To split or not to split: Capital allocation with convex risk measures

Convex risk measures were introduced by Deprez and Gerber [Deprez, O., Gerber, H.U., 1985. On convex principles of premium calculation. Insurance: Math. Econom. 4 (3), 179-189]. Here the problem of allocating risk capital to subportfolios is addressed, when convex risk measures are used. The Aumann-Shapley value is proposed as an appropriate allocation mechanism. Distortion-exponential measures are discussed extensively and explicit capital allocation formulas are obtained for the case that the risk measure belongs to this family. Finally the implications of capital allocation with a convex risk measure for the stability of portfolios are discussed. It is demonstrated that using a convex risk measure for capital allocation can produce an incentive for infinite fragmentation of portfolios.     

Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach

We present an approach for the transition from convex risk measures in a certain discrete time setting to their counterparts in continuous time. The aim of this paper is to show that a large class of convex risk measures in continuous time can be obtained as limits of discrete time-consistent convex risk measures. The discrete time risk measures are constructed from properly rescaled (‘tilted’) one-period convex risk measures, using a d-dimensional random walk converging to a Brownian motion. Under suitable conditions (covering many standard one-period risk measures) we obtain convergence of the discrete risk measures to the solution of a BSDE, defining a convex risk measure in continuous time, whose driver can then be viewed as the continuous time analogue of the discrete ‘driver’ characterizing the one-period risk. We derive the limiting drivers for the semi-deviation risk measure, Value at Risk, Average Value at Risk, and the Gini risk measure in closed form.     

Convex risk measures on Orlicz spaces: inf-convolution and shortfall

We focus on, throughout this paper, convex risk measures defined on Orlicz spaces. In particular, we investigate basic properties of inf-convolutions defined between a convex risk measure and a convex set, and between two convex risk measures. Moreover, we study shortfall risk measures, which are convex risk measures induced by the shortfall risk. By using results on inf-convolutions, we obtain a robust representation result for shortfall risk measures defined on Orlicz spaces under the assumption that the set of hedging strategies has the sequential compactness in a weak sense. We discuss in addition a construction of an example having the sequential compactness.     

On securitization,market completion and equilibrium risk transfer

We propose an equilibrium framework within which to price financial securities written on non-tradable underlyings such as temperature indices. We analyze a financial market with a finite set of agents whose preferences are described by a convex dynamic risk measure generated by the solution of a backward stochastic differential equation. The agents are exposed to financial and non-financial risk factors. They can hedge their financial risk in the stock market and trade a structured derivative whose payoff depends on both financial and external risk factors. We prove an existence and uniqueness of equilibrium result for derivative prices and characterize the equilibrium market price of risk in terms of a solution to a non-linear BSDE.     

Optimal static-dynamic hedges for exotic options under convex risk measures

We study the problem of optimally hedging exotic derivatives positions using a combination of dynamic trading strategies in underlying stocks and static positions in vanilla options when the performance is quantified by a convex risk measure. We establish conditions for the existence of an optimal static position for general convex risk measures, and then analyze in detail the case of shortfall risk with a power loss function. Here we find conditions for uniqueness of the static hedge. We illustrate the computational challenge of computing the market-adjusted risk measure in a simple diffusion model for an option on a non-traded asset.     

Living on the edge: how risky is it to operate at the limit of the tolerated risk?

This paper studies some of the implicit risks associated with strategies followed by a risk averse investor who maximizes the expected value of his final wealth, subject to a risk tolerance constraint characterized in terms of a convex risk measure such as Conditional Value-at-Risk. Embedded probability measures are uncovered using duality theory; these are used to assess the probability of surpassing a standard Value-at-Risk threshold. Using one of these embedded probabilities, a closed-form measure of the financial cost of hedging the loss exposure associated to the optimal strategies is derived and shown to be, under certain assumptions, a coherent measure of risk.     

New affine measures of symmetry for convex bodies

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with “enough symmetries”.To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are “far apart”. In particular, we give an example of a convex body whose centroid and Santaló point are “far apart”.     

