VaR and ES via Peaks-Over-Threshold
You are modeling extreme daily losses using Extreme Value Theory. Suppose daily losses $L_t$ above a high threshold $u$ follow a Generalized Pareto Distribution (GPD) with shape parameter $\xi$ and scale parameter $\sigma$:
$$P(L_t - u > y \mid L_t > u) = \begin{cases} \left(1 + \xi \frac{y}{\sigma}\right)^{-1/\xi} & \xi \neq 0 \\ e^{-y/\sigma} & \xi = 0 \end{cases}$$
for $y > 0$ (with $y < -\sigma / \xi$ when $\xi < 0$).
You observe $T$ total daily losses, of which $N_u$ exceed the threshold $u$.
- Derive closed-form expressions for VaR at confidence level $\alpha$ (close to 1) and Expected Shortfall (ES) at level $\alpha$, in terms of $\xi$, $\sigma$, $u$, $N_u$, and $T$.
- Explain how you would estimate $\xi$ and $\sigma$ from the $N_u$ exceedances, and how those estimates plug into your VaR and ES formulas.
- State the regularity conditions on the threshold $u$ and the shape parameter $\xi$ that are required for these formulas to be valid.
Hints
- Start from the unconditional tail probability: combine the empirical exceedance rate $N_u/T$ with the GPD survival function for losses above $u$.
- For VaR, invert $\bar{F}(q_\alpha) = 1 - \alpha$. For ES, use the GPD's linear mean excess property: if excesses above $u$ are $\text{GPD}(\xi, \sigma)$, then excesses above any higher threshold $v > u$ are $\text{GPD}(\xi, \sigma + \xi(v-u))$.
- Write $\text{ES}_\alpha = \text{VaR}_\alpha + E[L - \text{VaR}_\alpha \mid L > \text{VaR}_\alpha]$ and substitute the GPD mean excess at threshold $\text{VaR}_\alpha$ to get a closed form. Remember ES requires $\xi < 1$.
Worked Solution
How to Think About It: The whole point of POT is that you do not need to model the entire loss distribution -- you only need to get the tail right. The Pickands-Balkema-de Haan theorem tells you that for a broad class of distributions, the excess losses above a high threshold converge to a GPD. So the game is: pick a high threshold $u$, fit a GPD to the exceedances, then extrapolate into the far tail to get VaR and ES. This is the standard approach for risk management when you care about the 99th or 99.5th percentile and your data has fat tails.
Quick Estimate: Suppose you have $T = 1000$ daily observations, $N_u = 50$ exceed the threshold $u = 2\%$, and you fit $\xi = 0.25$, $\sigma = 0.8\%$. For $\alpha = 0.99$: the quantile $(T/N_u)(1-\alpha) = (1000/50)(0.01) = 0.2$. Then VaR $= 2\% + (0.8\%/0.25)(0.2^{-0.25} - 1) = 2\% + 3.2\%(0.2^{-0.25} - 1)$. Now $0.2^{-0.25} = (1/0.2)^{0.25} = 5^{0.25} \approx 1.495$. So VaR $\approx 2\% + 3.2\% \times 0.495 \approx 3.58\%$. ES is VaR/(1 - $\xi$) plus a correction: $\text{ES} \approx 3.58\%/(1-0.25) + (\sigma - \xi u)/(1-\xi) = 4.77\% + (0.8\% - 0.25 \times 2\%)/0.75 = 4.77\% + 0.40\% = 5.17\%$. Fat tails ($\xi > 0$) push ES well above VaR.
Approach: We combine the semi-parametric tail estimator $\bar{F}(x) = (N_u/T)(1 + \xi(x-u)/\sigma)^{-1/\xi}$ with standard quantile inversion for VaR, and then integrate the tail for ES.
Formal Solution:
*Step 1: Tail probability.* For $x > u$, the unconditional survival function is:
$$\bar{F}(x) = P(L_t > u) \cdot P(L_t > x \mid L_t > u) = \frac{N_u}{T}\left(1 + \xi\frac{x - u}{\sigma}\right)^{-1/\xi}$$
where we estimate $P(L_t > u) \approx N_u / T$.
*Step 2: VaR derivation.* VaR at level $\alpha$ is the quantile $q_\alpha$ such that $P(L_t > q_\alpha) = 1 - \alpha$. Setting $\bar{F}(q_\alpha) = 1 - \alpha$ and solving:
$$\frac{N_u}{T}\left(1 + \xi\frac{q_\alpha - u}{\sigma}\right)^{-1/\xi} = 1 - \alpha$$
$$1 + \xi\frac{q_\alpha - u}{\sigma} = \left(\frac{N_u}{T(1 - \alpha)}\right)^{\xi}$$
$$\boxed{\text{VaR}_\alpha = u + \frac{\sigma}{\xi}\left[\left(\frac{N_u}{T(1 - \alpha)}\right)^{\xi} - 1\right]}$$
This is the POT-GPD VaR formula.
