Yield Measures: Effective Annual Rate

James has a zero-coupon bond priced at 50 of 100 par that matures in 12 years. He realizes this means he'll earn a 100% rate of return spread over 12 years, but also knows dividing this 100% return by 12 will not give him the correct annual rate of return. He wants to find the annual compound rate of return which equates: $$\displaystyle 50(1 + r)^{12} = 100 $$

What rate of return should James find?

No, this is a string of numbers from an intermediate step. It's possible you didn't subtract one and convert to a decimal yet.

That's right! This rate of return which equates the cash flows with annual compounding is the __effective annual rate__, and this assumes a __periodicity__ of one. [[calc: 12 n 50 sign pv 0 pmt 100 fv cpt iy , 12 n 50 chs pv 0 pmt 100 fv i]]

No, this is the value of the simple division method, but not the compound rate.

It's certainly possible to have other periodicities, and in fact, it's also possible to convert from one to another. Suppose James didn't have the price of his bond, but he just knew the stated rate of return assuming a periodicity of four, or quarterly compounding, was 5.818%. This is calculated from the same bond price of 50, using $$4 \times 12 = 48$$ compounding periods: $$\displaystyle APR_4 = \left( \frac{100}{50} \right)^{1/48}- 1 = 0.014545, \times 4 = 5.818 \% $$

How would you interpret the difference between this stated rate and the effective annual rate?

No, the future value of the bond will be the price of 100, and any reinvestment will earn what it earns regardless of this measurement issue.

Yes! There is an inverse relationship between a bond's stated yield and the periodicity assumed. But of course James will get the same cash flows no matter what is calculated, so this choice doesn't affect his return at all. In fact, there is a general calculation to use for conversion between periodicity m and n is that: $$\displaystyle \left( 1 + \frac{APR_m}{m} \right)^m = \left( 1 + \frac{APR_n}{n} \right)^n $$ This also works for negative yielding debt, which has been issued by several countries.

Using this calculation, the effective annual rate can be found with a little algebra, starting with: $$\displaystyle \left( 1 + \frac{0.05818}{4} \right)^4 = \left( 1 + \frac{APR_n}{n} \right)^n $$ The effective annual rate assumes conversion to n=1, leaving this as the left-hand side minus one, which is calculated as: $$\displaystyle EAR = APR_1 = \left( 1 + \frac{0.05818}{4} \right)^{4} - 1 = 5.9462 \% $$ The slight difference is due to the rounding of the final decimal.

What is the effective annual rate of a bond with a stated rate of $$APR_{12} = 6.5124 \%$$?

To summarize this discussion: [[summary]]

No, it's possible to arrive at this answer by assuming a periodicity of four instead of 12 as in the example given.

No, it's possible to arrive at this answer by "going the wrong way" and converting from a periodicity of one to 12 instead of from 12 to one.

That's not it. That's not the difference between the stated rate and the effective annual rate.

Correct! $$\displaystyle EAR = APR_1 = \left( 1 + \frac{0.065124}{12} \right)^{12} - 1 = 0.0671035 $$ Just remember: any stated rate of return must have a periodicity attached to it, and any combination of stated rate and periodicity provides you with the same information about a bond's return. [[calc: iconv 6.5124 enter down down 12 enter up cpt , 6.5124 enter 12 n / i 0 pmt 100 chs enter pv fv +]]

1.0595%

5.9463%

8.3333%

The greater periodicity alone causes the stated rate to be lower

The bond pays James a lower rate of return than it otherwise would

The greater periodicity assumed for the bond will increase its future value

6.3257%

6.6732%

6.7104%

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