Executive Summary

Spectral measures of risk (SMRs) are a subclass of coherent measures of risk (CMRs). They are compliant with a list of general principles of ﬁnancial risk, providing a theoretically founded alternative to value at risk (VaR).

Any SMR turns out to be a mixture of quantiles, and, as such, they allow for immediate implementation in a VaR engine.

Any SMR is identiﬁed by a socalled risk aversion function, which models a subjective rational attitude toward all potential risks to which a portfolio is exposed.

Different SMRs turn out to be more appropriate to different speciﬁc circumstances, depending on the kind of risks to which the portfolio is subject. No measure among SMRs turns out to be preferable in general.

SMRs give rise to convex portfolio optimization problems, for which fast algorithms are available under any probabilistic assumptions and for portfolios of any size and complexity.
Introduction
In 1997, a groundbreaking paper (5) appeared with the explicit objective of establishing the properties that a portfolio risk measure should satisfy to comply with fundamental principles of ﬁnancial risk. The authors listed four such axioms and coined the phrase “coherent measures of risk” (CMRs) to denote those measures that satisfy them. They observed that there are an inﬁnite number of CMRs, actually a whole class, and they described this class completely by providing an explicit representation for any possible CMR (6).
Since then, a lively debate has arisen on whether this important theoretical achievement should inform the practice of ﬁnancial risk management, and, more importantly, the international standards of banking supervision on capital adequacy. In fact, it was immediately noticed that value at risk (VaR)—the benchmark measure in the practice of ﬁnancial risk management—is not a CMR, as it fails to satisfy the axiom of subadditivity (see section on characterization of SMRs). Researchers then turned to look for the subset of CMRs that are also versatile instruments for everyday market practice, sharing with VaR the good properties that made it so popular. It was clear that CMRs with these properties do, in fact, exist, as soon as expected shortfall (ES, also known as conditional value at risk, or CVaR), a natural evolution of VaR itself, was proved to be a CMR (1, 8). Yet, it was also clear that not all CMRs are candidate to become useful tools for ﬁnancial risk management, because, for instance, many of them are not even estimable from data. In other words, people realized that the class of CMRs, although extremely important from a theoretical point of view, is too large in view of potential applications. Additional conditions have to be added to the four axioms to end up with coherent measures that are also useful in practice.
A spectral measure of risk (SMR, 2) is a risk measure made of a weighted average of portfolio scenarios in which worse outcomes are given larger weights. SMRs represent a subclass of CMRs particularly appropriate for riskmanagement applications. They can be thought of as special mixtures of quantiles (the statistics on which VaR is based), and thus can be used straightforwardly in any existing VaR platform.
SMRs appeared in mathematical ﬁnance in 2001 (7, 2). They are closely connected to distortion risk measures, derived by Wang in 1996 (9) from premium principles, which may be seen as the actuarial forerunners of the axioms of coherency.
Deﬁnition and General Representation
To deﬁne SMRs, one must adopt a similar setting as for VaR, in the sense that one must ﬁrst model the probability distribution F_{x} : R → [0, 1] of the portfolio proﬁt–loss variable X on a speciﬁed time horizon T. Any inverse^{1 }F ^{← }: [0, 1] → R gives all p–quantiles xp ≡ F ^{←}(p) which are the possible future outcomes for X, ordered from the worstcase scenarios (when p tends to 0) to the bestcase scenarios (when p tends to 1). Note that hereafter the loss variable L = −X will also assume negative values, corresponding to portfolio proﬁts. This is different from actuarial approaches, where the loss variable takes on only positive values.
A function φ : [0, 1] → R will be said to be an admissible risk spectrum if it satisﬁes^{2 }

φ is positive or zero

the integral of φ is 1

φ is a monotonically decreasing function (locally ﬂat or decreasing)
Given an admissible risk spectrum φ, the associated spectral measure Mφ of the portfolio is deﬁned in plain English as the φ(p)−weighted average of all the losses l_{p} = −x_{p} (ordered decreasingly in p). In formulae, this becomes
Mφ(X)= ∫^{1 }φ(p) l_{p} dp
=“ ∑_{p}(weight φ(p)) × (loss outcome l_{p}) × (probability of l_{p})” (1)^{ }
which is clearly a mixture of loss quantiles, weighted by the risk spectrum φ. It can be shown (2) that the admissibility of φ(p) is a necessary and sufficient condition for the expression (1) to be a coherent measure. An important lesson then comes from condition 3, which tells us that a mixture of loss quantiles is a CMR only if we assign larger weights to worse losses. A measure is coherent only if it is “more worried” about more dangerous events.
The ﬁnancial interpretation of φ is clear. It proﬁles a speciﬁc risk aversion to all possible scenarios of the portfolio, assigning to them a particular weight. In this way, any investor can map his own attitude by choosing on a casebycase basis a speciﬁc φ, which for this reason is often called also the risk aversion function (RAF) of the SMR (see Figure 1). We see that SMRs Mφ are a class of risk measures in onetoone correspondence with all rational attitudes toward risk, represented by admissible RAFs φ. Subjectivity in the choice of a SMR allows for the design of appropriate measures for businessspeciﬁc issues affecting the portfolio under consideration.
Recalling that VaR with conﬁdence level α Є (0, 1), (typically a small percentage like α = 1% or 5%) is deﬁned as VaRα(X)= lα (2) we see that it admits a representation of type (1) but its risk spectrum φ^{V aR }= δ(p − α) is a socalled Dirac delta, namely a function sharply peaked in p = α (see Figure 2). This is clearly not an admissible RAF, because it is not decreasing on [0, 1]. VaRα, in other words, represents the α^{th }worst possible loss, and is not a CMR (and, hence, not a SMR) because it “fears” the αquantile more than the more dangerous tail pquantiles at p<α , which are, in fact, neglected altogether.
An important popular example of a SMR is, instead, ES, deﬁned as ESα(X)= 1 ∫0 l_{p} dp (3) which is, therefore, nothing but the average of the α worst losses. In this case, the risk spectrum φ^{ES }=1p≤α/α is simply a ﬂat function larger than zero, only between 0 and α (see Figure 3), which is easily shown to be admissible.
As an example of nonstandard SMRs, one may propose for, instance, a family of (putshaped) RAFs parameterized by γ Є (0, 1] given by φγ (p) = (2/γ^{2}) max(0; γ − p), which give a linearly decreasing weight to the worst cases in the quantile range p Є [0,γ] (see Figure 4). We call P utγ = Mφ_{γ }the corresponding SMR. Measures of this kind give more emphasis on tail events than, say, the ES, and could be interesting, for instance, in insurance portfolios, where prudent accounting of rare events is crucial.
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