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An important convention is the 𝑛𝑛-compounded annual interest rate, expressed as:

𝑟𝑟(𝑛𝑛) = 𝑛𝑛�𝑋𝑋𝑇𝑇 𝑋𝑋0�1𝑛𝑛𝑇𝑇− 𝑛𝑛

In Equation (3), we considered interest being applied each year, for some total number of years. Interest can instead be applied — or compounded — more often. If it were compounded twice a year, Equation (3) would need to be adjusted to:

𝑋𝑋1 = 𝑋𝑋0(1 + 𝑟𝑟(2) 2 )(1 + 𝑟𝑟(2) 2 ) = 𝑋𝑋0(1 + 𝑟𝑟(2) 2 )2 (8)

You may notice that the interest rate is divided by 2 in this equation (i.e. 𝑟𝑟(2) 2 ) — this new rate is still annual, it is just compounded more often. Instead of applying 𝑟𝑟 once, we cut it in half and apply it twice. The number 𝑛𝑛 is known as a compounding frequency and is sometimes given in qualitative terms.

For example, you may hear something like “12% p.a. compounded monthly”, where “p.a.” means per annum – confirming we are discussing annual rates; and “compounded monthly” indicates that we can apply Equation (7) if we set 𝑛𝑛= 12.

Note that an increased compounding frequency, all other things equal, will increase the final amount of the loan (for example, 12% p.a. compounded monthly end up charging more interest in total than 12% p.a. compounded semi-annually). This is because increased compounding involves interest being awarded to earlier interest payments. Also note that if n=1, Equation (7) is identical to Equation (2) — in other words, compounding once per year is exactly the same as giving a standard annual interest rate.

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