Credit spread formula(s)

[clement]

Hi David,

I have a fairly basic question but it's been bugging me. I have found that some people use a different formula to compute credit spread, namely:
1-(1+risk free yield)/(1+risky yield)
The difference in result with formula of -1/T*ln(D/F)-Rf is often not massive but it has already led me to choose wrong answers when looking at questions.

For instance, here was a question: zero coupon bond, 1 year remaining to maturity, currently trading at 85% of face value, Rf=2%.
-> Method 1: I compute the yield of the bond at 17.64%, and that gives a spread of 13.3% = 1-1.02/1.1764
-> Method 2: straightforward from the notes, -ln(0.85)-0.02=14.25%

So, which one is right? (The book I have says 13.3%) And where does the difference come from?

Many many thanks in advance!
Clement

 

[David Harper CFA FRM]

Hi @clement

Method #2 is more correct (can a thing be more correct; howabout I say that method #1 is less correct) because method #1 is inferring the default probability (PD) and relying on the fact that a PD approximates the spread if LGD = 100% per PD = spread/LG. Method #1 therefore contains a bit of a logical flaw: if we seek only the spread based on price, then we do not need to take a extended road trip to getting the PD and then coming back to spread by relying on an approximation (which, in turn contains a recovery assumption). The spread itself is a function of (i.e., contains the information) of PD and recovery/LGD, we do not necessarily need to parse them, if the spread is all we seek. Method #1 is not really solving for spread, it's solving for PD.

More specifically, if we are going to use annual compound frequency to infer the yield of 17.65% per (100/85)^(1/1) - 1 = RATE(1, 0, -85, 100) = 17.65%, then we can go directly to (what Malz calls) the yield spread as given by 17.65% - 2.0% = 15.65%. That's the problem of Method #1. Instead of retrieving yield, it goes on to inferring default probability per the assumption: (1-PD)(1+y) = (1+Rf); i.e., on the left, investing at risky yield should have weighted expectation to generate riskfree return, on the right.

BTW, there are discrete analogs to the continuous version (Method #1), which are valid alternatives. The continuous version in method #2 is simply solving for the spread, s, under assumption that P*exp[(r+s)*T] = F such that s = 1/T*ln(F/P) - r. But we can assume instead annual compound frequency per P*(1 + r + s)^T = F such that s = (F/P)^(1/T) - 1 - r; in this case, s = (100/85)^(1/1) - 1 - 2% = 15.65%. But notice that is the same as 17.65% - 2.0% = 15.65%. So this (15.65%) is a viable discrete alternative to the continuous 14.25%. It's all good stuff to think about, it's all foundational to the FRM. Touches on multiple reusable ideas. I hope that's helpful!

 

[clement]

Great, many thanks David. Very clear and useful!