Coherent and convex monetary risk measures for unbounded càdlàg processes

Assume that the random future evolution of values is modelled in continuous time. Then, a risk measure can be viewed as a functional on a space of continuous-time stochastic processes. In this paper we study coherent and convex monetary risk measures on the space of all càdlàg processes that are adapted to a given filtration. We show that if such risk measures are required to be real-valued, then they can only depend on a stochastic process in a way that is uninteresting for many applications. Therefore, we allow them to take values in ( −∞, ∞]. The economic interpretation of a value of ∞ is that the corresponding financial position is so risky that no additional amount of money can make it acceptable. The main result of the paper gives different characterizations of coherent or convex monetary risk measures on the space of all bounded adapted càdlàg processes that can be extended to coherent or convex monetary risk measures on the space of all adapted càdlàg processes. As examples we discuss a new approach to measure the risk of an insurance company and a coherent risk measure for unbounded càdlàg processes induced by a so called m-stable set.Due to errors during the typesetting process, this article was published incorrectly in Finance Stoch 9(3):369–387 (2005). The address of the first author was printed incorrectly, and in the whole paper the angular brackets were misprinted as [ ]. The complete corrected article is given here. The online version of the original paper can be found at: http://dx.doi.org/10.1007/s00780-004-0150-7     

Dynamic no-good-deal pricing measures and extension theorems for linear operators on L ∞

In an L ^∞-framework, we present majorant-preserving and sandwich-preserving extension theorems for linear operators. These results are then applied to price systems derived by a reasonable restriction of the class of applicable equivalent martingale measures. Our results prove the existence of a no-good-deal pricing measure for price systems consistent with bounds on the Sharpe ratio. We treat both discrete- and continuous-time market models. Within this study we present definitions of no-good-deal pricing measures that are equivalent to the existing ones and extend them to discrete-time models. We introduce the corresponding version of dynamic no-good-deal pricing measures in the continuous-time setting.     

The Iterated Cte

Abstract

In this paper we present a method for defining a dynamic risk measure from a static risk measure, by backwards iteration. We apply the method to the conditional tail expectation (CTE) risk measure to construct a new, dynamic risk measure, the iterated CTE (ICTE). We show that the ICTE is coherent, consistent, and relevant according to the definitions of Riedel (2003), and we derive formulae for the ICTE for the case where the loss process is lognormal. Finally, we demonstrate the practical implementation of the ICTE to an equity-linked insurance contract with maturity and death benefit guarantees.     

Correlation and the pricing of risks

Given a pricing kernel we investigate the class of risks that are not priced by this kernel. Risks are random payoffs written on underlying uncertainties that may themselves either be random variables, processes, events or information filtrations. A risk is said to be not priced by a kernel if all derivatives on this risk always earn a zero excess return, or equivalently the derivatives may be priced without a change of measure. We say that such risks are not kernel priced. It is shown that reliance on direct correlation between the risk and the pricing kernel as an indicator for the kernel pricing of a risk can be misleading. Examples are given of risks that are uncorrelated with the pricing kernel but are kernel priced. These examples lead to new definitions for risks that are not kernel priced in correlation terms. Additionally we show that the pricing kernel itself viewed as a random variable is strongly negatively kernel priced implying in particular that all monotone increasing functions of the kernel receive a negative risk premium. Moreover the equivalence class of the kernel under increasing monotone transformations is unique in possessing this property.      

Randomized observation periods for the compound Poisson risk model: the discounted penalty function

In the framework of collective risk theory, we consider a compound Poisson risk model for the surplus process where the process (and hence ruin) can only be observed at random observation times. For Erlang(n) distributed inter-observation times, explicit expressions for the discounted penalty function at ruin are derived. The resulting model contains both the usual continuous-time and the discrete-time risk model as limiting cases, and can be used as an effective approximation scheme for the latter. Numerical examples are given that illustrate the effect of random observation times on various ruin-related quantities.     

Recursive risk measures under regime switching applied to portfolio selection

In this paper, we define the conditional risk measure under regime switching and derive a class of time consistent multi-period risk measures. To do so, we describe the information process with regime switching in a product space associated with the product of two filtrations. Moreover, we show how to establish the corresponding multi-stage portfolio selection models using the time consistent multi-period risk measure for medium-term or long-term investments. Take the conditional value-at-risk measure as an example, we demonstrate the resulting multi-stage portfolio selection problem can be transformed into a second-order cone programming problem. Finally, we carry out a series of empirical tests to illustrate the superior performance of the proposed random framework and the corresponding multi-stage portfolio selection model.     

Iterated VaR or CTE measures: A false good idea?

