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Lecture 3:

Topic: Mathematics of Finance

Reading

H&P, Ch 5.

Reading for this lecture

H&P, Ch 5, Sect 5.1 - 5.2;

(also refer back to Ch 4, pp. 167-168)

Homework for Tutorial in Week 2

Ex 5.1, pp.200-201 , problems 1, 3, 9, 11.

Ex 5.2, pp. 204-205, problems 3, 11, 15, 19.2

Compound Interest

Suppose $100 is invested at 8 percent perannum. Interest is earned and reinvestedat the end of every quarter.

What is the compound amount after 1 year?

Principal =Nominal interest rate =

Quarterly interest rate =

Number of quarters =

3

Compound Interest

End of Qtr 1: S = 100(1 + 0.02) = 102

End of Qtr 2: S = 102(1 + 0.02) = 104.04

=

End of Qtr 3: S = 104.04(1 + 0.02) = 106.12

=

End of Qtr 4: S = 106.12(1 + 0.02) = 108.24

=4

Compound Interest Formula

S = P(1 + r)n

where

P is the original principal

n is the number of conversion periods

r is the rate of interest per conversion period

S is the compound amount after n periods

S P is amount the compound interest earned

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Find S, given P, r and nSuppose $1,000 is invested at 5%. What is thecompound amount after 4 years if interest isaccumulated (a) annually, (b) semi-annually?

S = P(1 + r)n

P =

(a)

(b)

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Find r, given S, P and nOver a three-year period, an original principal of$5,000 accumulated to $7,000 in an account inwhich interest was compounded monthly.Determine the nominal rate of interest.

S = P(1 + r)n

The periodic rate of interest isThe nominal rate of interest is

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Find n, given S, P and rHow long will it take to double a principal at a

rate of 10%?S = P(1 + r)n

It will take years to double a principalinvested at 10% per annum.

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The Effective Rate of Interest

What annual interest rate, re, would give the

same compounded amount in 4 years as 5%compounded 2 times per year?

Solve P(1 + re)4 = P(1 + 0.025)(4)(2)

The effective rate of interest is

An interest rate of compounded annuallywould give the same amount in 4 years as a5% interest rate compounded semi-annually.

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Definition: The Effective Rate of Interest, re ,is the per annum equivalent of a periodic interestrate (per annum rate/no. of conversion periods).

P(1 + re)n = P(1 + r/m)nm

(1 + re)n = (1 + r/m)nm

(1 + re) = (1 + r/m)m

re = (1 + r/m)m 1

wherer is the nominal rate of interest,n is the number of years,m is the number of conversion periods per year

(times per year that interest is compounded).10

Discounting

Find P, given S, n and r

Discounting is the method of finding the

current value of a sum of money that will be

available at a future date.

e.g. How much is $1,000 payable in a years

time worth today? In other words, what sum,

invested at the current rate of interest, would

compound to $1,000 after one year?

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The Present Value

S = P(1 + r)n

Find P, given S, r and n

P = S(1 + r)-n

where

P is the present value of the future sum, S

n is the number of conversion periods

r is the rate of interest per conversion period

S is the future sum after n periods12

Trust Fund

Ex 5.2, p. 204, problem 13.

A trust fund for a 10-year-old child is being

set up by a single payment so that by age 21

the child will receive $27,000. Find how much

the payment is if an interest rate of 6%

compounded semiannually is assumed.

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Trust Fund

Ex 5.2, p. 204, problem 13.

S =

r =

n =

P = S(1 + r)-n

=

=

=

invested now at 6% compounded semi-

annually will be worth $27,000 in 11 years time.14

Equations of Value

Ex 5.2, p. 205, problem 17.

A debt of $5,000 due five years from now and

$5,000 due ten years from now is to be

repaid by a payment of $2,000 in two years,

a payment of $4,000 in four years, and a final

payment at the end of six years. If the interest

rate is 2.5% compounded annually, how

much is the final payment?

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Ex 5.2, p. 205, problem 17.

year debt repayment12 200034 40005 50006 x

789

10 5000

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Ex 5.2, p. 205, problem 17.

1 + r =

Let x be the final payment at the end of 6 years.

Equate the PV of the payments with the PV of

the debt.

Isolate x by multiplying by

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Ex 5.2, p. 205, problem 17.

Solve for x

x =

= 5,125 + 4,529.75 2,207.63 4,202.5

=

The final payment at the end of six years must be

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Net Present Value

Example

An initial investment of $10,000 in a business

guarantees the following cash flows:

Year 2 3 4

Cash Flow 2,500 5,000 7,500

Find the NPV at an interest rate of 10%

Is the investment profitable?

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Net Present Value

1 + r =NPV =

=

=

The net present value of cash flows is

The investment is profitable because the NPV

of cash flows exceeds zero.