Harry Markowitz’s work in 1952, which introduced the notion that investors should consider both risk and return in their asset allocation decisions, has become one of the most influential works in investment theory. The authors highlight several shortcomings of Markowitz’s mean–variance optimization approach and introduce a more robust model for the growth of discretionary wealth that has practical applications for both individual and professional investors.
The motivation for an enhanced approach to portfolio risk–return optimization,
according to the authors, is a conceptual shift in the investment world that emphasizes the
need to refocus on the uncertainty in one’s future financial requirements. This shift is
grounded in the original approach to modeling the growth of discretionary wealth (DW),
introduced by Wilcox (
The authors make methodological improvements that allow them to use full probability distributions that contain all relevant information necessary to reduce the narrowness of point-estimate-based optimizing methods. Most importantly, they claim that the need to combine uncertainty in saving and spending plans with return risks is missing in setting investment policy. This factor can lead to improved long-run optimization of median wealth, and conveniently, it can be accomplished with a straightforward Bayesian approach.
The authors illustrate how the original DW approach, based on point estimates, can be applied to a simple investment policy problem for an investor with two investment opportunities (cash and a bond–equity balanced fund) to obtain the risk aversion parameter that is unspecified in the original Markowitz approach. The goal in the DW approach is to maximize the expected leveraged log return, in which leverage is defined as the value of the investor’s investment portfolio divided by DW. In contrast, the goal in the Markowitz approach is to maximize the expected return–variance relationship. The introduction of leverage as an objective risk aversion measurement in the DW approach is key. It formally incorporates a very intuitive measure of risk for households by focusing attention on the value of assets (either adequate or inadequate) to support future spending commitments.
The complexity is then increased by expanding the investment opportunity set to include cash, a bond, and two equity assets. The methodology is also refined to incorporate the point estimate for mean implied leverage, as opposed to the implied leverage calculated from mean lifetime. As the probability distribution for implied leverage is skewed to the right (i.e., higher leverage as DW is reduced), the implied ratio becomes somewhat larger, which reinforces implied leverage as an important discovery tool for risk aversion. The authors conclude this section by pointing out areas in which practical problems remain in the DW approach that are also evident in the Markowitz model.
The authors next use their full Bayesian-enhanced DW approach to derive an investor’s
appropriate asset allocation. The key difference between this approach and the original DW
approach is the retention of full probability distributions for intermediate calculations of
higher return moment values (e.g., mean, variance, kurtosis) and extended balance sheet items.
They use these to construct a Bayesian approach to model log returns on DW as a function of
portfolio weights
The authors construct tests for robustness and accuracy by using multiple assets in both long-only and long–short portfolios to formally ascertain the magnitude of any improvement in mean log returns that result from using the Bayesian DW model compared with the Markowitz mean–variance optimization model. The robustness tests confirm that in terms of mean (and median) deficits, the Bayesian approach outperforms the Markowitz approach by 111 bps (21 bps) for long-only portfolios and 334 bps (147 bps) for long–short portfolios. The prominent difference between the mean and median deficits for both portfolios highlights the efficacy of the Bayesian approach in avoiding very poor returns. The accuracy test results essentially mirror those of the robustness test, and in addition, the asset allocations for both portfolios using the Bayesian DW approach prove to be much more conservative than those for the Markowitz approach. This latter result was because of higher-moment effects of skewness on leverage.
After testing their theory on illustrative examples of investment policy problems, the authors conclude that improved results for expected growth in DW are obtained by applying the full Bayesian logic approach rather than limiting its application solely to the development of the risk-aversion parameter, which is absent from Markowitz’s conventional mean–variance optimization based on point estimates.