# Interest vs Discount

The difference between the value of a loan or investment today and its value at some future date can be referred to as both interest or discount depending on the point of reference. If we start with the value today and find its value at some future date, the difference is termed as interest. Alternatively, if we start with the value at some future date and arrive at a value today, the difference is called discount.

Let’s say you obtain a loan of $50,000 and you must pay $60,000 after 1 years. One way to calculate the rate is to divide the difference between two values and divide it by the initial value i.e. ($60,000 - $50,000)/$50,000. It equals a rate of 20%. This is the interest rate. However, another guy may calculate the return with reference to the future value—his calculation would be ($60,000 - $50,000)/$60,000. As per his calculation, the rate is 16.67%. This is the discount rate.

## Interest Rate

Interest rate is a rate applied to a value today (i.e. at time 0) to get a future value.

Given a present value PV and a future value FV, an interest rate is expressed by the following equation:

$$ Future\ Value=Present\ Value\ \times(1+Interest\ Rate\times Time) $$

or

$$ Interest\ Rate=\frac{Future\ Value-Present\ Value}{Present\ Value} $$

## Discount Rate

Discount rate is the rate by which a future value is reduced to get its present value.

A discount rate is expressed by the following equations:

$$ Future\ Value\ \times(1-Discount\ Rate\times Time)=Present\ Value $$

or

$$ Discount\ Rate=\frac{Future\ Value-Present\ Value}{Future\ Value} $$

INTRATE is an Excel function that calculate the interest rate while DISC is an Excel function that calculates discount rate.

The distinction between interest rate and discount rate is critical because different securities have different return conventions.

US T-bill are discount-based instruments. A recent US Government issue of 13-week T-bill with CUSIP 912796PC7 with issue date of 01 Feb 2018 and maturity date of 03 May 2018 had a price of $99.639792 per $100 face value. The US Government will pay back $100 per T-bill to the holder on the maturity date and his profit results from the difference in the maturity value and issue price of the T-bill which is $0.360208 (=$100 - $99.639792).

$$ Discount\ Rate=\frac{$100\ -\ $99.639792}{$100}=0.3602\% $$

The rate calculated above is the periodic discount rate. It can be converted to an annual rate by dividing it by total number of days between the issue date and maturity date of the T-bill and multiplying it with total number of days in a year.

Bank loans are based on interest. For example, if you raise $10 million as a short-term loan paying 5% per annum, you total loan liability at the end of one year will be $10.5 million:

$$ Loan\ Value\ after\ 1\ Year\ =$10\ million\ \times(1+5\%)=$10.5\ million $$