Monthly and effective interest rate calculator

A bank advertises a savings account paying 5% a year, but adds the interest every month. That means each month you get a twelfth of the rate, about 0.417%. Because the interest added in one month also earns interest in the following months, your money grows by a little more than 5% over the year, about 5.12%. The figure in the advert is called the nominal rate, and what you actually earn over a year is the effective rate.

So £100 in an account paying 5% a year becomes £105.12 after a year, not £105. The same applies to a loan, only it works against the borrower.

The other way: from effective rate to monthly rate

Sometimes we know the effective rate, for example when we want the money to grow by exactly 5% over a year, and we need the monthly rate that achieves this.

How it works

The calculator uses the formula

$$r_{ef}={(1+\frac{r}{12})}^{12}-1$$

Here r is the nominal annual rate and r_ef the effective rate, both as decimals (5% is 0.05).

Divide the nominal rate of 5% by twelve to get the monthly rate, $5÷12\approx 0.417$%. Interest is added twelve times over the year, so the money is multiplied by

$${(1+\frac{0.05}{12})}^{12}\approx 1.0512$$

The effective rate is therefore 5.12%. The difference grows with the rate: at 20% a year, the effective rate is already 21.94%. That is why loans in the UK quote an APR (annual percentage rate) alongside the interest rate, and the APR also includes fees.

To go the other way, when we know the effective rate and want the monthly rate, the twelfth root does the job:

$$i=\sqrt[12]{1+r_{ef}}-1$$

The nominal annual rate is then twelve times the monthly rate. For 5%, $\sqrt[12]{1.05}\approx 1.004074$, so the monthly rate is 0.407% and the nominal annual rate is 12 · 0.407 ≈ 4.89%. A bank that wants to pay an effective 5% exactly would have to advertise 4.89% with monthly compounding. The article on mortgage interest works through this derivation in more detail.