Das et al. (2010) develop a model where an investor divides his or her wealth among mental accounts with motives such as retirement and bequest. Nevertheless, the investor ends up selecting portfolios within mental accounts and an aggregate portfolio that lie on the mean–variance frontier. Importantly, they assume that the investor only faces portfolio risk. In practice, however, many individuals also face background risk. Accordingly, our paper expands upon theirs by considering the case where the investor faces background risk. Our contribution is threefold. First, we provide an analytical characterization of the existence and composition of the optimal portfolios within accounts and the aggregate portfolio. Second, we show that these portfolios lie away from the mean–variance frontier under fairly general conditions. Third, we find that the composition and location of such portfolios can differ notably from those of portfolios on the mean–variance frontier.

Most investors delegate the management of a fraction of their wealth to portfolio managers who are given the task of beating a benchmark. However, in an influential paper [Roll, R., 1992. A mean/variance analysis of tracking error. Journal of Portfolio Management 18, 13–22] shows that the objective functions commonly used by these managers lead to the selection of portfolios that are suboptimal from the perspective of investors. In this paper, we provide an explanation for the use of these objective functions based on the effect of background risk on investors’ optimal portfolios. Our main contribution is to provide conditions under which investors can optimally delegate the management of their wealth to portfolio managers.

Das et al. (2010) develop an elegant framework where an investor selects portfolios within mental accounts but ends up holding an aggregate portfolio on the mean–variance frontier. This investor directly allocates the wealth in each account among available assets. In practice, however, investors often delegate the task of allocating wealth among assets to portfolio managers who seek to beat certain benchmarks. Accordingly, we extend their framework to the case where the investor allocates the wealth in each account among portfolio managers. Our contribution is threefold. First, we provide an analytical characterization of the existence and composition of the optimal portfolios within accounts and the aggregate portfolio. Second, we present conditions under which such portfolios are not on the mean–variance frontier, and conditions under which they are. Third, we show that the aforementioned analytical characterization is also applicable within the framework of Das et al. and thus improves upon their numerical approach.

Active portfolio management often involves the objective of selecting a portfolio with minimum tracking error variance (TEV) for some expected gain in return over a benchmark. However, Roll (1992) shows that such portfolios are generally suboptimal because they do not belong to the mean-variance frontier and are thus overly risky. Our paper proposes an appealing method to lessen this suboptimality that involves the objective of selecting a portfolio from the set of portfolios that have minimum TEV for various levels of ex-ante alpha, which we refer to as the alpha-TEV frontier. Since practitioners commonly use ex-post alpha to assess the performance of managers, the use of this frontier aligns the objectives of managers with how their performance is evaluated. Furthermore, sensible choices of ex-ante alpha lead to the selection of portfolios that are less risky (in variance terms) than the portfolios that active managers would otherwise select.

When identifying optimal portfolios, practitioners often impose a drawdown constraint. This constraint is even explicit in some money management contracts such as the one recently involving Merrill Lynch’ management of Unilever’s pension fund. In this setting, we provide a characterization of optimal portfolios using mean–variance analysis. In the absence of a benchmark, we find that while the constraint typically decreases the optimal portfolio’s standard deviation, the constrained optimal portfolio can be notably mean–variance inefficient. In the presence of a benchmark such as in the Merrill Lynch–Unilever contract, we find that the constraint increases the optimal portfolio’s standard deviation and tracking error volatility. Thus, the constraint negatively affects a portfolio manager’s ability to track a benchmark.

In Das et al. (2010), an agent divides his or her wealth among mental accounts that have different goals and optimal portfolios. While the moments of the distribution of asset returns are exogenous in their normative model, they are endogenous in our corresponding positive model. We obtain the following results. First, there are multiple equilibria that we parameterize by the implied risk aversion coefficient of the agent’s aggregate portfolio. Second, equilibrium asset prices and the composition of optimal portfolios within accounts depend on this coefficient. Third, altering the goal of any given account affects the composition of each portfolio.

Financial institutions suffered large trading losses during the 2007–2009 global financial crisis. These losses cast doubt on the effectiveness of regulations and risk management systems based on a single Value-at-Risk (VaR) constraint. While some researchers have recommended using Conditional Value-at-Risk (CVaR) to control tail risk, VaR remains popular among practitioners and regulators. Accordingly, our paper examines the effectiveness of multiple VaR constraints in controlling CVaR. Under certain conditions, we theoretically show that they are more effective than a single VaR constraint. Furthermore, we numerically find that the maximum CVaR permitted by the constraints is notably smaller than with a single constraint. These results suggest that regulations and risk management systems based on multiple VaR constraints are more effective in reducing tail risk than those based on a single VaR constraint.

We examine the impact of adding either a VaR or a CVaR constraint to the mean–variance model when security returns are assumed to have a discrete distribution with finitely many jump points. Three main results are obtained. First, portfolios on the VaR-constrained boundary exhibit (K+2)-fund separation, where K is the number of states for which the portfolios suffer losses equal to the VaR bound. Second, portfolios on the CVaR-constrained boundary exhibit (K+3)-fund separation, where K is the number of states for which the portfolios suffer losses equal to their VaRs. Third, an example illustrates that while the VaR of the CVaR-constrained optimal portfolio is close to that of the VaR-constrained optimal portfolio, the CVaR of the former is notably smaller than that of the latter. This result suggests that a CVaR constraint is more effective than a VaR constraint to curtail large losses in the mean–variance model.