# Corporate Debt and the Crisis

This is more of a quick "research note" than a blog post, but it seems that this is what my personal blog has become. Here was my broad research question: How did corporate bond yields behave during the 2008 crisis and its aftermath? In particular, I was interested in modeling the behavior of these spreads over time by regression.

The reason why I am interested is because I've been thinking recently about financial stability -- as you can see here -- and the liquidity panic in the fall of 2008 through the winter of 2009 is just the sort of instability one would hope to avoid going forward. These are complex issues, but one way I thought I might begin to understand them was by looking at how the corporate-bond market treated risk, from a positive and empirical standpoint.

You can download all of the data I used here, from FRED. The FRED data comes from Bank of America Merrill Lynch's corporate bond indexes, which have been divided by credit rating. The ratings I have are: AAA, AA, A, BBB, BB, B, and CCC and below from 1997 to present. That's 4,315 observations, daily at the close of trading, and I am pretty pleased to have that size of a dataset.

My research method was simple. I wanted to fit a linear model to every day's worth of observations and then see how the model's parameters changed over time. I left a quadratic term in the model to capture any potential nonlinear behavior. The form of the model was:

Where

*i*is the interest rate,

*r*is the credit rating, and a, b, and c are fit by linearization and ordinary least squares.

Since credit ratings do not neatly translate into values, this requires one extra step. I initially assigned each rating from AAA to CC and below to the positive integers 1 to 7. Then I fit the model. The next step was to adjust the mapping such that the average value of the quadratic parameter was zero using Excel's solver. Then I refit the model.

The rationale for zeroing out the quadratic parameter is to monitor the concavity of the function over time. It would be interesting if concavity changed predictably. It is to also make sure that none of the slope parameter is accidentally captured in the quadratic parameter (which it would be, initially, because of the incorrect mapping).

I won't discuss the quadratic parameter much further, because the interesting result of it is a negative one: There has been basically no statistically significant behavior on concavity. 96 percent of the time, the parameter is not statistically different from zero with 95 percent confidence. None of that 4 percent of time it was significant was during the crisis.

This is an unexpected and perhaps important finding. Investors do not become especially averse to the riskiest bonds during a crisis, or at any time, which is something you might expect. Alternatively, investors don't decide that "all risk is bad risk" in a crisis and let concavity fall below normal. I will now put aside the quadratic part of the model.

The slope parameter and intercept are important in their own respects. I think of the former as the market price of risk and the latter as the nominal "riskless" interest rate. I use the word "riskless" with caution, because the intercept rate removes the risk associated with the lower credit-rating bonds.

Since there are many better ways to approximate a riskless rate, I will put that aside. The slope is more interesting -- here it is graphed:

The time series looks much like the CBOE volatility index, or "VIX," a measure of expected volatility from options prices, or indexes of financial stress. Which is not surprising, as this is a measure of stress or risk in corporate bond markets. It increases roughly tenfold during the recession.

The important conclusion here is that risk is, in a sense, homogeneous in corporate-bond markets. There is a "single price" for that risk, after adjusting for concavity. Let me know if you have any ideas as to how this might help us understand how financial crises work. I'm still thinking.

**Update: **The r-squared value of the linear fit is 0.960, for 4,315 observations.

**Update 2: **After some more tweaking of the model -- I realized the optimization goal should be mean squared deviation from the linear model, not zero for the quadratic term's coefficient -- I have the r-squared at 0.9906 and a root mean squared error of 45 basis points.