Aurora Borealis
1 November 2010 CFA Institute Journal Review

Portfolio Optimization with Mental Accounts (Digest Summary)

  1. Christopher J. Sullivan

The authors propose a new portfolio optimization method that incorporates the intuitively based risk and return parameters central to behavioral portfolio theory into Markowitz’s mean–variance approach. The result is a mental accounting framework that accommodates various levels of risk aversion by associating specific investor goals with unique subportfolios.

In economics, production decisions are typically distinguished from consumption decisions when making theoretical inferences. A similar separation exists in the mean–variance portfolio theory (MVT) of Markowitz (Journal of Finance 1952). In MVT, risk-averse individuals always maximize their consumption utility by selecting optimal quantities of risk and reward from portfolios residing along a production function—that is, the efficient frontier. Although favored for its tractability and rational results, MVT suffers from an inability to solve for such distinct multiperiod consumption goals as bequests, retirement funding, or education funding. The problem increases when investors are differentially risk averse with respect to their goals or even risk seeking when faced with an alluring potential (to become rich). Investors are also more adept at specifying risk in terms of threshold returns than they are at specifying risk in the variance terms of MVT.

Although MVT is limited by its lack of prescriptions for consumption goals, the behavioral portfolio theory (BPT) of Shefrin and Statman (Journal of Financial and Quantitative Analysis 2000) offers an alternative. BPT investors construct portfolios that resemble layers of sheltered mental accounts (MAs), with each layer dedicated to fulfilling a goal subject to a particular level of risk aversion. The authors define risk as the probability (α) of failing to obtain the return required to meet a goal’s threshold level (H), and each MA portfolio frontier measures the trade-off between those probabilities and expected returns.

Merging portfolio-production and investor-consumption goals, the authors propose an optimization method called the “MA Framework” that integrates the attractive properties of MAs and MVT. They construct a three-asset portfolio (bond, low-risk stock, high-risk stock) to illustrate their approach. The aggregate portfolio has a targeted allocation of 60:20:20 across three subportfolios (retirement, education, and bequest) and variable coefficients of risk aversion (γ). As γ declines, more wealth is distributed into riskier assets until the bequest portfolio becomes leveraged using a short position in the bond and increasing long positions in the stocks.

The authors establish the mathematical equivalence between MAs and MVT by modifying the original Markowitz formulation, wherein investors choose optimal weights for n assets, assuming returns are multivariate normal (μ, Σ) such that γ varies rather than expected return. They show that when MA investors give their threshold and probability preferences (H, α), they imply a γ for a given portfolio distribution (μ, Σ). Therefore, the MA optimization problem can be transformed into the standard mean–variance problem by using the analytic mapping function γ (μ, Σ; H, α) and solving for any feasible (H, α) critical pair. The portfolio example shows that expected return is convex in the MA probability space and that efficient portfolios occupy the frontier region where threshold risk is diminishing.

The authors also demonstrate that MAs and MVT are implicitly connected to the parametric VaR (value at risk) used by risk managers and to Tesler’s (Review of Economic Studies 1956) safety threshold procedure through the MA specifications. They conclude by applying inequality (short-selling) constraints to their approach; these modifications result in only small reductions (12 bps) in subportfolio efficiency, which are minor compared with reductions resulting from investors’ misspecified risk aversion (5–40 bps). These reductions are even smaller for more risk-averse investors.

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