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We can calculate another type of interest, called the annual interest rate or the annual effective interest rate, that does make this second correction. The annual effective rate is simply a way to describe how an investment has increased, however it is a more intelligent way than the effective interest rate, because it corrects for both the investment or loan’s size, and its duration. This type of interest is calculated according to the formula in below Equation.

𝑟𝑟= �𝑋𝑋𝑇𝑇 𝑋𝑋0� 1 𝑇𝑇 −1

There is important logic behind the definition of this term. The annual interest rate applies over the period of one year, as the name suggests. This means that T=1 (and therefore that 𝑋𝑋𝑇𝑇 = 𝑋𝑋1), denoting the annual increase you will get on one’s money.

𝑋𝑋1 = 𝑋𝑋0(1 + 𝑟𝑟)

If T=2 (i.e. if the period of investment was two years), the annual interest rate would need to be applied twice. This is to reflect that in the first year, 𝑋𝑋0 (the initial investment) grows as per the above equation, but then this amount (including the increase from the first year’s interest) increases similarly in the second year.

𝑋𝑋2 = 𝑋𝑋0(1 + 𝑟𝑟)(1 + 𝑟𝑟) = 𝑋𝑋0(1 + 𝑟𝑟)2

Accordingly, we then get the following expression for a general time period T, meaning that we can assign any value to T to reflect any period we want to consider.

𝑋𝑋𝑇𝑇 = 𝑋𝑋0(1 + 𝑟𝑟)𝑇𝑇

We can then rearrange the formula in Equation (5) to get the formula in Equation (2). Note that equation 2 coincides with equation 1 when T=1; in this special case, the effective interest rate already applied to an annual duration and does not need any further adjustment to apply to one year.

Calculating the annual effective interest rate enables us to compare different money market investments. To demonstrate this, let’s imagine you want to purchase one of two bills: 1. Bill 1 costs 2450 and will reach maturity in 3 months with a par value of 2500, 2. Bill 2 costs 1922 and will reach maturity in 6 months with a par value of 2000.

Upon hearing these prices, you calculate the interest payment using 𝑋𝑋𝑇𝑇−𝑋𝑋0 and see that these bills involve interest payments of 50 and 78, respectively. However, you cannot immediately say how the effective interest rate fares relative to the different periods. This can be done by calculating the annual effective interest rate using the formula in Equation (2):

For bill 1: 𝑟𝑟= �2500 2450� 1 0.25 −1 = 0.08416578473

For bill 2: 𝑟𝑟= �2000 1922� 1 0.5 −1 = 0.08281241032

Here we see that the annual interest rates offered by the two bills are 8,417% and 8,281%, respectively.

If we assume that default is not a material factor for this exercise, we see that the first bill offers a slightly better annual return on investment even though the gross amount of profit is less. The decreased gross amount of profit is more than compensated for by the short period of time that one’s money is lent for (which would enable you to re-lend your money after 3 months and continue to earn interest in a further loan).

This points to the inverse relationship between initial prices and interest rates, which is important to note. To demonstrate this, let’s consider a fixed repayment value, wherein a bill always pays 100 units in T-years’ time, so we can set 𝑋𝑋𝑇𝑇=100. If we were to apply the formula in Equation (2), we would see that a larger value for 𝑋𝑋0 results in a smaller annual interest rate while a smaller 𝑋𝑋0 results in a larger rate.

To ensure that you are comfortable with calculating the annual effective interest rate, it is a good idea to consider multiple examples of the two bills and experiment with different initial prices for the bills.

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