A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for d assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman’s principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.     

On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy

We consider a risk minimization problem in a continuous-time Markovian regime-switching financial model modulated by a continuous-time, observable and finite-state Markov chain whose states represent different market regimes. We adopt a particular form of convex risk measure, which includes the entropic risk measure as a particular case, as a measure of risk. The risk-minimization problem is formulated as a Markovian regime-switching version of a two-player, zero-sum stochastic differential game. One important feature of our model is to allow the flexibility of controlling both the diffusion process representing the financial risk and the Markov chain representing macro-economic risk. This is novel and interesting from both the perspectives of stochastic differential game and stochastic control. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution of the game is provided and some particular cases are discussed.     

Risk measuring under liquidity risk

We present a general framework for measuring the liquidity risk. The theoretical framework defines risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defined as the risk that a security or a portfolio of securities cannot be sold or bought without causing changes in prices. The risk measures are decomposed into two terms, one measuring the risk of the future value of a given position in a security or a portfolio of securities and the other the initial cost of this position. Within the framework of coherent risk measures, the risk measures applied to the random part of the future value of a position in a determinate security are increasing monotonic and convex cash sub-additive on long positions. The contrary, in certain situations, holds for the sell positions. By using convex risk measures, we apply our framework to the situation in which large trades are broken into many small ones. Dual representation results are obtained for both positions in securities and portfolios. We give many examples of risk measures and derive for each of them the respective capital requirement. In particular, we discuss the VaR measure.     

On Coherent Risk Measures Induced by Convex Risk Measures

We study the close relationship between coherent risk measures and convex risk measures. Inspired by the obtained results, we propose a class of coherent risk measures induced by convex risk measures. The robust representation and minimization problem of the induced coherent risk measure are investigated. A new coherent risk measure, the Entropic Conditional Value-at-Risk (ECVaR), is proposed as a special case. We show how to apply the induced coherent risk measure to realistic portfolio selection problems. Finally, by comparing its out-of-sample performance with that of CVaR, entropic risk measure, as well as entropic value-at-risk, we carry out a series of empirical tests to demonstrate the practicality and superiority of the ECVaR measure in optimal portfolio selection.     

On optimal control of capital injections by reinsurance and investments

This is a review paper on the optimal control of capital injections by reinsurance and investments. We will focus on the two most popular models for the surplus process of an insurer: a classical risk model and its diffusion approximation. Both models are modified by the possibility of reinsurance and investments into a risky or riskless asset. The insurer is allowed to change the amount to be invested and the retention level of the reinsurance continuously, i.e. we consider dynamic reinsurance and investment strategies. In addition, the cedent has to inject capital in order to keep the surplus positive. As a risk measure we choose the value of the expected discounted capital injections. The problem is to minimize the expected discounted capital injections over all admissible reinsurance and investments strategies and to find the optimal strategy if it exists. A detailed discussion of the topic can be found in my doctoral thesis “Optimal Control of Capital Injections by Reinsurance and Investments” (Eisenberg in Optimal control of capital injections by reinsurance and investments. PhD thesis, Universität zu Köln, 2010), which is the Gauss prize winning paper of 2009.     

Multivariate coherent risk measures induced by multivariate convex risk measures

In this paper, we study the close relationship between multivariate coherent and convex risk measures. Namely, starting from a multivariate convex risk measure, we propose a family of multivariate coherent risk measures induced by it. In return, the convex risk measure can be represented by its induced coherent risk measures. The representation result for the induced coherent risk measures is given in terms of the minimal penalty function of the convex risk measure. Finally, an example is also given.

