A financial advisor's primary objective in allocating clients' assets is to design and construct investment policies that meet their future economic needs while satisfying their risk preferences. This is accomplished by determining their total assets, helping them to formulate an objective, estimating their future financial needs and identifying their attitude toward risk.

The ultimate success of the resulting investment policy in achieving the client's objective will depend not only on the accuracy with which each of these steps is accomplished, but also, and more importantly, on the method used to determine the client's strategic asset allocation.

Most financial advisors are familiar with the "efficient frontier" first proposed by Nobel-prize winning economist Harry Markowitz in his doctoral dissertation, and later expanded upon in his 1959 book, "Portfolio Selection: Efficient Diversification of Investments." He argued that there is an "efficient frontier" that contains only those portfolios having maximum expected returns at given risk levels. Statistically, these portfolios are said to be mean-variance efficient, since the curvilinear shape of the frontier depends on the expected returns of all available investment alternatives, their variance (as a surrogate for risk) and the pair-wise covariance between them. Figure 1 contains the traditional representation of mean-variance (M-V) efficiency. The M-V Efficient Frontier is the uppermost part of the curve, bounded by the extreme left and right arrows.

Although mean-variance efficiency has been useful as a theoretical concept in promoting an understanding of market efficiency, it has not been of practical use in designing or constructing optimal portfolios. In fact, the concept has significant shortcomings, which severely limit its practical usefulness. First, M-V efficiency assumes that the relationship between return, risk and covariance is deterministic, that there is only one optimal portfolio at a specific level of return or risk. It makes no allowance for portfolios that may be statistically indistinguishable from the "one optimal M-V efficient portfolio."

Also, M-V efficiency usually employs historical average returns, standard deviations and covariances, but these observed historical values actually come from a universe of uncertain possibilities, any of which had some probability of occurring. For example, in recent months, it has been widely reported that the stock market's lackluster performance is the result of a combination of factors, such as a slowing economy, the number of high-visibility companies lowering their estimates of future sales growth and profitability, uncertainty over the future course of interest rates and doubts concerning whether the Bush administration's economic policies will be adopted by the Congress.

These factors combined resulted in levels of return and volatility that were observed and reported. Yet, absent one or more of these factors, each of which had some probability of occurring, other numbers would have been reported.

Also, historical observations frequently contain extreme values of return and volatility that are unlikely to occur in the foreseeable future. Extreme historical values do not accurately reflect the norm of the universe to which they belong. They can significantly distort the relationship between risk and return and the position of the efficient frontier over the investment horizon in which most financial advisors and their clients are interested. In fact, an efficient frontier derived from historical data is likely to overestimate returns at most risk levels.

Until recently, a financial advisor seeking to select a portfolio to optimally meet his or her client's needs was restricted to using M-V optimization procedures, usually with historical data. Depending on the client's risk preferences, the result would likely have been one of those shown in Figure 2, which contains the asset-allocation weights in six portfolios ranging in degrees of risk on a scale from 1 to 51, with 1 being the least risky and 51, the most. Annualized monthly data from January 1990 through December 1999 were used.

The results in Figure 2 are typical of those associated with M-V optimization, which ignores many asset classes that might otherwise improve a portfolio's diversification. For example, Portfolio No. 1, the least risky is, not surprisingly, composed mainly of T-bills, with only modest amounts of small-cap and international stocks. At the other extreme, Portfolio No. 51 is comprised of only large-cap stocks. Portfolios between these two extremes ignore two or more of the available asset classes. Concentrating, as it does, on few asset classes, M-V optimization results in portfolios that could be better diversified because the returns of these different asset classes do not move in unison. Hence, using a combination of all or most of the assets available is likely to reduce the overall risk of a portfolio through increased diversification.

A new approach to portfolio optimization has been developed, one that produces better-diversified portfolios than those selected by M-V methods. This new approach, "resampled efficiency optimization," recognizes the relationship between return, risk and covariance is stochastic, and there are a number of optimal portfolios at a specific risk level that are statistically equivalent.

Resampled efficiency is based on the simple notion that historical returns are merely observed elements in a probability distribution of possible outcomes, and the distribution from which they originate is most important in determining the optimality of a portfolio. In recognizing the statistical nature of the optimization process, one can infer that the M-V efficient frontier is overly restrictive and not very realistic. Instead, there is an efficient region of portfolio choices in which all portfolios are statistically indistinguishable from each other, irrespective of their asset allocation or weightings.

