added together or subtracted — this is one of the critical
issues conveyed in this chapter.
Copyright  2006 McGraw-Hill Australia Pty Ltd
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Prepared by Dr Buly Cardak
3–6
Simple Interest
•
Typically used when there is only a single
time period.
•
Interest is calculated on the original sum
invested:
Interest  Principal  P   periods  t   rate  r 
•
Where S is the lump sum payable:
S  P  Ptr  P 1  rt 
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3–7
Simple Interest: Present Value
•
Present cash equivalent of an amount to be paid
or received at some future date, calculated using
simple interest.
•
Formula:
where:
S
P
1  rt 
P  present value
S  payment at future date
r  applicable interest rate
t  number of periods before payment
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3–8
Compound Interest
•
Compounding involves accumulating interest
on previous interest payments.
•
This means that, unlike the case of simple
interest, previous interest payments will
generate further interest.
•
This earning of interest on interest is one of
the key differences between simple interest
and compound interest.
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3–9
Compound Interest (cont.)
•
The backbone of many time-value calculations are the
present value (PV) and future value (FV) based on
compound interest.
•
The sum or future value (S ) accumulated after
n periods is:
n
S  P 1  i 
where:
i = rate per period
n = number of periods
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3–10
Compound Interest (cont.)
• The future value formula can be manipulated to
provide a formula to determine the present value.
• The present value of a future sum is:
P
S
1  i 
n
• It is important to understand that the PV and FV
formulas are the inverse of each other — one is
derived from the other.
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3–11
Nominal and Effective Interest Rates
•
Nominal rate
–
•
Quoted interest rate where interest is charged or
calculated more frequently than the time period
specified in the interest rate.
Effective rate
–
Interest rate where interest is charged at the same
frequency as the interest rate quoted.
–
Used to convert different nominal rates so that they
are comparable.
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3–12
Nominal and Effective Interest
Rates (cont.)
•
The distinction is important when interest is
compounded over a period different from that
expressed by the interest rate, e.g. more than
once a year.
•
The effective interest rate can be calculated as:
m
j

i  1    1
 m
where:
j  nominal rate per period
m  number of compounding periods
which occur during a single nominal period
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3–13
Example: Effective Annual Interest
Rate
Example 3.7:
Calculate the effective annual interest rates
corresponding to 12% p.a., compounding:
(a) semi-annually.
Solution: Using equation 3.6
m
j

i  1    1
 m
2
2
 0.12 
 1 
  1  1.06   1  0.1236 (12.36%)
2 

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3–14
Example: Effective Annual Interest
Rate (cont.)
Example 3.7 (cont.):
Calculate the effective annual interest rates
corresponding to 12% p.a., compounding:
(b) quarterly.
Solution: Using equation 3.6
m
j

i  1    1
 m
4
4
 0.12 
 1 

1

1.03
   1  0.125509 (12.5509%)

4 

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3–15
Example: Effective Annual Interest
Rate (cont.)
Example 3.7 (cont.):
Calculate the effective annual interest rates
corresponding to 12% p.a., compounding:
(c) monthly.
Solution: Using equation 3.6
m
j

i  1    1
 m
12
12
 0.12 
 1 

1

1.01
   1  0.126825 (12.6825%)

12 

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3–16
Example: Effective Annual Interest
Rate (cont.)
Example 3.7 (cont.):
Calculate the effective annual interest rates
corresponding to 12% p.a., compounding:
(d) daily.
Solution: Using equation 3.6
m
j

