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PERIODICITY CONVERSIONS

A commonly used bond math technique is to convert an annual percentage rate from one periodicity to another. In the bond market, the need for this conversion arises when coupon interest cash flows have different payment frequencies. For example, interest payments on most fixed-income bonds are made semiannually, but on some the payments are quarterly or annually. Identifying relative value necessitates comparing yields for a common periodicity. In the money market, the need for the conversion arises when securities have different maturities. The 1-month, 3-month, and 6-month LIBOR have periodicities of about 12, 4, and 2, respectively, depending on the actual number of days in the time period.

The general periodicity conversion formula is shown in equation 1.9.

(1.9)

APRm and APRn are annual percentage rates for periodicities of m and n. Suppose that an interest rate is quoted at 5.25% for monthly compounding. Converted to a quarterly compounding basis, the new APR turns out to be 5.273%. This entails a periodicity conversion from m = 12 to n = 4 and solving for APR4.

The key idea is that the total return at the end of the year is the same whether one receives 5.25% paid and compounded monthly (at that same monthly rate) or 5.273% paid and compounded quarterly (at that same quarterly rate).

Suppose that another APR is 5.30% for semiannual compounding. Converting that rate to a quarterly basis (from m = 2 to n = 4) gives a new APR of 5.265%:

The general rule is that converting an APR from more frequent to less frequent compounding per year (e.g., from a periodicity of 12 to 4) raises the annual interest rate (from 5.25% to 5.273%). Likewise, converting an APR from less to more frequent compounding (2 to 4) lowers the rate (5.30% to 5.265%). Put on a common periodicity, we see that 5.25% with monthly compounding offers a slightly higher return than 5.30% semiannually.

Another periodicity conversion you are likely to encounter is from an APR to an effective annual rate (EAR) basis, which implicitly assumes a periodicity of 1.

(1.10)

For example, an APR of 5.25% having a periodicity of 12 converts to an EAR of 5.378% while the APR of 5.30% having a periodicity of 2 converts to 5.370%.

Some financial calculators have the APR to EAR conversion equation already programmed (note that “EFF” is sometimes used instead of “EAR”). The APR often is called a nominal interest rate in contrast to the effective rate. This is common in textbooks and in academic presentations. The idea is that the EAR represents the total return over a year, assuming replication and interest compounding at the same rate. The APR also assumes replication but merely adds up the rates per period and neglects the impact of compounding in obtaining the annualized rate of return.

An acronym used with U.S. commercial bank deposits is APY, standing for annual percentage yield. This is just another expression for the EAR. So, if the nominal rate on a 6-month bank deposit is quoted at 4.00%, its APY is displayed to be 4.04%. The higher the level of interest rates and the greater the periodicity of the nominal rate, the larger is the difference between an APR and its APY. If the APR on a 1-month bank deposit rate is 12.00%, its APY is 12.68%. It should be no surprise that banks like to display prominently the APY on time deposits and the APR on auto loans.

 
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