Miloš Kopa

General linear formulations of stochastic dominance criteria

We develop and implement linear formulations of general Nth order stochastic dominance criteria for discrete probability distributions. Our approach is based on a piece-wise polynomial representation of utility and its derivatives and can be implemented by solving a relatively small system of linear inequalities. This approach allows for comparing a given prospect with a discrete set of alternative prospects as well as for comparison with a polyhedral set of linear combinations of prospects. We also derive a linear dual formulation in terms of lower partial moments and co-lower partial moments. An empirical application to historical stock market data suggests that the passive stock market portfolio is highly inefficient relative to actively managed portfolios for all investment horizons and for nearly all investors. The results also illustrate that the mean–variance rule and second-order stochastic dominance rule may not detect market portfolio inefficiency because of non-trivial violations of non-satiation and prudence.

A Portfolio Optimality Test Based on the First-Order Stochastic Dominance Criterion

Existing approaches to testing for the efficiency of a given portfolio make strong parametric assumptions about investor preferences and return distributions. Stochastic dominance-based procedures promise a useful nonparametric alternative. However, these procedures have been limited to considering binary choices. In this paper we take a new approach that considers all diversified portfolios and thereby introduce a new concept of first-order stochastic dominance (FSD) optimality of a given portfolio relative to all possible portfolios. Using our new test, we show that the U.S. stock market portfolio is significantly FSD nonoptimal relative to benchmark portfolios formed on market capitalization and book-to-market equity ratios. Without appealing to parametric assumptions about the return distribution, we conclude that no nonsatiable investor would hold the market portfolio in the face of the attractive premia of small caps and value stocks.

Robustness of optimal portfolios under risk and stochastic dominance constraints

Solutions of portfolio optimization problems are often influenced by a model misspecification or by errors due to approximation, estimation and incomplete information. The obtained results, recommendations for the risk and portfolio manager, should be then carefully analyzed. We shall deal with output analysis and stress testing with respect to uncertainty or perturbations of input data for static risk constrained portfolio optimization problems by means of the contamination technique. Dependence of the set of feasible solutions on the probability distribution rules out the straightforward construction of convexity-based global contamination bounds. Results obtained in our paper [Dupacova, J., & Kopa, M. (2012). Robustness in stochastic programs with risk constraints. Annals of Operations Research, 200, 55–74.] were derived for the risk and second order stochastic dominance constraints under suitable smoothness and/or convexity assumptions that are fulfilled, e.g. for the Markowitz mean–variance model. In this paper we relax these assumptions having in mind the first order stochastic dominance and probabilistic risk constraints. Local bounds for problems of a special structure are obtained. Under suitable conditions on the structure of the problem and for discrete distributions we shall exploit the contamination technique to derive a new robust first order stochastic dominance portfolio efficiency test.

A general test for SSD portfolio efficiency

We develop and implement a Linear Programming test to analyze whether a given investment portfolio is efficient in terms of second-order stochastic dominance relative to all possible portfolios formed from a set of base assets. In case of efficiency, the primal model identifies a sub-gradient vector of a utility function that rationalizes the evaluated portfolio. In case of inefficiency, the dual model identifies a second, efficient portfolio that dominates the evaluated portfolio. The test gives a general necessary and sufficient condition, and can deal with general linear portfolio restrictions, inefficiency degree measures, and scenarios with unequal probabilities. We also develop a compact version of the test that substantially reduces computational burden at the cost of losing information about the dual dominating portfolio in case of inefficiency. An application to US investment benchmark data qualifies a broad stock market index as significantly inefficient, and suggests that no risk-averse investor would hold the market index in the face of attractive premiums offered by some more concentrated investment portfolios.

On relations between DEA-risk models and stochastic dominance efficiency tests

In this paper, several concepts of portfolio efficiency testing are compared, based either on data envelopment analysis (DEA) or the second-order stochastic dominance (SSD) relation: constant return to scale DEA models, variable return to scale (VRS) DEA models, diversification-consistent DEA models, pairwise SSD efficiency tests, convex SSD efficiency tests and full SSD portfolio efficiency tests. Especially, the equivalence between VRS DEA model with binary weights and the SSD pairwise efficiency test is proved. DEA models equivalent to convex SSD efficiency tests and full SSD portfolio efficiency tests are also formulated. In the empirical application, the efficiency testing of 48 US representative industry portfolios using all considered DEA models and SSD tests is presented. The obtained efficiency sets are compared. A special attention is paid to the case of small number of the inputs and outputs. It is empirically shown that DEA models equivalent either to the convex SSD test or to the SSD portfolio efficiency test work well even with quite small number of inputs and outputs. However, the reduced VRS DEA model with binary weights is not able to identify all the pairwise SSD efficient portfolios.

