27 Value at Risk test

Setup

library(dplyr)
library(latex2exp)
library(backports)
library(ggplot2)
library(knitr)

Let’s consider a ARMA(2,2)-GARCH(1,1) model defined as: $\begin{aligned} x_{t} = μ + ϕ_{1} x_{t - 1} + ϕ_{2} x_{t - 2} + θ_{1} e_{t - 1} + θ_{2} e_{t - 2} + e_{t} \\ e_{t} = σ_{t} u_{t} \\ σ_{t}^{2} = ω + α_{1} e_{t - 1}^{2} + β_{1} σ_{t - 1}^{2} \end{aligned}$ The Value at Risk (VaR) with confidence level $α$ depends on the distribution of $x_{t}$ that is implicitly defined from the distribution of $u_{t}$ . Therefore, the VaR at time $t$ , conditional to the information up to time $t - 1$ , is implicitly defined as: $P (X_{t} \leq {VaR}_{t ∣ t - 1}^{α}) = α .$

27.1 Normal distribution

Let’s consider independent and normally distributed residuals $u_{t}$ . Then, also the conditional distribution of $x_{t}$ given the information up to time $t - 1$ will be normal with conditional mean and variance given by: ${\begin{cases} E {X_{t} ∣ I_{t - 1}} = μ + ϕ_{1} x_{t - 1} + ϕ_{2} x_{t - 2} + θ_{1} e_{t - 1} + θ_{2} e_{t - 2} \\ V {X_{t} ∣ I_{t - 1}} = ω + α_{1} e_{t - 1}^{2} + β_{1} σ_{t - 1}^{2} = σ_{t}^{2} \end{cases}$ Hence, given the quantile $q^{α}$ of a standard normal with level $α$ , the VaR is computed as: ${VaR}_{t ∣ t - 1}^{α} = E {X_{t} ∣ I_{t - 1}} + q^{α} \sqrt{V {X_{t} ∣ I_{t - 1}}} .$

27.2 Gaussian Mixture distribution

Let’s consider independent and Gaussian mixture distributed residuals $u_{t}$ with 2 components. Then, also the conditional distribution of $x_{t}$ given the information up to time $t - 1$ will be a Gaussian mixture with conditional mean and variance given by: ${\begin{cases} E {X_{t} ∣ I_{t - 1}, B = 1} = μ + ϕ_{1} x_{t - 1} + ϕ_{2} x_{t - 2} + θ_{1} e_{t - 1} + θ_{2} e_{t - 2} + μ_{1} σ_{t} \\ E {X_{t} ∣ I_{t - 1}, B = 0} = μ + ϕ_{1} x_{t - 1} + ϕ_{2} x_{t - 2} + θ_{1} e_{t - 1} + θ_{2} e_{t - 2} + μ_{0} σ_{t} \\ V {X_{t} ∣ I_{t - 1}, B = 1} = (σ_{t} σ_{1})^{2} \\ V {X_{t} ∣ I_{t - 1}, B = 0} = (σ_{t} σ_{0})^{2} \end{cases}$ Hence, given the quantile $q^{α}$ of a Gaussian mixture with 2 components with level $α$ , that in general needs to be computed numerically, the VaR is defined as: ${VaR}_{t ∣ t - 1}^{α} = q^{α} .$

27.3 Test on the number of violations

Let’s define a violation of the ${VaR}_{t | t - 1}^{α}$ as $v_{t} = 1_{[x_{t} \leq {VaR}_{t ∣ t - 1}^{α}]} \sim Bernoulli (α),$

and let’s define the number of violations of the conditional VaR as follows, i.e. $N_{t} = \sum_{i = 1}^{t} v_{i} .$

