# What data transformations to use in regression of credit spreads on equity prices?

Clearly there is a strong relationship between credit spreads and equity prices (both theoretically and empirically). But how would one go about formulating a regression which seeks to explain this relationship?

To keep it narrow, let's say we have the following time series data;

• S&P500, daily % changes

• Generic 5 year swap yields, daily differences

• Generic 5 year high yield credit spread index, daily differences

The objective is to explain the daily movements in credit spreads by the changes in swap yields and equity prices.

I'm interested more in how one might think about structuring the regression (eg transforming the variables) to get the best fit from a linear regression, rather than the theoretical economic underpinnings or choice of data comprising this particular problem.

In particular i want to know how one might transform the inputs to be able to cover the case of when credit spreads are high (sensitivity to equity changes is also high) and when credit spreads are low (sensitivity to equity changes is also low), or if a different approach is warranted.

Among the potential other issues are that equity price changes are fat tailed, and have negative skew. Swap yields and credit spreads are both mean reverting (and lower bounded). Credit spreads depend on swap yields and equity prices, but swap yields also depend on equity prices.

Does it make sense to maybe do something like;

• transform each series by dividing by it's volatility (or it's implied volatility - additional non-linear issues with that?)

• regression on ranks rather than underlying data

• transform each series with a sigmoid function, possibly in combination any of the above

• other suggestions?

I appreciate this might be a wide topic, but am very interested to here how people with knowledge of time series statistics and finance might approach it.

• My first intuition is that more likely that it's either equity implied volatility and credit spreads that should be regressed, no? Jan 9, 2013 at 7:40
• @Strange. You mean along the lines of Merton model? That's more for a static valuation (using levels of equity, equity vol, face value of debt), but i'm looking at time series of changes in variables. Apologies if i've misunderstood your comment. Jan 9, 2013 at 9:34
• Not even thinking thus far, just simply pointing that equity options are more like the CDS rather then equity itself. If you look at the regression of VIX vs something like 5y CMT IG, it's a pretty reasonable regression, since both series are more or less stationary and are indicators of risk. Jan 9, 2013 at 17:01
• That's a fair point and while it probably holds over short horizons, it won't over longer periods. The coefficient on changes in vix also depends on the level of credit spreads. How would you think about transforming the data to stabilize the coefficient across both high a low credit spread periods? Jan 10, 2013 at 14:29

## 3 Answers

The most common transformations you see for these three variables on credit desks is to compute "returns" on the credit variables. So, rather than taking the straight daily differences $\Delta s_t$ of swap spreads and $\Delta H_t$ of the high yield index (by which I assume here you mean on-the-run CDX HY), practitioners will transform to $\frac{\Delta s_t}{s_t}$ and $\frac{\Delta H_t}{H_t}$. (This works in the case of CDX HY because of the 100-upfront quoting convention).

Now, there's no good evidence for any particular model of the $\Delta s_t$ and $\Delta H_t$, which means this lognormal-like approach is unsupported by the data, but not really much worse than anything else one might choose. It has the advantage of providing positive regression coefficients and never sending $s$ below zero, though it can send $H$ above it's theoretical maximum value.

What you'll find, however, is that over macroscopic periods the credit and equity markets exhibit regimes'' of correlation and anticorrelation between equity prices and credit spreads, not least because special dividends, share buybacks and mergers go in and out of fashion. This is easily observable from 2000-2010. Thus, no model you make is going to be particularly trustworthy.

I am a bit confused about your question in that you say at one point that you want to explain the relationship between credit spreads and equity prices. Is that what you really want to know? Why? I thought you already have empirical evidence that supports the relationship between the two? Or are you after something else?

Anyway, I would actually recommend you to run the principal components and only after that hack away with regression tests. You may also want to introduce other time series as Strange suggested (implied vols,...). You are severely limiting yourself if you only test the three mentioned time series.

But if you cant help it to dive into regressions then I would normalize the data rather than working with percent returns.

• I want to model the changes in credit spreads as a function of changes in equity prices and changes in swap yields, but it seems difficult to find a functional form that stabilizes the coefficients (in a rolling regression). Jan 9, 2013 at 9:30
• thats why I suggested you start with PC so you understand the explanatory power of each component first. Jan 9, 2013 at 9:36
• @Freddy 1) If both the equity returns and spreads are in percent, then the equity returns will dominate the first PC (if done on the covariance matrix). 2) I used to create factors when some variables were stationary and mean-reverting, but I have been less comfortable with that approach over time. Have you noticed any downside?
– John
Jan 9, 2013 at 15:55
• @John, just wanted to clarify what you mean, I recommended using normalized data not percent returns. I had standard deviations away from some sort of own mean in mind Jan 9, 2013 at 16:08
• That's effectively PCA on a correlation matrix, so what I said probably wouldn't apply.
– John
Jan 9, 2013 at 16:46

In short, I would not make any of the transformations you suggest and rely on traditional statistical tools like Vector Autoregressions and Garch.

• Would you not think you slightly aim to overshoot here? You basically suggest the entire arsenal of quant tools to look at some basic relationships and attributions. Jan 9, 2013 at 16:11
• @John. When the level of credit spreads is high, then the sensitivity of changes in credit spreads to changes equity prices is high, and vice versa when credit spreads are low. Doesn't this beg for some kind of transformation? Jan 10, 2013 at 12:10
• I agree with Freddy here. There's too much going on to expect any kind of reasonable predictive outcome from VAR or GARCH. Jan 13, 2013 at 17:46
• @Brian. In such cases does it make sense to instead convert all variables to ranks, or signed ranks, and run a linear regression using the ranked variables? My thinking that this might clear out some of the non-linearity. Can one use GARCH on ranked data? Jan 15, 2013 at 9:20