Figure 3 describes the configuration of the "Resampled Efficient Region." It has an upper boundary, similar in nature but not identical to an M-V efficient frontier, and a lower boundary, which defines a region in which all portfolios are statistically indistinguishable from each other at a specific risk level.

The resampled efficient region defines a new, more realistic efficient frontier, one that is bounded on both sides by possible random outcomes. By significantly reducing the distortions extreme historical observations impose on estimating return and risk, it facilitates locating a new, "true" efficient frontier, which lies within the resampled efficient region.

The results shown in Figure 4 are typical of those associated with resampled efficient optimization because they include most asset classes available to improve a portfolio's diversification. For example, in contrast to Portfolio No. 1 derived in the M-V optimization, the resampled efficient Portfolio No. 1 contains five asset classes instead of three. Its Portfolio No. 51 comprises all six asset classes, rather than only large-cap stocks, as in the case of M-V efficiency. Moreover, portfolios in between the two extremes also contain all six of the available asset classes. In sum, resampled efficient optimization results in portfolios that are better diversified than those using M-V optimization.

The M-V optimization results illustrated in Figure 2 relate to only five specific risk levels contained in a range of risk from 1 to 51. Figure 5 maps the asset allocations over the entire spectrum of risk. Each asset's weighting in a portfolio is vertically indicated by color above the horizontal axis. The combined weightings of all assets in a portfolio, at any risk level, sum to 100 percent along the vertical axis.

A common characteristic shared by all such composition maps related to M-V optimization is the sharp change in the weighting that each asset undergoes in moving from one risk level to another. For example, Treasury bills clearly dominate the lowest-risk portfolios and then drop out halfway across the risk spectrum. Large-cap stocks are omitted from the lowest-risk portfolio, but they totally dominate the high-risk aggressive portfolio.

In contrast, the resampled optimization results illustrated in Figure 6 contain some portion of all six asset classes as the portfolios incrementally increase in risk. This is a common characteristic of composition maps related to resampled efficient optimization: a gradual change in the weighting of each asset in moving from one risk level to another with many assets used in the portfolio.

A basic tenet of modern portfolio theory states that returns to underdiversified portfolios possess two elements of risk. The first, systematic risk, is related to the returns of the overall market and cannot be further reduced through diversification. The second, unsystematic risk, is unrelated to returns of the market but is related to a specific investment. Unsystematic risk, therefore, can be eliminated through further diversification. Since unsystematic risk can be eliminated, investors holding underdiversified portfolios are not generally compensated for assuming the additional risk they bear. Consequently, the more diversified portfolios resulting from resampled efficient optimization represent superior investments to M-V derived portfolios.

Estimates other than those based on historical data can result in dramatically different asset allocations. Many financial advisors do not fully understand the limitations of using raw historical data in estimating the returns necessary to create optimized portfolios. This is unfortunate, since modern estimation methods are theoretically superior and can often result in better-performing portfolios. There are essentially two methods for enhancing performance: Stein estimators, which efficiently forecast future returns using historical data, and Bayesian procedures, which use proprietary estimates of future events to forecast returns.

Charles Stein, a pioneer in modern multivariate estimation, developed a method for refining inputs for asset allocation. His method uses information about a group of return estimates as a reference point to determine whether the estimate for a given asset is normal or extreme. Using this method, a return estimate far from the global mean that is highly volatile is assumed to be likely to revert to less extreme returns, while an extreme estimate with a small variance is less likely to revert to the group's mean.

Stein estimators are not only useful in modifying return estimates, but also can significantly improve estimation of the covariance matrix and enhance the stability of the optimization process in both M-V or resampled efficient portfolios. Oliver Ledoit developed a covariance estimation method in which the number of assets can be very large and the number of historic periods of useful data very small, making it eminently suitable for applications involving short-term asset allocation.

The most potentially rewarding inputs to a portfolio optimization process are those that contain accurate proprietary estimates of future events. Modern portfolio theory cautions that such clairvoyance is rare in efficient markets. Yet financial advisors who believe they can add value to the optimization process may use a Bayesian approach to forecasting returns.

