Dr. P.V. Viswanath

pviswanath@pace.edu

A brief description of the CAPM:
The CAPM provides a way of estimating the required return on an asset, using an equilibrium model. According to this model,

E(r_i) - r_f = b_i [E(r_m) - r_f],

where r_f is the risk-free rate, [E(r_m) - r_f] is the risk premium on the market portfolio and b_i is the asset's beta -- that is, the sensitivity of the asset return to changes in the return on the market portfolio. Even though the assumptions required to derive the CAPM are pretty stringent, the model's pricing relationship is very intuitive and sensible. Since investors can diversify, they only need to be compensated for risk that cannot be diversified away. Hence, in the limit, they would hold the market portfolio. Consequently, the risk-premium that they need to be attracted to hold a particular asset is simply that asset's beta with respect to the market portfolio.

Problems with the CAPM:
The model does seem to assume that all investors hold the market portfolio -- this is clearly not true; however, one would expect investors to compensate in their holdings of traded assets to adjust for assets that they cannot trade, such as their human capital. Of course, to the extent that their human capital does not capture market-portfolio-like features, investors would also seek to be compensated for risk that is correlated with changes in the value of their human capital. This is also the conclusion that Zhenyu Wang and Ravi Jagannathan come to.

There is another problem with the CAPM; this time, not with the theoretical basis for the model, but rather with the way in which it is implemented. In other words, researchers and practitioners usually assume that betas are constant over time and use historical betas to estimate current betas. This is not necessarily true. For example, over the business cycle, leverage goes up and down and this affects equity betas. Asset betas are also affected to the extent that different businesses are affected differently by the business cycle. To understand this better, consider the following example from Wang and Jagannathan:

Suppose there are two stocks and two periods. Suppose the beta of the first stock is 0.5 in the first period and 1.25 in the second period, while the beta of the second stock is 1.5 in the first period and 0.75 in the second period. Under these circumstances, the beta of stock 1 estimated as the average over the two periods is 0.875, while that of the second stock is 1.125. Note also, that the cross-sectional betas are consistent, if both stocks have equal weight -- the market beta is 1, as required.

Now suppose the CAPM holds in each period, but that the market risk premium is 10% in the first period and 20% in the second period. Then the expected risk premium on the first stock would be 0.5(10) = 5% and 1.25(20) = 25% in the first and second period respectively, while the same quantities would be 15% the second stock in both periods. Now, the average return on both stocks over both periods is 15%, while their average betas differ. Hence an ex-post test of the CAPM would reject the model!

In fact, when Wang and Jagganathan relax these two assumptions, the CAPM works pretty well. Here's what they say as a description of their work:

In empirical studies of the CAPM, it is commonly assumed that, (a) the return to the value-weighted portfolio of all stocks is a reasonable proxy for the return on the market portfolio of all assets in the economy, and (b) betas of assets remain constant over time. Under these assumptions, Fama and French (1992) find that the relation between average return and beta is flat. We argue that these two auxiliary assumptions are not reasonable. We demonstrate that when these assumptions are relaxed, the empirical support for the CAPM is very strong. When human capital is also included in measuring wealth, the CAPM is able to explain 28% of the cross sectional variation in average returns in the 100 portfolio studied by Fama and French. When, in addition, betas are allowed to vary over the business cycle, the CAPM is able to explain 57%. More important, relative size does not explain what is left unexplained after taking sampling errors into account.

What this means is that investors should continue to use the CAPM as an intuitive tool to think about required rates of return. However, they should not rely blindly on measures of beta using historical information, and they should also take into account asset return sensitivity to changes in the value of human capital.

Enter the Fama-French Factors:
All well and good at the intuitive level. However, this does not mean that purely empirical methods cannot be used -- just that they need to be used with caution. The Fama-French expected returns are an empirical tool that can be used in this way, viz. with caution. Fama and French (1992) found that size and book-to-market value were able to explain expected returns. Wang and Jagganathan's results suggest that size and book-to-market might proxy for human capital and time-variation in betas. Thus, we can still use Fama-French betas under that interpretation. So how do we go about estimating the Fama-French betas?

Empirical estimation of discount rates using the Fama-French factors:
Fama-French factors can be found from Ken French's website; variable definitions can also be found there. For our purposes, I have downloaded Fama-French factors into an excel file (data from 1990 to Sept. 2007). To compute expected return, we first regress returns on the asset of interest on the three factors -- the excess return on the market portfolio, the size factor and the book-to-market factors -- to get estimates of the three betas. The expected return on the asset is simply the risk-free rate plus the market beta times the market risk premium plus the size beta times the expected return on the size factor plus the book-to-market beta times the expected return on the book-to-market factor. The last two premiums are not adjusted for the risk-free rate, the way the market risk premium is. This is because the size and book-to-market factors are actually returns on self-financing portfolios (see Jonathan Berk and Peter DeMarzo, Corporate Finance, for a very easy read on these issues).

