Shanken Asset Pricing
Just how bad are traditional pricing factors?
Jay Shanken has made enormous contributions to asset pricing. The standard 2-pass OLS method of estimating the risk premium associated with risk factors goes like this. Begin by suspecting that some risk factors (size, momentum, shocks to the risk-bearing capacity of market-based intermediaries, or whatever) are priced into the cross-section of expected excess returns. Take a bunch of assets — one usually takes the 25 Fama-French portfolios sorted on size (“SMB”) and value (“HML”). In the first pass, regress the excess returns on each of these assets separately onto your risk factors in the time-series. The slopes of the asset returns against the risk factors are their betas — the sensitivity of the returns to your risk factors. In the second pass, regress the expected excess returns (ie, mean returns in excess of the risk-free rate) of the assets onto their betas in the cross-section. The slopes of the expected excess returns on the assets against the betas are the prices of risk, denoted lambda — the risk premium associated with the risk, or how much extra compensation investors can expect from an additional unit of exposure to the risk. If the pricing errors (the intercept plus the error terms) are small, if the fit of the cross-sectional regression is good (say, as measured by the adjusted-R^2), then you have a decent asset pricing model — you can say that your risk factors are priced-in.
So far so good. The problem is that this standard procedure is plagued by problems. In the first place, the error terms of the cross-sectional regression are nearly guaranteed to be heteroskedastic, meaning that their variances are different. In the 25 Fama-French portfolios sorted on size and value, for instance, the variance of the asset returns range from 1.66 to 4.58. Because OLS assumes that the variances are all equal, the standard errors are deflated — you are likely to overestimate the statistical significance of the lambdas. So you must use robust standard errors. But even robust standard errors are guaranteed to be too small. The reason is that OLS assumes that the predictors are measured without error. But this is certainly not true of the betas, since we estimate them from the time-series regressions. Shanken (1992) solved this problem by deriving a formula for standard errors corrected for measurement error, also called errors-in-variable (EIV).
Another major contribution by Shanken and his coauthors was to derive a test of asset pricing models based on what we may call the first law of asset pricing — there should be no compensation without risk. The idea in Gibbons, Ross, and Shanken (1989) is that the intercepts in the time-series regressions, called alphas, correspond to expected excess returns that are independent of systematic risk — as captured by the risk factors. If your model is good, they should all vanish. The null hypothesis of the GRS test statistic is that all the alphas are simultaneously zero. I just implemented GRS test in python. You can find the function on my GitHub. This is the reason for this dispatch.
In what follows, I will document a number of results. First, traditional asset pricing models suck. No, really, as we shall see, they’re just awful. Second, we will see that systematic risk in asset markets is a function of fluctuations in the risk-bearing capacity of market-based intermediaries — risk appetite for short. I have previously isolated the signal for risk appetite in the term structure of systematic volatility. Here I use an even simpler risk factor that contains a strong signal of risk appetite. Our risk factor will be the daily high of the CBOE’s 3-month VIX. We obtain the Fama-French factors and the portfolios sorted on size and value from Kenneth French’s website. We obtain our proxy for risk appetite from the CBOE’s website. All data is at the daily frequency from Dec 4, 2007 to Nov 30, 2020. Then we carry out 2-pass OLS regressions, taking care to report EIV-corrected robust standard errors.
We start with the famous Capital Asset Pricing Model. The CAPM sucks. Not only is the price associated with exposure to market risk not significant, it bears the wrong sign! This the source of the “betting-against-beta” trade.
What about the Fama-French 3-factor model? It should do well right? After all, these portfolios are specifically sorted on the Fama-French factors. Well, it sucks too. The prices of risk associated with size and market vanish. The price of risk associated with the value factor does not. But it bears the wrong sign! This means that investors lose money by gaining exposure to value. So the benchmark models are completely broken.
By comparison, our proxy for risk appetite, the daily high of the 3-month VIX, sports a large and highly significant risk premium. This means that exposure to this risk factor is handsomely rewarded. If you want to make high risk-adjusted returns, you want exposure to this risk factor. That is, you want to hold a portfolio that is very sensitive to fluctuations in this variable.
We also find that controlling for market returns improves the performance of our pricing model. The price of risk becomes even larger, although measured with slightly less precision. The adjusted R-squared goes up. We can now explain about half the cross-sectional variation in expected excess returns. The mean absolute pricing error (MAPE) falls dramatically, as does the GRS test statistic. In fact, we cannot reject the null that all the alphas jointly vanish at the standard 5 percent level of significance. This is what an excellent asset pricing model looks like.
We also carry out a “kitchen-sink” regression. That is, we throw in all the risk factors together and see how they perform against each other. This is not really kosher because it introduces severe multicollinearity. But worth eye-balling anyway. The price of our risk factor is still very large, although now only marginally significant. Market and Size are again statistically indistinguishable from zero. Meanwhile, Value continues to bear the wrong sign! Since we’ve thrown in all possible risk factors, the mean absolute pricing error is small.
Finally, we compare the model fit statistics. While the 3-factor Fama-French benchmark sports the lowest MAPE, the adjusted R-squared and the GRS test statistic shows that it is inferior to the 2-factor intermediary model. In fact, the 2-factor intermediary model is the only model that passes the Shanken test — no investor compensation without bearing any systematic risk.
Anyway, I just wanted to celebrate Shanken’s contributions and demonstrate the power of intermediary asset pricing. Here’s a scatter plot of expected excess returns against the betas for our proxy of risk appetite.
