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 $z_\alpha$ is the so-called standard normal $z$-score of the significance level $\alpha$ such that $$ \frac 1 {\sqrt{2\pi}}\int_{-\infty}^{z_\alpha} e^{-\xi^2/2}d\xi=\alpha $$ and we assume normality, (ignoring skewness and kurtosis,) then we can estimate the $\alpha$ quantile of a distribution with cdf $\Phi$ as $$\Phi^{-1}(\alpha)=\mu + \sigma z_\alpha.$$ The Cornish-Fisher expansion is an attempt to estimate this more accurately directly in terms of the first few cumulants as $$\Phi^{-1}(\alpha)=y_\alpha,$$ where (before we have applied any scaling) $$y_\alpha= \kappa_1 + \frac {z_\alpha}2 + \frac {z_\alpha \kappa_2 }2 + \frac {(z_\alpha^2-1) \kappa_3}6 + \frac {(z_\alpha^3-3z_\alpha) \kappa_4}{24} - \frac {(2z_\alpha - 5z_\alpha) \kappa_3^2}{36}. $$ (Note that $\mu=\kappa_1$ and $\sigma^2=\kappa_2$.) 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. $$y_\alpha[t] = \kappa_1t + \frac {z_\alpha}2 + \frac {z_\alpha \kappa_2 t}2 + \frac {(z_\alpha^2-1) \kappa_3 t}6 + \frac {(z_\alpha^3-3z_\alpha) \kappa_4 t}{24} - \frac {(2z_\alpha - 5z_\alpha) \kappa_3^2 t^2}{36}. $$ but we want $y_\alpha[t] = \mu t + (\sigma\sqrt t )x_\alpha[t]$ where $x_\alpha[t]$ is the quantile function of a random variable with zero mean and unit variance. First the term $\kappa_1t$ drops off as our $\mu t$ since all the other cumulants are shift-invariant. Second we need to divide each remaining cumulant $\kappa_kt$ by $(\sigma\sqrt t)^k=(\kappa_2t)^{k/2}$ (because the $k$th cumulant is homogeneous of order $k$.) So: $$y_\alpha[t] = \mu t + \sigma\sqrt t \left[ z_\alpha + \frac {(z_\alpha^2-1) \kappa_3 t}{6(\kappa_2t)^{3/2}} + \frac {(z_\alpha^3-3z_\alpha) \kappa_4 t}{24(\kappa_2t)^2} - \frac {(2z_\alpha - 5z_\alpha) \kappa_3^2 t^2}{36(\kappa_2t)^3}\right]. $$ $$y_\alpha[t] = \mu t + \sigma\sqrt t \left[ z_\alpha + \frac {(z_\alpha^2-1) \kappa_3}{6\sigma^3t^{1/2}} + \frac {(z_\alpha^3-3z_\alpha) \kappa_4}{24\sigma^4 t} - \frac {(2z_\alpha - 5z_\alpha) \kappa_3^2}{36\sigma^6t}\right]. $$ but generally we write $\gamma_1=\kappa_3/\sigma^3$ and $\gamma_2=\kappa_4/\sigma^4$ for the skewness and the kurtosis respectively, so that $$y_\alpha[t] = \mu t + \sigma\sqrt t \left[ z_\alpha + \frac {(z_\alpha^2-1) \gamma_1}{6\sqrt t} + \frac {(z_\alpha^3-3z_\alpha) \gamma_2}{24t} - \frac {(2z_\alpha - 5z_\alpha) \gamma_1^2}{36t}\right]. $$
The Value-at-Risk is then $$\mathrm{VaR} = K_0 \left( 1 - {\exp (y_{\alpha}[t]-rt})\right),$$ where $K_0$ is the initial capital, $\alpha$ 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 $\mu>0$.)
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