Compatibility between pricing rules and risk measures: the CCVaR

Research partially supported by “RD Sistemas SA”, “Comunidad Autónoma de Madrid” (Spain), Grant s-0505/tic/000230, and “MEyC” (Spain), Grant SEJ2006-15401-C0

Good deals in markets with frictions

This paper studies a portfolio choice problem such that the pricing rule may incorporate transaction costs and the risk measure is coherent and expectation bounded. We will prove the necessity of dealing with pricing rules such that there are essentially bounded stochastic discount factors, which must be also bounded from below by a strictly positive value. Otherwise good deals will be available to traders, i.e., depending on the selected risk measure, investors can build portfolios whose (risk, return) will be as close as desired to (- infinite, + infinite) or (0, infinite). This pathologic property still holds for vector risk measures (i.e., if we minimize a vector valued function whose components are risk measures). It is worthwhile to point out that essentially bounded stochastic discount factors are not usual in financial literature. In particular, the most famous frictionless, complete and arbitrage free pricing models imply the existence of good deals for every coherent and expectation bounded measure of risk, and the incorporation of transaction costs will no guarantee the solution of this caveatRisk measure, Perfect and imperfect markets, Stochastic discount factor, Portfolio choice model, Good deal

CAPM and APT-like models with risk measures.

The paper deals with optimal portfolio choice problems when risk levels are given by coherent risk mea sures, expectation bounded risk measures or general deviations. Both static and dynamic pricing models may be involved. Unbounded problems are characterized by new notions such as (strong) compatibility between prices and risks. Surprisingly, the lack of bounded optimal risk and/or return levels arises for important pricing models (Black and Scholes) and risk measures (VaR, CVaR, absolute deviation, etc.). Bounded problems present a Market Price of Risk and generate a pair of benchmarks. From these bench marks we introduce APT and CAPM like analyses, in the sense that the level of correlation between every available security and some economic factors explains the security expected return. The risk level non correlated with these factors has no influence on any return, despite the fact that we are dealing with risk functions beyond the standard deviation.Risk measure; Compatibility between prices and risks; Efficient portfolio; APT and CAPM-like models;

Minimizing measures of risk by saddle point conditions.

The minimization of risk functions is becoming a very important topic due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Many types of risk function may be involved. A general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions. This new problem involves Banach spaces but a mean value theorem for risk measures is stated, and this simplifies the dual problem. Then, optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable. This characterization is significantly different if one compares it with previous literature. Furthermore, the saddle point condition very easily applies in practice. Four applications in finance and insurance are presented.Risk minimization; Saddle point condition; Actuarial and finantial aplications;

Martingales and arbitrage: a new look

This paper addresses the equivalence between the absence of arbitrage and the existence of equivalent martingale measures. The equivalence will be established under quite weak assumptions since there are no conditions on the set of trading dates (it may be finite or infinite, with bounded or unbounded horizon, etc.) or on the trajectories of the price process (for instance, they do not have to be right-continuous). Besides we will deal with arbitrage portfolios rather than free-lunches. The concept of arbitrage is much more intuitive than the concept of free lunch and has more clear economic interpretation. Furthermore it is more easily tested in theoretical models or practical applications. In order to overcome the usual mathematical difficulties arising when dealing with arbirage strategies, the set of states of nature will be widened by drawing on projective systems of Radon probability measures, whose projective limit will be the martingale measure. The existence of densities between the "real" probabilities and the "risk-neutral" probabilities will be guaranteed by introducing the concept of "projective equivalence". Hence some classical counter-examples will be solved and a complete characterization of the absence of arbitrage will be provided in a very general framework

Capital requirements: Are they the best solution?

General risk functions are becoming very important in finance and insurance. Many risk functions are interpreted as initial capital requirements that a manager must add and invest in a risk-free security in order to protect his clients wealth. Nevertheless, until now it has not been proved that an alternative investment will be outperformed by the riskless asset. This paper deals with a complete arbitrage free market and a general expectation bounded risk measure and analyzes whether the investment in the riskless asset of the capital requirements is optimal. It is shown that it is not optimal in many important cases. For instance, if the risk measure is the CVaR and we consider the assumptions of the CAPM or the Black and Scholes model. Furthermore, the Black and Scholes model the explicit expression of the optimal strategy is provided, and it is composed of several put options. If the confidence level of the CVaR is close to 100% then the optimal strategy becomes a classical portfolio insurance strategy. This may be a surprising and important finding for both researchers and practitioners. In particular, managers can discover how to reduce the level of initial capital requirements by trading options

