اقدامات منسجم ریسک از دیدگاه تعادل عمومی

Coherent measures of risk defined by the axioms of monotonicity, subadditivity, positive homogeneity, and translation invariance are recent tools in risk management to assess the amount of risk agents are exposed to. If they also satisfy law invariance and comonotonic additivity, then we get a subclass of them: spectral measures of risk. Expected shortfall is a well-known spectral measure of risk. We investigate the above mentioned six axioms using tools from general equilibrium (GE) theory. Coherent and spectral measures of risk are compared to the natural measure of risk derived from an exchange economy model, which we call the GE measure of risk. We prove that GE measures of risk are coherent measures of risk. We also show that spectral measures of risk are GE measures of risk only under stringent conditions, since spectral measures of risk do not take the regulated entity’s relation to the market portfolio into account. To give more insights, we characterize the set of GE measures of risk via the pricing kernel property.

Risk management is of crucial importance considering the enormous financial risk our economy is exposed to. The risks of many economic agents are regulated by various institutions. For example, if a financial trader wants to sell options, which give the buyer rights of buying or selling at a given price during a specified time horizon (or at a given time), he has to fulfil margin requirements, i.e. he has to deposit some cash or some other riskless and liquid instrument. An exchange’s clearing firm, which is responsible for the promises to all parties of transactions being securely completed, requires margin deposits. A measure of risk can be used to determine the margin requirement. The riskier the trader’s portfolio, the more the margin requirement should be. Other external regulators, at an international level, are the International Actuarial Association (IAA) and the International Accounting Standards Board (IASB), who determine the capital requirements for insurance companies. Similarly, the Basel Committee gives guidelines for the acceptable level of capital on banking supervision. Since a government or central bank could be a lender of last resort for these institutions, and the default of them could cause serious problems, they are regulated as well. As an internal regulator, a portfolio manager has to regulate the risk of its traders. In the context of a multi-division firm setting, the head-office may also set risk-limits for the divisions. Internally the risk values can also be used for planning and performance evaluation. It is therefore crucial to measure risk in an appropriate way. We will use the term portfolio when referring to a risky entity (portfolio, firm, insurance company, bank, etc.). The value of a portfolio might change due to all kinds of uncertain events. We relate risk to the probability distribution of the future value of the portfolio. For the sake of simplicity in this paper we use discrete random variables. Our approach can be extended to the case of continuous risks and risks with unbounded support. All this requires is an analysis of competitive equilibrium in such an environment. The interested reader is referred to Chapter 10 of Duffie (2001). A measure of risk assigns a real number to a random variable. It is the minimal amount of cash the regulated agent has to add to his portfolio, and to invest in a zero coupon bond. Coherent measures of risk ( Artzner et al., 1999) are defined by four axioms: monotonicity, subadditivity, positive homogeneity and translation invariance. When adding two more axioms: law invariance and comonotonic additivity we get a subclass of coherent measures of risk, namely spectral measures of risk ( Acerbi, 2002). Expected shortfall is a well-known spectral measure of risk ( Acerbi and Tasche, 2002). For an introduction to risk measures and the above mentioned axioms see for instance Chapter 4 of Föllmer and Schied (2002). Our approach is to model the situation at hand as an exchange economy in a general equilibrium (GE) setting, and determine which axioms are compatible with this model, and whether other axioms emerge as natural. This approach has the advantage that it recognizes the fact that the risk of a portfolio depends on the other assets present in the economy (the market portfolio), an insight that is generated immediately by the Capital Asset Pricing Model as developed by Sharpe, 1964 and Lintner, 1965. By doing so we would like to contribute to the research agenda that connects finance to GE theory, see for instance Geanakoplos and Shubik, 1990, Magill and Quinzii, 1996 and Leroy and Werner, 2001, or Jaschke and Küchler (2001). The corresponding measure of risk of a portfolio would be the amount of cash needed to sell the risk involved in the portfolio to the market. More precisely, the so-called GE measure of risk of a portfolio would be the negative of its equilibrium market price. We prove that GE measures of risk are coherent and comonotonic additive measures of risk. However, GE measures of risk fail to satisfy law invariance, i.e. they are functions of not only the probability distributions of the portfolios, since they also take the regulated entity’s relation to the market portfolio into account. Nevertheless we show that GE measures of risk satisfy a generalized notion of law invariance. To check on which domain spectral measures of risk are GE measures of risk, we consider a general domain for the measures of risk. We find that the corresponding domain is very small. To give more insights, we characterize GE measures of risk as the only measures of risk satisfying the property that we call the pricing kernel property. The structure of the paper is as follows. In Section 2 we discuss coherent measures of risk. In Section 3 spectral measures of risk are considered. Using the exchange economy model of Section 4 the properties of GE measures of risk are investigated in Section 5. In Section 6 we show that spectral measures of risk are GE measures of risk only under stringent conditions, and we characterize GE measures of risk via the pricing kernel property. We conclude in Section 7.

In this paper we discussed coherent and spectral measures of risk from a general equilibrium (GE) perspective. Coherent measures of risk can be defined by four axioms: monotonicity, subadditivity, positive homogeneity, translation invariance. Adding two more axioms, law invariance and comonotonic additivity leads to spectral measures of risk. We considered the discrete setting and a general domain V⊆RSV⊆RS. We proved that it is also true in the discrete setting with unrestricted domain, i.e. if V=RSV=RS that spectral measures of risk are the only coherent measures of risk satisfying law invariance and comonotonic additivity. However, we have shown examples where on a general domain V this may not hold. We defined a natural measure of risk coming out of a general equilibrium model. The GE measure of risk of a portfolio is the negative of its equilibrium market price. Checking the properties of GE measures of risk enabled us to assess the above mentioned six axioms. We found that GE measures of risk are coherent measures of risk. This way the four axioms of coherent measures of risk are supported from a general equilibrium perspective. Thus Value at Risk and other non-coherent risk measures cannot be associated with our general equilibrium framework. However, GE measures of risk do not satisfy law invariance, but only a generalized version of it, in which the market portfolio is also taken into account. Since spectral measures of risk are law invariant, we can conclude that in general, when calculating the risk of a regulated entity, spectral measures of risk do not take into account its relation to the market portfolio, leading to an under- or overestimation of risk. The same idea is shown by our result that spectral measures of risk are GE measures of risk if and only if all the regulated entities are comonotonic with the market portfolio, i.e. their values go up and down together event by event. Finally, we showed that GE measures of risk are the only measures of risk satisfying the pricing kernel property, which means that any nonnegative pricing kernel can induce them as the negative of the equilibrium market price.