Читать книгу Risk Management and Financial Institutions - Hull John C. - Страница 9

How do investors decide on the expected returns they require for individual investments? Based on the analysis we have presented, the market portfolio should play a key role. The expected return required on an investment should reflect the extent to which the investment contributes to the risks of the market portfolio.

A common procedure is to use historical data and regression analysis to determine a best-fit linear relationship between returns from an investment and returns from the market portfolio. This relationship has the form:

(1.3)

where R is the return from the investment, R_M is the return from the market portfolio, a and β are constants, and ε is a random variable equal to the regression error.

Equation (1.3) shows that there are two uncertain components to the risk in the investment's return:

1. A component βR_M, which is a multiple of the return from the market portfolio.

2. A component ε, which is unrelated to the return from the market portfolio.

The first component is referred to as systematic risk. The second component is referred to as nonsystematic risk.

Consider first the nonsystematic risk. If we assume that the ε variables for different investments are independent of each other, the nonsystematic risk is almost completely diversified away in a large portfolio. An investor should not therefore be concerned about nonsystematic risk and should not require an extra return above the risk-free rate for bearing nonsystematic risk.

The systematic risk component is what should matter to an investor. When a large well-diversified portfolio is held, the systematic risk represented by βR_M does not disappear. An investor should require an expected return to compensate for this systematic risk.

We know how investors trade off systematic risk and expected return from Figure 1.4. When β = 0 there is no systematic risk and the expected return is R_F. When β = 1, we have the same systematic risk as the market portfolio, which is represented by point M, and the expected return should be E(R_M). In general

(1.4)

This is the capital asset pricing model. The excess expected return over the risk-free rate required on the investment is β times the excess expected return on the market portfolio. This relationship is plotted in Figure 1.5. The parameter β is the beta of the investment.

FIGURE 1.5 The Capital Asset Pricing Model

EXAMPLE 1.1

Suppose that the risk-free rate is 5 % and the return on the market portfolio is 10 %. An investment with a beta of 0 should have an expected return of 5 %. This is because all of the risk in the investment can be diversified away. An investment with a beta of 0.5 should have an expected return of

or 7.5 %. An investment with a beta of 1.2 should have an expected return of

or 11 %.

The parameter, β, is equal to ρσ/σM where ρ is the correlation between the return from the investment and the return from the market portfolio, σ is the standard deviation of the return from the investment, and σM is the standard deviation of the return from the market portfolio. Beta measures the sensitivity of the return from the investment to the return from the market portfolio. We can define the beta of any investment portfolio as in equation (1.3) by regressing its returns against the returns from the market portfolio. The capital asset pricing model in equation (1.4) should then apply with the return R defined as the return from the portfolio. In Figure 1.4 the market portfolio represented by M has a beta of 1.0 and the riskless portfolio represented by F has a beta of zero. The portfolios represented by the points I and J have betas equal to βI and βJ, respectively.

Assumptions

The analysis we have presented leads to the surprising conclusion that all investors want to hold the same portfolios of assets (the portfolio represented by M in Figure 1.4.) This is clearly not true. Indeed, if it were true, markets would not function at all well because investors would not want to trade with each other! In practice, different investors have different views on the attractiveness of stocks and other risky investment opportunities. This is what causes them to trade with each other and it is this trading that leads to the formation of prices in markets.

The reason why the analysis leads to conclusions that do not correspond with the realities of markets is that, in presenting the arguments, we implicitly made a number of assumptions. In particular:

1. We assumed that investors care only about the expected return and the standard deviation of return of their portfolio. Another way of saying this is that investors look only at the first two moments of the return distribution. If returns are normally distributed, it is reasonable for investors to do this. However, the returns from many assets are non-normal. They have skewness and excess kurtosis. Skewness is related to the third moment of the distribution and excess kurtosis is related to the fourth moment. In the case of positive skewness, very high returns are more likely and very low returns are less likely than the normal distribution would predict; in the case of negative skewness, very low returns are more likely and very high returns are less likely than the normal distribution would predict. Excess kurtosis leads to a distribution where both very high and very low returns are more likely than the normal distribution would predict. Most investors are concerned about the possibility of extreme negative outcomes. They are likely to want a higher expected return from investments with negative skewness or excess kurtosis.

