Adjusting yields for periodicity

[Submariner]

“ABC corp is quoted with a YTM of 4% on a semi-annual bond basis. What yields should be used to compare it with a quarterly pay bond and an annual pay bond.”
4% on semianual bond basis is an effective yield (i.e. compound yield) of 2% per 6 months or 1.02^2
“For the annual YTM on the quarterly pay bond we need to calculate the effective quarterly yield. In this case, the quarterly yield is equivelant to a yield of 2% per 6 months which = 1.02^.5.”
My question is, why don’t we divide the 2% by 2 to get the quarterly yield when we divided 4% by 2 to get the semiannual yield? I think I understand why but I’m not 100% convinced I am correct.
Thanks!

 

[oktavian]

I don’t know where you get this from but the wording is very confusing. Try this simplification: when you have two bonds, with different coupon periods, and you want them to have the same effective annual yield, use this formula:
(1 + YTM1 / n ) ^ n = (1 + YTM2 / m ) ^ m
So if n = 2 (semi-annual) then YTM1 = 4%. If m = 4 (quarterly), we can rearrange to find YTM2:
(1 + 0.04 / 2 ) ^ 2 = (1 + YTM2 / 4 ) ^ 4
(1 + 0.04 / 2 ) ^ (2/4) = 1 + YTM2 / 4
[ (1 + 0.04 / 2 ) ^ (1/2) - 1 ] * 4 = YTM2
annual YTM of the quartlery bond = 3.98%.
You need think when the payment is made. After the payment is made, it can be reinvested to earn interest so this point is where compount interest occurs. If you hold a bond for half its coupon period, you earned half of that interest but it can’t be reinvested (simple interest)
The YTM of the quarterly is slightly lower than the YTM of teh semi-annual bond because you get the first payment sooner (after 3 months, instead of 6). So this can be reinvested sooner. Over one year your income from both bonds will be the same (i.e. the have the same effective annual yield)

Submariner wrote:4% on semianual bond basis is an effective yield (i.e. compound yield) of 2% per 6 months or 1.02^2

Are you sure this part is right? YTM = Effective yield for annual-pay bonds only! The effective yield 4.04%, not 4%.

 

[S2000magician]

You can compare an effective yield directly to an effective yield, or a nominal yield directly to a nominal yield, but you cannot compare an effective yield directly to a nominal yield.
If you’re given a 4% BEY, that means that you have a 2% effective semiannual yield. To compare a quarterly-pay bond to that, you need to convert the quarterly-pay bond’s yield to an effective semiannual yield, and to get from a quarterly (effective) yield to an effective semiannual yield, you compound.