27.3.1 Asymptotic variance

Applying the central limit theorem (CLT) it is possible to prove that the statistic test converges in distribution to a standard normal, i.e. $T_{1}^{α} = \frac{1}{\sqrt{t}} \sum_{i = 1}^{t} (\frac{v_{i} - α}{\sqrt{α (1 - α)}}) \underset{n \to \infty}{\overset{d}{⟶}} N (0, 1) .$ Hence, given the null hypothesis $H_{0} : P {X_{t} \leq {VaR}_{t | t - 1}^{α} | I_{t - 1}) = α$ , that is equivalent to $E {e_{t}} = α$ we define the critical values at a confidence level $α^{*}$ as $\begin{aligned} α & = P {| T_{1}^{α} | > t_{α^{*} / 2}} \\ ⇕ \\ t_{α^{*} / 2} & = P^{- 1} {P {| T_{1}^{α} | > t_{α^{*} / 2}}} \end{aligned}$

where $P$ and $P^{- 1}$ are respectively the distribution and the quantile of a standard normal. Therefore, the null hypothesis is rejected at a confidence level $α^{*}$ if: ${\begin{cases} [T_{1}^{α} < - t_{α^{*} / 2}] \cup [T_{1}^{α} > t_{α^{*} / 2}] & ⟹ H_{0} rejected \\ [- t_{α^{*} / 2} < T_{1}^{α} < t_{α^{*} / 2}] & ⟹ H_{0} not rejected \end{cases}$

27.3.2 Empirical variance

Instead of using the theoretical variance of $e_{t}$ , namely $α (1 - α)$ , let’s substitute it with the empirical one, i.e. $α (1 - α) ⟶ \frac{N_{t}}{t} (1 - \frac{N_{t}}{t}) .$ Hence, the new statistic test $N V_{2}$ converges to $N V_{1}$ in probability, therefore also in distribution, i.e. $T_{2}^{α} = \frac{1}{\sqrt{t}} \sum_{i = 1}^{t} (\frac{v_{i} - α}{\sqrt{\frac{N_{t}}{t} (1 - \frac{N_{t}}{t})}}) \underset{n \to \infty}{\overset{p}{⟶}} T_{1}^{α} \underset{n \to \infty}{\overset{d}{⟶}} N (0, 1) .$

For small samples, the following relation between the two statistics should be used: $T_{2}^{α} = \frac{\sqrt{α (1 - α)}}{\sqrt{\frac{N_{t}}{t} (1 - \frac{N_{t}}{t})}} T_{1}^{α} .$

27.4 Example: $H_{0}$ is not rejected

Instead of simulating exactly $u_{t} \sim N (0, 1)$ , let’s simulate residuals that are close to the normal distribution, i.e. $u_{t} \sim t (25)$ .

AR(2)-GARCH(1,2) simulation and VaR

# ===================== Setups ======================
set.seed(1)   # random seed 
ci <- 0.05    # confidence level 
t_bar <- 5000 # number of simulations 
# parameters
parAR <- c(mu=0.5, phi1 = 0.34, phi1 = 0.14)
parGARCH <- c(omega=0.4, alpha1=0.25, 
              beta1=0.25, beta2=0.15) 
# long-term std deviation of residuals 
sigma_eps <- sqrt(parGARCH[1]/(1-sum(parGARCH[-1])))
# ================== Simulation =====================
# Initial points
Xt <- rep(parAR[1]/(1-sum(parAR[-1])), 3)
sigma <- rep(sigma_eps, 3)
# Simulated residuals 
eps <- rt(t_bar, 25)
eps[1:3] = eps[1:3]*sigma_eps
# Value at Risk
q_alpha = qnorm(ci)
VaR = c(0)
for(t in 3:t_bar){
  # AR component 
  Xt[t] <- parAR[1] + parAR[2]*Xt[t-1] + parAR[3]*Xt[t-2]
  # ARCH component 
  sigma[t] <- parGARCH[1] + parGARCH[2]*eps[t-1]^2 
  # GARCH component 
  sigma[t] <- sigma[t] + parGARCH[3]*sigma[t-1]^2 + parGARCH[4]*sigma[t-2]^2
  sigma[t] <- sqrt(sigma[t])
  # Simulated residuals 
  eps[t] <- sigma[t]*eps[t]
  # Simulated value at risk
  VaR[t] <- Xt[t] + sigma[t]*q_alpha
  # Simulated time series 
  Xt[t] <- Xt[t] + eps[t]
}
# ===================== Plot ======================
# GARCH(1,1) simulation
ggplot()+
  geom_line(aes(1:t_bar, Xt), size = 0.2)+
  geom_line(aes(1:t_bar, VaR), size = 0.2, color = "red")+
  theme_bw()+
  labs(x = NULL, y = TeX("$X_t$"),
       subtitle = TeX(paste0("$\\mu:\\;", parAR[1], 
                             "\\;\\; \\phi_{1}:\\;", parAR[2],
                             "\\;\\; \\phi_{2}:\\;", parAR[3],
                             "\\;\\; \\omega:\\;", parGARCH[1], 
                             "\\;\\; \\alpha_{1}:\\;", parGARCH[2],
                             "\\;\\; \\beta_{1}:\\;", parGARCH[3],
                             "\\;\\; \\beta_{2}:\\;", parGARCH[4],
                             "$")))

Figure 27.1: AR(2)-GARCH(1,2) simulation with theoric (red) VaR at $α = 0.05$ .

