Which is a higher annual rate and by how much: 18.2 percent per year compounded weekly or 18.5 percent per year compounded quarterly?
On the face of it, this is a very simple problem. We have a choice of 18.2 percent interest per year and 18.5 percent interest per year. Surely, it’s a no-brainer. You go for the higher interest rate. However, things aren’t quite that simple. The plan with an 18.2 percent interest rate per year is compounded weekly. This means that interest is added weekly to the account. And as there are 52 weeks in the year, near enough, this means that interest is added 52 times per year. On the other hand, the plan with an 18.5 percent interest rate is compounded quarterly. That means interest is added every quarter or four times per year.
We might now think that it would be better to have interest added 52 times per year than only four times per year. But of course, this depends on how much interest is added each time. There’s a formula for the balance in the bank account in terms of the amount initially invested, called the principal; the annual rate, 𝑟; the number of times the interest is compounded per year, 𝑛; and the amount of time in years that the money is invested, 𝑡. We can see that setting 𝑛 equal to one, we get the normal compound interest formula where interest is added once every year.
Let’s substitute into this formula to find the annual rate for the first plan. Our balance 𝐴 will be equal to the principal we invest times one plus the interest rate per year, which we convert to a decimal, divided by the number of times the interest is compounded by year. We’re compounding weekly and so this is 52 times per year. We also have to raise to the power of 52. And as we’re looking for the effective annual rate, we’ll find the balance after one year. So 𝑡 is one. To make things clearer, perhaps we should call 𝑟 the interest rate per year and not the annual rate which means something else in this context.
If we now put this value into our calculators, we get 1.19923 dot dot dot. So after a year, the balance in our account is 1.19923 dot dot dot times the principal value that we invested. The interest rate is given by this fractional part. We’ve got an extra 0.19923 times our principal 𝑃 of the year. So we have 19.923 dot dot dot percent more money than we invested at the start of the year. And an effective annual rate of 19.923 dot dot dot percent. This is the interest rate that you would need for an account where the interest was compounded once per year to be equivalent to the 18.2 percent account compounded weekly.
It’s exactly the same process for the other plan, 18.5 percent compounded quarterly. Our value of 𝑟 is 0.185 because the interest rate per year for this account is 18.5 percent. And our value of 𝑛 is four because we’re compounding quarterly. That is, four times per year as mentioned before. 𝑡, the time in years that we’re investing our principal value for, is still one. So we don’t need to multiply the exponent four by anything. Four times one is four.
We put this expression into our calculators. And we get 1.19823 dot dot dot. We therefore have 19.823 dot dot dot percent more money than we had at the start of the year for an effective annual rate of 19.823 dot dot dot percent.
Now that we have the effective annual rates of the two plans, we can compare them to see which plan is better. 19.923 dot dot dot percent is a higher effective annual rate than 19.823 dot dot dot percent. So the plan with the 18.2 percent interest rate compounded weekly is better. How much better is it? It’s not much. The difference in the effective annual rates is about 0.1 percent. But it is, nonetheless, better. Which might be surprising, if you thought that the 18.5 percent per year plan would be automatically better than the 18.2 percent plan.
We’ve seen that how good a bank account or investment plan is, depends not only on the interest rate advertised but also on how often that interest is compounded. Plans with a lower advertised rate can actually be better than those with a higher advertised rate, if they are compounded more frequently.