# Compound Interest

Compound interest is where the interest on a loan or investment is added to the principal balance such that the interest for future periods is calculated based on the amount including any previous interest. This results in an ever-increasing interest expense or income for each progressive period of the loan or investment term.

Let’s say you loan out $100,000 on 1 January 2017 paying interest at 6% compounded semi-annually (i.e. twice in one year). Your interest expense for the first six months is $3,000 (=$100,000 × 6% × 1/2). Since the interest is compounded, the loan balance for calculation of interest in the next six months (i.e. second half of the year) equals $103,000 (initial principal of $100,000 plus interest in the previous period of $3,000). Hence, interest expense for the next six months i.e. from 1 July 2017 to 31 December 2017 shall be $3,090 (=$103,000 × 6% × 1/2). Interest expense in the six months from 1 January 2018 to 30 June 2017 shall be $3,183 and so on.

## Formula

Interest for a Period (compound interest) = P × Interest Rate × Time Period

Where P is the opening loan/investment balance including all accrued interest

$$ Future\ Value\ of\ a\ Loan/Investment\ (F) \\ = Initial\ Balance\ (P) × (1\ +\ Interest\ Rate)^{number\ of\ periods} $$

Where there are more than one compounding periods in a year, the formula can be modified as follows:

Future value of loan/investment (F) = initial balance (P) × (1 + interest rate/m)^{(m×n)}

Where n is the time period in years and m is the compounding periods per year.

The principle to remember is that the time period must be expressed in the same units for which the interest rate is included in the formula.

$$ Present\ Value\ of\ a\ Loan/investment \\ = \frac{Future\ Value}{\left(1+ \frac{i}{m}\right)^{m×n}} $$

While a loan or investment under simple interest grows linearly, they grow exponentially under compound interest method.

## Example

Tom has a friend, Jerry, who is allergic to banks but nevertheless likes the idea of earning a fixed guaranteed amount each period on his savings. He gave Tom $50,000 on 1 January 2011 for 5 years and Tom agreed to pay 3% per annum without any compounding. Tom put the money in a bank paying 5% per annum compounded quarterly. At the end of 5th year, Tom paid Jerry $57,500 worked out as follows:

Payable to Jerry after 5 years (simple interest) = $50,000 × (1 + 3% × 5) = $57,500

He made $6,601.86 because the value of his investment in the bank is $64,101.86 which is $6,601.86 higher than the amount payable to Jerry i.e. $57,500.

Value of Tom’s investment after 5 years = $50,000 × (1 + 5%/4) ^ (4 × 5) = 64,101.86

The first equation is for the future value under simple interest method and the second one is for future value under compound interest method. You can see that since there are four compounding periods in one year, the interest rate used in the equation for investment future value is (5%/4).

After a week Tom receives a letter from Jerry. He is angry at Tom for duping him. He tells Tom to pay him interest under the compound interest method else he will sue him. Tom must pay Jerry $464 more because future value of $50,000 Jerry gave to Tom under 3% compound interest for 5 years exceeds the amount under 3% paid under simple interest for 5 years by $464.

Payable to Jerry (compound interest) = $50,000 × (1 + 3%)^5 = $57,964

Additional amount to be paid to Jerry = $57,964 - $57,500 = $464.

Written by Obaidullah Jan