Adjusting yields for periodicity

CFA Level I Fixed Income
#1

I am reading from kaplan’s fixed income session and confused with converting yields for periodicity.

A bond is quoted with YTM of 4% on a semiannual bond basis. What yields should be used to compare it with a quarterly-pay bond and an annual-pay bond?

4% on a semiannual basis is an effective yield of 2% per 6 month period (I understand this)

1.02 ^ 2 - 1 = 4.04% (I understand this because we are taking *1 + Period Rate) ^ # of periods per year - 1

The quarterly yield (yield per quarter) that is equivalent to a yield of 2% per six months is

1.02 ^ 1/2 - 1 = 0.995%

I need clarification on the exponent of this quarterly calculation. I clearly see that we are taking 2/4, 2 from semiannual, and 4 for number of annual periods. I fail to understand why, though.

For the first calculation, we simply put it to the # of periods we’re examining. Why isn’t our quarterly calculation (1.01)^4 - 1 ?

#2

This looks more confussing than it actually is. To convert a yield from one periodicity to another just take this formula:

(1+ i/m)^m = (1 + i/n)^n, where n and m indicate the different periodicities.

From your example:

  1. 4% semiannual (m=2) to annual (n=1):

(1+0.04/2)^2 = (1+i/1)^1 => solving for i will give you 4.04 for the annual rate

  1. 4% semiannual (m=2) to quarterly (n=4):

(1+ 0.04/2)^2 = (1+ i/4)^4 => solving for i will give you 3.98 for the annual rate Taking the equation from above you’ll be able to tranfer every given rate to any other periodicity. Regards and happy easter! Oscar

1 Like
#3

Thank you Oscar. I will have to spend some time playing with this algebra in order to be confident for the exam. Appreciate the quick reply.

#4

This is simply a confusion of nominal rates and effective rates. I wrote an article that may be of some help: http://financialexamhelp123.com/nominal-vs-effective-interest-rates/.

The upshot is that effective interest rates compound; nominal interest rates do not.

Here, you’re told that the annual rate is a nominal rate: twice the semiannual effective rate. So it doesn’t compound: you divide by 2 to get the semiannual effective rate, which does compound. Therefore, to go from the semiannual rate to the quarterly rate, you compound

(1 + q)² = 1 + s

When you have a nominal rate, the only thing you can do with it is to transform it into the specific effective rate on which it is based by dividing (or multiplying). Once you have the effective rate, it’s always compounding to get other effective rates.