Benchmarking to the Market Portfolio

All our work to this point made no reference to the market portfolio, in this case, the S&P 500 return. It is common, however, to compare the return to an asset or a portfolio of assets to the return on the market. There are many reasons, foremost of which is that the return to the market represents the opportunity cost of portfolio under management. For example, the return to Port 2 in the monthly data (weekly data) is 3 percent (0.742 percent) with risk 5.1 percent (2.99 percent). Had we invested in the market portfolio instead, we would have a return of 1 percent (0.3 percent) with risk 2.6 percent (1.3 percent). Does this look like a favorable risk-return trade-off to you? It depends on your risk preferences, but the return per unit risk is 3 percent/5.1 percent on the monthly Port 2 versus 1 percent/2.6 percent on the monthly S&P. This is a Sharpe ratio and it favors Port 2, at least from a risk-adjusted return perspective.

What about risk at the security level? Obviously, we can compare volatilities (standard deviations). But is there a measure of security risk relative to benchmark risk? The CAPM provides this measure in the form of the asset's β:

Beta is a measure of the covariation in the asset's return to the market return, specifically:

Actually, it is a very intuitive concept; a β of one means that the two return series are perfectly correlated, that is, a 1 percent change in the market excess return is matched by a 1 percent return in the asset's excess return. A beta of two means that the asset's return is twice as variable as the market return, and a beta of zero means that the asset's return is uncorrelated with the market. If β > 1, then we say the asset is riskier than the market. When portfolio managers talk about their beta exposure, they are generally referring to how correlated their portfolio is to the market or benchmark. The betas for the 10 firms under study here range as we would expect (you can find them in Chapter 7 Examples.xlsx on the mthly_r and wkly_r spreadsheets). I estimate betas using Excel's slope function. Citigroup (C) has a beta that is close to two in the monthly data, suggesting that is twice as risky as holding the market. It is no wonder that the optimizer tries to find risk-diversifying combinations of these firms in its search for the minimum variance portfolio.

We can tie our β measures into the CAPM as a pricing model. Recalling the dividend discount model that we developed in Chapter 4 and applied to the CAPM in Chapter 5, and using, say, Citigroup as a case in point, then the discount rate for C would be . To get at the point directly here, if we assume that the risk-free rate is zero, then suggests that Citigroup's dividends should be discounted at twice the market rate. This is clearly a far cry from discounting at a bond rate of interest or the market rate of interest—the correct discount factor should account for the risk on the security, and the CAPM is a pricing model that accounts for that extra-market risk.

What about the portfolio's exposure to market risk? How is that risk compensated? I estimate the betas for Port 1 and Port 2 (see mthly_r) over the five-year period under scrutiny to be 0.85 and 1.24, respectively. Thus, the minimum variance portfolio has lower risk than the market portfolio, while Port 2 has higher risk relative to the market. Both results make intuitive sense.

Beta measures systematic risk because it shows the degree to which the asset's return covaries with the market return. Thus, the asset's return should reflect this risk, which we can see from the CAPM we developed in Chapter 5. In its empirical form, we add an error term to capture asset-specific pricing error:

Therefore, the expected return on the asset should be:

The term ε_i is the asset's idiosyncratic risk and is also referred to as nonsystematic risk, or the firm's specific risk. It is separate from and independent of the market return and is therefore not compensated. Since β is estimated by least squares and the standard assumptions of the classical linear regression model require that ε be independent of r_m with expected value equal to zero (meaning that on average these risks are zero), then is simply linear in β. We will return to this point momentarily, but we must first understand that while market risk is compensated, idiosyncratic risk is not and, in fact, idiosyncratic risk can only be diversified away. We demonstrated this point in Chapter 5. Diversification rests on the assumption of idiosyncratic risks being distributed independently across assets. If the CAPM is specified correctly in the sense that the market excess return is the single factor sufficient to explain systematic movements in asset returns, then the ε_i must be distributed independently across assets. Adding assets to the portfolio will increase the chances that the average of the ε_i converge to zero. Thus, in the limit, the portfolio will contain only systematic risk, which is compensated.

Let's now return to the assertion that the CAPM posits a linear relationship between asset returns and beta. What this means is that higher beta stocks should have higher returns. We can demonstrate this relationship using the betas and observed mean returns from the 10 firms and the S&P 500 from the Chapter 7 Examples spreadsheet. Certainly, this sample is too small but it does give you something to fix ideas upon.

Figure 7.1 Security Market Line

If we fit a trend line to these data points, we get the security market line (SML), as shown in Figure 7.1. This suggests a trade-off between return and risk as measured by beta. Assets (firms) with higher betas should also have higher returns. The CAPM is not only a pricing model—it also says something about the cross-section of returns. Tests of the CAPM focus on empirical tests of the slope and intercept of the SML, a topic that we explore in more detail in Chapter 8.