i  1    1
 m
 0.12 
 1 

365 

365
 1  0.127475 (12.7475%)
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3–17
Real Interest Rates
•
The ‘real interest rate’ is the interest rate after taking
out the effects of inflation.
•
The ‘nominal interest rate’ is the interest rate before
taking out the effects of inflation.
•
The real interest rate (i*) can be found as follows:
 1 i 
i*  
 1
 1 p 
where:
i*  real interest rate
i  nominal interest rate
p  expected inflation rate
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3–18
Continuous Interest Rates
•
‘Continuous interest’ is a method of calculating interest in
which it is charged so frequently that the time period
between each charge approaches zero.
•
Continuous interest is an example of exponential growth:
where:
S  Pe
jn
S  future sum
P  principal
j  continuously compounding
interest rate per period
n  number of periods
e  2.718 281 828 46 (constant)
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A Generalisation: Geometric Rates
of Return
•
The rate of return between two dates, measured
by the change in value divided by the earlier value.
•
The average of a sequence of geometric rates
of return is found by a process that resembles
compounding.
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A Generalisation: Geometric Rates
of Return (cont.)
•
‘Average geometric rate of return’ is also referred
to as the ‘average compound rate of return’.
1
n
 Pn 
i    1
 P0 
where:
Pn  final value or price
P0  initial value or price
n  number of periods
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3–21
Valuation of Contracts with
Multiple Cash Flows
•
Value additivity
–
Cash flows occurring at different times cannot be
validly added without accounting for timing.
–
Only cash flows occurring at the same time can
be added.
–
Therefore, it is necessary to convert multiple cash
flows into a single equivalent cash flow.
–
Cash flows can be carried either forward in time
(accumulated) or back in time (discounted).
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3–22
Valuation of Contracts with
Multiple Cash Flows (cont.)
•
Where a cash flow of C dollars occurs on a date t,
the value of that cash flow at a future valuation
date t* is given by:
Vt  Ct 1  i 
*
•
t *-t
This formula takes a cash flow of $C and converts
it into a future value.
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3–23
Valuation of Contracts with
Multiple Cash Flows (cont.)
•
Measuring the rate of return
–
Where there are n cash inflows Ct (t = 1, ..., n), following
an initial outflow of C0 , the internal rate of return is that
value of r that solves the equation:
n
 1  r 
t 1
Ct
t
 C0  0
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Example: Internal Rate of Return
•
Consider three cash flows:
–$1000 today, +$1120 in 1 year, +$25 in 2 years
•
What is the average rate of return on the initial
investment of $1000, taking into account
compounding, that is, the IRR?
•
The IRR is the r that satisfies the following equation:
$ 1120
$25

 $1000  0
2
1  r  1  r 
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Example: Internal Rate of Return
(cont.)
•
The answer can be solved for precisely, as the
equation is a quadratic equation.
•
Alternatively, and more generally, trial and error
can be used, substituting different values for r.
•
In practice, this would be done with a computer,
using a program such as Excel or Lotus.
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Example: Internal Rate of Return
(cont.)
•
The solution is, IRR = 14.19%.
•
This can be confirmed by substituting r = 0.1419.
$1120

1  0.1419 
$25
1  0.1419 
2
 $ 1000
 $980.82  $19.17  $1000
 0
•
The result is zero, confirming that the IRR = 14.19%
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3–27
Annuities
•
An annuity is a stream of equal cash flows,
equally spaced in time.
•
We consider four types of annuities:
–
Ordinary annuity
–
Annuity due
–
Deferred annuity
–
Ordinary perpetuity
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Ordinary Annuities
•
Annuities in which the time period from the date of
valuation to the date of the first cash flow is equal to
the time period between each subsequent cash flow.
•
Assume that the first cash flow occurs at the end of
the first time period:
0
1
2
3
4
5
6
$C
$C
$C
$C
$C
$C
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Valuing Ordinary Annuities
•
Present value of an ordinary annuity:
C
1 
P  1 
  C  A  n, i 
n
i 
1  i  

where:
C  annuity cash flow
i  interest rate per compound period
n  number of annuity cash flows
•
Using the present value of annuity tables, values of
A(n,i ) for different values of n and i can be found.
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Example: Ordinary Annuities
Example 3.16:
• Find the present value of an ordinary annuity
of $5000 p.a. for 4 years if the interest rate
is 8% p.a. by:
• (a) Discounting each individual cash flow.
C
C
C
C
P



2
3
4
1  i 1  i  1  i  1  i 
$5000 $5000 $5000 $5000




2
3
1.08 1.08 1.08 1.084
 $16 560.63
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Example: Ordinary Annuities (cont.)
Example 3.16 (cont.):
• Find the present value of an ordinary annuity
of $5000 p.a. for 4 years if the interest rate
is 8% p.a. by:
• (b) Using equation 3.19.
C
1  $5000 
1 
P  1 

1 
n
4
i  1  i   0.08  1.08 
 $5000  3.31212684
 $16 560.63
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Example: Ordinary Annuities (cont.)
Example 3.16 (cont):
• Find the present value of an ordinary annuity
of $5000 p.a. for 4 years if the interest rate
is 8% p.a. by:
• (c) Using Table 4, Appendix A and equation 3.20.
P  C  A  n, i 
 $5000  3.3121
 $16560.50
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Annuity Due
•
An annuity where the first cash flow is to
occur immediately:
0
1
$C $C
•
2
$C
3
$C
4
$C
5
$C
6
$C
An annuity due of n cash flows is simply an
ordinary annuity of (n – 1) cash flows, plus an
immediate cash flow of C.
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Annuity Due (cont.)
•
The present value of an annuity due:

