We present a framework for inverse optimization in a Markowitz portfolio model that is extended to include a third criterion. The third criterion causes the traditional nondominated frontier to become a surface. Until recently, it had not been possible to compute such a surface. But by using a new method that is able to generate the nondominated surfaces of tri-criterion portfolio selection problems, we are able to compute via inverse optimization the implied risk tolerances of given funds that pursue an additional objective beyond risk and return. In applying this capability to a broad sample of conventional and socially responsible (SR) mutual funds, we find that after the screening process there appears to be no significant difference between how assets are allocated in socially responsible and conventional mutual funds, which is likely to be different from what most SR investors would expect.

In standard portfolio theory, an investor is typically taken as having one stochastic objective, to maximize the random variable of portfolio return. But in this paper, we focus on investors whose purpose is to build, more broadly, a "suitable portfolio" taking additional concerns into account. Such investors would have additional stochastic and deterministic objectives that might include liquidity, dividends, number of securities in a portfolio, social responsibility, and so forth. To accommodate such investors, we develop a multiple criteria portfolio selection formulation, corroborate its appropriateness by examining the sensitivity of the nondominated frontier to various factors, and observe the conversion of the nondominated frontier to a nondominated surface. Furthermore, multiple criteria enable us to provide an explanation as to why the "market portfolio," so often found deep below the nondominated frontier, is roughly where one would expect it to be with multiple criteria. After commenting on solvability issues, the paper concludes with the idea that what is the "modern portfolio theory" of today might well be interpreted as a projection onto two-space of a real multiple criteria portfolio selection problem from higher dimensional space.

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Computing the nondominated set of a multiple objective mathematical program has long been a topic in multiple criteria decision making. In this paper, motivated by the desire to extend Markowitz portfolio selection to an additional linear criterion (dividends, liquidity, sustainability, etc.), we demonstrate an exact method for computing the nondominated set of a tri-criterion program that is all linear except for the fact that one of its objectives is to minimize a convex quadratic function. With the nondominated set of the resulting quad-lin-lin program being a surface composed of curved platelets, a multiparametric algorithm is devised for computing the platelets so that they can be graphed precisely. In this way, graphs of the tri-criterion nondominated surface can be displayed so that, as in traditional portfolio selection, a most preferred portfolio can be selected while in full view of all other contenders for optimality. Finally, by giving an example for socially responsible investors, we demonstrate that our algorithm can outperform standard portfolio strategies for multicriterial decision makers.

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