chapter 3 the time value of money: an introduction to financial mathematics

Copyright 2006 McGraw-Hill Australia Pty LtdPPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and PinderPrepared by Dr Buly Cardak

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Chapter 3

The Time Value of Money:

An Introduction to Financial Mathematics


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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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Learning Objectives (cont.)

• Apply knowledge of annuities to solve a range of problems, including problems involving principal-and-interest loan contracts.

• Distinguish between ordinary and general annuities and make basic calculations involving general annuities.


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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.

1 0

0

C Cr

C

1

0

where:

= cash inflow at time 1

= cash inflow at time 0

= rate of return per period

C

C

r


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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.


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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.


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Simple Interest

• Typically used when there is only a single time period.

• Interest is calculated on the original sum invested:

• Where S is the lump sum payable:

Interest Principal periods rateP t r

1S P Ptr P rt


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Simple Interest: Present Value

• Present cash equivalent of an amount to be paid or received at some future date, calculated using simple interest.

• Formula:

1

SP

rt

where:

present value

payment at future date

applicable interest rate

number of periods before payment

P

S

r

t


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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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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: 1

nS P i

where:

= rate per period

= number of periods

i

n


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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:

• It is important to understand that the PV and FV formulas are the inverse of each other — one is derived from the other.

1n

SP

i


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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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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:

1 1m

ji

m where:

nominal rate per period

number of compounding periods

which occur during a single nominal period

j

m


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Example: Effective Annual Interest RateExample 3.7:

Calculate the effective annual interest rates corresponding to 12% p.a., compounding:

(a) semi-annually.Solution: Using equation 3.6

2

2

1 1

0.121 1 1.06 1 0.1236 (12.36%)

2

mj

im


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Example: Effective Annual Interest Rate (cont.)Example 3.7 (cont.):


(b) quarterly.Solution: Using equation 3.6

4

4

1 1

0.121 1 1.03 1 0.125509 (12.5509%)

4

mj

im


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(c) monthly.Solution: Using equation 3.6

12

12

1 1

0.121 1 1.01 1 0.126825 (12.6825%)

12

mj

im


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(d) daily.Solution: Using equation 3.6

365

1 1

0.121 1 0.127475 (12.7475%)

365

mj

im


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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* 1

1

ii

p

where:

* real interest rate

nominal interest rate

expected inflation rate

i

i

p


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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:

jnS Pe

where:

future sum

principal

continuously compounding

interest rate per period

number of periods

2.718 281 828 46 (constant)

S

P

j

n

e


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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’.

0

where:

final value or price

initial value or price

number of periods

nP

P

n

1

0

1n

nPiP


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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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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:

• This formula takes a cash flow of $C and converts it into a future value.

*-* 1t t

t tV C i


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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:

0

10

1

Cr

Cn

tt

t


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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:

2

$ 1120 $25$1000 0

1 1r 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.

• The result is zero, confirming that the IRR = 14.19%

2

$1120 $25 $ 1000

1 0.1419 1 0.1419

$980.82 $19.17 $1000

0


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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:

• Using the present value of annuity tables, values of A(n,i ) for different values of n and i can be found.

1

1 ,1

n

CP C A n i

i i

where:

annuity cash flow

interest rate per compound period

number of annuity cash flows

C

i

n


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Example: Ordinary AnnuitiesExample 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.

63.56016$

08.1

5000$

08.1

5000$

08.1

5000$

08.1

5000$

1111

432

432

i

C

i

C

i

C

i

CP


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Example: Ordinary Annuities (cont.)Example 3.16 (cont.):• Find the present value of an ordinary annuity


• (b) Using equation 3.19.

63.56016$

31212684.35000$

08.1

11

08.0

5000$

1

11 4

nii

CP


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Example: Ordinary Annuities (cont.)

Example 3.16 (cont):• Find the present value of an ordinary annuity


• (c) Using Table 4, Appendix A and equation 3.20.

,

$5000 3.3121

$16560.50

P C A n i


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Annuity Due

• An annuity where the first cash flow is to occur immediately:

• An annuity due of n cash flows is simply an ordinary annuity of (n – 1) cash flows, plus an immediate cash flow of C.

0 1 2 3 4 5 6

$C $C $C $C $C $C $C


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Annuity Due (cont.)

• The present value of an annuity due:

1

11

1

1 1,

n

CP C

i i

C A n 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:

1

,

1k

C A n iP

i

where:

annuity cash flow



number of time periods until the first cash flow

C

i

n

k


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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:

• The present value of an ordinary perpetuity:

i

CP

0 1 2 3 4 5 6

$C $C $C $C $C

where:

cash flow per period

interest rate per period

C

i


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Valuing Ordinary Annuities

• Future value of an ordinary annuity:

• Using the future value of annuity tables, values of S(n,i) for different values of n and i can be found.

1 1 ,nC

S i C S n ii

where:

annuity cash flow



C

i

n


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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:

11 nii

CS


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Example: Ordinary Annuities (cont.)

• Substituting the values we have:

• Thus, at the end of 2 years, Harold will have saved $5206.99.

99.5206$

03492507.26200$

1007.1007.0

200$ 24

S


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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:

1

11

n

CP

i i


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Principal-and-Interest Loans (cont.)Example 3.22 (cont):• Substituting values:

• Thus, annual repayments on this loan are $27 398.18.

5

1$100,000 1

0.115 1.115

$100,000

3.64988

C

C


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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:

i

PiCCn

1log

log


3–47

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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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)

11 mSL ii


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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.

chapter 3 the time value of money: an introduction to financial mathematics

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