Dynamic coherent risk measures

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)).

Dynamic Risk Measures for Processes via Backward Stochastic Differential Equations Associated with Lévy Processes

In this paper, we study the dynamic risk measures for processes induced by backward stochastic differential equations driven by Teugel’s martingales associated with Lévy processes (BSDELs). The representation theorem for generators of BSDELs is provided. Furthermore, the time consistency of the coherent and convex dynamic risk measures for processes is characterized by means of the generators of BSDELs. Moreover, the coherency and convexity of dynamic risk measures for processes are characterized by the generators of BSDELs. Finally, we provide two numerical examples to illustrate the proposed dynamic risk measures.

Continuous-time limits of multi-period cost-of-capital margins

We consider multi-period cost-of-capital valuation of a liability cash flow subject to repeated capital requirements that are partly financed by capital injections from capital providers with limited liability. Limited liability means that, in any given period, the capital provider is not liable for further payment in the event that the capital provided at the beginning of the period turns out to be insufficient to cover both the current-period payments and the updated value of the remaining cash flow. The liability cash flow is modeled as a continuous-time stochastic process on [ 0 , T ] {[0,T]} . The multi-period structure is given by a partition of [ 0 , T ] {[0,T]} into subintervals, and on the corresponding finite set of times, a discrete-time cost-of-capital-margin process is defined. Our main objective is the analysis of existence and properties of continuous-time limits of discrete-time cost-of-capital-margin processes corresponding to a sequence of partitions whose meshes tend to zero. Moreover, we provide explicit expressions for the limit processes when cash flows are given by Itô diffusions and processes with independent increments.

Down-Side Risk Metrics as Portfolio Diversification Strategies across the Global Financial Crisis

This paper features an analysis of the effectiveness of a range of portfolio diversification strategies, with a focus on down-side risk metrics, as a portfolio diversification strategy in a European market context. We apply these measures to a set of daily arithmetically-compounded returns, in U.S. dollar terms, on a set of ten market indices representing the major European markets for a nine-year period from the beginning of 2005 to the end of 2013. The sample period, which incorporates the periods of both the Global Financial Crisis (GFC) and the subsequent European Debt Crisis (EDC), is a challenging one for the application of portfolio investment strategies. The analysis is undertaken via the examination of multiple investment strategies and a variety of hold-out periods and backtests. We commence by using four two-year estimation periods and a subsequent one-year investment hold out period, to analyse a naive 1/N diversification strategy and to contrast its effectiveness with Markowitz mean variance analysis with positive weights. Markowitz optimisation is then compared to various down-side investment optimisation strategies. We begin by comparing Markowitz with CVaR, and then proceed to evaluate the relative effectiveness of Markowitz with various draw-down strategies, utilising a series of backtests. Our results suggest that none of the more sophisticated optimisation strategies appear to dominate naive diversification.

Conditional risk and acceptability mappings as Banach-lattice valued mappings

Conditional risk and acceptability mappings quantify the desirability of random variables (e.g. financial returns) by accounting for available information. In this paper the focus lies on acceptability mappings, concave translation-equivariant monotone mappings L p (Ω,F,ℙ) →  L p´(Ω,F´,ℙ) with 1 ≤ p´ ≤ p ≤ ∞, where the σ-algebras F´ ⊂ F describe the available information. Based on the order completeness of L p (Ω,F,ℙ)-spaces, we analyze superdifferentials and concave conjugates of conditional acceptability mappings. The related results are used to show properties of two important classes of multi-period valuation functionals: SEC-functionals and additive acceptability compositions. In particular, we derive a chain rule for superdifferentials and use it for characterizing the conjugates of additive acceptability compositions and SEC-functionals.

Continuous Time Portfolio Selection under Conditional Capital at Risk

Portfolio optimization with respect to different risk measures is of interest to both practitioners and academics. For there to be a well-defined optimal portfolio, it is important that the risk measure be coherent and quasiconvex with respect to the proportion invested in risky assets. In this paper we investigate one such measure—conditional capital at risk—and find the optimal strategies under this measure, in the Black-Scholes continuous time setting, with time dependent coefficients.

Convex risk measures and the dynamics of their penalty functions

We study various properties of a dynamic convex risk measure for bounded random variables which describe the discounted terminal values of financial positions. In particular we characterize time-consistency by a joint supermartingale property of the risk measure and its penalty function. Moreover we discuss the limit behavior of the risk measure in terms of asymptotic safety and of asymptotic precision, a property which may be viewed as a non-linear analogue of martingale convergence. These results are illustrated by the entropic dynamic risk measure.