## Comparative Analysis of Risk and Return Models

All the risk and return models developed in this chapter have common ingredients. They all assume that only market-wide risk is rewarded, and they derive the expected return as a function of measures of this risk. Figure 3.7 presents a comparison of the different models:

12 Adding to the confusion, researchers in recent years have taken to describing proxy models also as multi factor models.

Figure 3.7: Competing Models for Risk and Return in Finance

Step 1: Defining Risk

The risk in an investment can be measured by the variance in actual returns around an expected return

Riskless Investment Low Risk Investment High Risk Investment

Step 2: Differentiating between Rewarded and Unrewarded Risk

Risk that is specific to investment (Firm Specific) Risk that affects all investments (Market Risk)

Can be diversified away in a diversified portfolio Cannot be diversified away since most assets

1. each investment is a small proportion of portfolio are affected by it.

2. risk averages out across investments in portfolio

The marginal investor is assumed to hold a "diversified" portfolio. Thus, only market risk will be rewarded and priced.

The CAPM If there is ## 1. no private information2. no transactions cost the optimal diversified portfolio includes every traded asset. Everyone will hold this market portfolio Market Risk = Risk added by any investment to the market portfolio: |
The APM If there are no arbitrage opportunities then the market risk of any asset must be captured by betas relative to factors that affect all investments. Market Risk = Risk exposures of any asset to market factors |
Multi-Factor Models Since market risk affects most or all investments, it must come from macro economic factors. Market Risk = Risk exposures of any asset to macro economic factors. |
Proxy Models In an efficient market, differences in returns across long periods must be due to market risk differences. Looking for variables correlated with returns should then give us proxies for this risk. Market Risk = Captured by the Proxy Variable(s) |

Beta of asset relative to Market portfolio (from a regression) |
Betas of asset relative to unspecified market factors (from a factor analysis) |
Betas of assets relative to specified macro economic factors (from a regression) |
Equation relating returns to proxy variables (from a regression) |

The capital asset pricing model makes the most assumptions but arrives at the simplest model, with only one risk factor requiring estimation. The arbitrage pricing model makes fewer assumptions but arrives at a more complicated model, at least in terms of the parameters that require estimation. In general, the CAPM has the advantage of being a simpler model to estimate and to use, but it will under perform the richer multi factor models when the company is sensitive to economic factors not well represented in the market index. For instance, oil companies, which derive most of their risk from oil price movements, tend to have low CAPM betas. Using a multi factor model, where one of the factors may be capturing oil and other commodity price movements, will yield a better estimate of risk and higher cost of equity for these firms13.

13 Weston, J.F. and T.E. Copeland, 1992, Managerial Finance, Dryden Press. They used both approaches to estimate the cost of equity for oil companies in 1989 and came up with 14.4% with the CAPM and 19.1% using the arbitrage pricing model.

The biggest intuitive block in using the arbitrage pricing model is its failure to identify specifically the factors driving expected returns. While this may preserve the flexibility of the model and reduce statistical problems in testing, it does make it difficult to understand what the beta coefficients for a firm mean and how they will change as the firm changes (or restructures).

Does the CAPM work? Is beta a good proxy for risk, and is it correlated with expected returns? The answers to these questions have been debated widely in the last two decades. The first tests of the model suggested that betas and returns were positively related, though other measures of risk (such as variance) continued to explain differences in actual returns. This discrepancy was attributed to limitations in the testing techniques. In 1977, Roll, in a seminal critique of the model's tests, suggested that since the market portfolio (which should include every traded asset of the market) could never be observed, the CAPM could never be tested, and that all tests of the CAPM were therefore joint tests of both the model and the market portfolio used in the tests, i.e., all any test of the CAPM could show was that the model worked (or did not) given the proxy used for the market portfolio.14 He argued that in any empirical test that claimed to reject the CAPM, the rejection could be of the proxy used for the market portfolio rather than of the model itself. Roll noted that there was no way to ever prove that the CAPM worked, and thus, no empirical basis for using the model.

The study by Fama and French quoted in the last section examined the relationship between the betas of stocks and annual returns between 1963 and 1990 and concluded that there was little relationship between the two. They noted that market capitalization and book-to-market value explained differences in returns across firms much better than did beta and were better proxies for risk. These results have been contested on two fronts. First, Amihud, Christensen, and Mendelson, used the same data, performed different statistical tests, and showed that betas did, in fact, explain returns during the time period.15 Second, Chan and Lakonishok look at a much longer time series

14 Roll, R., 1977, A Critique of the Asset Pricing Theory's Tests: Part I: On Past and Potential Testability of Theory, Journal of Financial Economics, v4, 129-176.

15 Amihud, Y., B. Christensen and H. Mendelson, 1992, Further Evidence on the Risk-Return Relationship, Working Paper, New York University.

of returns from 1926 to 1991 and found that the positive relationship between betas and returns broke down only in the period after 1982.16 They attribute this breakdown to indexing, which they argue has led the larger, lower-beta stocks in the S & P 500 to outperform smaller, higher-beta stocks. They also find that betas are a useful guide to risk in extreme market conditions, with the riskiest firms (the 10% with highest betas) performing far worse than the market as a whole, in the ten worst months for the market between 1926 and 1991 (See Figure 3.8).

Figure 3.8: Returns and Betas: Ten Worst Months between 1926 and 1991

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