CAT / FIA FFM Syllabus E. Investment Decisions - Simple and compound interest - Notes ^{1 / 3}

Simple interest

Simple interest is calculated on the original principal only.

Illustration

You invest $100 for 3 years and you receive a simple interest rate of 10% a year on the $100. 

This would be $10 each year. 

Simply $100 x 10% = $10.

Compound interest

The important thing to remember is that you get interest on top of the previous interest. 

This is called compound interest.

Illustration

Suppose that a business has $100 to invest and wants to earn a return of 10%. 

What is the future value at the end of each year using compound interest?

Solution

Yr 1 - 100 x 1.10 = $110
Yr 2 - 110 x 1.10 = $121  or  100 x (1.10) ^ 2
Yr 3 - 121 x 1.10 = $133 or 100 x (1.10) ^ 3

This future value can be calculated as:

FV = PV (1+r) ^ n

Where  

FV is the future value of the investment with interest
PV is the initial or ‘present’ value of the investment
r is the compound annual rate of return or rate of interest  expressed as a proportion
n is the number of years

e.g. $100 x 1.1 ^3 = $133

Nominal interest rate = Real interest rate + Inflation

The nominal interest rate is given as a percentage. 

A compounding period is also given. In the above example, the 10% is the nominal rate and the compounding period is a year.

The compounding period is important when comparing two nominal interest rates, for example 10% compounded semi-annually is better than 10% compounded annually. 

In the exam, unless told otherwise, presume the compounding period is a year.

Effective annual rate of interest (annual percentage rate – APR)

The effective interest rate, on the other hand, can be compared with another effective rate as it takes into account the compounding period automatically, and expresses the percentage as an annual figure.

In fact, when interest is compounded annually the nominal interest rate equals the effective interest rate.

To convert a nominal interest rate to an effective interest rate, you apply the formula:
 
= (1 + i/m) ^ m – 1

Where ‘m’ is the number of compound periods
‘i’ is the interest rate

Illustration

What is the effective rate of return of a 15% p.a. monthly compounding investment?

Solution

Effective rate = (1 + (0.15/12)) ^ 12 - 1 = (1 + 0.0125) ^ 12 - 1 = 0.1608 = 16.08%

Illustration

What effective rate will a stated annual rate of 6% p.a. yield when compounded semi-annually?

Solution

Effective Rate = (1 + (0.06/2)) ^ 2 - 1 =  0.0609 = 6.09%