Causal Insight into CAPM Class Models (Part III)
Interpreting the CAPM Equation
Social scientists often interpret regression results in terms of causal processes, even if the mathematics of regression is really just about quantifying relationships observed in data. Since active managers often look for models to predict returns, I initially thought of CAPM as mechanism to predict returns, as the name ”expected return“ implies. In causal modeling terms, CAPM is simply a regression model where a dependent variable (expected asset returns) is assumed to be predicted by an independent variable (market or systemic returns).
So, if one were to ask, ”what caused the return on asset A to be Y%?“ the answer under CAPM would be:
”The Market as a whole returned X% above the Rfr, and that caused the Asset to return (Beta*X%) more than the Rfr. The difference, [Y% - (Beta*X%)-Rfr] was caused by other [presumably unpredictable] idiosyncratic factors.“
This is all true, provided that you accept that the relationship between A and the Market is a causal one. This is fairly defensible on logical grounds, since if Beta is statistically significant, there is most likely a relationship between the variables and that relationship is much more likely to run from the Market to the asset than the other way around (it would be a magical Asset to drive the rest of the Market). Technically, it is possible that some third factor could drive both asset A and the Market, but if we don’t actually know what that other factor is, the Market is probably just as good a proxy for that factor as anything else. So the causal interpretation that the Market made the Asset return (Beta*X%) is plausible on first brush.
This logic works, but there is a big practical problem. If we are using the return on the Market to predict the return of Asset A, the problem is that we still do not know what the Market will actually return in the same time period. From an asset manager’s forecasting viewpoint, both the Asset return and the Market return are in the future and essentially unpredictable, other than (perhaps) their long term averages. Essentially CAPM is asserting that one thing that we don’t know yet (Asset A’s return in the future) is consistently caused by something else that we don’t know yet (the Market’s return in the future). It may be true, technically, but it doesn’t really help us if we don’t have a way to predict what the Market is going to return.
While there are no solutions to the problem of not knowing the future, there are some ways to approach it:
1) If one i) has a long time horizon, and ii) is interested in long term averages, and iii) believes that the historical past is reasonably representative of the long term future, it is possible to use CAPM-type models to estimate long-term average returns for an asset. For many purposes, this is simple and sufficient.
2) If one can develop a separate model that predicts overall market returns reasonably well - such as a time series regression or ARIMA - one can use that model as an input into the asset return model. This would increase the apparent overall risk of the asset pricing model, since the uncertainties of the asset model’s output uncertainty would be compounded by the the fact that one of the inputs (expected market return) is also uncertain. However, the total risk of using both models together is likely to be lower, because the market expectations model - if useful - should reduce the uncertainty in market expectations over the uncertainty of not having any expectations at all. With this approach, there is also model risk from two separate models: the asset pricing model and the market return model.
3) If one has an asset model and a market expectations model as in approach (2), it might be more efficient to try to combine the independent variables into one single model to predict returns on the asset. This would improve predicted asset returns if there is any correlation between an asset’s beta parameter and any of the factors that determine market expected returns.
4) One can try to eliminate the effect of market uncertainty by taking appropriately balanced and opposite positions in the asset and the market as a whole. For example, for every X% of long exposure in an asset, one can take Beta*X% short exposure in the market index, through ETFs, index futures, or in some cases the index itself. Provided that the returns model is a good one and is still appropriate, this has the effect of removing the uncertainty that comes with the market performance, leaving an investor with only the non-market or non-systematic component of an asset’s risk.
Causality, CAPM, and beyond
Approach (4) has some interesting implications for returns modeling. If one believes wholeheartedly in CAPM, then taking opposing positions in an asset and the market index such that net Beta=0 will effectively remove all systemic risk from a position and leave the investor with asset-specific (a.k.a diversifiable or idiosyncratic) risk only. In a pure CAPM world, there isn’t much advantage to this, because the expected value of asset-specific risk is zero (as discussed in Part 1), and the volatility created by assuming this risk will actually eat away at a portfolio over time.