The Size factor is computed by taking the returns on a self-financing portfolio constructed by going long small stocks and going short large stocks (this is commonly called the SMB factor). The Size beta is obtained as the slope coefficient of asset returns on the size factor in a multiple regression. A positive value for the SMB beta is assumed to indicate additional risk, for which the investor needs to be compensated. A SMB beta of 1 has, on average, yielded a premium of about 1.9% over the time period 1/1990-9/2007. Others (see Koller, Goedhart and Wessels, "Valuation" p. 316) estimate this SMB premium to be on the order of 3%.

The Market-to-Book factor is obtained as the return on a self-financing portfolio constructed by going long stocks that have high book-to-market ratios and financing this long position by going short stocks that have low book-to-market ratios (this is commonly called the HML factor). A positive value for the HML factor is assumed to indicate additional risk, for which the investor needs to be compensated. An HML beta of 1 has, on average, yielded a premium of about 4.18% over the period 1/1990-9/2007. Others estimate this HML premium to be on the order of 4.4%.

Here's what Ken French's website says about the factors:

The Fama/French factors are constructed using the 6 value-weight portfolios formed on size and book-to-market. SMB (Small Minus Big) is the average return on the three small portfolios minus the average return on the three big portfolios. HML (High Minus Low) is the average return on the two value portfolios minus the average return on the two growth portfolios.

SMB=1/3 (Small Value + Small Neutral + Small Growth) - 1/3 (Big Value + Big Neutral + Big Growth).
HML = 1/2 (Small Value + Big Value) - 1/2 (Small Growth + Big Growth)

Rm-Rf, the excess return on the market, is the value-weight return on all NYSE, AMEX, and NASDAQ stocks (from CRSP) minus the one-month Treasury bill rate (from Ibbotson Associates).

See Fama and French, 1993, "Common Risk Factors in the Returns on Stocks and Bonds," Journal of Financial Economics, for a complete description of the factor returns.

Why do the Fama-French factors work:
What do the SMB and HML factors represent? Why do stocks that load high on these factors (that is, have high SMB and HML betas) earn higher returns on average that stocks that do not load high on these factors? There are various suggestions that have been made. I present one based on work published by Ralitsa Petkova in the April 2006 Journal of Finance, "Do the Fama–French Factors Proxy for Innovations in Predictive Variables?"

We start with one extension of the CAPM, the ICAPM or the Intertemporal Capital Asset Pricing Model, proposed by Robert Merton in 1973. The ICAPM could be interpreted as saying the following: the CAPM is a good description of asset prices if the market portfolio in a distributional sense does not change over time -- that is, its distribution at a future point in time is the same as its probability distribution today. However, if it does, then in valuing an asset, an investor would want to know, not just how an asset contributes to the riskiness of the (current) market portfolio which does not fully capture the riskiness of his future wealth -- the portfolio that would capture his future wealth is the future market portfolio; rather, he'd want to know how an asset contributes to the risk of the market portfolio that he would hold in the future. And the return on this future market portfolio can be thought of, roughly, as comprising the return on the current market portfolio plus innovations in the current market portfolio. Hence the riskiness of an asset can be measured by its covariance not only with the return on the market portfolio with its current distribution, but also with innovations to the return on the current market portfolio.

Petkova finds that the HML factor is a proxy for changes in the term spread (that is the spread between the yield on short-term Treasuries and long-term Treasuries), while the SMB factor is a proxy for changes in the default spread (that is, the spread between the YTM on more risky corporates and Treasuries). And since the changes in the term spread and the default spread are indicators of how the market portfolio is going to change over time, asset betas with respect to the SMB and HML factors measure the risk of an asset in terms of its covariance with the innovations in the market portfolio.

We now have to explain why the term spread and the default spread are relevant in explaining the development of the market portfolio. The return on the market portfolio at any point in time depends on the characteristics of the macro-economy at any point in time. The characteristics of the macro-economy can be represented by variables that economists call state variables. Two natural variables are the interest rate and the risk propensity of investors. The term spread is a measure of where the interest rate is moving, while the default spread can be thought of as measuring the willingness of investors to take risk. Hence changes in these two variables, the term spread and the risk spread predict the future movements of the return on the market portfolio. Hence, an asset's covariance with changes in the term spread and changes in the default spread measure its riskiness. The SMB and HML factors mimic the changes in the term spread and the default spread. And this is why the Fama-French betas are good measures of an asset's riskiness.