## Jbi

Figure 14.4 The location of portfolios with return R'F.

efficient segment of the minimum variance frontier, and the slope at this point must be positive. Thus, as we move along the line tangent to RM toward the vertical axis, we lower return. Since R7 is the intercept of the tangency line and the vertical axis, it has a return less than RM. Second, as we prove below, the minimum variance zero Beta portfolio cannot be efficient.

Proof Denote by s the portfolio that has the smallest possible variance. This portfolio can be formed as a combination of the market portfolio and the zero Beta portfolio.

al=Xla2z+(l-Xxfol

There is no covariance term since the covariance between these two assets is zero. To find the weights in each portfolio that minimize variance, take the derivative with respect to Xz and set it equal to zero, or

Solving for Xz

Since both <j2m and ct| must be positive numbers, that portfolio with the smallest possible variance must involve positive weights on both the zero Beta and market portfolio. Since liy < Rm, portfolios of Z and M with positive weights must have higher expected returns ilian Z. Since the minimum variance portfolio has higher return and smaller variance than / Z cannot be on the efficient portion of the minimum variance frontier.

We can locate portfolios Z, M, and s on the minimum variance frontier of all portfolios in expected return standard deviation space.6 This is done in Figure 14.5. This figure pres-nts the location of all efficient portfolios in expected return standard deviation space. All investors will hold some portfolio that lies along the efficient frontier (SMC). Investors ■ lio hold portfolios offering returns between s and RM will hold combinations of the zero Beta portfolio and the market portfolio.7 Investors who choose to hold portfolios to the right of M (choose returns above RM) will hold a portfolio constructed by selling portfolio / short and buying the market portfolio. No investor will choose to hold only portfolio Z ior this is an inefficient portfolio. Furthermore, since investors in the aggregate hold the market portfolio, the aggregate holding of portfolio Z (long positions minus short positions) must be exactly zero. Note also that we still have a two mutual fund theorem. All investors can be satisfied by transactions in two mutual funds: the market portfolio and the minimum variance zero Beta portfolio.

We started out this section by assuming that the market portfolio is efficient. While we do not intend to provide a rigorous proof of its efficiency, a few comments should convince the reader of its truth. Those interested in a rigorous proof are referred to Fama [29],

With homogeneous expectations, all investors face the same efficient frontier. Recall that with short sales allowed all combinations of any two minimum variance portfolios are

' T lie minimum variance curve or minimum variance frontier contains the set of portfolios that offers the lowest I|,J< at any obtainable level of return. The efficient set (frontier) is a subset of these minimum variance portfolios. Kecall from Chapter 6 that the entire efficient frontier can be generated as portfolios of any two portfolios on ! efficient frontier.

Figure 14.5 The minimum variance frontier.

minimum variance. Thus, if we combine any two investors' portfolios, we have a minimum variance portfolio. The market portfolio is a weighted average or portfolio of each investor's portfolio where the weights are the proportion each investor owns of the total of all risky assets. Thus, it is minimum variance. Since each investor's portfolio is efficient and since the return on the market is an average of the return on the portfolios of individual investors, the return on the market portfolio is the return of a portfolio on the efficient segment of the minimum variance frontier. Thus, the market portfolio is not only minimum variance but efficient.

### Riskless Lending But No Riskless Borrowing

We have gone too far in changing our assumptions. As we agreed earlier, while it is unrealistic to assume that individuals can borrow at the riskless rate, it is realistic to assume that they can lend at a rate that is riskless. Individuals can place funds in government securities that have a maturity equal to their time horizon and, thus, be guaranteed of a riskless payoff at the horizon.

If we allow riskless lending, then the investor's choice can be pictured as in Figure 14.6.8 As we argued in earlier chapters, all combinations of a riskless asset and a risky portfolio lie on the straight line connecting the asset and the portfolio. The preferred combination lies on the straight line passing through the risk-free asset and tangent to the efficient frontier. This is the line RfT in Figure 14.6.

