## Capital Allocation and the Separation Property

Now that we have the efficient frontier, we proceed to step two and introduce the risk-free asset. Figure 8.14 shows the efficient frontier plus three CALs representing various

234 PART II Portfolio Theory

Figure 8.14 Capital allocation lines with various portfolios from the efficient set.

portfolios from the efficient set. As before, we ratchet up the CAL by selecting different portfolios until we reach Portfolio P, which is the tangency point of a line from F to the efficient frontier. Portfolio P maximizes the reward-to-variability ratio, the slope of the line from F to portfolios on the efficient frontier. At this point our portfolio manager is done. Portfolio P is the optimal risky portfolio for the manager's clients. This is a good time to ponder our results and their implementation.

The most striking conclusion is that a portfolio manager will offer the same risky portfolio, P, to all clients regardless of their degree of risk aversion.14 The degree of risk aversion of the client comes into play only in the selection of the desired point along the CAL. Thus the only difference between clients' choices is that the more risk-averse client will invest more in the risk-free asset and less in the optimal risky portfolio than will a less risk-averse client. However, both will use Portfolio P as their optimal risky investment vehicle.

This result is called a separation property; it tells us that the portfolio choice problem may be separated into two independent tasks. The first task, determination of the optimal risky portfolio, is purely technical. Given the manager's input list, the best risky portfolio is the same for all clients, regardless of risk aversion. The second task, however, allocation of the complete portfolio to T-bills versus the risky portfolio, depends on personal preference. Here the client is the decision maker.

The crucial point is that the optimal portfolio P that the manager offers is the same for all clients. This result makes professional management more efficient and hence less costly. One management firm can serve any number of clients with relatively small incremental administrative costs.

14 Clients who impose special restrictions (constraints) on the manager, such as dividend yield, will obtain another optimal portfolio. Any constraint that is added to an optimization problem leads, in general, to a different and less desirable optimum compared to an unconstrained program.

### CHAPTER 8 Optimal Risky Portfolios 235

In practice, however, different managers will estimate different input lists, thus deriving different efficient frontiers, and offer different "optimal" portfolios to their clients. The source of the disparity lies in the security analysis. It is worth mentioning here that the rule of GIGO (garbage in-garbage out) also applies to security analysis. If the quality of the security analysis is poor, a passive portfolio such as a market index fund will result in a better CAL than an active portfolio that uses low-quality security analysis to tilt portfolio weights toward seemingly favorable (mispriced) securities.

As we have seen, optimal risky portfolios for different clients also may vary because of portfolio constraints such as dividend-yield requirements, tax considerations, or other client preferences. Nevertheless, this analysis suggests that a limited number of portfolios may be sufficient to serve the demands of a wide range of investors. This is the theoretical basis of the mutual fund industry.

The (computerized) optimization technique is the easiest part of the portfolio construction problem. The real arena of competition among portfolio managers is in sophisticated security analysis.

Suppose that two portfolio managers who work for competing investment management houses each employ a group of security analysts to prepare the input list for the Markowitz algorithm. When all is completed, it turns out that the efficient frontier obtained by portfolio manager A dominates that of manager B. By dominate, we mean that A's optimal risky portfolio lies northwest of B's. Hence, given a choice, investors will all prefer the risky portfolio that lies on the CAL of A.

a. What should be made of this outcome?

b. Should it be attributed to better security analysis by A's analysts?

c. Could it be that A's computer program is superior?

d. If you were advising clients (and had an advance glimpse at the efficient frontiers of various managers), would you tell them to periodically switch their money to the manager with the most northwesterly portfolio?

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