Equilibrium and Corporate Spread Reversion

One way to  show the validity of a financial model’s equilibrium is to demonstrate mean reversion about that value. From an investment management perspective, evidence of mean reversion is an opportunity for outperformance. Here, we will suggest evidence of mean reversion in corporate bond spreads for the bonds in the S&P 500 energy sector, from December 2015 to June 2016.

The basic elements of the corporate spread model are:

• Credit default swap spreads
• Secondary credit, reflecting the asymmetry of rating transition outcomes
• Liquidity compensation
• Residual term

The model is described in greater detail below. The framework is not heavily calibrated, but it does bring out central tendencies in the movements of corporate spreads. As for the factors themselves, liquidity compensation is bond specific. Secondary credit is mainly market specific, through the calibration of market spreads and rating transition. Ultimately though, secondary credit represents broad market exposure.  Merton and residual are the interesting factors. 

Let us consider Merton first.  After deriving the comparable credit default swap (CDS) value for a particular bond, fit a regression of:

LN(CDS) = α + β · LN(leverage) + β * LN(equity implied vol) + β ∙ LN(time to maturity)

Following that, compute the residuals. Rank the change in the residuals each month, and, one month later, compare the change in CDS. Next, consider the residuals versus the overall model and repeat the exercise. The third test considers the possibility that large positive CDS residuals in turn cause large negative spread residuals, and we are measuring two sides of the same phenomena. In this instance, the sum of the CDS and spread residuals are ranked, and the change in spread is computed. (Please note, tested but not reported are results based on percentage of spread instead of absolute levels.  Results show strong but slightly different outperformance in this case as well, though with some different bonds.) 

The results of the three tests are shown here:

All three show clear evidence that extreme changes in one month are followed by reversals in the following month. While the model is not exact, it does indicate a search for equilibrium in the corporate bond spread market. In addition, the first and third tests are actionable investment strategies.

Model for Corporate Bond Spreads

The Merton model for credit spreads views equity as an option on the assets of a company, and thus corporate bond spreads as compensation for that optionality. The key variables are the company’s leverage and its asset volatility. However, both variables are not observed in the market, and are revealed via a set of relations to the stock price and the implied volatility of its equity options. To make this more difficult, the leverage and the implied volatility are interconnected. If Merton is a complete view of corporate spreads, then the credit default swap market should completely explain individual corporate bond spreads.

However, there is more to corporate spreads than CDS. Residual plots quickly reveal the impact of term and rating, effects which should have been removed via Merton. An interesting idea is that there is more to go wrong to a corporate bond than simply default. For instance, if one owns a high grade bond, only bad things can happen to it. Similarly, if ones owns a high yield bond, assuming it doesn’t default, only good things can happen to it. If long run rating transition studies are consulted, it is possible to estimate the probability of:

• A good thing happening, and the market implied spread compensation of that transition
• A non-default related bad thing happening to that bond, and its associated spread compensation

The simple difference between bad things minus good things does an effective job of reducing the residual and the rating and term influences. This term is further adjusted to reflect that the transition studies include an estimation of probability of default, which is different than the probability implied by Merton. This effect is called Secondary Credit.

Finally, corporate bonds are less liquid than stocks, so some additional compensation for that illiquidity is warranted. A simple model is to view a corporate bond as trading among participants and eventually ending up in a held to maturity portfolio, where it will never be traded again. The two drivers of that transition are the:

• Size of the issue, where larger issues take longer to find a home
• Time since issuance, where the longer the time since issuance the more likely the bond has found a home

These two variables can be used to produce a hazard rate.  From a hazard rate it is a short step to a spread.  While very simplistic, this model provides a starting point for more nuanced models.

There is another widely accepted approach to corporate spreads, called Duration Times Spread (DTS), which relates corporate spread movements to common sector and rating curves.  These two models have diametrically opposite views of corporate spreads and could appear to be in contradiction to each other.  As discussed earlier, Merton is a bottom up model, with the source of risk being the individual company’s leverage and asset volatility. 

DTS is a top down model, based on the relationship:

σ(spread returns)= Duration · spread · σ(relative spread moves)

From this perspective, all bonds have the same source of relative spread risk. 

In terms of similarity, one characterization of Merton asserts that spread is:

LN(probability of survival)/time to maturity

Phrasing in terms of DTS, and assuming the duration of a zero coupon bond, the relationship becomes:

Time to maturity · LN(probability of survival)/time to maturity

Thus, Merton and DTS together imply that the market is pricing lines of equal probability of survival so that long bonds with high probability of survival are priced equivalently to short bonds with low probability of survival. This view of DTS is consistent with the CDS residual test studied above.