Estimating Risk Measures for \(X\sim NVM_1(\mu, \sigma, F_W)\)
In the following, assume without loss of generality that \(X\sim \operatorname{NVM}_1(0, 1, F_W)\), the general case follows from a location-scale argument.
Value-at-risk
The value-at-risk of \(X\)
at confidence level \(\alpha\in(0,1)\)
is merely the \(\alpha\)-quantile of
\(X\). That is, \[ \operatorname{VaR}_\alpha(X) =
\inf\{x\in[0,\infty):F_X(x)\ge \alpha\},\] where \(F_X(x)=\mathbb{P}(X\le x)\) for \(x\in\mathbb{R}\) is the distribution
function of \(X\). Such quantile can be
estimated via the function qnvmix(), or equivalently, via
the function VaR_nvmix() of the R package
nvmix.
As an example, consider \(W\sim\operatorname{IG}(\nu/2, \nu/2)\) so
that \(X\) follows a \(t\) distribution with \(\nu\) degrees of freedom. In this case, the
quantile is known. If the argument qmix is provided as a
string, VaR_nvmix() calls qt(); if
qmix is provided as a function or list, the quantile is
internally estimated via a Newton algorithm where the univariate
distribution function \(F_X()\) is
estimated via randomized quasi-Monte Carlo methods.
set.seed(1) # for reproducibility
qmix <- function(u, df) 1/qgamma(1-u, shape = df/2, rate = df/2)
df <- 3.5
n <- 20
level <- seq(from = 0.9, to = 0.995, length.out = n)
VaR_true <- VaR_nvmix(level, qmix = "inverse.gamma", df = df)
VaR_est <- VaR_nvmix(level, qmix = qmix, df = df)
stopifnot(all.equal(VaR_true, qt(level, df = df)))
## Prepare plot
pal <- colorRampPalette(c("#000000", brewer.pal(8, name = "Dark2")[c(7, 3, 5)]))
cols <- pal(2) # colors
if(doPDF) pdf(file = (file <- "fig_VaR_nvmix_comparison.pdf"),
width = 7, height = 7)
plot(NA, xlim = range(level), ylim = range(VaR_true, VaR_est),
xlab = expression(alpha), ylab = expression(VaR[alpha]))
lines(level, VaR_true, col = cols[1], lty = 2, type = 'b')
lines(level, VaR_est, col = cols[2], lty = 3, lwd = 2)
legend('topleft', c("True VaR", "Estimated VaR"), col = cols, lty = c(2,3),
pch = c(1, NA))
if(doPDF) dev.off()
Expected Shortfall
Another risk measure of great theoretical and practical importance is the expected-shortfall. The expected shortfall of \(X\) at confidence level \(\alpha\in(0,1)\) is, provided the integral converges, given by \[ \operatorname{ES}_\alpha(X) = \frac{1}{1-\alpha} \int_\alpha^1 \operatorname{VaR}_u(X)du.\] If \(F_X()\) is continuous, one can show that \[ \operatorname{ES}_\alpha(X) = \operatorname{E}(X \mid X > \operatorname{VaR}_\alpha(X)).\]
The function ES_nvmix() in the R package
nvmix can be used to estimate the expected shortfall for
univariate normal variance mixtures. Since these distributions are
continuous, we get the following:
\[ (1-\alpha) \operatorname{ES}_\alpha(X) = \operatorname{E}\left(X \mathbf{1}_{\{X>\operatorname{VaR}_\alpha(X)\}}\right)= \operatorname{E}\left( \sqrt{W} Z \mathbf{1}_{\{\sqrt{W} Z > \operatorname{VaR}_\alpha\}}\right) \]
\[= \operatorname{E}\Big( \sqrt{W} \operatorname{E}\big(Z \mathbf{1}_{\{Z> \operatorname{VaR}_\alpha(X)/\sqrt{W}\}} \mid W\big)\Big)= \operatorname{E}\left(\sqrt{W} \phi(\operatorname{VaR}_\alpha(X) / \sqrt{W})\right) \]
Here, \(\phi(x)\) denotes the
density of a standard normal distribution and we used that \(\int_k^\infty x\phi(x)dx = \phi(k)\) for
any \(k\in\mathbb{R}\). Internally, the
function ES_nvmix() estimates \(\operatorname{ES}_\alpha(X)\) via a
randomized quasi-Monte Carlo method by exploiting the displayed
identity.
In case of the normal and \(t\)
distribution, a closed formula for the expected shortfall is known; this
formula is then used by ES_nvmix() if qmix is
provided as string.
ES_true <- ES_nvmix(level, qmix = "inverse.gamma", df = df)
ES_est <- ES_nvmix(level, qmix = qmix, df = df)
## Prepare plot
if(doPDF) pdf(file = (file <- "fig_ES_nvmix_comparison.pdf"),
width = 7, height = 7)
plot(NA, xlim = range(level), ylim = range(ES_true, ES_est),
xlab = expression(alpha), ylab = expression(ES[alpha]))
lines(level, ES_true, col = cols[1], lty = 2, type = 'b')
lines(level, ES_est, col = cols[2], lty = 3, lwd = 2)
legend('topleft', c("True ES", "Estimated ES"), col = cols, lty = c(2,3),
pch = c(1, NA))
if(doPDF) dev.off()