Calculate and compare effective annual rates

Effective annual rates

An effective annual rate measures the actual proportional change over one year. It is found by applying the correct period multiplier for every compounding period in that year

1. Find the period rate Divide a nominal annual rate by the number of compounding periods per year \(\displaystyle r=8,\quad p=4\quad\Rightarrow\quad i=\frac{8}{100\times4}=0.02\)
2. Form the period multiplier Add 1 to the period rate so that the multiplier includes the original balance \(\displaystyle R=1+i=1+0.02=1.02\)
3. Compound for one year Raise the period multiplier to the number of periods in one year, then subtract 1 and express the result as a percentage \(\displaystyle E=\left(R^p-1\right)\times100=\left(1.02^4-1\right)\times100=8.243216\%\)
4. Use the role when comparing A borrower prefers the lower effective annual rate, while an investor prefers the higher effective annual rate \(\displaystyle E_A=7.8\%,\ E_B=8.1\%\quad\Rightarrow\quad\text{loan: choose A; investment: choose B}\)

Worked example

A term deposit earns 10% nominal annual interest compounded quarterly. Find its effective annual rate to 2 decimal places

1. \(\displaystyle i=\frac{10}{100\times4}=0.025\)The nominal annual rate is divided by four to obtain the quarterly rate
2. \(\displaystyle R=1+i=1.025\)Adding 1 to the quarterly rate gives the multiplier for one quarter
3. \(\displaystyle E=\left(1.025^4-1\right)\times100=10.3812890625\%\)Compounding the quarterly multiplier four times gives the exact one-year percentage change
4. \(\displaystyle E\approx10.38\%\)The effective annual rate is rounded to 2 decimal places only after the full-year calculation

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