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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.

Das et al. (2010) develop an appealing model that incorporates features of both behavioral and mean–variance models. Consistent with Shefrin and Statman (2000), Das et al. consider an investor who divides his or her wealth among mental accounts (hereafter, ‘accounts’) with motives such as retirement and bequest.1 Within each account, the investor seeks to select the portfolio with maximum expected return subject to a constraint that reflects the account’s motive. This constraint precludes the probability that the account’s return is less than or equal to some threshold return from exceeding some threshold probability. Consistent with Markowitz (1952), optimal portfolios within accounts are on the mean–variance frontier.2 Therefore, the corresponding aggregate portfolio is also on it.3 Importantly, Das et al. assume that the investor only faces portfolio risk. In practice, however, many individuals also face background risk (i.e., risk that is not fully insurable in financial markets). Sources of background risk include, for example, labor income and real estate. Due to the practical prominence of background risk, its recognition is of particular interest. Indeed, Das et al. suggest an extension of their results to the case where background risk is present. In this paper, we provide such an extension. In doing so, we develop a portfolio selection model with accounts and background risk. Like Das et al., we consider an investor who divides his or her wealth among accounts. Unlike Das et al., however, we assume that the investor faces background risk in each account. For any given account, the investor seeks to select the portfolio that, taking into consideration background risk, maximizes the account’s expected return subject to a constraint that reflects the account’s motive. This constraint precludes the probability that, again taking into consideration background risk, the account’s return is less than or equal to some threshold return from exceeding some threshold probability. The motivation for our model can be seen by combining two literatures. The first literature involves behavioral portfolio theory. As noted earlier, our model incorporates the idea that investors view their aggregate portfolios as collections of portfolios within accounts. The second literature involves portfolio selection with background risk. As also noted earlier, our model incorporates the idea that investors face background risk. We begin by characterizing the existence and composition of optimal portfolios within accounts. We show that the optimal portfolio within a given account exists if and only if the threshold probability is sufficiently low and the threshold return is sufficiently small. Moreover, we find that the composition of optimal portfolios within accounts can differ notably from those of portfolios on the mean–variance frontier. Hence, the former portfolios can lie considerably away from this frontier. Similarly, we next characterize the existence and composition of the aggregate portfolio. The aggregate portfolio exists if and only if the threshold probability of each account is sufficiently low and the threshold return of each account is sufficiently small. Furthermore, we find that the composition of the aggregate portfolio can differ notably from those of portfolios on the mean–variance frontier. Hence, the former portfolio can lie considerably away from this frontier. The finding that the optimal portfolios within accounts and the aggregate portfolio generally lie away from the mean–variance frontier contrasts with the results of Das et al. who show that such portfolios lie on it. However, we should emphasize that our finding is driven by the presence of background risk, not by mental accounting. Indeed, an investor with a mean–variance objective function who faces background risk and has a single account optimally selects a portfolio that generally lies away from the mean–variance frontier (see, e.g., Baptista, 2008 and Jiang et al., 2010). Hence, an investor who faces background risk and has multiple accounts ends up selecting optimal portfolios within accounts and an aggregate portfolio that generally also lie away from it. There is an extensive literature recognizing the effect of background risk on portfolio selection. For example, some papers provide conditions on utility functions under which the presence of background risk makes investors less willing to bear other risks (see, e.g., Pratt and Zeckhauser, 1987, Kimball, 1993 and Gollier and Pratt, 1996). Other papers examine the effect of background risk on the optimal portfolios of investors who use an expected utility model (see, e.g., Heaton and Lucas, 2000). There are also papers that investigate the effect of background risk on the optimal portfolios of investors who use a mean–variance model (see, e.g., Flavin and Yamashita, 2002, Baptista, 2008 and Jiang et al., 2010). Our paper differs from this literature in two respects. First, while we consider an investor with multiple accounts, the literature considers an investor with a single account. Second, we assume that for each account the investor seeks to maximize the account’s expected return subject to a constraint that reflects the account’s motive, whereas the literature assumes that the investor maximizes either expected utility or a mean–variance objective function. We proceed as follows. Section 2 describes the model, and characterizes the optimal portfolios within accounts and the aggregate portfolio when short sales are allowed. Section 3 provides an example to illustrate these portfolios. Section 4 examines the case when short sales are disallowed. Section 5 concludes.4

Das et al. (2010) develop an appealing model that incorporates features of both behavioral and mean–variance models. Consistent with Shefrin and Statman (2000), Das et al. consider an investor who divides his or her wealth among mental accounts with different motives such as retirement and bequest. Consistent with Markowitz (1952), optimal portfolios within accounts are on the mean–variance frontier. Therefore, the corresponding aggregate portfolio is also on it. Importantly, Das et al. assume that the investor only faces portfolio risk. In practice, however, many individuals also face background risk from sources such as labor income and real estate. Indeed, Das et al. suggest an extension of their results to the case where background risk is present. In this paper, we provide such an extension. 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. Our results are of practical relevance to both investors and financial advisers. In regard to investors, we highlight the need for them to either: (1) recognize background risk while determining their optimal portfolios within accounts; or (2) instruct others (e.g., financial advisers) to recognize it on their behalf. In regard to financial advisers, two aspects are worth noting. First, our results are useful to identify the optimal portfolios of their clients when they have mental accounts and face background risk. Second, financial advisers should be aware that the composition of their clients’ optimal portfolios notably depends on background risk.