Worst-Case Higher Moment Coherent Risk Based on Optimal Transport with Application to Distributionally Robust Portfolio Optimization

The tail risk management is of great significance in the investment process. As an extension of the asymmetric tail risk measure—Conditional Value at Risk (CVaR), higher moment coherent risk (HMCR) is compatible with the higher moment information (skewness and kurtosis) of probability distribution of the asset returns as well as capturing distributional asymmetry. In order to overcome the difficulties arising from the asymmetry and ambiguity of the underlying distribution, we propose the Wasserstein distributionally robust mean-HMCR portfolio optimization model based on the kernel smoothing method and optimal transport, where the ambiguity set is defined as a Wasserstein “ball” around the empirical distribution in the weighted kernel density estimation (KDE) distribution function family. Leveraging Fenchel’s duality theory, we obtain the computationally tractable DCP (difference-of-convex programming) reformulations and show that the ambiguity version preserves the asymmetry of the HMCR measure. Primary empirical test results for portfolio selection demonstrate the efficiency of the proposed model.

Weighted Scoring Rules and Convex Risk Measures

In Weighted Scoring Rules and Convex Risk Measures, Dr. Zachary J. Smith and Prof. J. Eric Bickel (both at the University of Texas at Austin) present a general connection between weighted proper scoring rules and investment decisions involving the minimization of a convex risk measure. Weighted scoring rules are quantitative tools for evaluating the accuracy of probabilistic forecasts relative to a baseline distribution. In their paper, the authors demonstrate that the relationship between convex risk measures and weighted scoring rules relates closely with previous economic characterizations of weighted scores based on expected utility maximization. As illustrative examples, the authors study two families of weighted scoring rules based on phi-divergences (generalizations of the Weighted Power and Weighted Pseudospherical Scoring rules) along with their corresponding risk measures. The paper will be of particular interest to the decision analysis and mathematical finance communities as well as those interested in the elicitation and evaluation of subjective probabilistic forecasts.

Market and Accounting Measures of Risk: The Case of the Frankfurt Stock Exchange

The main purpose of this study was to explore the relationship between market and accounting measures of risk and the profitability of companies listed on the Frankfurt Stock Exchange. An important aspect of the study was to employ accounting beta coefficients as a systematic risk measure. The research considered classical and downside risk measures. The profitability of a company was expressed as ROA and ROE. When determining the downside risk, two approaches were employed: the approach by Bawa and Lindenberg and the approach by Harlow and Rao. In all the analyzed companies, there is a positive and statistically significant correlation between the average value of profitability ratios and the market rate of return on investment in their stocks. Additionally, correlation coefficients are higher for the companies included in the DAX index compared with those from the MDAX or SDAX indices. A positive and in each case a statistically significant correlation was observed for all DAX-indexed companies between all types of market betas and corresponding accounting betas. Likewise, for the MDAX-indexed companies, these correlations were positive but statistical significance emerged only for accounting betas calculated on ROA. As regards the DAX index, not every correlation was positive and significant.

Downside Beta and Downside Gamma: In Search for a Better Capital Asset Pricing Model

In the financial world, the importance of “downside risk” and “higher moments” has been emphasized, predominantly in developing countries such as Pakistan, for a substantial period. Consequently, this study tests four models for a suitable capital asset pricing model. These models are CAPM’s beta, beta replaced by skewness (gamma), CAPM’s beta with gamma, downside beta CAPM (DCAPM), downside beta replaced by downside gamma, and CAPM with downside gamma. The problems of the high correlation between the beta and downside beta models from a regressand point of view is resolved by constructing a double-sorted portfolio of each factor loading. The problem of the high correlation between the beta and gamma, and, similarly, between the downside beta and downside gamma, is resolved by orthogonalizing each risk measure in a two-factor setting. Standard two-pass regression is applied, and the results are reported and analyzed in terms of R2, the significance of the factor loadings, and the risk–return relationship in each model. The risk proxies of the downside beta/gamma are based on Hogan and Warren, Harlow and Rao, and Estrada. The results indicate that the single factor models based on the beta/downside beta or even gamma/downside gamma are not a better choice among all the risk proxies. However, the beta and gamma factors are rejected at a 5% and 1% significance level for different risk proxies. The obvious choice based on the results is an asset pricing model with two risk measures.

INSURANCE VALUATION: A TWO-STEP GENERALISED REGRESSION APPROACH

Current approaches to fair valuation in insurance often follow a two-step approach, combining quadratic hedging with application of a risk measure on the residual liability, to obtain a cost-of-capital margin. In such approaches, the preferences represented by the regulatory risk measure are not reflected in the hedging process. We address this issue by an alternative two-step hedging procedure, based on generalised regression arguments, which leads to portfolios that are neutral with respect to a risk measure, such as Value-at-Risk or the expectile. First, a portfolio of traded assets aimed at replicating the liability is determined by local quadratic hedging. Second, the residual liability is hedged using an alternative objective function. The risk margin is then defined as the cost of the capital required to hedge the residual liability. In the case quantile regression is used in the second step, yearly solvency constraints are naturally satisfied; furthermore, the portfolio is a risk minimiser among all hedging portfolios that satisfy such constraints. We present a neural network algorithm for the valuation and hedging of insurance liabilities based on a backward iterations scheme. The algorithm is fairly general and easily applicable, as it only requires simulated paths of risk drivers.

Time consistency for scalar multivariate risk measures

In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)).

A pessimistic bilevel stochastic problem for elastic shape optimization

We consider pessimistic bilevel stochastic programs in which the follower maximizes over a fixed compact convex set a strictly convex quadratic function, whose Hessian depends on the leader’s decision. This results in a random upper level outcome which is evaluated by a convex risk measure. Under assumptions including real analyticity of the lower-level goal function, we prove the existence of optimal solutions. We discuss an alternate model, where the leader hedges against optimal lower-level solutions, and show that solvability can be guaranteed under weaker conditions in both, a deterministic and a stochastic setting. The approach is applied to a mechanical shape optimization problem in which the leader decides on an optimal material distribution to minimize a tracking-type cost functional, whereas the follower chooses forces from an admissible set to maximize a compliance objective. The material distribution is considered to be stochastically perturbed in the actual construction phase. Computational results illustrate the bilevel optimization concept and demonstrate the interplay of follower and leader in shape design and testing.