Fixed income - when to use continuous compounding

[afterworkguinness]

Can you shed some light on when it is correct to use continuous compounding for bonds ? I was under the impression that you use continuous compounding if you are given a series of spot rates for different maturities ... is this correct ?

As always, thank you very much in advance for all the help (David, team and fellow forum members)

 

[David Harper CFA FRM]

Hi afterwork,

I don't perceive that a "correct" assumption exists. I believe that the correct compound frequency is a property (feature) of a well-specified interest rate; and, exam-wise, therefore, the burden is on the question to be precise. For example, "a six-month spot rate of 4.0%" is actually imprecise. This is where the Hull text is very strong, all of his questions include compound frequency as a property of the rate.

For example, if a $1,000 face bond with a 4.0% coupon rate (always per annum) pays semi-annual coupons, then the next semi-annual coupon pays $20. Say that $20 coupon pays in six months.

• It is insufficient to say "the six-month zero rate is 4.0%"
• If, however, the six-month zero (spot) rate is 4.0% with continuous compounding, then it's PV = exp(-4%*0.5) * 20 = $19.6040
• If the six-month zero (spot) rate is 4.0% with annual compounding, then it's PV = 20/1.04^0.5= $19.6116
• And, if we translate the continuous 4.0% into annual, its equivalent annual rate = exp(4%)-1 = 4.0811%,
  such that we get the same PV if we say "the six-month zero rate is 4.0811% with annual compounding"; i.e., 20/(1.040811)^0.5 = $19.6040

in short, compound frequencies are a property of the rate. For any given maturity, we can express the same exact zero rate in various frequencies, but an "interest rate" without a frequency can map to several zero rates and is therefore imprecise.

Note this does not include two rates that two include, by definition, their compound frequency:

• a bond-equivalent yield is, by definition, semi-annual discrete
• an effective annual yield (EAY or EAR) is, by definition, annual discrete.

I hope that helps,

 

[afterworkguinness]

That helps a lot thanks