Last time we saw that our $1000 invested at 3% compounded annually will result in $1343.92 in 10 years. To get even better results, the interest rate can be compounded more frequently than annually. Let’s say the interest is applied mid-year. Then the interest earned during the first half of the year will be added to the principal amount and that total will be used to apply the second half interest to.

This will change the formula *A*_{n} = *A*_{0}(1 + *r*)^{n} a bit though. First of all, we can’t apply the full per annum interest rate to the initial investment as only half a year has passed. So only half the interest rate will be used. Also, the period length is now half a year so the number of periods refers to how many half-years have passed.

So in our example, the annual interest rate is 3%, so the 6 month interest rate, *r*, is 3/2 = 1.5% since there are 2 six-month periods in a year. If we want to know how much we will have after 10 years, the number of periods, *n*, is now 10×2 = 20 since there are 20 half -year periods in 10 years. So we can use the same formula with *r* = 1.5% = 0.015 and *n* = 20:

*A*_{20} = *A*_{0}(1 + *r*)^{20} = 1000(1 + 0.015)^{20} = $1346.86

So the extra compounding has made us a bit more money.

You might ask (OK – I’ll ask for you), would we make more money by compounding more frequently? Yes we would!

Let’s compound every quarter-year. this means the interest rate we apply each quarter is 3/4 = 0.75% and the number of periods after 10 years is 10×4 = 40:

*A*_{40} = *A*_{0}(1 + *r*)^{40} = 1000(1 + 0.0075)^{40} = $1348.35

Looks like we want more yet. What about monthly? Here *r* = 3/12 = 0.25% = 0.0025 and *n* = 10×12 = 120:

*A*_{120} = *A*_{0}(1 + *r*)^{120} = 1000(1 + 0.0025)^{120} = $1349.35

Better, but notice that this is not much better that quarterly. It appears that we will reach a limit as to how much we can make. Let’s try compounding daily. Here *r* = 3/365 = 0.0082% = 0.000082 and *n* = 10×365 = 3650:

*A*_{3650} = *A*_{0}(1 + *r*)^{3650} = 1000(1 + 0.000082)^{3650} = $1349.84

Well that’s disappointing. There is only a 0.49 difference between compound monthly and daily after 10 years.

There is a maths formula that computes the amount of interest if the investment is compounded continuously. This is the limit of what you can make by compounding. For our example, after 10 years, the most compounding will get us after 10 years is $1349.86, just 2 cents more than compounding daily.