Other Compounding Periods

In the real world, interest rates are often compounded more often than once per year. By convention, interest rates are quoted on an annual basis. An interest rate, quoted on an annual basis, which is compounded more often than once per year is called a nominal rate, stated rate, quoted rate, or annual percentage rate (APR). For example, mortgages typically require monthly payments and, therefore, the interest rates quoted on mortgages are compounded monthly. Thus, the nominal interest rate on a mortgage might be 12% compounded monthly. However, the relevant rate for valuations is the periodic rate. The periodic rate is computed by dividing the nominal rate by the number of compounding periods per year.

where

• r = the rate per period,
• r_nom = the nominal rate, and
• m = the number of compounding periods per year.

Thus a 12% nominal rate compounded monthly is equivalent to a periodic rate of 1% per month.

The following sections of this page demonstrate how to convert a nominal rate into an equivalent rate that is compounded annually and provide versions of the Present Value and Future Value formulas for use with interest rates compounded more often than once per year. The page concludes with a discussion of continuous compounding.

EAR - Effective or Equivalent Annual Rate

The Effective or Equivalent Annual Rate (EAR) is the interest rate compounded annually that is equivalent to a nominal rate compounded more than once per year. In other words, present and future values computed using the EAR will be the same as those computed using the nominal rate. The EAR is computed as follows:

• EAR = the Equivalent or Effective Annual Rate,
• r_nom = the nominal interest rate,
• m = the number of compounding periods per year, and

Moreover, it is not proper to directly compare interest rates which have a particular compounding frequency with those that have a different compounding frequency, e.g.,, comparing 10.1% compounded semiannually with 10% compounded quarterly. This problem can be overcome by finding the EAR for each of the rates and then comparing the EARs.

First, let's find the EAR for 10.1% compounded semiannually. Here, m equals 2.

EAR for 10.1% compounded semiannually

Now, let's find the EAR for 10% compounded quarterly. Here m = 4.

EAR for 10% compounded quarterly

Thus, we see that 10% compounded quarterly is actually a higher interest rate than 10.1% compounded semiannually. Given a choice, we would prefer to invest at 10% compounded quarterly.

Present Value

The Present Value of a future cash flow when the interest rate is compounded m times per year can be calculated as follows:

where

• PV = the Present Value,
• CF_t = the cash flow which occurs at the end of year t,
• r_nom = the nominal interest rate,
• m = the number of compounding periods per year, and
• t = the number of years.
• Thus, mt = the number of compounding periods in t years.

In the earlier discussion of Present Value the interest rate was compounded annually and there was one compounding period per year. In that case m = 1. Thus, our earlier Present Value formula is actually just a special case of this formula since under annual compounding the rate per period is the same as the nominal rate.

Future Value

The Future Value of a future cash flow when the interest rate is compounded m times per year can be calculated as follows:

where

• FV = the Future Value,
• CF₀ = the cash flow which occurs at time 0,
• r_nom = the nominal interest rate,
• m = the number of compounding periods per year, and
• t = the number of years.
• Thus, mt = the number of compounding periods in t years.

Thus, the earlier Future Value formula is actually just a special case of this formula since under annual compounding (i.e., when m = 1) the rate per period is the same as the nominal rate.

Present Value of an Annuity

The Present Value of an Annuity when the payments occur m times per year and the interest rate is compounded m times per year can be calculated as follows:

where

• PVA = the Present Value,
• PMT = the Annuity Payment which occurs m times per year,
• r_nom = the nominal interest rate,
• m = the number of compounding periods per year, and
• t = the number of years.
• Thus, mt = the number of payments and compounding periods in t years.

This formula can only be applied when the frequency of the annuity payments is the same as the compounding period for the interest rate. For example, if the annuity has quarterly payments the interest rate must be compounded quarterly (m = 4).

Thus, the earlier Present Value on an Annuity formula is actually just a special case of this formula since under annual compounding (i.e., when m = 1) the rate per period is the same as the nominal rate.

Future Value of an Annuity

The Future Value of an Annuity when the payments occur m times per year and the interest rate is compounded m times per year can be calculated as follows:

where

• FVA_t = the Future Value of the annuity at the end of year t,
• PMT = the Annuity Payment which occurs m times per year,
• r_nom = the nominal interest rate,
• m = the number of compounding periods per year, and
• t = the number of years.
• Thus, mt = the number of payments and compounding periods in t years.

This formula can only be applied when the frequency of the annuity payments is the same as the compounding period for the interest rate. For example, if the annuity has quarterly payments the interest rate must be compounded quarterly (m = 4). As with the earlier formula, the Future Value is computed at the end of the period in which the last annuity payment occurs.

Thus, the earlier Future Value on an Annuity formula is actually just a special case of this formula since under annual compounding (i.e., when m = 1) the rate per period is the same as the nominal rate.