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# implied LGD from bond rates (or spread)?

#### ajsa

##### New Member
Hi David,

"Full E1.09. Suppose the rate on Company A’s one-year zero-coupon bond is 10.0% and the one-year T-bill rate is 8.0%. Assume the T-bill is riskless and the probability of default of Company A’s bond is 10%. What is the LGD of Company A’s bond?

a. 18.18%
b. 81.82%
c. 20.01%
d. 79.99%

1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]"

I wonder when I should instead use the spread to calculate LGD:
LGD = spead/PD = (10-2)/10 = 20% ?

For example Full II question 2:
The risk-free rate is 5% per year and a corporate bond yields 6% per year. Assuming a recovery rate of 75% on the corporate bond, what is the approximate market implied one-year probability of default of the corporate bond?
a. 1.33% b. 4.00% c. 8.00% d. 1.60%
CORRECT: B Using the approximation method, the 1-year probability of default is (6%-5%)/(1-0.75) = 4%

Does it mean we need to VERY careful about “approximate”? or does it mean that as long as the question provides 2 rates, i should not use spread to approximate?

Thanks

#### ajsa

##### New Member
Hi David,

Could you help me with this one please? I feel we will 100% have similar questions in the exam

Many thanks!!

#### hsuwang

##### Member
oh wow I did not recognize it until you mention it, i think there are 2 ways to solving this.. but yea I'm not sure which method to use either..

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Asja & Jack -

Asja has been very good about spotting these sort of issues ... I would *love* to tell you that the difference hinges on the use of "approximation" or not. But GARP has not demonstrated that sort of precision, frankly. Rather, clearly, what happened here is: the Hull approximation (i.e., PD ~ spread / LGD) was only formally added this year and this has created (yet another) definitional conflict. It is frustrating: I feel this is the #1 issue for GARP, to resolve definitional conflicts (e.g., alpha, VaR, SaR...)...in their defense, it is an *inherent* challenge of a dynamic curriculum (as fresh readins are assigned, definitional/semantic conflicts are inevitable. Although, IMO, the thing to do here is simply to avoid using Hull for credit risk and keep him for his strength, market risk...)

But (since i am being candid), in this case, the issue is also the AIM (Ch 22, Hull): "Estimate the probability of default for a company from its bond price." So, technically, the AIM refers not to the approximation, but the "more exact" calculation in Table 22.3 (and which i tutored in the video, and XLS, due to the AIM!). However, it is highly highly unlikely this AIM will be tested due to the tedium involved (?!)

but to the practical issue at hand, I do see one way to "infer" what the question is looking for:
The question here does not give the spread, it gives the riskless rate *and* the risky rate. What's the difference? The Saunders' approach (i.e., 1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]) varies even as the spread is constant. Given a 2% spread, as asja shows, the est. LGD (under Hull) is always 20%; however, the implied LGD under the Saunder's method will be higher if, say, the risky = 8% and the riskless = 6% (spread is still 2%). So, my point is: by giving the both yields instead of the spread, that (maybe) is a "tip" that we should use Saunders' method. And, giving further support, to agree with ajsa's semantic point, it is more accurate than Hull's "approximation." (although, GARP has never shown the sort of precision that would give a hint to the method, with a single word, so I don't believe the term "approximation" by itself should be the guidance. In fact, the majority of the models we use are already approximations that each have more precise cousins, incl the Saunders approach to LGD, so "approximation" is itself an "approximation"!)

I hope that helps...David