Mini Chapter Two
Hazard Rate/Default Intensity
Hazard Rate/Default Intensity – is the common academic reference to the Rate of probability of default (Pd) over a certain time period that’s conditional on the underlying credit/issuer having survived till the beginning of that time period. In theory, to calculate the present value of CDS payments and expected pay-offs pre and post default we use the unconditional counterpart of Pd (let’s call it UPd,t to denote unconditional probability of default in time t) denoted as:
Let’s illustrate the computation of the credit default swap spread. Few things to remember: a) Hazard rates apply ‘during’ a time period and determine the probability of survival at the end of that time period b) unconditional probability of default is computed as in the notation above c) defaults are assumed to occur at the end of every year, and for year 1 the unconditional probability is the same as the hazard rate.
Table 1 – Present Value of Expected payments for CDS = s per annum
| Time period | Hazard rate | Survival probability | Discount Factor | PV of Expected payment |
|---|---|---|---|---|
| 00 | 100% | |||
| 1 | 2.0% | 98.0% | 95.2% | 1.0s |
| 2 | 2.8% | 95.3% | 90.7% | 0.9s |
| 3 | 3.5% | 92.0% | 86.4% | 0.8s |
| 4 | 2.5% | 89.7% | 82.3% | 0.8s |
| 5 | 1.0% | 88.8% | 78.4% | 0.7s |
| Total | 4.1s |
Table 2 – Present Value of Expected pay-off in the event of default
| Time period | Probability of default | Expected pay-off | Discount Factor | PV of expected pay-off |
|---|---|---|---|---|
| 1 | 2.0% | 0.4 | 95.2% | 0.0 |
| 2 | 2.7% | 0.4 | 90.7% | 0.0 |
| 3 | 3.3% | 0.4 | 86.4% | 0.0 |
| 4 | 2.3% | 0.4 | 82.3% | 0.0 |
| 5 | 0.9% | 0.4 | 78.4% | 0.0 |
| Total | 0.1 |
Contractually then, payments made in case of no default should equal (or be an insurance for) the pay-off that’s received at the time of default. Hence:
4.12s = 0.0589, i.e. s = 142.96 basis points
Real world and Risk neutral probability of default
Real world and Risk neutral probability of default – is an important differentiation to consider when thinking of what default probability estimate to use. Pd that’s backed out from the market traded credit spreads (that have a recovery value assumption) is the risk neutral probability of default. But these can be different from the historical pattern of default rates or Real world Pd for the same obligor as markets and macro environment changes. The term ‘Risk-Neutrality’ can also be understood to suggest the default metric that would imply a credit spread to neutralise the current credit risk exposure. Hence traders use the risk-neutral Pd to value their credit portfolios/pnl for a fair reflection of default adjusted cash flows. But estimating let’s say the Credit VAR (Value at Risk – to be covered in future write-ups on Risk) which entails figuring out the potential future losses based on a historical pattern of default, Real world Pd comes in handy.
- Connecting the Dots
You can think of the risk neutral Pd metric as what’s implied by the live market credit spread curve, quite analogous to the implied volatility backed out of the market traded price of an option. Or extending the analogy further, real world Pd can be thought of as realised vol while risk neutral Pd is akin to the implied vol.
As for the differences between Real World Pd and Risk-Neutral Pd, the following reasons can be attributed: a) illiquidity of corporate bonds which warrants a higher credit spread b) recency bias i.e. traders might be more inclined to price in a credit blow-up depending on how recently it has happened vs history c) default correlation risks or contagion risks where credit risk from a similar industry/sector are prone to defaults as they react similarly to changing market variables.
