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?
If zα is the so-called standard normal z-score of the significance level α such that 1√2π∫zα−∞e−ξ2/2dξ=α
and we assume normality, (ignoring skewness and kurtosis,) then we can estimate the
α quantile of a distribution with cdf
Φ as
Φ−1(α)=μ+σzα.
The Cornish-Fisher expansion is an attempt to estimate this more accurately directly in terms of the first few cumulants as
Φ−1(α)=yα,
where (before we have applied any scaling)
yα=κ1+zα2+zακ22+(z2α−1)κ36+(z3α−3zα)κ424−(2zα−5zα)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+(z2α−1)κ3t6+(z3α−3zα)κ4t24−(2zα−5zα)κ23t236.
but we want
yα[t]=μt+(σ√t)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
(σ√t)k=(κ2t)k/2 (because the
kth cumulant is homogeneous of order
k.) So:
yα[t]=μt+σ√t[zα+(z2α−1)κ3t6(κ2t)3/2+(z3α−3zα)κ4t24(κ2t)2−(2zα−5zα)κ23t236(κ2t)3].
yα[t]=μt+σ√t[zα+(z2α−1)κ36σ3t1/2+(z3α−3zα)κ424σ4t−(2zα−5zα)κ2336σ6t].
but generally we write
γ1=κ3/σ3 and
γ2=κ4/σ4 for the skewness and the kurtosis respectively, so that
yα[t]=μt+σ√t[zα+(z2α−1)γ16√t+(z3α−3zα)γ224t−(2zα−5zα)γ2136t].
The Value-at-Risk is then VaR=K0(1−exp(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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