The purpose of this paper is twofold. Firstly, we consider different risk measures in order to determine the solvency capital requirement of a pension fund. Secondly, we illustrate the impact of the time horizon of long-term guarantee products on these capital. We consider a financial market modelled by a common Black–Scholes–Merton model. We neglect the mortality and underwriting risks by assuming that the pension fund is fully hedged against these risks, which allows us to keep understandable and tractable formulæ (the longevity risk will be a part of future researches). A portfolio is built in this market according to different strategies and the pension fund offers a fixed guaranteed rate on a certain time horizon. We begin with well-known static risk measures (value at risk and conditional tail expectation measures) and then we consider their natural dynamic generalization. In order to be time consistent, we consider their iterated versions by a backward iterations scheme. Within the dynamic setting, we show that solvency capital can be expensive and that attention must be paid to the safety level considered.     

Optimal portfolio choice for a behavioural investor in continuous-time markets

The aim of this work consists in the study of the optimal investment strategy for a behavioural investor, whose preference towards risk is described by both a probability distortion and an S-shaped utility function. Within a continuous-time financial market framework and assuming that asset prices are modelled by semimartingales, we derive sufficient and necessary conditions for the well-posedness of the optimisation problem in the case of piecewise-power probability distortion and utility functions. Finally, under straightforwardly verifiable conditions, we further demonstrate the existence of an optimal strategy.     

Performance ratio-based coherent risk measure and its application

Utilizing a specific acceptance set, we propose in this paper a general method to construct coherent risk measures called the generalized shortfall risk measure. Besides some existing coherent risk measures, several new types of coherent risk measures can be generated. We investigate the generalized shortfall risk measure’s desirable properties such as consistency with second-order stochastic dominance. By combining the performance evaluation with the risk control, we study in particular the performance ratio-based coherent risk (PRCR) measures, which is a sub-class of generalized shortfall risk measures. The PRCR measures are tractable and have a suitable financial interpretation. Based on the PRCR measure, we establish a portfolio selection model with transaction costs. Empirical results show that the optimal portfolio obtained under the PRCR measure performs much better than the corresponding optimal portfolio obtained under the higher moment coherent risk measure.     

A note on generalized distortion risk measures

A generalized distortion risk measure is introduced as power of the mean absolute deviation power of a distorted random variable with respect to a location parameter. This class of risk measures extends both the distortion risk measure by Wang and Denneberg and the class of financial risk measures by Pedersen and Satchell, which itself contains the class of Stone. Integral representations and a stop–loss order preserving property of a special up-side risk measure are derived.     

Decision making with Expected Shortfall and spectral risk measures: The problem of comparative risk aversion

We analyze spectral risk measures with respect to comparative risk aversion following Arrow (1965) and Pratt (1964) for deterministic wealth, and Ross (1981) for stochastic wealth. We argue that the Arrow–Pratt-concept per se well matches with economic intuition in standard financial decision problems, such as willingness to pay for insurance and simple portfolio problems. Different from the literature, we find that the widely-applied spectral Arrow–Pratt-measure is not a consistent measure of Arrow–Pratt-risk aversion. Instead, the difference between the antiderivatives of the corresponding risk spectra is valid. Within the framework of Ross, we show that the popular subclasses of Expected Shortfall, and exponential and power spectral risk measures cannot be completely ordered with respect to Ross-risk aversion. Thus, for all these subclasses, the concept of Ross-risk aversion is not generally compatible with Arrow–Pratt-risk aversion, but induces counter-intuitive comparative statics of its own. Compatibility can be achieved if asset returns are jointly normally distributed. The general lesson is that these restrictions have to be considered before spectral risk measures can be applied for the purpose of optimal decision making and regulatory issues.     

Dynamic equilibrium and the structure of premiums in a reinsurance market

In this paper we present an economic equilibrium analysis of a reinsurance market. The continuous-time model contains the principal components of uncertainty; about the time instants at which accidents take place, and about claim sizes given that accidents have occurred. We give sufficient conditions on preferences for a general equilibrium to exist, with a Pareto optimal allocation, and derive the premium functional via a representative agent pricing theory. The marginal utility process of the reinsurance market is represented by the density process for random measures, which opens up for numerous applications to premium calculations, some of which are presented in the last section. The Hamilton-Jacobi-Bellman equations of individual dynamic optimization are established for proportional treaties, and the term structure of interest rates is found in this reinsurance syndicate. The paper attempts to reach a synthesis between the classical actuarial risk theory of insurance, in which virtually no economic reasoning takes place but where the net reserve is represented by a stochastic process, and the theory of equilibrium price formation at the heart of the economics of uncertainty.     

Risk measures based on behavioural economics theory

Coherent risk measures (Artzner et al. in Math. Finance 9:203–228, 1999) and convex risk measures (Föllmer and Schied in Finance Stoch. 6:429–447, 2002) are characterized by desired axioms for risk measures. However, concrete or practical risk measures could be proposed from different perspectives. In this paper, we propose new risk measures based on behavioural economics theory. We use rank-dependent expected utility (RDEU) theory to formulate an objective function and propose the smallest solution that minimizes the objective function as a risk measure. We also employ cumulative prospect theory (CPT) to introduce a set of acceptable regulatory capitals and define the infimum of the set as a risk measure. We show that the classes of risk measures derived from RDEU theory and CPT are equivalent, and they are all monetary risk measures. We present the properties of the proposed risk measures and give sufficient and necessary conditions for them to be coherent and convex, respectively. The risk measures based on these behavioural economics theories not only cover important risk measures such as distortion risk measures, expectiles and shortfall risk measures, but also produce new interesting coherent risk measures and convex, but not coherent risk measures.     

