# Compound Interest

Compound interest is a method in which interest is calculated based on principal plus any interest already accrued. This results in an ever-increasing interest expense/income.

Let us say you loan out $100,000 on 1 January 20X7 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

If P is the value of a loan at time 0 and r is the periodic interest rate, the interest expense for the first year is as follows:

Interest Expense (First Period)

= P × r

The carrying value of loan for the next period is the sum of initial principal P and interest for first period:

Loan Balance (1st Period)

= P + P × r

= P × (1 + r)

Interest expense for the second period is calculated by applying the interest rate to the opening loan balance (inclusive of first period interest):

Interest Expense (Second Period)

= P × (1 + r) × r

The loan balance at end of the second period is hence:

Loan Balance (2nd Period)

= P × (1 + r) + P × (1 + r) × r

= P × (1 + r) × 1 + P × (1 + r) × r

= P × (1 + r) × (1 + r)

= P × (1 + r)^{2}

This can be generalized as follows:

Future Value (Compound Interest) = P × (1 + r)^{n}

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

Future Value (Compound Interest)

= P × (1 + i/m)^{(m×n)}

Where future value is the value of loan/investment including all compounded interest, **i** is the annual percentage rate, **m** is the compounding periods per year and **n** is the number of years.

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.

We can rearrange the above equation to obtain a formula for present value.

Present Value (Compound Interest) = | Future Value |

(1 + r/m)^{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, ACA, CFA and last modified on