# Simple Interest

Simple interest is where interest is charged only on the principal balance of a loan or an investment excluding any interest previously accrued resulting in a constant interest expense or income for the whole duration of the loan or investment if there is no change in the rate of interest. Under the simple interest method, interest is charged/earned only on the principal amount and there is no interest on interest.

Let’s say you loan $10,000 from a very generous friend for 2 years. You will pay interest at 6% per annum payable at the end of loan term i.e. 2 years. Interest expense calculation for the first year is easy, just apply the rate (6%) to the principal balance of $10,000 to get $600. This $600 is the accrued interest which you will pay at the end of the second year. Before calculating interest expense for the second year, we have an important question to answer: i.e. whether the rate (6%) should be applied to the principal balance ($10,000) or the new loan outstanding balance ($10,000 plus $600). Under the simple interest method, the interest rate is applied to the principal balance only (excluding any interest expense). Your interest expense obligation for the second year under simple interest will be $600 too. At the end of the second year, you will owe $11,200 (principal balance of $10,000 plus first year interest of $600 plus second year interest of $600). Had you applied the interest rate to the total outstanding loan balance (inclusive of first year interest) while calculating interest expense for second year, your outstanding obligation related to the loan would have been $11,236 (principal of $10,000 plus first year interest of $600 and second year interest of $636 (=6%×$10,600)). This second method is called compound interest.

## Formula

If there is no change in the principal balance of the loan and interest rate, the interest expense for all equivalent periods is the same under simple interest.

Interest (simple interest) = Principal × interest rate × number of periods

Where i is the interest rate for the period m and t is the time period in units of m

Cumulative value i.e. future value of a loan or investment under simple interest is calculated by adding the total interest to the principal balance

Future value (simple interest) = P × (1 + interest rate × number of periods)

Simple algebraic manipulation of the above equation gives us a formula to calculate the money needed today to get a specific future value after a specific period of time:

$$ Present\ Value\ (Simple\ Interest) \\ = \frac{Future\ Value}{1 + Interest\ Rate × Number\ of\ Periods} $$

Value of a loan or an investment under simple interest grows linearly.

## Example

Promissory notes and Canadian Treasury Bills are a couple of financial instruments that are based on simple interest. On 1 January 2017 you enter into two transactions: (a) write a 2-year promissory note for $10,000 paying interest at 5% per annum and (b) purchase Canadian Treasury Bills that will pay $100,000 in 105 days yielding 2.40%. We will attempt to calculate the interest expense on the promissory note for both the year, the value of the promissory note at its maturity and the amount at which you can buy the Canadian Treasury Bills today.

Interest expense on the promissory note – 2017 = $10,000 × 5% × 1 = $500

Interest expense on the promissory note – 2018 = $10,000 × 5% × 1 = $500

Please note that since there no change in the interest rate, the interest expense is the same in both years and there is no interest on interest under the simple interest method.

Future value i.e. maturity balance of the promissory note = $10,000 × (1 + 5% × 2) = $11,000

The interest rate and the time unit used must be consistent. If the interest rate is for the full year, the number of periods must be in years too.

In case of the Canadian Treasury Bills, we are effectively working out the present value using the accumulation function of simple interest. We know that the future value that the bills pay, i.e. $100,000 after 105 days must equal the money we invest in the bills at time 0 i.e. 1 January 2017 multiplied by one plus product of interest rate and number of periods. Mathematically, this would be written as:

$100,000 = initial principal × (1 + 2.40% × 105/365)

Initial principal as at 1 January 2017 = $100,000/(1 + 2.40% × 105/365)

Amount of money to be invested in Treasury Bills = $99,314.

Please note that since the yield i.e. interest rate on the Treasury Bills is the annual rate the time component must be expressed in years too, hence the 105/365 fraction.