Optimizing Measures of Risk: A Simplex-like Algorithm

The minimization of general risk or dispersion measures is becoming more and more important in Portfolio Choice Theory. There are two major reasons. Firstly, the lack of symmetry in the returns of many assets provokes that the classical optimization of the standard deviation may lead to dominated strategies, from the point of view of the second order stochastic dominance. Secondly, but not less important, many institutional investors must respect legal capital requirements, which may be more easily studied if one deals with a risk measure related to capital losses. This paper proposes a new method to simultaneously minimize several risk or dispersion measures. The representation theorems of risk measures are applied to transform the general risk minimization problem in a minimax problem, and later in a linear programming problem between infinite-dimensional Banach spaces. Then, new necessary and sufficient optimality conditions are stated and a simplex-like algorithm is developed. The algorithm solves the dual (and therefore the primal) problem and provides both optimal portfolios and their sensitivities. The approach is general enough and does not depend on any particular risk measure, but some of the most important cases are specially analyzed.Risk Measure. Deviation Measure. Portfolio Selection. Infinite-Dimensional Linear Programming. Simpl

Stability of the optimal reinsurance with respect to the risk measure

The optimal reinsurance problem is a classic topic in Actuarial Mathematics. Recent approaches consider a coherent or expectation bounded risk measure and minimize the global risk of the ceding company under adequate constraints. However, there is no consensus about the risk measure that the insurer must use, since every risk measure presents advantages and shortcomings when compared with others. This paper deals with a discrete probability space and analyzes the stability of the optimal reinsurance with respect to the risk measure that the insurer uses. We will demonstrate that there is a “stable optimal retention” that will show no sensitivity, insofar as it will solve the optimal reinsurance problem for many risk measures, thus providing a very robust reinsurance plan. This stable optimal retention is a stop-loss contract, and it is easy to compute in practice. A fast algorithm will be given and a numerical example presented.Optimal reinsurance, Risk measure, Sensitivity, Stable optimal retention, Stop-loss reinsurance

Portfolio choice and optimal hedging with general risk functions: a simplex-like algorithm.

The minimization of general risk functions is becoming more and more important in portfolio choice theory and optimal hedging. There are two major reasons. Firstly, heavy tails and the lack of symmetry in the returns of many assets provokes that the classical optimization of the standard deviation may lead to dominated strategies, from the point of view of the second order stochastic dominance. Secondly, but not less important, many institutional investors must respect legal capital requirements, which may be more easily studied if one deals with a risk measure related to capital losses. This paper proposes a new method to simultaneously minimize several general risk or dispersion measures. The representation theorems of risk functions are applied to transform the general risk minimization problem in a minimax problem, and later in a linear programming problem between infinite-dimensional Banach spaces. Then, new necessary and sufficient optimality conditions are stated and a simplex-like algorithm is developed. The algorithm solves the dual problem and provides both optimal portfolios and their sensitivities. The approach is general enough and does not depend on any particular risk measure, but some of the most important cases are specially analyzed. A final real data numerical example illustrates the practical performance of the proposed methodology.Risk measures; Deviation measure; Portfolio selection; Infinite dimensional linear programming; Simplex like method;

Stable solutions for optimal reinsurance problems involving risk measures.

The optimal reinsurance problem is a classic topic in actuarial mathematics. Recent approaches consider a coherent or expectation bounded risk measure and minimize the global risk of the ceding company under adequate constraints. However, there is no consensus about the risk measure that the insurer must use, since every risk measure presents advantages and shortcomings when compared with others. This paper deals with a discrete probability space and analyzes the stability of the optimal reinsurance with respect to the risk measure that the insurer uses. We will demonstrate that there is a ‘‘stable optimal retention’’ that will show no sensitivity, insofar as it will solve the optimal reinsurance problem for many risk measures, thus providing a very robust reinsurance plan. This stable optimal retention is a stop-loss contract, and it is easy to compute in practice. A fast linear time algorithm will be given and a numerical example presented.Optimal reinsurance; Risk measure; Sensitivity; Stable optimal retention; Stop-loss reinsurance;