A Bayesian approach uses reasonable estimates, or a set of assumptions, to impose an exogenous structure on the optimization process. It transforms the optimization by reducing dependence on pure statistically estimated data, as is done by using raw historical data or Stein estimators. A wide range of applications is available using Bayesian procedures. For example, one could forecast returns of a best-case, worst-case and maximum-likelihood case to determine the sensitivity of the resulting asset allocations to each of those scenarios. The suitability of results, of course, will depend on the quality and accuracy of the inputs.

There is a significant difference between mean-variance and resampled efficient optimization in the area of portfolio rebalancing, one which clearly favors using resampling. Since M-V efficiency assumes that there is a deterministic relationship between return, risk and covariance, there is only one optimal portfolio at any specified level of return or risk.

Portfolios that are M-V re-optimized rarely contain the same asset exposures from one rebalancing period to the next, and therefore almost always incur trading costs. To mitigate these costs, financial advisors have adopted a number of heuristic policies to avoid rebalancing clients' portfolios. Among these policies are:

1) "Set it and forget it," implying that the initial strategic allocation is correct and no further action is warranted.

2) "Rebalance at fixed calendar periods" to ensure the portfolio is reviewed regularly so that any necessary tactical adjustments may be made to the strategic allocation. This policy, however, ignores portfolio misalignment between periods.

3) "Rebalance at fixed trigger points" to ensure that the portfolio's allocations do not drift too far from their desired levels. During periods of market volatility this policy may require frequent rebalancing to bring the portfolio's allocations to their exact strategic levels, thereby incurring large trading costs.

4) "Rebalance to an allowed range within a set limit" is intended to reduce the number of necessary rebalancings, and the degree to which the portfolio must be adjusted, but still is costly.

Each of these widely practiced rebalancing policies is unable to objectively determine whether the existing portfolio is statistically equivalent to the target portfolio. Thus, any attempt at rebalancing is likely to result in costs that may, or may not, be warranted.

A major advantage of resampled efficient optimization is that it provides financial advisors with a new approach to rebalancing, one that can greatly reduce trading costs and related operational problems. This new approach entails specifying an appropriate confidence interval bounding the efficient region, within which the advisor considers all portfolios at specific risk levels to be statistically indistinguishable from each other. Once the resampled efficient optimization is completed, it is then possible to determine the statistical equivalence of the existing and target portfolios. If they are significantly equivalent, rebalancing is unnecessary. If they are not equivalent, then rebalancing is necessary.

The availability of a rebalancing test gives financial advisors significant control over trading costs, and reduced trading costs mean higher returns for clients. While the prospective savings will depend on the width of the confidence interval, it is estimated that an approximately 60% reduction in trading costs is possible, on average.

Perhaps the greatest economy provided by the rebalancing test accrues to the advisor. An advisor responsible for 100 portfolios is required to spend, on average, at most 1% of his or her time analyzing each portfolio, irrespective of whether it needs rebalancing. With a simple computerized rebalancing test, he or she can scan all 100 portfolios and readily identify those that require further attention. Assuming that only 10 of the portfolios need to be rebalanced, the advisor then has the luxury of spending an average of 10% of his or her time on each. In other words, the advisor is able to direct his or her attention to those portfolios that need it and avoid those that don't. Such a system enables the advisor to provide equivalent oversight to client portfolios with less effort.

Conclusion

New developments in efficient asset allocation now provide financial advisors with valuable tools for estimating returns and optimizing and rebalancing portfolios. These developments, which promise to increase advisors' effectiveness and productivity, are based on the following:

1) The Resampled Efficient Region has rendered the concept of a deterministic Efficient Frontier obsolete for practical purposes.

2) It is advisable to adjust historical observations of return, volatility and covariance to produce usable forecasts of the performance of a wide range of asset allocations.

3) Proprietary forecasts (Bayesian estimates) of return, risk or covariance may significantly improve the performance of the asset-allocation process.

4) A new rebalancing test can significantly reduce trading costs, improve returns and strengthen the oversight function, while reducing the effort to administer it.

C. Michael Carty is Principal and Chief Investment Officer for New Millennium Advisors, Inc., a New York City-based investment advisor. He can be reached at [email protected].