Monthly interest from an annual rate

In practice banks often quote an annual interest rate but apply it month by month. How do we work out the monthly interest rate?

A savings account at Earth Bank

Earth Bank boasts one of the best savings accounts, with an annual interest rate of 2%. Although it quotes an annual rate, it adds the interest every month, so over a year it adds interest 12 times in total. How high is the monthly interest rate? That is, how much money is added to our account each month?

We can derive the formula from the formula for compound interest. But first let's work out how much money we will have in the account after exactly one year. Say we paid £1,000 into the account. Then after a year we will have +2%, which gives

$$\mathrm{1,000}\cdot 1.02=\mathrm{1,020}$$

After a year we will have £1,020 in the account – we have gained twenty pounds. From the compound interest formula we know that

$$P_t=P_0\cdot (1+r{)}^t,$$

where P₀ is the initial deposit, r is the interest rate and t is the number of times we added interest to the amount. In this formula the unknown is precisely the letter r – we only know the annual interest rate, but we are looking for the monthly one. We added interest 12 times in total, so t = 12. We also know that P_t equals 1,020: after a year we have saved £1,020, which we already worked out from the annual interest rate. This gives us the equation:

$$\mathrm{1,020}=\mathrm{1,000}\cdot 1.02=\mathrm{1,000}\cdot (1+r{)}^{12}$$

We can see that the value of (1 + r)¹² must equal 1.02. If we work out the twelfth root of 1.02, we get exactly the value of 1 + r. The twelfth root is approximately 1.0016515813. If we add interest to the amount every month at the rate 0.0016515813, we get a total compound interest of 0.02, with a final amount of £1,020.