## Info

8Assuming the mean return is above R[

8Assuming the mean return is above R[

This is the equation of a straight line with an intercept of RL and a slope of K. Thus, all points of equal desirability (i.e., constant K) plot on a straight line and the preferred line is one with the highest slope. This is shown in Figure 11.1 where the ICs are ordered such tli.it > K3> K2> Ky The Roy criterion with normally distributed returns produces a decision problem of exactly the same form as the portfolio problem with riskless lending and borrowing. In this case, rl serves the role of the riskless rate, rp. The desired portfolio is the feasible portfolio lying on the line in the most counterclockwise direction and is easy to find, utilizing the standard techniques discussed earlier. Notice that the portfolio that maximizes Roy's criterion must lie along the efficient frontier in mean standard deviation space.9

Although the analysis was performed assuming normally distributed returns, a similar result holds for any distribution that has first and second moments. The very same maximization problem follows from the use of Tchebyshev's inequality.10

'The location of portfolios that satisfy Roy's criterion when lending and borrowing are allowed is discussed in V pendix A at the end of this chapter. This conclusion assumes Rp > RL.

'"One of the ways to determine the probability of some outcome is the use of Tchebyshev's inequality. 11 tebyshev's inequality allows one to determine the maximum probability of obtaining an outcome less than some value. It does not assume any distribution for returns. If a distribution was assumed, a more precise statement about probability could be made. Rather, it is a general statement applicable for all distributions. TheTchebyshev inequality is

Prob

K is the outcome is the mean return <rp is the standard deviation K is a constant we are interest

,s than Rp. Therefore, the term in the absolute value sign is negative. Noting this, we can write the term in the rentheses as

id the expression as

iVe can express the lower limit in Roy's criterion as the number of standard deviations K lies below the mean.

Since Tchebyshev's inequality holds for any value of K, we can substitute the expression for K shown in !quation (11.2) into the left-hand side of Equation (11.1). Doing so, and simplifying, yields

Since this is precisely Roy's criterion, we want to maximize ji or maximize Eqaation (11.2). But this is exactly it we did in the case of the normal distribution.

Figure 11.1 Lines of constant preference—Roy's criterion.

The Tchebyshev inequality makes very weak assumptions about the underlying distribution. It gives an expression that allows the determination of the maximum odds of obtaining a return less than some number. The use of this inequality leads to the same maximization problem and the same analysis as previously discussed. Thus, mean-variance analysis follows from the Roy safety first criterion.

The second safety first criterion was developed by Kataoka. Kataoka suggests the following criterion: maximize the lower limit subject to the constraint that the probability of a return less than, or equal to, the lower limit is not greater than some predetermined value. For example, maximize Rr subject to the constraint that the chance of a return below RL is less than or equal to 5%. If a is the probability (in the example 5%), then in symbols this is maximize Rh subject to (1) Prob^ <RL)<a

If returns are normally distributed, we can analyze this criterion in mean standard deviation space. Earlier, we noted that if returns are normally distributed, then the probability of obtaining returns below some number depends on the number of standard deviations below the mean that the number lies. Thus the odds of obtaining a return more than 3 standard deviations below the mean is 0.13% while the odds of obtaining a return more than 2 standard deviations below the mean is 2.28%. As an example set a = 0.05. From the table of the normal distribution, we see that this is met as long as the lower limit is at least 1.65 standard deviations below the mean. With a = 0.05, the constraint becomes

RL<Rp-1.65op

Since we want to make RL as large as possible, this inequality can be written as an equality. Writing it as an equality and rearranging, we obtain for a constant Rt

This is the equation of a straight line. Since the intercept is RL as RL changes, the line shifts in a parallel fashion. Figure 11.2 illustrates this for various values of RL. The objective is

Figure 11.2 The portfolio choice problem with Kataoka's safety first rule.

to maximize RL or to move as far up as possible (in the direction of the arrow). If there is no lending or borrowing, then a unique maximum exists and it is the tangency point on the highest Rl line (R[s in the example). Note that, as in the case of Roy's criterion, the optimum portfolio must be on the efficient frontier in mean standard deviation space. Once again, the same analysis follows if one chooses to use the Tchebyshev inequality, rather than assuming normally distributed returns.

