Hazard Function & Yield Degradation

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Fixed-income assets come with default risk. For example, the default probability statistics for commercial mortgages by years since loan origination look like this:

Source: Commercial Real Estate Analysis and Investments by D. M. Geltner, N. G. Miller, J. Clayton, P. Eichholtz

In the event of default, it is often impossible to recover the full amount of the invested capital… which ultimately leads to a difference between the contractually expected IRR and the actual IRR — this is called “Yield Degradation.”

In the example below, a 5-year loan defaults in year 4, and only 80% of the expected amount is recovered. As a result, instead of the agreed IRR of 10%, the investor ends up with just 4.89%. The difference between these two figures is what is referred to as Yield Degradation:

An interesting point here is that the later the default occurs, the lower the yield degradation tends to be.

Understanding this reality leads to the need to calculate the probability of default and incorporate its impact into the pricing of contracts in a meaningful way. In other words, investors should understand what kind of return they can realistically expect on their investment, given the likelihood of default.

To achieve this, historical statistical analysis is conducted, credit rating systems are developed, and default probabilities are determined. For example (ChatGPT):

In order to calculate the expected return based on default probabilities, certain mathematical steps are necessary. First, based on statistical data, we define the Hazard Function, then compute the Survival Probability, Cumulative Survival Probability, and Unconditional Default Probability by year. Finally, we arrive at the expected IRR (it’s a bit complex, but manageable in Excel).

Definitions:

• Hazard Function – The conditional probability of default in each period (year). “Conditional” means the probability that default happens in the given period assuming it hasn’t happened in any prior period.
• Conditional Survival Probability = 1 – Hazard Function
  The probability of not defaulting in the current period, assuming survival through all previous periods.
  If we know the default rates by period, we can calculate this.
• Cumulative Survival Probability [t] = Cumulative Survival [t-1] × Conditional Survival [t]
  The probability that default has not occurred up to a specific period.
  Since the survival in any period depends on survival in the prior periods as well, we multiply the survival probabilities together to get the cumulative value.
• Unconditional Default Probability = Cumulative Survival Probability × Hazard
  This is the actual probability that default will occur in a specific period.

Mortgage Default Probability Excel

p.s.

The diagram below shows both Yield Degradation and Default Risk. These are two distinct components. The first reflects the expectation of default, while the second shows the range of deviation from that expectation — one is the average, the other is the potential scale of deviation.

Source: Commercial Real Estate Analysis and Investments, D. M. Geltner, N. G. Miller, J. Clayton, P. Eichholtz