Notice that we have drawn T below and to the left of the market portfolio M and, hence, Rz > Rp. This was not an accident. Let us examine why this must hold. Before we introduced the ability to lend at the riskless rate, all investors held portfolios along the efficient frontier SMC (portfolios along the line RZM do not exist). With riskless lending the investor can hold portfolios of riskless and risky assets along the line RFT. If the investor chooses to hold an investment on the line RpT, he would be placing some of his funds in

8Once again we are assuming short sales are allowed. This is a necessary assumption.

Figure 14.6 The opportunity set with riskless lending.

the portfolio of risky assets denoted by T and some in the riskless asset. The choice to hold .iny portfolio of risky assets other than T would never be made. Now, why can't T and M be the same portfolio? As long as any investor has a risk-return trade-off such that he or she chooses to hold a portfolio of investments to the right of T, the market must lie to the right of T. For example, assume that all investors but one choose to lend money and hold portfolio T. Now this one investor who does not choose Tmust hold a portfolio to the right of T on the efficient frontier STC. If the investor did not, then he or she would be better off holding a portfolio on the line RpT and, hence, holding portfolio T. Since the market portfolio is an average of the portfolios held by all investors, the market portfolio must be a combination of the investor's portfolio and T. Thus, it lies to the right of T. M, being to the right of T, leads directly to R? being larger than Rp. Rp is the intersection of the vertical axis and a line tangent to the efficient frontier at T. Similarly, R? is the intersection of the vertical axis and a line tangent at M. Since the slope of the efficient frontier at M is less than at T and since M lies above T, the line tangent at M must intersect the vertical axis above the line tangent at T.9 Thus, Rz must be greater than Rp.

The efficient frontier is given by the straight line segment RfT and curve TMC.10 Notice that, in the case of no lending and borrowing, combinations of all efficient portfolios were efficient. In the case where riskless lending is allowed, not all combinations of efficient portfolios are efficient. It should be obvious to the reader that combinations of a portfolio from the line segment RpT and a portfolio from the curve TMC are dominated by a portfolio lying along the curve TMC.

Portfolio T can be obtained by combining portfolios Z and M. Examining the efficient frontier we see that investors who select a portfolio along the line segment RpT are placing some of their money in portfolio T (which is constructed from the market portfolio plus

The property of the two slopes follows directly from the concavity of the efficient frontier proved in Chapter 5.

^The reader might note that portfolio T is a corner portfolio, a portfolio whose composition is different from those immediately adjacent to it. All portfolios to the right of T on the efficient frontier are made up of combinations of portfolios M and Z. while those to the left of T are made up of portfolios M and Z plus the riskless security.

portfolio Z) and some in the riskless asset. (Those that select a portfolio on the segment TM are placing some of their money in portfolio M and some in Z.) Those that select a portfolio on MC are selling portfolio Z short and investing all of the proceeds in M. (Notice that our two mutual fund theorem has been replaced with a three mutual fund theorem.) All investors can be satisfied by holding (long or short) some combination of the market portfolio, the minimum variance zero Beta portfolio, and the riskless asset.11

Having examined all efficient portfolios in expected return standard deviation space, let us turn our attention to the location of securities and portfolios in expected return Beta space. Let us develop the security market line.

The market portfolio M is still an efficient portfolio. Thus, the analysis of the last section holds. All securities contained in M have an expected return given by

Similarly, all portfolios composed solely of risky assets have their return given by Equation (14.5). This splits as a straight line in expected return Beta space and is the line RZTMC in Figure 14.7. This equation holds only for risky assets and for portfolios of risky assets. It does not describe the return on the riskless asset or the return on portfolios that contain the riskless asset.

In the previous chapter we examined combinations of the riskless asset and a risky portfolio and found that they lie on the straight line connecting the two points in expected return Beta space. Since investors who lend all hold risky portfolio T, the relevant line segment is RfT in Figure 14.7.

