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 Markowitz approach, the DW approach incorporates current and implied assets and
liabilities to create a superior balance sheet for the investor by including his or her
finances. The DW approach also includes an objective risk aversion component that considers
higher return moments (i.e., skewness and kurtosis), which is absent in the Markowitz
approach.
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
incorporates a more diverse set of potential return characteristics, each with varying degrees
of risk and thus resulting in a better-informed final optimized portfolio.
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.