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Chapter10: Risk and Return: The Capital Asset Pricing Model

10.8 Market Equilibrium

Definition of the Market Equilibrium Portfolio

The preceding analysis concerns one investor. His estimates of the expected returns and variances for individual securities and the covariances between pairs of securities are his and his alone. Other investors would obviously have different estimates of these variables. However, the estimates might not vary much because everyone would be forming expectations from the same data about past price movements and other publicly available information.

Financial economists often imagine a world where all investors possess the same estimates of expected returns, variances and covariances. Though this can never be literally true, it can be thought of as a useful simplifying assumption in a world where investors have access to similar sources of information. This assumption is called homogeneous expectations.13

If all investors had homogeneous expectations, Fig. 10.9 would be the same for all individuals. That is, all investors would sketch out the same efficient set of risky assets, because they would be working with the same inputs. This efficient set of risky assets is represented by the curve XAY. Because the same risk-free rate would apply to everyone, all investors would view point A as the portfolio of risky assets to be held.

This point A takes on great importance, because all investors would purchase the risky securities that it represents. Investors with a high degree of risk aversion might combine A with an investment in the riskless asset, achieving point 4, for example. Others with low aversion to risk might borrow to achieve, say, point 5. Because this is a very important conclusion, we restate it:

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In a world with homogeneous expectations, all investors would hold the portfolio of risky assets represented by point A.

If all investors choose the same portfolio of risky assets, it is possible to determine what that portfolio is. Common sense tells us that it is a market-value-weighted portfolio of all existing securities. It is the market portfolio.

In practice, economists use a broad-based index such as the FTSE 100, Dow Jones Euro Stoxx 50 or Standard & Poor’s (S&P) 500 as a proxy for the market portfolio, depending on the country they analyse. Of course, in practice all investors do not hold the same portfolio. However, we know that many investors hold diversified portfolios, particularly when mutual funds or pension funds are included. A broad-based index is a good proxy for the highly diversified portfolios of many investors.

Definition of Risk when Investors Hold the Market Portfolio

The previous section states that many investors hold diversified portfolios similar to broad-based indexes. This result allows us to be more precise about the risk of a security in the context of a diversified portfolio.

Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta of the security. We illustrate beta by an example.

EXAMPLE 10.4Beta

Consider the following possible returns, both on the equity of Jelco plc and on the market:

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Though the return on the market has only two possible outcomes (15% and 5%), the return on Jelco has four possible outcomes. It is helpful to consider the expected return on a security for a given return on the market. Assuming each state is equally likely, we have

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Jelco plc responds to market movements because its expected return is greater in bullish states than in bearish states. We now calculate exactly how responsive the security is to market movements. The market’s return in a bullish economy is 20 per cent [= 15% − (−5%)] greater than the market’s return in a bearish economy. However, the expected return on Jelco in a bullish economy is 30 per cent [= 20% − (−10%)] greater than its expected return in a bearish state. Thus Jelco plc has a responsiveness coefficient of 1.5 (= 30%/20%).

This relationship appears in Fig. 10.10. The returns for both Jelco and the market in each state are plotted as four points. In addition, we plot the expected return on the security for each of the two possible returns on the market. These two points, each of which we designate by an X, are joined by a line called the characteristic line of the security. The slope of the line is 1.5, the number calculated in the previous paragraph. This responsiveness coefficient of 1.5 is the beta of Jelco.

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The interpretation of beta from Fig. 10.10 is intuitive. The graph tells us that the returns of Jelco are magnified 1.5 times over those of the market. When the market does well, Jelco’s equity is expected to do even better. When the market does poorly, Jelco’s equity is expected to do even worse. Now imagine an individual with a portfolio near that of the market who is considering the addition of Jelco to her portfolio. Because of Jelco’s magnification factor of 1.5, she will view this security as contributing much to the risk of the portfolio. (We shall show shortly that the beta of the average security in the market is 1.) Jelco contributes more to the risk of a large, diversified portfolio than does an average security, because Jelco is more responsive to movements in the market.

Figure 10.10 Performance of Jelco plc and the market portfolio
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Further insight can be gleaned by examining securities with negative betas. One should view these securities as either hedges or insurance policies. The security is expected to do well when the market does poorly, and vice versa. Because of this, adding a negative-beta security to a large, diversified portfolio actually reduces the risk of the portfolio.14

Table 10.7 presents empirical estimates of betas for individual securities. As can be seen, some securities are more responsive to the market than others. For example, Siemens has a beta of 1.51. This means that for every 1 per cent movement in the market, Siemens is expected to move 1.51 per cent in the same direction. Conversely, SAP has a beta of only 0.56. This means that for every 1 per cent movement in the market, SAP is expected to move 0.56 per cent in the same direction.

We can summarize our discussion of beta by saying this:

Beta measures the responsiveness of a security to movements in the market portfolio.

The Formula for Beta

Our discussion so far has stressed the intuition behind beta. The actual definition of beta is

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where Cov(Ri, RM) is the covariance between the return on asset i and the return on the market portfolio, and σ2(RM) is the variance of the market.

Table 10.7Estimates of beta for selected individual equities
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One useful property is that the average beta across all securities, when weighted by the proportion of each security’s market value to that of the market portfolio, is 1. That is,

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where Xi is the proportion of security i’s market value to that of the entire market, and N is the number of securities in the market.

Equation (10.16) is intuitive, once you think about it. If you weight all securities by their market values, the resulting portfolio is the market. By definition, the beta of the market portfolio is 1. That is, for every 1 per cent movement in the market, the market must move 1 per cent – by definition.

A Test

We have put these questions on past corporate finance examinations:

  1. What sort of investor rationally views the variance (or standard deviation) of an individual security’s return as the security’s proper measure of risk?

  2. What sort of investor rationally views the beta of a security as the security’s proper measure of risk?

A good answer might be something like the following:

A rational, risk-averse investor views the variance (or standard deviation) of her portfolio’s return as the proper measure of the risk of her portfolio. If for some reason the investor can hold only one security, the variance of that security’s return becomes the variance of the portfolio’s return. Hence the variance of the security’s return is the security’s proper measure of risk.

If an individual holds a diversified portfolio, she still views the variance (or standard deviation) of her portfolio’s return as the proper measure of the risk of her portfolio. However, she is no longer interested in the variance of each individual security’s return. Rather, she is interested in the contribution of an individual security to the variance of the portfolio.

Under the assumption of homogeneous expectations, all individuals hold the market portfolio. Thus we measure risk as the contribution of an individual security to the variance of the market portfolio. This contribution, when standardized properly, is the beta of the security. Although few investors hold the market portfolio exactly, many hold reasonably diversified portfolios. These portfolios are close enough to the market portfolio that the beta of a security is likely to be a reasonable measure of its risk.

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