Where’s the real hazard in Hazard Rates?

All fixed income securities carry inherent default risk, and investors should be compensated for the potential loss of capital that is associated with such investments. The spread received or demanded by investors should have a logical mapping to the combination of the integral of the time weighted probability of default of the entity, as well as the loss given default. Spreads should at least compensate investors for the potential loss that would be realised, as well as some other risks that are associated with those instruments, such as mark-to-market risk and liquidity risk.

I often hear puzzling comments made about default risk, and these normally revolve around perceived risks associated with governments and large banks. One argument is that these instruments carry so little risk that they can even be considered “risk-free”, and thus the very concept of associating a probability of default for securities issued by these entities is as useful as a pair of sunglasses for a man with one ear. I would never argue about the fact that the instantaneous probability of near risk-free entities is exceptionally low, but I would contend that the instantaneous probability of a very risky security is also very low. What is of much more interest to the practitioner is the integral of the default probability over time. The cumulative probability of default.  

If we consider a continuous probability of default, aptly named a hazard rate (HR), a simple model of the probability of default (PD) between time 0 and time t would look like this:

Hazard rates themselves are derived from the spread (excess coupon) of a risky cash flow (coupon) and the expected loss of that security upon a default occurring. A simple example of this would be a one-year investment in a company that would realise 50% of its value upon default. If the spread of the coupon is 1.5%, we can derive that the probability of default over one year is 3%. Following on, for a 5% spread (excess coupon), the probability of default for the same period would be 10% in this simple model.   

Figure 1: Modelled cumulative probability of default for differing hazard rates

Source: Fairtree

Figure 1 shows the cumulative probability of default for a firm with a hazard rate of 10% (Company A) and another with 3% (Company B). When looking at the figure, one can observe that the implied probability of default of the entity with the higher hazard rate dominates the trace of the entity with the lower hazard rate. The implied probability is always higher for any period.

Does this make the investment into Company A inferior to a similar investment in Company B? Absolutely not. Essentially, the practitioner has (hopefully) priced in this higher probability of default and demands a much higher spread to compensate for this risk. The increased risk is inherently priced in to reflect the higher likelihood of Company A defaulting. Company A has a 63% probability of defaulting within 10 years, whilst Company B has a 26% probability. Arguably, if either of the companies does default after 10 years, what has not been priced is that probability that is above the traces – in this case 37% for Company A and 74% for Company B. Which investment suffered higher losses? Well, obviously, the one with the lower coupon, in this case, Company B. Arguably, the lower the implied probability of default, the lower the unpriced risk or the “surprise” of default. It remains obvious that unpriced risk is the real risk in portfolios of defaultable securities, as the eventuality of default (which will happen in the model) has not been adequately compensated.   

What is also observable is that the difference between these two curves actually decreases with time. In other words, the model implies that the relative riskiness of the entity with the lower hazard rate is actually increasing with time. Figure 2 below shows the difference in cumulative probabilities for different time periods, effectively computing the discrete probability of a default in that period.

Figure 2: Modelled discrete probability of default for differing hazard rates

Source: Fairtree

The favouring of one of the securities over the other could be construed as a risk swap between different time periods. Favouring the entity with the lower hazard rate merely swaps risk from the short run to the long run. Entities or companies with higher implied hazard rates are taking a lot of short-term risk, but lower relative long-term risk. One could use an analogy of stating, “If the firm can succeed in the short run, it will be fine in the long run”. The corollary is also true for the low implied hazard rate firm. Firms that are already successful, although having low short-term risk, probably have much more that can go wrong in the longer run, thanks to changes in their operating environment, new technologies, changes in legislation, etc. One can even use a basic investment analogy for a youngster who wants to avoid risk completely in their investment portfolio by only investing in money market portfolios. Yes, they avoid near-term risks but switches that for higher probabilities of missing performance targets in the long term. Unfortunately, as the saying goes, one cannot focus on the short run and the long run simultaneously. 

So hazard rates are a measure of the instantaneous probability of default, whilst integrating those probabilities gives us a tool to consider the cumulative probability of default for a given period. It should be obvious to all concerned that the cumulative probability of default of any entity should increase with time and that the cumulative integral will always end at 100%. When one makes this statement, one has to acknowledge that the ex-ante terminal probability of any entity, irrespective of its creditworthiness or being deemed “risk-free” is 100%. It’s not a question of “if” a company or entity will default, but more of a question of “when” it will default. When one finally accepts this, one realises that the adequate diversification of any portfolio of defaultable securities becomes of paramount importance. One certainly cannot state that because your preferred issuing entity has an anticipated low probability of default that you do not need to diversify. This could also be deemed imprudent, as your assessment of that probability may have been completely wrong. “Oops, sorry for that”. Even the smartest of fund managers make errors in their assessments of the probability of default. A manager’s only protection against making excessive errors is adequate diversification. Well-diversified credit portfolios with (literally) hundreds of entities are the best protection against getting things wrong. I would argue that the Dunning-Kruger effect, where overconfidence or “smartest man in the room syndrome” is definitely in play. 

The European Undertakings for Collective Investment in Transferable Securities (UCITS) take this concept into consideration in setting out obligor limits for portfolios and thus protecting investors against overconfident investment managers. They have determined that a simple 5/10/40 rule would go a long way in minimising the losses associated with an entity defaulting in a UCITS fund. So what is this 5/10/40 rule? Well firstly, the legislation makes no mention of creditworthiness or government versus corporate versus Special Purpose Vehicle, but rather sets issuing entity exposure limits in an easy-to-understand (and monitor) way. The unconstrained maximum exposure to any entity is 5%. A fund manager can increase the exposure to up to 10% but the summation of all exposures larger than 5% must be less than 40%. So the maximum exposure that an investor can have through a UCITS fund to German Bunds, or UK Gilts, or Swiss Government bonds is 10%. It also makes no mention of tenor or even where the specific instrument sits on the issuing entity’s balance sheet – senior and subordinated instruments are treated the same.

Perhaps the question that we should be asking our credit managers is not how many instruments they hold, but what capital exposure do they have to issuing entities?

Indeed with its particular nuances, the South African market could never adopt a UCITS-type framework – the very index that South African bond managers use would be non-UCITS compliant. However, when moving to the multi-asset income category, rather than looking at priced risk, one should perhaps look at the consequences of a tail event when an entity with a low implied hazard rate unexpectedly defaults. I really don’t believe that a response of “That will never happen and if it does, we are all doomed” is a respectable answer.

Unpriced risk remains the real risk to investors in defaultable (read “all fixed income”) portfolios. Couple this with oversized entity exposures, or not adequately diversified portfolios, and the investor is left with risk exposures they would rather not be taking. Adequate risk pricing to begin with and then portfolio diversification remain key to the production of stable long-term returns. It’s either that or get prepared for yet another “big surprise” sometime in the future.