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WIFE Week in Financial Education (2012-05-17)

David Harper CFA FRM

David Harper CFA FRM
Staff member
Welcome to another Week in Financial Education! This week saw some great questions and fascinating insights. I will just highlight the instructive example of a flawed solution in the 2021 practice paper (as usual, the impetus is candidates' justifiable confusion). This one is tricky because the correct answer is coincident to two implicit assumptions, but the approach will break under other conditions! One thing we've always cared about here (at Bionic Turtle) is mastery of the fundamentals. We don't just want you to pass the exam. We want you to takeaway a robust toolkit. Part of that toolkit is fluency in the fundamental building blocks of finance, including its vocabulary. What exactly is meant by an "interest rate factor" and what are the possible instances of such a factor? The question (below) that we analyzed refers to bond yield, spread and the implied hazard rate. Yield and hazard rate do have specific definitions, but spread requires clarification. (As another example, the question ultimately asks for "risk-neutral" cumulative default probability; what step in the solution does this risk-neutral assumption enable?). I hope you appreciate that we support your aspirations as a Professional by seeking to assist in the clarifications that constitute a robust and coherent toolkit.

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David Harper CFA FRM

David Harper CFA FRM
Staff member
Note: On the methodological error as pertains to hazard rate in GARP's 2021 Practice Exam (P2 Pre-Study #17 )

Here is the detail on the GARP's methodological mistake in their solution to the hazard/PD problem linked above (GARP 2021 Practice Exam Part 1 Question 17). My illustrated XLS is here https://www.dropbox.com/s/czpzv2al2nw0xcz/garp-2021-PE2-Q17-hazard.xlsx?dl=0 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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