This is a good sample question from GARP: first, it is not too easy and not too hard; this IMO is about typical of the exam’s difficulty and (ii) it does not favor formula memorization: the cited reference is de Servigny, but I do not think you will find the formula in the referenced reading. A formula is not needed, we can apply logic and yet another instance of a no-arbitrage idea - David
Question
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%
[my adds next, let’s beat this up!]
9b. [hard] Assume instead we are given that the LGD of Company’s A’s bond is 75% (using Basel II LGD for junior debt under foundation IRB approach). Assuming the credit spread remains 2% (10% - 8%), what is the implied probability of default (PD)?
9c. The answer assumes annual compounding. If we instead assume continuous compounding (i.e., risky bond returns 10% continuous and T-bill returns 8% continuous), what is the implied loss given default (LGD)?. Notice that compounding matters; semi-annual compounding would give a slightly different result.
9d. [hard] As Saunders writes, “Collateral requirements are a method of controlling default risk; they act as a direct substitute for risk premiums in setting required loan rates.” Let collateral (recovery) = 1 - Loss given default (LGD) and solve for the credit spread as a function of the probability of default (PD) and the recovery (c).
9e. Which continuous probability distribution—that is reviewed in the assigned Rachev (Fat-tailed and Skewed Asset Return Distributions)—are we most likely to find in use to characterize the loss given default (LGD) or recovery function?
9f. How are recovery rates (1-LGD) estimated?
9g. What are the most important determinants of recovery/LGD?
Answers:
9a.
An investor should be indifferent between:
(i) invest in riskless T-bill, or
(ii) invest in risky bond
Such that, if k = risky return and i = riskless return (using Saunders’ notation)
1+i = (probability of repayment * return if repayment) + (probability of default * return if default); i.e., weighted average return
1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]; i.e., we recover 1-LGD
solve for LGD:
1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]
PD*(1+k)*(1-LGD) = (1+i) - [(1-PD) * (1+k)]
(1-LGD) = (1+i) - [(1-PD) * (1+k)] / (PD*(1+k))
LGD = 1- [(1+i) - [(1-PD) * (1+k)] / (PD*(1+k))]
and here:
LGD = 1- [(1+8%) - [(1-10%) * (1+10%)] / 10%*(1+10%)] = 18.18%
9b. [hard] Assume instead we are given that the LGD of Company’s A’s bond is 75% (using Basel II LGD for junior debt under foundation IRB approach). Assuming the credit spread remains 2% (10% - 8%), what is the implied probability of default (PD)?
Instead solve for PD:
1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]
I get PD = 2.42%
9c. The answer assumes annual compounding. If we instead assume continuous compounding (i.e., risky bond returns 10% continuous and T-bill returns 8% continuous), what is the implied loss given default (LGD)?. Notice that compounding matters; semi-annual compounding would give a slightly different result.
The no-arbitage indifference is now:
EXP(i) = (1-PD)*EXP[k] + PD*EXP[k]*(1-LGD)
solving for PD, I get 19.8%
9d. [hard] As Saunders writes, “Collateral requirements are a method of controlling default risk; they act as a direct substitute for risk premiums in setting required loan rates.” Let collateral (recovery) = 1 - Loss given default (LGD) and solve for the credit spread as a function of the probability of default (PD) and the recovery (c).
1+i = [(1-PD) * (1+k)] + [PD*(1+k)*(1-LGD)]—>
1+i = (1+k)*[(1-PD) + [PD*(1-LGD)]
...divide both sides by [(1-PD) + [PD*(1-LGD)]:
(1+i)/[(1-PD) + [PD*(1-LGD)] = (1+k)
...since we want the spread (k-i), subtract (1-i) from both sides:
(1+i)/[(1-PD) + [PD*(1-LGD)] - (1+ i) = (1+k) - (1+i), or
(1+k) - (1+i) = k-i = (1+i)/[(1-PD) + [PD*(1-LGD)] - (1+i)
note: Saunders gives an equivalent expression where p = prob of repayment = 1- PD and gamma = recovery = 1 - LGD
so that that above: (1+k) - (1+i) = k-i = (1+i)/[(1-PD) + [PD*(1-LGD)] - (1+i)
= k-i = (1+i)/[gamma + p - p*gamma] - (1+i); Saunder’s version
9e. Which continuous probability distribution—that is reviewed in the assigned Rachev (Fat-tailed and Skewed Asset Return Distributions)—are we most likely to find in use to characterize the loss given default (LGD) or recovery function?
Beta distribution (flexible)
9f. How are recovery rates (1-LGD) estimated?
Postdefault prices (if possible) or estimated recovery (e.g., PV of estimated cash flows)
9g. What are the most important determinants of recovery/LGD?
Seniority (arguably most important, but the others are not in any order)
Collateral
Macroeconomy & Business cycle (systemic risks)
Industry
