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## A technical note on inferring cumulative default probability from credit spreads

I wanted to write a note about a mistake in one of GARP®’s Pre-Study Practice Exams. Unfortunately, although we’ve given plenty of feedback over the years, technical mistakes persist. Especially in regard to hazard rate properties (especially when applied to counterparty CVA), our members continue to spot solutions that are misleading. Ironically, some misunderstandings are more likely under the new material, which often simplifies without corresponding illustrations, than in prior years when far more detailed readings (with copious examples) were provided.

The impetus for this note was understandable confusion surfaced in our forum here at https://trtl.bz/3huQOD1. The source Q&A is GAPR’s 2021 Pre-Study Part 2 Question #17. The topic is a subtle methodological mistake in their solution to this hazard/PD problem. The core concept is remarkably simple and highly testable: we are given credit spreads and asked to infer a cumulative default probability from the spreads (conveniently assuming zero recovery). So elegant and non-tedious is the exponential in continuous time that you are almost guaranteed to see some variation on the Part 2 exam. My illustrated XLS is here https://www.dropbox.com/s/czpzv2al2nw0xcz/garp-2021-PE2-Q17-hazard.xlsx and the screenshot is below.

The question’s assumptions are shown in yellow, as usual. The XLS snippet below has four quadrants. The upper panel assumes continuous compounding, the lower panel assumes annual compounding. The upper-left is the problem as given: a flat riskfree (Treasury) yield curve at 2.0% per annum; a corporate bond with an upward-sloping yield curve (4.0% @ 1 year, 7.0% @ 2 years, 10.0% @ 3 years). The question asks: if recovery is zero, what is the risk-neutral 3-year cumulative default probability (cumulative PD) of the corporate bond?

The given solution infers the 3-year spread of 8.0% as the hazard rate and solves for 1 – exp(-8.0%*3) = 21.34%. This is the correct number but it is a coincidence that is obligated to the combination of a flat Treasury curve and the continuous compound frequency. This incorrect approach suffers from a misunderstanding (Before GARP replaced superior source material with their own P1 material, Saunders was assigned and careful readers of Saunders are far less likely to make this mistake! Although GARP always struggled with the hazard/PD question pattern; it’s one of their historical question types with errors in most of their instances, which has been a perennial source of confusion.)

The correct approach is to infer the forward rate curve, in this case, given by {2.0%, 8.0%, and 14.0% in blue} and let this impute a conditional default probability in each year. The hazard rate, after all, is an instantaneous conditional default probability. In the upper right panel, my only change is to render the Treasury curve non-flat and the problem is revealed. Notice that continuous compounding is elegant: we can solve for 3-year cumulative PD = 1 – EXP(- ʎ1 – ʎ2 – ʎ3). But, you might be wondering, what about the familiar T-year cumulative PD = 1 – EXP(- ʎ*T) which is an essential FRM® formula in P2.T6? This convenient expression assumes a constant hazard rate (notice the hazards are not here constant). The bottom panel illustrates the same idea but it assumes the yields are “per annum with annual compounding”. I hope that’s interesting!

P.S. Okay, I will answer here my own question:

• What is meant by “risk neutral” when the question asks, “If the recovery rate on the 3-year BBB-rated discount bond is expected to be 0% in the event of default, which of the following is the best estimate of the risk-neutral probability that the BBB-rated discount bond defaults within the next 3 years?”

Answer: The risk-neutral modifier is relevant here. It enables the step that solves for the forward probability of repayment, p, given the forward spreads. For example, the 3rd year conditional probability of repayment is given by p(3) = exp(2.0%)/exp(16.0%) =86.94%. That’s elegant but why can we do it? Because we’re assuming that p(3)*exp(16%) = exp(2%). On the right-side is the future value (i.e., at the end of one year) of \$1.00 invested at 2.0% per annum with continuous compounding: exp(2.0%) = \$1.0202; it has no multiplier because it is considered safe. On the left-hand side is the future value of \$1.00 if it returns 16.0% but weighted by the probability of repayment: p(3)*exp(16.0%) = p(3)*\$1.17351. Solving for p(3) as a function of these respective forward rates depends on the equality:

• certain \$1.0202 = [p(3) = 86.94%] * uncertain \$1.17351

Those are equal because we said we want the risk-neutral probability: we assume the expected return on the risky (Corporate) bond equals the expected return on the safe (Treasury) bond. But if we are risk averse, then

• certain \$1.0202 < [p(3) > 86.94%] * uncertain \$1.17351 … so maybe the following but my 90.0% is arbitrary:
• certain \$1.0202 < [p(3) = 90.00%] * uncertain \$1.17351; i.e., \$1.0202 < \$1.05616

Because the Corporate bond could default, and with zero recovery, the risky route could leave us with nothing! The greater our risk aversion the higher the expected return needs to compensate us for the risk of default. Put another way, given the same forward spread of 14%, risk aversion says the implied survival probability is higher than exp(-14%) = 86.94% because risk-averse investors demand higher expected yield as compensation for the uncertainty (the possibility of losing the entire principle). The “risk neutral” assumption enables the “=” that solves for the survival probability. Here is my YouTube video on risk-neutral probabilities.

In the lower panel, annual compounding makes this logic a bit more explicit (because we can’t apparently “skip” the p due to log properties). Here p(3) = (1+2.0%)/(1+10%) because the risk-neutral assumption enables (1+2.0%) = p(3)*(1+10%). Then the 3-year cumulative probability of repayment is given by 98.08% * 92.65% * 87.74% = 79.73%, and the 3-year cumulative PD = 1 – 79.73% = 20.27%. I hope that’s helpful!

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