Abstract
A spectrum of upper bounds (Qα(X ; p) αε[0,∞] on the (largest) (1-p)-quantile Q(X;p) of an arbitrary random variable X is introduced and shown to be stable and monotonic in α, p, and X , with Q0(X ;p) = Q(X;p). If p is small enough and the distribution of X is regular enough, then Qα(X ; p) is rather close to Q(X ; p). Moreover, these quantile bounds are coherent measures of risk. Furthermore, Qα(X ; p) is the optimal value in a certain minimization problem, the minimizers in which are described in detail. This allows of a comparatively easy incorporation of these bounds into more specialized optimization problems. In finance, Q0(X;p) and Q1(X ; p) are known as the value at risk (VaR) and the conditional value at risk (CVaR). The bounds Qα(X ; p) can also be used as measures of economic inequality. The spectrum parameter α plays the role of an index of sensitivity to risk. The problems of the effective computation of the bounds are considered. Various other related results are obtained.
Highlights
The most common measure of risk is apparently the value at risk, VaRp pXq, defined as the largest p1 ́pq-quantile of a random variable (r.v.) X, which represents an uncertain future loss on an investment portfolio
If X „ N pμ, σ 2 q and H “ κ id for some positive constant κ, RH pXq “ μ ?2κπ σ, a linear combination of the mean and the standard deviation, so that in such a case we find ourselves in the realm of the Markowitz mean-variance risk-assessment framework; cf. (3.12)
Let us summarize some of the advantages of the risk/inequality measures Pα pX; xq and Qα pX; pq:
Summary
The most common measure of risk is apparently the value at risk, VaRp pXq, defined as the largest p1 ́pq-quantile of (the distribution of) a random variable (r.v.) X, which represents an uncertain future loss on an investment portfolio. A very serious shortcoming of VaR, in addition, is that it provides no handle on the extent of the losses that might be suffered beyond the threshold amount indicated by this measure. It is incapable of distinguishing between situations where losses that are worse may be deemed only a little bit worse, and those where they could well be overwhelming. It merely provides a lowest bound for losses in the tail of the loss distribution and has a bias toward optimism instead of the conservatism that ought to prevail in risk management. Related to these two kinds of instability is the inherent instability in the computation of VaRp pXq
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