“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 1.022
“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 = 1.020.5.”
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.
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)
Are you sure this part is right? YTM = Effective yield for annual-pay bonds only! The effective yield 4.04%, not 4%.
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.
why cant I use the formula (1+I/y)m-1 to calculate the ytm of the quarterly?
according to this method we got 4.04%, 4% and 3.98% respectivley, which the ytm is decreasing, however, when ytm is decreasing means the yield is less, right? If we think about compound interest, the quarter ytm should be higher because we get the pmt more frequently and we can reinvest to earn more, so ytm of the quartely should be higher than semiannually than annually. Maybe I got a wrong point to understand this, but your method is completely right, please correct me, Thank you.
Hope you found the answer. I’m having the exact issue
The base formula for finding nominal rates is as follows:
Nominal rate = m * [ (1 + effective annual) ^ (1/m) - 1] where the nominal rate is compounded m-thly.
You can derive it from oktavian’s equation above and setting it equal to 1 + effective annual. In simple terms, after 1 year, I will have the exact same accumulated amount under annual compounding and m-thly compounding.
You can also use the BAII to due all the grunt arithmetic:
NOM 4 C/Y 2 CPT EFF 4.04 (use 4.04 as your “anchor” to calculate other nominal rates - and it’s also the annually compounded rate!!! )
C/Y 4 CPT NOM 3.980198
Happy you responded. Maybe you can assist with the rationale behind the calculations. What’s confusing me is that normally when periodicity increases the effective yield increases due to compounding.
In this case however, it seems to be the opposite where instead, when periodicity increases there is a decrease in the yield. What’s the rationale behind this?
It’s the magic of compounding (interest credited on previously earned interest) that allows you to set a LOWER nominal rate as the compounding frequency increases.