Portfolio Choice Based on Third-Degree Stochastic Dominance

We develop an optimization method for constructing investment portfolios that dominate a given benchmark portfolio in terms of third-degree stochastic dominance. Our approach relies on the properties of the semivariance function, a refinement of an existing ‘super-convex’ dominance condition and quadratic constrained programming. We apply our method to historical stock market data using an industry momentum strategy. Our enhanced portfolio generates important performance improvements compared with alternatives based on mean-variance dominance and seconddegree stochastic dominance. Relative to the CSRP all-share index, our portfolio increases average out-of-sample return by almost seven percentage points per annum without incurring more downside risk, using quarterly rebalancing and without short selling.

Individual optimal pension allocation under stochastic dominance constraints

An individual investor has to decide how to allocate his/her savings from a retirement perspective. This problem covers a long-term horizon. In this paper we consider a 40-year horizon formulating a multi-criteria multistage program with stochastic dominance constraints in an intermediate stage and in the final stage. As we are dealing with a real problem and we have formulated the model in cooperation with a commercial Italian bank, the intermediate stage corresponds to a possible withdrawal allowed by the Italian pension system. The sources of uncertainty considered are: the financial returns, the interest rate evolution, the investor’s salary process and a considerable withdrawal event. We include a set of portfolio constraints according to the pension plan regulation. The objective of the model is to minimize the Average Value at Risk Deviation measure and to satisfy wealth goals. Three different wealth target formulations are considered: a deterministic wealth target (i.e. a comparison between the accumulated average wealth and a fixed threshold) and two stochastic dominance relations—the first order and the second order—introducing a benchmark portfolio and then requiring the optimal portfolio to dominate the benchmark. In particular, we prove that solutions obtained under stochastic dominance constraints ensure a safer allocation while still guaranteeing good returns. Moreover, we show how the withdrawal event affects the solution in terms of allocation in each of the three frameworks. Finally, the sensitivity and convergence of the stochastic solutions and computational issues are investigated.

Evaluation of aspirin's effect on platelet function early after coronary artery bypass grafting.

OBJECTIVE Aspirin therapy decreases mortality and ischemic complication rates after coronary artery bypass grafting (CABG). However, platelet inhibition after oral aspirin seems to be insufficient in the early postoperative period. There are incomplete data reporting aspirin efficacy early after CABG. The aim of this study was to assess the pharmacologic effect of aspirin on platelets in the first postoperative days using the most specific laboratory tests for the evaluation of aspirin efficacy. DESIGN A prospective study. SETTING A clinical study in one cardiac surgery center and measurements in two pharmacologic institutions. PARTICIPANTS Thirty patients. INTERVENTIONS Postoperative aspirin efficacy (200 mg/d) was assessed by the suppression of serum thromboxane B(2) (TxB(2)) and by arachidonic acid-induced aggregometry using the MULTIPLATE analyzer. Samples were collected before surgery and on postoperative days 1-5. METHODS AND MAIN RESULTS The median baseline value (range) of serum TxB(2) was 1.6 ng/mL (1.4-1.9). The median TxB(2) inhibition >90% (the value required for full platelet inhibition) was not achieved until day 5 (-91%, 0.13 ng/mL [0.08-0.22], p < 0.001) and in only 55% of patients. The median baseline ASPI value was 805 (640-975) aggregation units (AU)*min. A significant decrease in aspirin insufficiency was not seen before postoperative day 5 (390 [243-621], p < 0.003) and only 34% of patients reached an effective platelet inhibition on day 5 (cutoff < 300 AU*min). CONCLUSIONS The effect of aspirin on inhibition of TxB(2) production and arachidonic acid-induced platelet aggregation is impaired during the first postoperative days after CABG. A more effective antiplatelet strategy presumably could increase early graft patency and improve clinical outcomes after CABG.

Linear Tests for DARA Stochastic Dominance

We develop and implement linear formulations of convex stochastic dominance relations based on decreasing absolute risk aversion for discrete and polyhedral choice sets. Our approach is based on a piecewise-exponential representation of utility and a local linear approximation to the exponentiation of log marginal utility. An empirical application to historical stock market data suggests that a passive stock market portfolio is DSD inefficient relative to concentrated portfolios of small-cap stocks. The mean-variance rule and N-th order stochastic dominance rules substantially underestimate the degree of market portfolio inefficiency, because they do not penalize the unfavorable skewness of diversified portfolios, in violation of DARA.

DEA models equivalent to general N th order stochastic dominance efficiency tests

We introduce data envelopment analysis (DEA) models equivalent to efficiency tests with respect to the N th order stochastic dominance (NSD). In particular, we focus on strong and weak variants of convex NSD efficiency and NSD portfolio efficiency. The proposed DEA models are in relation with strong and weak Pareto-Koopmans efficiencies and employ N th order lower and co-lower partial moments.