VaR test Student-

t_{25}

# ======================================
# Violation of the VaR
vt <- ifelse(Xt < VaR, 1, 0)
vt[1:3] <- 0
# Theoric variance
v_theoric <- ci*(1-ci)
# Empiric variance
v_empiric <- (sum(vt)/t_bar)*(1 - sum(vt)/t_bar)
# Standardized number of violations
Nt <- sum((vt - ci)/sqrt(v_theoric))
# Statistic test (NV_1)
T1 <- (1/sqrt(t_bar))*Nt
# Statistic test (NV_2)
T2 <- (sqrt(v_theoric)/sqrt(v_empiric))*T1
# Rejection level 
t_alpha <- qnorm(ci/2)
# =============== Kable =============== 
kab <- dplyr::tibble(
  n = t_bar,
  alpha = paste0(format(ci*100, digits = 3), "%"),
  alpha_hat = paste0(format(sum(vt)/t_bar*100, digits = 3), "%"),
  t_alpha_dw = t_alpha,
  T1 = T1,
  T2 = T2,
  t_alpha_up = -t_alpha,
  H01 = ifelse(T1 > t_alpha_up | T1 < t_alpha_dw, "Rejected", "Non-Rejected"),
  H02 = ifelse(T2 > t_alpha_up | T2 < t_alpha_dw, "Rejected", "Non-Rejected")
)  %>%
  dplyr::mutate_if(is.numeric, format, digits = 4, scientific = FALSE)
colnames(kab) <- c("$$n$$", "$$\\alpha$$", "$$\\frac{N_n}{n}$$", "$$t_{\\alpha/2}$$", 
                   "$$\\text{T}_1^{\\alpha}$$", "$$\\text{T}_2^{\\alpha}$$", "$$t_{\\alpha/2}$$", "$$H_0(\\text{T}_1)$$", "$$H_0(\\text{T}_2)$$")

knitr::kable(kab)

Table 27.1: Test for a Student-

t

with 25 degrees of freedom at

α^{*} = 0.05

on the number of violations of the theoric VaR at

α = 0.05

$n$	$α$	$\frac{N_{n}}{n}$	$t_{α / 2}$	$T_{1}^{α}$	$T_{2}^{α}$	$t_{α / 2}$	$H_{0} (T_{1})$	$H_{0} (T_{2})$
5000	5%	5.6%	-1.96	1.947	1.845	1.96	Non-Rejected	Non-Rejected

27.5 Example: $H_{0}$ is rejected

Instead of simulating exactly $u_{t} \sim N (0, 1)$ , let’s simulate residuals that are not close to the normal distribution, i.e. $u_{t} \sim t (5)$ .

