Risk Management and Financial Institutions. Hull John C.

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.

      1.4 ARBITRAGE PRICING THEORY

      Arbitrage pricing theory can be viewed as an extension of the capital asset pricing model. In the capital asset pricing model, an asset's return depends on just one factor. In arbitrage pricing theory, the return depends on several factors. (These factors might involve variables such as the gross national product, the domestic interest rate, and the inflation rate.) By exploring ways in which investors can form portfolios that eliminate exposure to the factors, arbitrage pricing theory shows that the expected return from an investment is linearly dependent on the factors.

      The assumption that the ε variables for different investments are independent in equation (1.3) ensures that there is just one factor driving expected returns (and therefore one source of systematic risk) in the capital asset pricing model. This is the return from the market portfolio. In arbitrage pricing theory there are several factors affecting investment returns. Each factor is a separate source of systematic risk. Unsystematic (i.e., diversifiable) risk in arbitrage pricing theory is the risk that is unrelated to all the factors.

      1.5 RISK VS. RETURN FOR COMPANIES

      We now move on to consider the trade-offs between risk and return made by a company. How should a company decide whether the expected return on a new investment project is sufficient compensation for its risks?

      The ultimate owners of a company are its shareholders and a company should be managed in the best interests of its shareholders. It is therefore natural to argue that a new project undertaken by the company should be viewed as an addition to its shareholder's portfolio. The company should calculate the beta of the investment project and its expected return. If the expected return is greater than that given by the capital asset pricing model, it is a good deal for shareholders and the investment should be accepted. Otherwise it should be rejected.

      The argument just given suggests