Optimal Portfolios with Credit Default Swaps

Using a continuous-time, stochastic, and dynamic framework, this study derives a closed-form solution for the optimal investment problem for an agent with hyperbolic absolute risk aversion preferences for maximising the expected utility of his or her final wealth. The agent invests in a frictionless, complete market in which a riskless asset, a (defaultable) bond, and a credit default swap written on the bond are listed. The model is calibrated to market data of six European countries and assesses the behaviour of an investor exposed to different levels of sovereign risk. A numerical analysis shows that it is optimal to issue credit default swaps in a larger quantity than that of bonds, which are optimally purchased. This speculative strategy is more aggressive in countries characterised by higher sovereign risk. This result is confirmed when the investor is endowed with a different level of risk aversion. Finally, we solve a static version of the optimisation problem and show that the speculative/hedging strategy is definitely different with respect to the dynamic one.     

A supermartingale relation for multivariate risk measures

The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk measure. Examples of multivariate risk measures satisfying the supermartingale property are given. Crucial for obtaining the results are dual representations of scalarizations of set-valued dynamic risk measures, which are of independent interest in the fast growing literature on multivariate risks.     

Identity Theft and Consumer Payment Choice: Does Security Really Matter?

In their paper “Spectral Risk Measures: Properties and Limitations”, Dowd et al. (J Financ Serv Res 341:61–75, 2008) introduce exponential and power spectral risk measures as subclasses of spectral risk measures (SRMs) to the literature, and claim that they are subject to three serious limitations: First, for these subclasses, the spectral risk may be counterintuitively decreasing when the user’s risk aversion is increasing. Second, these subclasses, and power SRMs in particular, become completely insensitive to market volatility when the respective parameters of risk aversion tend to their lower and upper boundaries. Third, exponential SRMs exhibit constant absolute risk aversion, while constant relative risk aversion better meets the empirical evidence. Consequently, “users of spectral risk measures must be careful to select utility functions that fit the features of the particular problems they are dealing with, and should be especially careful when using power SRMs.” (p. 61). In this comment, we show that the findings of Dowd et al. (J Financ Serv Res 341:61–75, 2008) suffer from misinterpretations and wrong conclusions.     

Hedging under multiple risk constraints

Motivated by the asset–liability management problems under shortfall risk constraints, we consider in a general discrete-time framework the problem of finding the least expensive portfolio whose shortfalls with respect to a given set of stochastic benchmarks are bounded by a specific shortfall risk measure. We first show how the price of this portfolio may be computed recursively by dynamic programming for different shortfall risk measures, in complete and incomplete markets. We then focus on the specific situation where the shortfall risk constraints are imposed at each period on the next-period shortfalls, and obtain explicit results. Finally, we apply our results to a realistic asset–liability management problem of an energy company, and show how the shortfall risk constraints affect the optimal hedging of liabilities.     

Översikt av utländska aktuarietidskrifter

Abstract

In this paper a continuous-time model of a reinsurance market is presented, which contains the principal components of uncertainty transparent in such a market: Uncertainty about the time instants at which accidents take place, and uncertainty about claim sizes given that accidents have occurred.

Due to random jumps at random time points of the underlying claims processes, the absence of arbitrage opportunities is not sufficient to give unique premium functionals in general. Market preferences are derived under a necessary condition for a general exchange equilibrium. Information constraints are found under which premiums of risks are determined. It is demonstrated how general reinsurance treaties can be uniquely split into proportional contracts and nonproportional ones.

Several applications to reinsurance markets are given, and the results are compared to the corresponding theory of the classical one-period model of a reinsurance syndicate.

This paper attempts to reach a synthesis between the classical actuarial risk theory of insurance, in which virtually no economic reasoning takes place but where the net reserve is represented by a stochastic process, and the theory of partial equilibrium price formation at the heart of the economics of uncertainty.     

Methods of Partial Hedging

In this article we survey methods of dealing with the following problem: A financial agent is trying to hedge a claim C, without having enough initial capital to perform a perfect (super) replication. In particular, we describe results for minimizing the expected loss of hedging the claim C both in complete and incomplete continuous-time financial market models, and for maximizing the probability of perfect hedge in complete markets and markets with partial information. In these cases, the optimal strategy is in the form of a binary option on C, depending on the Radon-Nikodym derivative of the equivalent martingale measure which is optimal for a corresponding dual problem. We also present results on dynamic measures for the risk associated with the liability C, defined as the supremum over different scenarios of the minimal expected loss of hedging C. This revised version was published online in August 2006 with corrections to the Cover Date.