AR(2)-GARCH(1,2) simulation and VaR

# ==================== Setups =====================
set.seed(1)   # random seed 
ci <- 0.05    # confidence level 
t_bar <- 1000 # number of simulations 
# parameters
parAR <- c(mu=0.5, phi1 = 0.34, phi1 = 0.14)
parGARCH <- c(omega=0.4, alpha1=0.25,
              beta1=0.25, beta2=0.15) 
# quasi long-term std deviation of residuals 
sigma_eps <- sqrt(parGARCH[1]/(1-sum(parGARCH[-1])))
# ================== Simulation ===================
set.seed(1)
# Initial points
Xt <- rep(parAR[1]/(1-sum(parAR[-1])), 3)
mu <- rep(parAR[1]/(1-sum(parAR[-1])), 3)
sigma <- rep(sigma_eps, 3)
# Simulated residuals 
eps <- rt(t_bar, 5) 
eps[1:3] <- eps[1:3]*sigma_eps
# Value at Risk
q_alpha = qnorm(ci)
VaR = c(0)
for(t in 3:t_bar){
  # AR component 
  mu[t] <- parAR[1] + parAR[2]*Xt[t-1] + parAR[3]*Xt[t-2]
  # ARCH component 
  sigma[t] <- parGARCH[1] + parGARCH[2]*eps[t-1]^2 
  # GARCH component 
  sigma[t] <- sigma[t] + parGARCH[3]*sigma[t-1]^2 + parGARCH[4]*sigma[t-2]^2
  sigma[t] <- sqrt(sigma[t])
  # Simulated residuals 
  eps[t] <- sigma[t]*eps[t]
  # Simulated value at risk
  VaR[t] <- mu[t] + sigma[t]*q_alpha
  # Simulated time series 
  Xt[t] <- mu[t] + eps[t]
}
# Empirical quantile 
q_alpha_emp <- quantile(eps/sigma, probs = ci)
VaR_emp <- mu + sigma*q_alpha_emp

# ===================== Plot ======================
# GARCH(1,1) simulation
ggplot()+
  geom_line(aes(1:t_bar, Xt), size = 0.2)+
  geom_line(aes(1:t_bar, VaR), size = 0.2, color = "red")+
  geom_line(aes(1:t_bar, VaR_emp), size = 0.2, color = "blue")+
  theme_bw()+
  labs(x = NULL, y = TeX("$X_t$"),
       subtitle = TeX(paste0("$\\mu:\\;", parAR[1], 
                             "\\;\\; \\phi_{1}:\\;", parAR[2],
                             "\\;\\; \\phi_{2}:\\;", parAR[3],
                             "\\;\\; \\omega:\\;", parGARCH[1], 
                             "\\;\\; \\alpha_{1}:\\;", parGARCH[2],
                             "\\;\\; \\beta_{1}:\\;", parGARCH[3],
                             "\\;\\; \\beta_{2}:\\;", parGARCH[4],
                             "$")))

Figure 27.2: AR(2)-GARCH(1,2) simulation with theoric (red) and empirical (blue) VaR at $α = 0.05$ .

Computing the test on the normal quantile gives a rejection of the null hypothesis $H_{0}$ , i.e. the deviation from the VaR is not stochastic and it is not an adequate measure of risk.

VaR test Student-

t_{5}

# ======================================
# Violation of the VaR
vt <- ifelse(Xt < VaR, 1, 0)
vt[1:3] <- 0
# Theoric variance
v_theoric <- ci*(1-ci)
# Empiric variance
v_empiric <- (sum(vt)/t_bar)*(1 - sum(vt)/t_bar)
# Standardized number of violations
Nt <- sum((vt - ci)/sqrt(v_theoric))
# Statistic test (NV_1)
T1 <- (1/sqrt(t_bar))*Nt
# Statistic test (NV_2)
T2 <- (sqrt(v_theoric)/sqrt(v_empiric))*T1
# Rejection level 
t_alpha <- qnorm(ci/2)
# =============== Kable =============== 
kab <- dplyr::tibble(
  n = t_bar,
  alpha = paste0(format(ci*100, digits = 3), "%"),
  alpha_hat = paste0(format(sum(vt)/t_bar*100, digits = 3), "%"),
  t_alpha_dw = t_alpha,
  T1 = T1,
  T2 = T2,
  t_alpha_up = -t_alpha,
  H01 = ifelse(T1 > t_alpha_up | T1 < t_alpha_dw, "Rejected", "Non-Rejected"),
  H02 = ifelse(T2 > t_alpha_up | T2 < t_alpha_dw, "Rejected", "Non-Rejected")
)  %>%
  dplyr::mutate_if(is.numeric, format, digits = 4, scientific = FALSE)
colnames(kab) <- c("$$n$$", "$$\\alpha$$", "$$\\frac{N_n}{n}$$", "$$-t_{\\alpha/2}$$", 
                   "$$\\text{T}_1^{\\alpha}$$", "$$\\text{T}_2^{\\alpha}$$", "$$t_{\\alpha/2}$$", 
                   "$$H_0(\\text{T}_1)$$", "$$H_0(\\text{T}_2)$$")
knitr::kable(kab)

Table 27.2: Test for a Student-

t

with 5 degrees of freedom at

α^{*} = 0.05