Chapter 3
The Time Value of
Money:
An Introduction to
Financial Mathematics
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3-1
Learning Objectives
• Understand 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.
• Calculate and distinguish present and future
values of different annuities. Apply knowledge
of annuities to solve a range of problems,
including problems involving principal-andinterest loan contracts.
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PPTs t/a Business Finance 10e by Peirson
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3-2
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. (Equation 3.1)
• Interest rate — special case of rate of return
(used when the financial agreement is in the
form of debt).
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3-3
Time Value of Money
• Money received now can be invested to
earn additional cash in the future.
• Relates to opportunity cost of giving up
money or resources for a period of time —
either forgone investments or consumption.
• Consider time significance — significant
amount of time may elapse between cash
outflows and inflows.
• 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.
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3-4
Simple Interest
• Typically used only for a single time period. Interest
is calculated on the original sum invested:
– Where S is the lump sum payable
Interest  Principal  P   periods t   rate  r 
S  P  Ptr  P 1  rt 
Present Value:
• Typically present cash equivalent of an amount to
be paid or received at some future date, calculated
using simple interest.
S
P
1  rt 
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3-5
Compound Interest
• Compounding involves accumulating interest on previous
interest payments, which will generate further interest.
• This earning of interest on interest is one of the key differences
between simple interest and compound interest.
• 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 accumulated after n periods is:
S  P 1  i 
n
• The present value of a future sum is:
P
S
1  i 
n
• Note: The PV and FV formulas are the inverse of each other!
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3-6
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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3-7
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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3-8
Example: Effective Annual
Interest Rate
• Example 3.7:
– Calculate the effective annual interest rates corresponding
to 12% p.a., compounding:
(a) Semi-annually =m
 j
i  1    1
 m
2
2
 0.12 
 1 
  1  1.06   1  0.1236 (12.36%)
2 

(b) Quarterly = 0.125509 (12.55%)
(c) Monthly = 0.126825 (12.68%)
(d) Daily = 0.127475 (12.7475%)
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3-9
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:
where:
 1 i 
i*  real interest rate
i*  
 1
i  nominal interest rate
 1 p 
p  expected inflation rate
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3-10
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  future sum
jn
P  principal
j  continuously compounding
interest rate per period
n  number of periods
e  2.718 281 828 46 (constant)
S  Pe
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3-11
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.
• 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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3-12
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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3-13
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
• 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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3-14
Example: Internal Rate of Return (IRR)
• 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, i.e. 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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3-15
Example: Internal Rate of Return
(cont.)
• The answer cannot 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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3-16
Example: Internal Rate of Return
(cont.)
• The solution is: IRR = 14.19%, substitute back into
the equation to confirm:
$1120

1  0.1419 
$25
1  0.1419 
2
 $ 1000
 $980.82  $19.17  $1000
 0
• The result is zero, confirming that the IRR is 14.19%.
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3-17
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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3-18
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
$C
2
$C
3
$C
4
$C
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5
$C
6
$C
3-19
Valuing Ordinary Annuities
• Present value (PV) of an ordinary annuity:
C
1
P
1 
n
i 
1  i 


  C  A  n, 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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3-20
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:
(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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3-21
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.
(b) Using Equation 3.19.

1  $5000 
1 

1 
1 
n 
4
0.08  1.08 
 1  i  
 $5000  3.31212684  $16 560.63
C
P
i
(c) Using Table 4, Appendix A and Equation 3.20.
P = C x A(n,i) = $16 560.50
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3-22
Annuity Due
• 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. 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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3-23
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
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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3-24
Deferred Annuity (cont.)
• The present value (PV) 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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3-25
Ordinary Perpetuity
• An ordinary annuity where the cash flows are to
continue forever:
0
1
2
3
4
5
$C
$C
$C
$C
$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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3-26
Example: Ordinary Annuities
• Future value of an ordinary annuity:
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
1  i   1
S
i
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3-27
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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3-28
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 loans.
– Amortised loans.
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3-29
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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3-30
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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3-31
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 
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3-32
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.
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3-33
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
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3-34
Summary
• Fundamental concepts in financial mathematics
include rates of return, simple interest 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.
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3-35