Abstract In this paper, we compare two of the main paradigms of portfolio theory: mean variance analysis and expected utility. In particular, we show empirically that mean variance efficient portfolios are typically sub-optimal for non satiable and risk averse investors. We illustrate that the second order stochastic dominance (SSD) efficient set is the solution of a multi-objective optimization problem. We further show that the market portfolio is not necessarily a solution to this optimization problem. We also conduct an empirical analysis, examining the ex ante and ex post performance of SSD and mean variance efficient portfolios, using a bootstrap approach. In an ex ante analysis, we compare empirical moments, the level of diversification and set distances of mean variance and SSD efficient sets. We also show that the global minimum variance (GMV) portfolio and the part of the mean variance efficient frontier (MVEF) composed of highly diversified portfolios is second order stochastically dominated. This result also provides a possible alternative explanation for the diversification puzzle. Conducting an ex post analysis, we construct second order stochastic dominating strategies that outperform the GMV portfolio in terms of wealth and various other performance measures, producing a positive ex post opportunity cost.

We develop a theory of decision making and General Equilibrium for contingent markets when incomplete preferences are generated by second-order stochastic dominance (SSD). Demand, Pareto-optima and equilibria dominance are fully characterized. Demands and equilibrium allocations are non-increasing functions of the pricing density and Pareto-optimal allocations are comonotone. They generalize mean–variance demands and CAPM equilibrium allocations which are non-increasing affine functions of the pricing density. They are not observationally distinguishable from those of von-Neumann–Morgenstern decision makers with increasing strictly concave utilities nor from those of strict risk averse non-expected utility maximizers. We also show that expenditure functions associated to second-order stochastic dominance, provide microeconomic foundations for a class of law invariant risk-measures used in mathematical finance.

We develop and implement methods for determining whether relaxing sparsity constraints on portfolios improves the investment opportunity set for risk-averse investors. We formulate a new estimation procedure for sparse second-order stochastic spanning based on a greedy algorithm and Linear Programming. We show the optimal recovery of the sparse solution asymptotically whether spanning holds or not. From large equity datasets, we estimate the expected utility loss due to possible under-diversification, and find that there is no benefit from expanding a sparse opportunity set beyond 45 assets. The optimal sparse portfolio invests in 10 industry sectors and cuts tail risk when compared to a sparse mean-variance portfolio. On a rolling-window basis, the number of assets shrinks to 25 assets in crisis periods, while standard factor models cannot explain the performance of the sparse portfolios.

This work analyses the impact of incorporating risk measures such as conditional value-at-risk, (second-order) stochastic dominance constraints, and a concave utility function, into equilibrium energy markets. It defines two illustrative problems, with price-making firms facing stochastic demand and stochastic costs respectively, and derives risk-averse equilibrium models based upon these problems. Analytic and numeric results from these models are discussed, highlighting how risk aversion leads to modified expected prices and profits. Specific attention is paid to how the risk measures compare to each other, and how their incorporation alters the complexity of the problem. The problems are defined as mixed complementarity problems using the Karush-Kuhn-Tucker conditions, and the potential to recast them to equivalent convex optimization problems is detailed. The work conducts a numerical analysis of the models to gauge the computational expense associated with each risk measure. The impact of incorporating risk measures into equilibrium energy markets is hitherto underexplored in the literature, as are comparative analyses of the various risk measures and their respective advantages in terms of applicability and computational cost. The results indicate that risk aversion causes the variability of profits to decrease regardless of the chosen measure, and highlight the influence which hedging instruments have on risk averse players and price dynamics. They additionally demonstrate that CVaR has a significantly higher computational cost than other risk measures. With increasing stochasticity in energy markets owing to the transition to renewables and geopolitical turmoil, it is imperative that models account for risk-averse decision making accurately, and that the impact of risk measures is understood—this work addresses this gap.

In order to rank investments under uncertainty, the most widely used method is mean variance analysis. Stochastic dominance is an alternative concept which ranks investments by using the whole distribution function. There exist three models: first-order stochastic dominance is used when the distribution functions do not intersect, second-order stochastic dominance is applied to situations where the distribution functions intersect only once, while third-order stochastic dominance solves the ranking problem in the case of double intersection. Almost stochastic dominance is a special model. Finally we show that the existence of arbitrage opportunities implies the existence of stochastic dominance, while the reverse does not hold.