everything in portfolio M.
Consider what happens when we form an investment I by putting βI of the funds we have available for investment in the risky portfolio, M, and 1 − βI in the risk-free investment F (0 < βI < 1). From equation (1.1) the expected return from the investment, E(RI), is given by
and from equation (1.2), because the risk-free investment has zero standard deviation, the return RI has standard deviation
where σM is the standard deviation of return for portfolio M. This risk-return combination corresponds to the point labeled I in Figure 1.4. From the perspective of both expected return and standard deviation of return, point I is βI of the way from F to M.
All points on the line FM can be obtained by choosing a suitable combination of the investment represented by point F and the investment represented by point M. The points on this line dominate all the points on the previous efficient frontier because they give a better risk-return combination. The straight line FM is therefore part of the new efficient frontier.
If we make the simplifying assumption that we can borrow at the risk-free rate of RF as well as invest at that rate, we can create investments that are on the continuation of FM beyond M. Suppose, for example, that we want to create the investment represented by the point J in 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 RF 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(RJ) 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, RM 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 βRM, 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 βRM 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 RF. 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(RM). 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
