Starting from the requirement that risk measures of financial portfolios should be based on their losses, not their gains, we define the notion of loss-based risk measure and study the properties of this class of risk measures. We characterize loss-based risk measures by a representation theorem and give examples of such risk measures. We then discuss the statistical robustness of estimators of loss-based risk measures: we provide a general criterion for qualitative robustness of risk estimators and compare this criterion with sensitivity analysis of estimators based on influence functions. Finally, we provide examples of statistically robust estimators for loss-based risk measures.
A main focus of quantitative modeling in finance has been to measure the risk of financial portfolios. In connection with the widespread use of Value-at-Risk (VaR) and related risk measurement methodologies, a considerable theoretical literature (Acerbi, 2002(Acerbi, , 2007;;Artzner et al., 1999;Cont et al., 2010;Föllmer and Schied, 2002;Föllmer and Schied, 2011;Frittelli and Rosazza Gianin, 2002;Kou and Heyde, 2013;McNeil et al., 2005) has focused on the design of appropriate risk measures for financial portfolios. In this approach, a risk measure is represented as a real-valued map assigning to each random variable X-representing the payoff of a portfolio-a number which measures the risk of this portfolio. A framework often used as a starting point is the axiomatic setting of Artzner et al. (1999), which defines a coherent risk measure as a map ρ : L ∞ (Ω, F , P) → R that is 1. monotone (decreasing): ρ(X) ≤ ρ(Y ) provided X ≥ Y ; 2. cash-additive (additive with respect to cash reserves): ρ(X + c) = ρ(X) -c for any c ∈ R;
positive homogeneous: ρ(λX) = λρ(X) for any λ ≥ 0;
sub-additive: ρ(X + Y ) ≤ ρ(X) + ρ(Y ). Artzner et al. (1999) argue that these axioms correspond to desirable properties of a risk measure, such as the reduction of risk under diversification. These axioms have provided an elegant mathematical framework for the study of coherent risk measures, but fail to take into account some key features encountered in the practice of risk management, as illustrated by the following (important) example. Consider a central clearing facility or an exchange, in which various market participants clear portfolios of financial instruments. Any participant of the clearing house must deposit a margin requirement for the purpose of covering the potential cost of liquidating the clearing participant’s portfolio in case of default. The risk measurement problem facing the exchange is therefore to determine the margin requirement for each portfolio, which is in this case the risk measure of the portfolio as seen by the exchange. Unlike the situation of an investor evaluating his/her own risk, the exchange is affected by the gains and losses of the market participants in an asymmetric way. As long as the market participant’s positions results in a gain, the gain is kept by the participant, but if the participant suffers a loss, the exchange may have to step in and cover the loss in case the participant default. It follows that, when measuring the risk posed to the exchange by a participant’s portfolio, it is only relevant to consider the losses of this portfolio, not the gains. Indeed, the well known Standard Portfolio ANalysis (SPAN) method introduced by the Chicago Merchantile Exchange and used by many other exchanges, computes the margin requirement of a financial portfolio with profit and loss (P&L) X as the maximum loss of the portfolio over a set of pre-selected stress scenarios ω 1 , . . . , ω n : ρ(X) = max{-min(X(ω 1 ), 0), …, -min(X(ω n ), 0)}.
(1)
As we can see in this example, the risk measure of a portfolio is only based on the loss min(X, 0) that is, the negative part of X.
The argument that a risk measure should be based on losses, not gains, is not restricted to the problem of computing margin requirements for a central clearing facility. Indeed, a regulator faces a similar issue when evaluating the cost of a bank failure: these costs materialize only in scenarios when a bank undergoes large losses resulting in its default, whereas the trading gains of a bank do not positively affect the regulator’s position. Thus, the risk of a bank’s portfolio, as viewed by the regulator, should also be based on the magnitude of the bank’s loss, not its potential gains.
These examples show that, a risk measure used for determining capital (or margin) requirements (called ’external’ risk measure in Kou and Heyde (2013)) should be solely based on the loss of a portfolio. This property can be formulated by requiring the risk measure ρ(X) to depend only on the negative part of X, representing the loss: ρ(X) = ρ( min(X, 0) ).
(
This property is clearly not contained in the axioms of coherent risk measures. In fact, the cash-additivity property implies that coherent risk measures must depend on gains as well, which clearly contradicts (2). So, one may not simply add the loss-dependence property (2) to the axioms of coherent risk measures without reconsidering the other axioms. In fact, the CME SPAN method does not verify the cash-additivity axiom and therefore is not a coherent risk measure. 1Many revisions to the axioms of coherent risk measures have been proposed
and studied in the literature, replacing in particular positive homogeneity and subadditivity with the more general convexity property (Föllmer and Schied, 2002;Föllmer and Schied, 2011;Frittelli and Rosazza Gianin, 2002), co-monotonic sub-additivity (Kou and Heyde, 2013) or co-monotonic convexity (Song and Yan, 2009). But these alternative fr
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