*Step 3: ES derivation.* Expected Shortfall is the conditional expectation of losses given they exceed VaR:
$$\text{ES}_\alpha = E[L_t \mid L_t > \text{VaR}_\alpha] = \text{VaR}_\alpha + E[L_t - \text{VaR}_\alpha \mid L_t > \text{VaR}_\alpha]$$
The mean excess function of the GPD with shape $\xi < 1$ and threshold $v$ is $(\sigma_v)/(1 - \xi)$, where $\sigma_v = \sigma + \xi(v - u)$ is the GPD scale at the new threshold $v = \text{VaR}_\alpha$. So:
$$E[L_t - \text{VaR}_\alpha \mid L_t > \text{VaR}_\alpha] = \frac{\sigma + \xi(\text{VaR}_\alpha - u)}{1 - \xi}$$
Substituting $\text{VaR}_\alpha - u = (\sigma/\xi)[(N_u/(T(1-\alpha)))^\xi - 1]$:
$$\text{ES}_\alpha = \text{VaR}_\alpha + \frac{\sigma + \xi \cdot \frac{\sigma}{\xi}\left[\left(\frac{N_u}{T(1-\alpha)}\right)^\xi - 1\right]}{1 - \xi}$$
$$= \text{VaR}_\alpha + \frac{\sigma\left(\frac{N_u}{T(1-\alpha)}\right)^\xi}{1 - \xi}$$
This simplifies to:
$$\boxed{\text{ES}_\alpha = \frac{\text{VaR}_\alpha}{1 - \xi} + \frac{\sigma - \xi u}{1 - \xi}}$$
*Step 4: Estimation.* Given the $N_u$ exceedances $y_1 = L_{(1)} - u, \ldots, y_{N_u} = L_{(N_u)} - u$, estimate $\xi$ and $\sigma$ by maximizing the GPD log-likelihood:
$$\ell(\xi, \sigma) = -N_u \ln \sigma - \left(1 + \frac{1}{\xi}\right)\sum_{i=1}^{N_u}\ln\left(1 + \xi\frac{y_i}{\sigma}\right)$$
This is solved numerically (e.g., Newton-Raphson or profile likelihood). The MLEs $\hat{\xi}$ and $\hat{\sigma}$ are then plugged directly into the VaR and ES formulas above. Confidence intervals can be obtained via the delta method or profile likelihood.
*Step 5: Regularity conditions.*
- Threshold $u$: Must be high enough that the GPD approximation is accurate (bias-variance trade-off). In practice, use a mean residual life plot or the Hill plot to select $u$. As $u \to \infty$, the GPD approximation improves but $N_u$ shrinks, increasing variance.
- Shape parameter $\xi$: For ES to be finite, we need $\xi < 1$. For the mean excess function to exist, we need $\xi < 1$. For variance of exceedances to be finite, we need $\xi < 1/2$. The MLE is consistent and asymptotically normal for $\xi > -1/2$.
- Support constraint: When $\xi < 0$ (short-tailed case), the GPD has a finite right endpoint at $u - \sigma/\xi$, and exceedances must satisfy $y_i < -\sigma/\xi$.
- Sample size: Need $N_u$ sufficiently large for reliable MLE estimation -- typically $N_u \geq 30$ is a practical minimum.
Answer: The POT-GPD risk measures are:
$$\text{VaR}_\alpha = u + \frac{\sigma}{\xi}\left[\left(\frac{N_u}{T(1-\alpha)}\right)^{\xi} - 1\right]$$
$$\text{ES}_\alpha = \frac{\text{VaR}_\alpha}{1 - \xi} + \frac{\sigma - \xi u}{1 - \xi}$$
with $\xi$ and $\sigma$ estimated by MLE on the $N_u$ exceedances. The key regularity conditions are: $\xi < 1$ for finite ES, $u$ chosen high enough for the GPD limit to hold, and $\xi > -1/2$ for MLE consistency.
Intuition
The Peaks-Over-Threshold method is the workhorse of tail risk estimation in practice because it lets you focus modeling effort exactly where it matters -- the extreme tail -- without committing to a parametric form for the entire distribution. The Pickands-Balkema-de Haan theorem is the theoretical backbone: it says that for essentially any reasonable distribution, the excesses over a high threshold converge to a Generalized Pareto. This is the tail analog of the Central Limit Theorem, and it is why GPD-based VaR and ES formulas are so widely used in risk management.
The practical tension is in choosing the threshold $u$. Too low and the GPD approximation is poor (you are fitting a limit theorem to non-limit data). Too high and you have so few exceedances that your parameter estimates are noisy. This bias-variance trade-off is the real art in EVT applications. The shape parameter $\xi$ is what drives everything: $\xi > 0$ means Pareto-type (fat) tails where ES can be many multiples of VaR, $\xi = 0$ means exponential tails, and $\xi < 0$ means bounded tails. Financial return data almost always has $\xi > 0$, typically in the range 0.1-0.4, which is why normal-distribution VaR systematically underestimates true tail risk.