The above assumes that the idiosyncratic component of asset risk is completely unpredictable, however. In Bob Litterman’s recent discussion of ”exotic beta“, he suggests that the returns to non-systemic factors such as size or value might average out to zero over the long term, yet still offer predictive value in the short term. Similarly, fundamental or competitive analysis might identify one company as riskier or less risky depending on something like competitive position, cash reserves, a valuation figure, or some other temporary and observable factor. In these cases, although the expected return on asset-specific risk is zero over the long term, it can be related to observable predictors in the short or medium term.
Believing that there are observable predictors for short term idiosyncratic returns requires believing that markets are not fully efficient - that it takes time for observable predictors to be disseminated and incorporated into market prices, and therefore presently known factors can predict some level of future returns. Moreover, there are empirical reasons to believe that markets are not fully efficient, such as the presence of momentum / autocorrelation in some markets.
If we assume that markets are not fully efficient, and that some observable variables can help in predicting short-term returns, then the use of a market factor in a returns model has another key role. Under these conditions, we have a model that states that some parts of an asset’s return are predictable and some parts are not. The effect of the predictable component may be small compared to the effect of the unpredictable component, but we would still like a way to incorporate this into an investment decision. If we have a variable, X, that we think can explain some part of a return, the model now looks like:
R_a = Rfr + (Beta_m) * (R_m) + (Beta_X) * X(t-1) + noise
Where R_a is the return on the asset, R_m is the return on the market, Rfr is the risk-free rate, and X(t-1) is the variable we think can predict returns. Variable X is measured at time=(t-1) because we know its value at the start of the investment period, but we can only receive the returns from an asset or the market at the end of this period.
The equation form can obscure the causal structure of the model. A causal diagram helps to make the model clearer.
[If not embedded, click here for causal diagram]
The model says that the asset return R_a is the result of a knowable variable X, and an unpredictable market return R_m. This means that the model says that variable X has a (Beta_X) effect on the asset’s return after controlling for the impact of (more or less) unpredictable market moves.
Isolating the value of investment insights
Including the (unpredictable) market return into the causal diagram has two implications, one statistical, the other financial. Statistically speaking, if variable X is uncorrelated with market returns, it is not strictly necessary to include the market return in the model, because a regression process should be able to pick up the statistical significance of X and estimate it’s beta in any case. However, the market return is still useful as a method of reducing statistical uncertainty and getting a more accurate estimate of X’s effect. The technical term for this is that the market variable functions as an instrumental (rather than causal) variable: we may not be able to predict the market return, but in the regression it reduces the quantity of unexplained variance.
In addition, it is statistically helpful to include the market return in the predictive model because there is still some chance that X is mildly correlated with the market return and so including the market return in the same period will control for that possibility.
Financially speaking, the advantage of this formulation is that the effects of the market return can be hedged so that a portfolio produces only the returns achieved from knowing X and not returns from assuming market risk. This is useful because the returns to variable X may be quite small compared to the overall effect of the entire market and might otherwise be difficult to capture.
For example, if variable X suggests that asset A is relatively attractive, that may be true, comparatively speaking, but even if the asset is better than its alternatives, the market as a whole may still go down, taking asset A with it and destroying any benefit of your insight into X. However, by shorting the correct number of market futures or shares of an index ETF, the effects of market uncertainty can be eliminated from the portfolio, and the effects of variable X isolated. This can be an attractive way of ensuring that your investment positions capture your insight into variable X independent of what the market as a whole does and maximizes the value of your insight. It does, of course, depend on your model of returns being both accurate and stable. In times of great market turmoil -- like the present -- models based on past performance data become less reliable and many fail completely.
Recap
What I hoped to show in this blog piece is how the specification of returns models connects to causal analysis, and why -- even if we cannot predict what the market return will be -- a regression-based returns model should still include a contemporaneous market factor. The main reasons are:
1) It produces more efficient (i.e. precise) estimates of the effects of other variables that you can (potentially) use for predictive modeling.
2) It controls for the possibility that your predictive variables might be somewhat correlated with future market wide returns.
3) The beta estimate for the market return factor can be used to hedge out the effects of the market return on your investment, leaving you with a position that capitalizes on the quality of your insight. Since the market return is often the dominant factor in the performance of individual assets, this can be very useful for managing the total risk one takes by acting on a specific insight (in this case, the predictive value of variable X).