Thus, while the straight line R?M can be thought of as the security market line for all risky assets and for all portfolios composed entirely of risk assets, it does not describe the return on portfolios (and, of particular note, on those efficient portfolios) that contain the

Figure 14.7 The location of investments in expected return Beta space.

nNote that while we continually speak of using the market portfolio and the minimum variance zero Beta portfolio to obtain the efficient frontier, any other two minimum variance portfolios would serve equally well.

riskless asset. Efficient portfolios have their return given by the two line segments RpT and TC in Figure 14.7. The fact that efficient portfolios have lower return for a given level of Beta than individual assets may seem startling. But remember that securities or portfolios on RZT have a higher standard deviation than portfolios with the same return on segment RfT. (In order to understand this, remember that the return on portfolio Z is uncertain, even though it has a zero Beta, while the return on the riskless asset is certain.)

Before moving on to other models, it is well worth reviewing certain characteristics of those we have been discussing, particularly insofar as they resemble or are different from the characteristics of the simple capital asset pricing model.

First, note that, under either of these models, all investors no longer hold the same portfolio in equilibrium. This is comforting for it is more consistent with observed behavior. Of less comfort is that investors still hold most securities (either long or short) and hold many securities short. In the case where neither lending nor borrowing is allowed, we have a two mutual fund theorem. In the case where riskless lending is allowed, we have a three mutual fund theorem.

As in the case of the simple CAPM, we still get a security market line. In addition, many of the implications of this relationship are the same. For risky assets or portfolios expected return is still a linearly increasing function of risk as measured by Beta. It is only market risk that affects the return on individual risky securities and portfolios of risky securities. On these securities the investor gains no extra return from bearing diversifiable risk. In fact, the only difference lies in the intercept and slope of the security market line.12

### Other Lending and Borrowing Assumptions

Brennan [10] has analyzed the situation where riskless lending and borrowing is available, but at different rates. The efficient frontier for the individual when riskless borrowing and lending at different rates is possible was analyzed in Chapter 5. If all investors face the same efficient frontier, this efficient frontier must appear as in Figure 14.8.

f ¡(jure 14.8 The opportunity set with a differential lending and borrowing rate.

In ail models the efficient frontier itself is affected by diversifiable risk. Since the shape of the frontier affects location of the tangency portfolio, diversifiable risk has some effect on security returns.

portfolio 2) and some in the riskless asset. (Those that select a portfolio on the segment TM are placing some of their money in portfolio M and some in Z.) Those that select a portfolio on MC are selling portfolio Z short and investing all of the proceeds in M. (Notice that our two mutual fund theorem has been replaced with a three mutual fund theorem.) All investors can be satisfied by holding (long or short) some combination of the market portfolio, the minimum variance zero Beta portfolio, and the riskless asset.11

Having examined all efficient portfolios in expected return standard deviation space, let us turn our attention to the location of securities and portfolios in expected return Beta space. Let us develop the security market line.

The market portfolio M is still an efficient portfolio. Thus, the analysis of the last section holds. All securities contained in M have an expected return given by

Similarly, all portfolios composed solely of risky assets have their return given by Equation (14.5). This splits as a straight line in expected return Beta space and is the line RZTMC in Figure 14.7. This equation holds only for risky assets and for portfolios of risky assets. It does not describe the return on the riskless asset or the return on portfolios that contain the riskless asset.

In the previous chapter we examined combinations of the riskless asset and a risky portfolio and found that they lie on the straight line connecting the two points in expected return Beta space. Since investors who lend all hold risky portfolio T, the relevant line segment is RFT in Figure 14.7.

Thus, while the straight line R^M can be thought of as the security market line for all risky assets and for all portfolios composed entirely of risk assets, it does not describe the return on portfolios (and, of particular note, on those efficient portfolios) that contain the

## Lessons From The Intelligent Investor

If you're like a lot of people watching the recession unfold, you have likely started to look at your finances under a microscope. Perhaps you have started saving the annual savings rate by people has started to recover a bit.

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