C
1
P  C  1 

n 1
i  1  i  


 C 1  A  n  1, i  
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Deferred Annuity
•
Annuity in which the first cash flow is to occur
after a time period that exceeds the time period
between each subsequent cash flow:
0
1
2
3
4
5
6
7
8
$C $C $C $C $C $C
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Deferred Annuity (cont.)
•
Present value of a deferred annuity:
P
C  A  n, i 
1  i 
k 1
where:
C  annuity cash flow
i  interest rate per compound period
n  number of annuity cash flows
k  number of time periods until the first cash flow
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Deferred Annuity (cont.)
•
The present value of a deferred annuity involves
taking the present value of an ordinary annuity.
•
This figure is a present value but, as the annuity
is deferred, we need to discount the PV further.
•
If the first cash flow is k periods into the future,
we discount the PV by (k – 1) periods.
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Ordinary Perpetuity
•
An ordinary annuity where the cash flows are to
continue forever:
0
•
1
$C
2
$C
3
$C
4
$C
5
$C
6

The present value of an ordinary perpetuity:
P
C
i
where:
C  cash flow per period
i  interest rate per period
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Valuing Ordinary Annuities
•
Future value of an ordinary annuity:
C
n
S
1  i   1  C  S  n, i 


i 
where:
C  annuity cash flow
i  interest rate per compound period
n  number of annuity cash flows
•
Using the future value of annuity tables, values of
S(n,i) for different values of n and i can be found.
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Example: Ordinary Annuities
Example 3.20:
• Starting with his next monthly salary, Harold intends
to save $200 each month.
• If the interest rate is 8.4% p.a., payable monthly,
how much will Harold have saved after 2 years?
• Solution: Monthly interest rate is 0.4/12 = 0.7%.
Using equation 3.28, Harold’s savings will
amount to:


C
n
S  1  i   1
i
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Example: Ordinary Annuities (cont.)
•
Substituting the values we have:


$200
24
1.007   1
S
0.007
 $200  26 .03492507
 $5206 .99
•
Thus, at the end of 2 years, Harold will have
saved $5206.99.
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Principal-and-Interest Loans
•
An important application of annuities is to loans
involving a sequence of equal cash flows, each
of which is sufficient to cover the interest accrued
since the previous payment and to reduce the
current balance owing.
•
Such loans can be referred to as:
–
Principal-and-interest loans
–
Credit foncier loans
–
Amortised loans
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Principal-and-Interest Loans (cont.)
Example 3.22:
• Borrow $100 000.
• Make 5 years of annual repayments at a
fixed interest rate of 11.5% p.a.
• What is the annual repayment?
•
Use the PV of annuity formula:
C
1 
P  1 

n
i  1  i  


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Principal-and-Interest Loans (cont.)
Example 3.22 (cont):
• Substituting values:

C 
1
$100, 000 
1 

5
0.115  1.115  


$100, 000
C 
3.64988
•
Thus, annual repayments on this loan
are $27 398.18.
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Principal-and-Interest Loans (cont.)
•
Balance owing at a given date
–
•
Equals the present value of the then-remaining
repayments
Loan term required
–
Solving for the required loan term n:
log C C  Pi 
n
log 1  i 
Copyright  2006 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and Pinder
Prepared by Dr Buly Cardak
3–46
Principal-and-Interest Loans (cont.)
•
Changing the interest rate:
–
•
In some loans (usually called variable interest rate loans),
the interest rate can be changed at any time by the lender.
Two alternative adjustments can be made:
–
The lender may set a new required payment which will
be calculated as if the new interest rate is fixed for the
remaining loan term.
–
The lender may allow the borrower to continue making
the same repayment and, instead, alter the loan term
to reflect the new interest rate.
Copyright  2006 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and Pinder
Prepared by Dr Buly Cardak
3–47
General Annuities
•
Annuity in which the frequency of charging interest
does not match the frequency of payment; thus,
repayments may be made either more frequently
or less frequently than interest is charged.
•
Link between short period interest rate (iS) and
long period interest rate (iL)
iL  1  iS   1
m
Copyright  2006 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and Pinder
Prepared by Dr Buly Cardak
3–48
Summary
•
Fundamental concepts in financial mathematics
include rates of return, simple and compound
interest.
•
Valuation of cash flows:
•
–
Present value of a future cash flow
–
Future value of a current payment/deposit.
Annuities are a special class of regularly spaced
fixed cash flows.
Copyright  2006 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and Pinder
Prepared by Dr Buly Cardak
3–49