Yield Measures: Semiannual Bond Basis Yield

James has just purchased a $1,000 face value bond with six years until maturity for $1,006. This bond has a semiannual 6% coupon with a __semiannual bond basis yield__ of 5.88%. This is a US bond and is quite typical for US bonds as they normally pay a semiannual coupon. So the semiannual bond yield is quite a popular measure. He now wants to compare this yield directly with his zero-coupon bond, which has an effective annual rate of 5.9463%, since right now that zero-coupon bond looks like a better deal.

To do this, James has to recognize that an effective annual rate assumes a periodicity of _m_ = 1. What periodicity do you think he would need to use in his conversion?

No, a periodicity of 4 implies quarterly compounding, but James wants a semiannual yield.

No, a periodicity of 12 implies monthly compounding. You might also be thinking in terms of there being 12 remaining payments for this bond, which is true, but this is not the periodicity of the bond.

Yes! A periodicity of 2 implies semiannual compounding. He can use the conversion formula of: $$\displaystyle \left( 1 + \frac{APR_m}{m} \right)^m = \left( 1 + \frac{APR_n}{n} \right)^n $$ He can solve explicitly for the semiannual rate using the conversion formula and a little algebra, finding that: $$\displaystyle APR_2 = [(1 + APR_1)^{1/2} - 1] \times 2 $$

What semiannual bond basis yield do you think he will find for this zero-coupon bond, starting with the annual effective rate given?

No, it's possible to obtain this yield from "going the wrong way" from a periodicity of 2 to 1 instead of from 1 to 2.

That's right! $$\displaystyle APR_2 = [(1 + 0.059463)^{1/2} - 1] \times 2 = 0.05860 $$ This converts the effective annual rate of the zero-coupon bond to a semiannual bond basis yield of 5.86%, which can now be compared with the semiannual bond basis yield of the coupon bond. It looks like his zero-coupon bond doesn't quite offer the same return as his new purchase after all. [[calc: iconv down 5.9463 enter down 2 enter down cpt , 1.059463 enter 0.5 pwr 1 - 2 *]]

No, it's possible to calculate this value by performing a conversion to daily compounding rather than semiannual compounding.

How would you best characterize the consistent relationship between the stated rate and the frequency of compounding referred to here as periodicity?

Absolutely! That's a logical relationship that can provide you with some reassurance when the correct numbers come out of your calculator. A greater periodicity, or more frequent compounding, increases the effective rate calculated, so a lower rate with a higher periodicity corresponds to a higher rate with a lower periodicity.

No, consider the fact that a higher compounding frequency will turn a given annual percentage rate into a higher annual percentage yield. A lower rate with a higher periodicity corresponds to a higher rate with a lower periodicity.

To summarize this discussion: [[summary]]

2

4

12

5.777%

5.860%

6.035%

A lower rate with a lower periodicity corresponds to a higher rate with a higher periodicity

A lower rate with a higher periodicity corresponds to a higher rate with a lower periodicity

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