LECTURE 9

Financial Mathematics: Introduction

1.) Revision: Indices

When aⁿ = b; then nloga = log b and n = logb/loga

Example 1:

Find n, given 1.02ⁿ = 1000

Exercise 1

a.) 1.12ⁿ = 2000

b.) 1.085ⁿ = 5000

2.) Simple interest:

FORMULA 1: I = Pin

I = Simple interest

P = principal

i = interest rate

n = number of periods

FORMULA 2: A = P + Pin = P(1 + in)

A = amount due

Note: Amount due = Principal + Interest = P + Pin

Example 2

a.) A firm borrows RM1000 at a simple interest rate of 8% for 2 years

(i) Find the simple interest after 2 years

(ii) Find the full amount to be repaid after 2 years

Exercise 2

a.) A student borrows $500 at a simple interest rate of 6%

for three years

i) Find the simple interest after 3 years

ii) Find the amount due after 3 years

b.) A firm pays RM175 simple interest on a loan of RM5000

after six months. Find the interest rate.

c.) A company gives a loan of RM20,000 at a simple interest

rate of 12%. After how many months can it collect

interest amounting to RM3600?

d.) A firm collects simple interest of RM450, paid at an

interest rate of 9%, after 10 months. Find the principal

e.) A businessman pays RM16,875 for a loan plus

simple interest borrowed for 15 months at 10%

How much did he borrow initially?

3.) Compound interest:

If you were to deposit an amount P at an interest rate i for 1

year, the interest plus principal at the end of the year is:

A₁ = P (1 + in) = P (1 + i (1)) = P (1 + i)

If you were to let this amount remain for another year, then A

becomes the principal for the following year and the interest plus

principal at the end of the 2^nd year is:

A₂ = A₁ (1+i) = P (1+i)(1+i) = P(1+i)²

If you were to allow A₂ to continue to earn interest i for another

year, the amount at the end of the third year, A₃ is:

A₃ = A₂ (1+i) = P(1+i)²(1+i) = P(1+i)³

Thus, after n years, A_n = P(1+i) ⁿ

This concept, where interest earned in a certain period acts as

part of the principal for the subsequent period, is called

compounding.

Compound interest rate is given by:

S = P(1+i) ⁿ

where,

S = compound amount

P = principal

i = interest rate per compounding period

n = number of compounding periods

Example 3:

You invest RM2000 in a savings account which earns interest

at a rate of 4% per year compounded annually.

Find the compound amount after 8 years.

Answer:

S = P(1+i) ⁿ

where S = ?, P = 2000, i = 4%, n = 8

S = 2,000 (1+0.04) ⁸ = $2737.14

Financial calculator solution:

MODE: FIN

2ndF CA (to clear memory)

2ndF Tab 2 (To set decimal to 2 places)

8 = n

2,000 +/-= PV (note: negative – money going out)

4 = i (note: enter direct %, do not change to decimal)

COMP FV

In the following exercises, pay close attention to how the

values of i and n are adapted.

Exercise 3a:

Tim invests RM5000 in a savings account which earns

interest at a rate of 6% per year compounded annually.