     

A convex analytic approach to Markov decision processes

Summary This paper develops a new framework for the study of Markov decision processes in which the control problem is viewed as an optimization problem on the set of canonically induced measures on the trajectory space of the joint state and control process. This set is shown to be compact convex. One then associates with each of the usual cost criteria (infinite horizon discounted cost, finite horizon, control up to an exit time) a naturally defined occupation measure such that the cost is an integral of some function with respect to this measure. These measures are shown to form a compact convex set whose extreme points are characterized. Classical results about existence of optimal strategies are recovered from this and several applications to multicriteria and constrained optimization problems are briefly indicated.Research supported by NSF Grant CDR-85-00108     

A control approach to robust utility maximization with logarithmic utility and time-consistent penalties

We propose a stochastic control approach to the dynamic maximization of robust utility functionals that are defined in terms of logarithmic utility and a dynamically consistent convex risk measure. The underlying market is modeled by a diffusion process whose coefficients are driven by an external stochastic factor process. In particular, the market model is incomplete. Our main results give conditions on the minimal penalty function of the robust utility functional under which the value function of our problem can be identified with the unique classical solution of a quasilinear PDE within a class of functions satisfying certain growth conditions. The fact that we obtain classical solutions rather than viscosity solutions facilitates the use of numerical algorithms, whose applicability is demonstrated in examples.     

On efficient portfolio selection using convex risk measures

In this article we systematically revisit the classic portfolio selection theory in both of its branches, the determination of the efficient financial positions among such a choice set and the selection of the financial position which maximizes some utility function whose functional form involves some ‘measure of risk’. We study these problems by considering certain classes of convex risk measures and we show that for these classes the solution of the utility maximization problems in reflexive spaces take the form of a zero-sum game between the investor and the market.     

Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders

This paper proposes some new classes of risk measures, which are not only comonotonic subadditive or convex, but also respect the (first) stochastic dominance or stop-loss order. We give their representations in terms of Choquet integrals w.r.t. distorted probabilities, and show that if the physical probability is atomless then a comonotonic subadditive (resp. convex) risk measure respecting stop-loss order is in fact a law-invariant coherent (resp. convex) risk measure.     

Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders

This paper proposes some new classes of risk measures, which are not only comonotonic subadditive or convex, but also respect the (first) stochastic dominance or stop-loss order. We give their representations in terms of Choquet integrals w.r.t. distorted probabilities, and show that if the physical probability is atomless then a comonotonic subadditive (resp. convex) risk measure respecting stop-loss order is in fact a law-invariant coherent (resp. convex) risk measure.     

Multivariate stress scenarios and solvency

We show how the probabilistic concepts of half-space trimming and depth may be used to define convex scenario sets Q_α for stress testing the risk factors that affect the solvency of an insurance company over a prescribed time period. By choosing the scenario in Q_α which minimizes net asset value at the end of the time period, we propose the idea of the least solvent likely event (LSLE) as a solution to the forward stress testing problem. By considering the support function of the convex scenario set Q_α, we establish theoretical properties of the LSLE when financial risk factors can be assumed to have a linear effect on the net assets of an insurer. In particular, we show that the LSLE may be interpreted as a scenario causing a loss equivalent to the Value-at-Risk (VaR) at confidence level α, provided the α-quantile is a subadditive risk measure on linear combinations of the risk factors. In this case, we also show that the LSLE has an interpretation as a per-unit allocation of capital to the underlying risk factors when the overall capital is determined according to the VaR. These insights allow us to define alternative scenario sets that relate in similar ways to coherent measures, such as expected shortfall. We also introduce the most likely ruin event (MLRE) as a solution to the problem of reverse stress testing.     

Efficient hedging with coherent risk measure

The idea of efficient hedging has been introduced by Föllmer and Leukert. They defined the shortfall risk as the expectation of the shortfall weighted by a loss function, and looked for strategies that minimize the shortfall risk under a capital constraint. In this paper, to measure the shortfall risk, we use the coherent risk measures introduced by Artzner, Delbaen, Eber and Heath. We show that, for a given contingent claim H, the optimal strategy consists in hedging a modified claim ?H for some randomized test ?. This is an analogue of the results by Föllmer and Leukert.