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

Question

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 1

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 $\mu>0$.)

Answer 2

Skewness decays with time, but the rate of that skewness decay will vary based on the instruments and how they are traded, so a simple estimator such as the square root of time rule is not appropriate.

I typically recommend that to scale VaR or ES it makes more sense to lower your confidence level (raise the alpha parameter) to one that makes sense for your holding period.

So, for example, assume that we are working with daily returns, as in your question. Now assume that I have a one-month holding period, and I want a 'monthly VaR'.

I would argue that a rational confidence level for this is 95%, or 1 in 20, corresponding approximately to the loss that will be exceeded about 1 day a month.

For monthly returns, as in a hedge fund portfolio, a confidence of 92% may be most appropriate, to specify the VaR that will be exceeded, on average, once a year.

I think that this is a much more rational approach than asking 'what loss level will be exceeded once in 10000 years?', as many papers and standards bodies recommend. These numbers aren't very useful, as many other authors have pointed out.

Also, extension to Cornish Fisher Expected Shortfall (also called CVaR or Expected Tail Loss) with the same approach as above, helps scale these numbers in a rational way, asking what the mean loss is when the loss exceeds the VaR.

More information on this is available in our published work including this paper from the Journal of Risk which also covers additive/coherent portfolio decomposition of Cornish Fisher VaR and Expected Shortfall.

Answer 3

The time scaling of higher moments for ordinary (discrete) returns as per the Wingender paper is illustrated in Excel and VBA in the following spreadsheet demonstration files:

Terminal-Wealth-Time-Horizon-Calcs-Normal-and-Modified-VBA and;

Liqudity-VaR-With-Correct-Time-Scaling-of-Higher-Moments

Available here

For more on the weaknesses of the Cornish Fisher expansion see the presentation on Why-Distributions-Matter-16-Jan-2012