Estimating risk measures for normal variance mixture distributions

library(nvmix)
library(RColorBrewer)
doPDF <- FALSE

Introduction

A random vector X = (X1, …, Xd) follows a normal variance mixture, in notation X ∼ NVMd(μ, Σ, FW), if, in distribution, $$ \mathbf{X} = \mathbf{\mu}+\sqrt{W}A\mathbf{Z}, $$ where μ ∈ ℝd denotes the location (vector), Σ = AA⊤ for A ∈ ℝd × k denotes the scale (matrix) (a covariance matrix), and the mixture variable W ∼ FW is a non-negative random variable independent of Z ∼ Nk(0, Ik) (where Ik ∈ ℝk × k denotes the identity matrix). Both the Student’s t distribution with degrees of freedom parameter ν > 0 and the normal distribution are normal variance mixtures; in the former case, W ∼ IG (ν/2, ν/2) (inverse gamma) and in the latter case W is almost surely constant (taken as 1 so that Σ is the covariance matrix of X in this case).

It follows readily from the stochastic representation that linear combinations of multivariate normal variance mixtures are univariate normal variance mixtures. Let a ∈ ℝd. If X ∼ NVMd(μ, Σ, FW), then a⊤X ∼ NVM1(a⊤μ, a⊤Σa, FW).

If X models, for instance, financial losses, a⊤X is the loss of a portfolio with portfolio weights a. It is then a common task in risk management to estimate risk measures of the loss a⊤X. We consider the two prominent risk measures value-at-risk and expected shortfall.

Estimating Risk Measures for X ∼ NVM1(μ, σ, FW)

In the following, assume without loss of generality that X ∼ NVM1(0, 1, FW), the general case follows from a location-scale argument.

Value-at-risk

The value-at-risk of X at confidence level α ∈ (0, 1) is merely the α-quantile of X. That is, VaRα(X) = inf {x ∈ [0, ∞) : FX(x) ≥ α}, where FX(x) = ℙ(X ≤ x) for x ∈ 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 ∼ IG (ν/2, ν/2) so that X follows a t distribution with ν 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 FX() 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 α ∈ (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 FX() is continuous, one can show that ESα(X) = E (X ∣ X > VaRα(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, ϕ(x) denotes the density of a standard normal distribution and we used that ∫k∞xϕ(x)dx = ϕ(k) for any k ∈ ℝ. Internally, the function ES_nvmix() estimates ESα(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()