APR v. EAR and Continuous Compounding

APR (annual percentage rate) and EAR (effective annual rate, sometimes called annualized rate of return) are two different rates that are frequently mentioned. The APR is equal to the number of interest periods per year times the interest rate per period, whereas the EAR is the total rate of return over the entire year.

Consider a one-year loan with semiannual compounding (that means interest is charged every 6 months) that is charged 5% interest per period. Since interest is charged twice per year, the APR is 10% (2 * 5% = 10%). However, the EAR is 10.25% (1.05 * 1.05 - 1 = 10.25%). The general formulas for calculating APR and EAR are:

APR = rate-per-period * periods-per-year

EAR = (1 + rate-per-period)periods-per-year - 1

Now, if you have a one-year loan with quarterly compouding that is charged 2.5% per period. The APR is still 10% (4 * 2.5% = 10%), but the EAR is 10.38%. The effective annual rate increases since interest starts collecting on interest at a earlier time. The following table shows the EARs for different compounding intervals given a constant 10% APR. As you may notice, the values start converging as the compounding intervals shrink.

Compounding Periods EAR
Yearly 1 10.000000%
Semiannually 2 10.250000%
Quarterly 4 10.381289%
Monthly 12 10.471307%
Daily 365 10.515578%
Hourly 8760 10.517029%
Per Minute 525600 10.517091%
Per Second 31536000 10.517092%

Continuous compounding is where the compounding intervals are infinitesimally small. The annualized rate of return for continuous compounding is calculated with the following formula:

EAR = eAPR - 1
APR = ln (EAR + 1)

For a 10% APR, the EAR with continuous compounding is 10.517092% which is where the values converge for shrinking compounding intervals. When calculating present and future values with continuous compounding, the following formulas are used:

PV * e(APR * t) = FV
PV =    FV   
e(APR * t)