The final safety first criterion was put forth by Telser. He suggested that a reasonable criterion would be for an investor to maximize expected return, subject to the constraint that the probability of a return less than, or equal to, some predetermined limit was not greater than some predetermined number. In symbols, we have maximize Rp subject to (1) Prob(i?/; </?J<a

Once again, it is convenient to rearrange the constraint. In the discussion of the Kataoka criterion, it was shown that if returns are normally distributed, this constraint becomes

Rearranging yields

In the last section, the constant was set equal to 1.65 for the example. In general, it depends on the value of a. As discussed earlier, when the equality holds, this expression is the equation of a straight line. Consider Figure 11.3. The efficient frontier and the constraint are plotted in that figure. All points above the line meet the constraint. In Figure 11.3 the feasible set is bounded by the straight line and the efficient frontier (the shaded area). In this case the optimum is point A. If the portfolio with the overall highest return lies above the hne, it will be selected. If it does not, the constraint line excludes part of the efficient set. Eti this case, the feasible portfolio with the highest mean return will lie at the highest intersection of the efficient frontier and the constraint. In either case, the point selected will be on the efficient set. It is possible that there are no feasible points that meet the constraint.

Figure 11.3 The investor's choice problem—Teiser's criterion.

For example, in Figure 11.4 the constraint lies above the efficient set. In this case, there is no feasible portfolio lying above the constraint and the criterion fails to select any portfolios.

Note that with the Telser criterion, the optimum portfolio either lies on the efficient frontier in mean standard deviation space or it does not exist

As with the other two criteria, the same analysis follows if we use the Tchebyshev inequality rather than assuming normal returns.

The safety first criteria were originally suggested as appealing decision making and an alternative to the expected utility framework of traditional analysis. We see in this section that, under reasonable sets of assumptions, they lead to mean-variance analysis and to the selection of a particular portfolio in the efficient set. As shown in Appendix A, at the end of this chapter, with unlimited lending and borrowing at a riskless rate, the analysis may lead to infinite borrowing, an unreasonable prescription for managers. However, the difficulties lie, not with the criteria, but with the original assumption that investors can borrow

Figure 11.4 No feasible portfolio—Teiser's criterion.

unlimited amounts at a riskless rate of interest. Whether the safety first criteria are reasonable criteria can be answered only by the readers themselves. To some, they seem sensible as a description of reality. To others, the fact that they may be inconsistent with expected utility maximization leads to their rejection. If one accepts one of the safety first criteria and believes that the probability distribution of returns is normal or sufficiently well behaved that the Tchebyshev inequality holds, then the discussion in all previous chapters concerning the generation of the efficient frontier is useful in finding the optimal portfolio.

stochastic dominance

A third set of alternatives to mean-variance analysis that has been advocated in the literature is stochastic dominance. The most general form of stochastic dominance makes no assumptions about the form of the probability distribution of returns. Furthermore, when we employ stochastic dominance we do not have to assume the specific form of investors' utility functions. Rather, we can define efficient sets under alternative assumptions about the general characteristics of investors' utility functions. These characteristics are consistent with whole families of utility functions. There are three progressively stronger assumptions about investor behavior that are employed in the stochastic dominance literature. They lead directly to first-, second-, and third-order stochastic dominance. First-order stochastic dominance assumes an investor prefers more to less. Second-order stochastic dominance assumes that, in addition to investors preferring more to less, they are risk averse. Finally, third-order stochastic dominance adds to the two assumptions of second-order dominance the assumption that investors have decreasing absolute risk aversion.11

Associated with each level of stochastic dominance is a theorem that allows the investor to eliminate many portfolios from consideration. Appendix B at the end of this chapter contains a proof of each of the theorems. In the chapter proper we intend to state the theorem and illustrate its use with a simple example.

Consider the example shown in Table 11.3. If an investor preferred more to less, then investment in asset A is preferable to investment in asset B because no matter which outcome occurs, A will always yield a higher return than B. For example, if market conditions are good, then A will return 10% and B will return 9%. The theorem of first-order stochastic dominance would, in fact, show that A dominated B. Now, consider the choices shown in Table 11.4. Investment A has the better outcomes. However, it is no longer certain that an investor will do worse by investing in B. For example, it is possible the return

Table 11.3 Outcomes Associated with Alternative Market Conditions