Risk Management and Financial Institutions. Hull John C.

Figure 1.4 where the distance of J from F is βJ times the distance of M from F (βJ > 1). We borrow βJ − 1 of the amount that we have available for investment at rate R_F and then invest everything (the original funds and the borrowed funds) in the investment represented by point M. After allowing for the interest paid, the new investment has an expected return, E(R_J) given by

      and the standard deviation of the return is

      This shows that the risk and expected return combination corresponds to point J. (Note that the formulas for the expected return and standard deviation of return in terms of beta are the same whether beta is greater than or less than 1.)

      The argument that we have presented shows that, when the risk-free investment is considered, the efficient frontier must be a straight line. To put this another way there should be linear trade-off between the expected return and the standard deviation of returns, as indicated in Figure 1.4. All investors should choose the same portfolio of risky assets. This is the portfolio represented by M. They should then reflect their appetite for risk by combining this risky investment with borrowing or lending at the risk-free rate.

      It is a short step from here to argue that the portfolio of risky investments represented by M must be the portfolio of all risky investments. Suppose a particular investment is not in the portfolio. No investors would hold it and its price would have to go down so that its expected return increased and it became part of portfolio M. In fact, we can go further than this. To ensure a balance between the supply and demand for each investment, the price of each risky investment must adjust so that the amount of that investment in portfolio M is proportional to the amount of that investment available in the economy. The investment represented by point M is therefore usually referred to as the market portfolio.

      1.3 THE CAPITAL ASSET PRICING MODEL

      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