Question

APR and EAR: What is meant by the ”frequency of compounding”? Illustrate the meaning of the “frequency of compounding” for a one-year investment of $100 with an APR of 10% and both annual and semiannual compounding. For a monthly rate of 1%, what is the APR and EAR? What is the relationship--positive or negative--between the APR and EAR and the frequency of compounding? Use the EAR formula and make up a specific numerical example to illustrate that (a) the EAR and APR would be equal for annual compounding; and (b) that for frequencies of compounding greater than annual, the EAR would be greater than the APR.

Answer 1

Frequency of compounding refers to the number of times interest is compounded in a year.

For a one year investment of $100 , with annual compounding at 10% , the final amount will be $100+ 10%*100

= $110

With semi-annual compounding

The Semi-annual accumulated amount will be $100+ 5%*100 = $105

The final amount will be $105+ 5%*105 = $110.25

Thus the final amount is greater with greater compounding.

For a monthly rate of 1%,with annual compounding, both APR and EAR are same

APR= 1%

EAR= (1+APR/m)^m-1

= (1+1%/2)^2-1

=1.0025%

APR does not change with Frequency of compounding

EAR has positive relationship with Frequency of compounding.

Example:

Suppose the annual rate is 20%

For Annual compounding

EAR= (1+APR/m)^m-1 (where m=frequency of compounding)

= (1+ 20%/1)^1-1 = 20%

So the EAR and APR are the same.

For semi-annual compounding, APR will be 20%

EAR = (1+ 20%/2)^2-1

= 21%

Thus EAR is greater than APR if the frequency of compounding is higher.