Wednesday, April 22, 2015

value at risk - How does Cornish-Fisher VaR (aka modified VaR) scale with time?


I am thinking about the time-scaling of Cornish-Fisher VaR (see e.g. page 130 here for the formula).


It involves the skewness and the excess-kurtosis of returns. The formula is clear and well studied (and criticized) in various papers for a single time period (e.g. daily returns and a VaR with one-day holding period).



Does anybody know a reference on how to scale it with time? I would be looking for something like the square-root-of-time rule (e.g. daily returns and a VaR with d days holding period). But scaling skewness, kurtosis and volatility separately and plugging them back does not feel good. Any ideas?



Answer



If zα is the so-called standard normal z-score of the significance level α such that 12πzαeξ2/2dξ=α

and we assume normality, (ignoring skewness and kurtosis,) then we can estimate the α quantile of a distribution with cdf Φ as Φ1(α)=μ+σzα.
The Cornish-Fisher expansion is an attempt to estimate this more accurately directly in terms of the first few cumulants as Φ1(α)=yα,
where (before we have applied any scaling) yα=κ1+zα2+zακ22+(z2α1)κ36+(z3α3zα)κ424(2zα5zα)κ2336.
(Note that μ=κ1 and σ2=κ2.) Expressed directly in terms of cumulants, let us try to scale this directly with time as the convolution of infinitely divisible, independent identically distributed random variables. Cumulants of all orders scale linearly with time in this case, since they are simply additive under convolution. yα[t]=κ1t+zα2+zακ2t2+(z2α1)κ3t6+(z3α3zα)κ4t24(2zα5zα)κ23t236.
but we want yα[t]=μt+(σt)xα[t] where xα[t] is the quantile function of a random variable with zero mean and unit variance. First the term κ1t drops off as our μt since all the other cumulants are shift-invariant. Second we need to divide each remaining cumulant κkt by (σt)k=(κ2t)k/2 (because the kth cumulant is homogeneous of order k.) So: yα[t]=μt+σt[zα+(z2α1)κ3t6(κ2t)3/2+(z3α3zα)κ4t24(κ2t)2(2zα5zα)κ23t236(κ2t)3].
yα[t]=μt+σt[zα+(z2α1)κ36σ3t1/2+(z3α3zα)κ424σ4t(2zα5zα)κ2336σ6t].
but generally we write γ1=κ3/σ3 and γ2=κ4/σ4 for the skewness and the kurtosis respectively, so that yα[t]=μt+σt[zα+(z2α1)γ16t+(z3α3zα)γ224t(2zα5zα)γ2136t].


The Value-at-Risk is then VaR=K0(1exp(yα[t]rt)),

where K0 is the initial capital, α is some level of significance, say 1 to 5% or so, and r is some instantaneous risk-free rate, appropriate discount rate, or required rate of return, however one chooses to define it. (This expression ought to become negative for a long enough time, because in the long run one will almost surely make money if μ>0.)


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