2. We assumed that the ε variables for different investments in equation (1.3) are independent. Equivalently we assumed the returns from investments are correlated with each other only because of their correlation with the market portfolio. This is clearly not true. Ford and General Motors are both in the automotive sector. There is likely to be some correlation between their returns that does not arise from their correlation with the overall stock market. This means that the ε for Ford and the ε for General Motors are not likely to be independent of each other.

3. We assumed that investors focus on returns over just one period and the length of this period is the same for all investors. This is also clearly not true. Some investors such as pension funds have very long time horizons. Others such as day traders have very short time horizons.

4. We assumed that investors can borrow and lend at the same risk-free rate. This is approximately true in normal market conditions for a large financial institution that has a good credit rating. But it is not exactly true for such a financial institution and not at all true for small investors.

5. We did not consider tax. In some jurisdictions, capital gains are taxed differently from dividends and other sources of income. Some investments get special tax treatment and not all investors are subject to the same tax rate. In practice, tax considerations have a part to play in the decisions of an investor. An investment that is appropriate for a pension fund that pays no tax might be quite inappropriate for a high-marginal-rate taxpayer living in New York, and vice versa.

6. Finally, we assumed that all investors make the same estimates of expected returns, standard deviations of returns, and correlations between returns for available investments. To put this another way, we assumed that investors have homogeneous expectations. This is clearly not true. Indeed, as mentioned earlier, if we lived in a world of homogeneous expectations there would be no trading.

In spite of all this, the capital asset pricing model has proved to be a useful tool for portfolio managers. Estimates of the betas of stocks are readily available and the expected return on a portfolio estimated by the capital asset pricing model is a commonly used benchmark for assessing the performance of the portfolio manager, as we will now explain.

Alpha

When we observe a return of R_M on the market, what do we expect the return on a portfolio with a beta of β to be? The capital asset pricing model relates the expected return on a portfolio to the expected return on the market. But it can also be used to relate the expected return on a portfolio to the actual return on the market:

where R_F is the risk-free rate and R_P is the return on the portfolio.

EXAMPLE 1.2

Consider a portfolio with a beta of 0.6 when the risk-free interest rate is 4 %. When the return from the market is 20 %, the expected return on the portfolio is

or 13.6 %. When the return from the market is 10 %, the expected return from the portfolio is

or 7.6 %. When the return from the market is −10 %, the expected return from the portfolio is

or − 4.4 %. The relationship between the expected return on the portfolio and the return on the market is shown in Figure 1.6.

FIGURE 1.6 Relationship between Expected Return on Portfolio and the Actual Return on the Market When Portfolio Beta Is 0.6 and Risk-Free Rate Is 4%

Suppose that the actual return on the portfolio is greater than the expected return:

The portfolio manager has produced a superior return for the amount of systematic risk being taken. The extra return is

This is commonly referred to as the alpha created by the portfolio manager.2

EXAMPLE 1.3

A portfolio manager has a portfolio with a beta of 0.8. The one-year risk-free rate of interest is 5 %, the return on the market during the year is 7 %, and the portfolio manager's return is 9 %. The manager's alpha is

or 2.4 %.

Portfolio managers are continually searching for ways of producing a positive alpha. One way is by trying to pick stocks that outperform the market. Another is by market timing. This involves trying to anticipate movements in the market as a whole and moving funds from safe investments such as Treasury bills to the stock market when an upturn is anticipated and in the other direction when a downturn is anticipated. Chapter 4 explains other strategies used by hedge funds to try to create positive alpha.

Although the capital asset pricing model is unrealistic in some respects, the alpha and beta parameters that come out of the model are widely used to characterize investments. Beta describes the amount of systematic risk. The higher the value of beta, the greater the systematic risk being taken and the greater the extent to which returns are dependent on the performance of the market. Alpha represents the extra return made from superior portfolio management (or perhaps just good luck). An investor can make a positive alpha only at the expense of other investors who are making a negative alpha. The weighted average alpha of all investors must be zero.