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Home > Financial Risk Management Best Practice > Spectral Measures of Risk—Where Risk Theory and Risk Practice Overlap

Financial Risk Management Best Practice

Spectral Measures of Risk—Where Risk Theory and Risk Practice Overlap

by Carlo Acerbi

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 so-called 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.


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 risk-management 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 Fx : R → [0, 1] of the portfolio profit–loss variable X on a specified time horizon T. Any inverse1 F : [0, 1] → R gives all p–quantiles xp ≡ F (p) which are the possible future outcomes for X, ordered from the worst-case scenarios (when p tends to 0) to the best-case 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 satisfies2

  1. φ is positive or zero

  2. the integral of φ is 1

  3. 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 lp = −xp (ordered decreasingly in p). In formulae, this becomes

Mφ(X)= 1 φ(p) lp dp

=“ ∑p(weight φ(p)) × (loss outcome lp) × (probability of lp)” (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 case-by-case 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 one-to-one 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 business-specific 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 φV aR = δ(p − α) is a so-called 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 αth worst possible loss, and is not a CMR (and, hence, not a SMR) because it “fears” the α-quantile more than the more dangerous tail p-quantiles 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 lp dp (3) which is, therefore, nothing but the average of the α worst losses. In this case, the risk spectrum φES =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 non-standard SMRs, one may propose for, instance, a family of (put-shaped) RAFs parameterized by γ Є (0, 1] given by φγ (p) = (2/γ2) 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 = Mφγ 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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