Extrapolating TVaR With Generalized Pareto Tail Fits

Why Extrapolate a Loss Tail?

A finite loss data set may contain too few extreme observations to estimate a very high quantile directly. For example, the largest observed loss does not by itself reveal the 99.5th percentile of the underlying loss distribution. A tail model can address this limitation by using observations above a high threshold to estimate the shape of losses farther into the tail.

The generalized Pareto distribution (GPD) is commonly used for this purpose. The method is a model-based extrapolation, not a way to create additional observed information. Its results depend on whether the chosen threshold is high enough for the GPD approximation to be reasonable, whether the exceedances are adequately modeled, and whether the fitted parameters are stable enough for the intended use. The methods below are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let $X$ be a loss and $u$ a threshold. An exceedance is $Y=X-u$, conditional on $X>u$. A generalized Pareto distribution with scale parameter $\beta>0$ and shape parameter $\xi$ has survival function $$ P(Y>y\mid X>u)=\left(1+\frac{\xi y}{\beta}\right)^{-1/\xi} $$ for $y\geq 0$ such that $1+\xi y/\beta>0$. When $\xi=0$, the limiting survival function is $\exp(-y/\beta)$. For $\xi<0$, exceedances have a finite upper endpoint, $-\beta/\xi$; for $\xi\geq 0$, there is no finite upper endpoint.

From Exceedances to Tail Quantiles

Suppose $n$ losses are observed, $k$ of them exceed $u$, and the fitted GPD parameters are $\hat{\xi}$ and $\hat{\beta}$. The empirical estimate of the probability of exceeding the threshold is $\hat{p}_u=k/n$. Treating the fitted GPD as an approximation to the conditional exceedance distribution gives the following tail approximation for $x\geq u$:

P(X>x)\approx \hat{p}_u\left(1+\frac{\hat{\xi}(x-u)}{\hat{\beta}}\right)^{-1/\hat{\xi}}.

This expression combines two probabilities: the chance of reaching the threshold and the conditional chance of exceeding $x$, given that the threshold has been exceeded. For a target confidence level $\alpha$, let $q=1-\alpha$ be the target upper-tail probability. When $q<\hat{p}_u$, the target quantile lies above the threshold and can be obtained by solving the fitted tail equation for $x$.

\widehat{\operatorname{VaR}}_{\alpha} = u+\frac{\hat{\beta}}{\hat{\xi}} \left[ \left(\frac{\hat{p}_u}{1-\alpha}\right)^{\hat{\xi}}-1 \right], \qquad \hat{\xi}\ne 0.

If $\hat{\xi}=0$, the limiting formula is $u+\hat{\beta}\ln(\hat{p}_u/(1-\alpha))$. The ratio is threshold exceedance probability divided by target tail probability. It is greater than one when the target quantile is beyond $u$. If $1-\alpha\geq \hat{p}_u$, this extrapolation formula is not the appropriate way to estimate the quantile: the target is at or below the threshold, so the body of the loss distribution is needed.

Calculating TVaR from the Fitted Tail

For a continuous loss distribution, TVaR at level $\alpha$ is the mean loss conditional on exceeding $\operatorname{VaR}_{\alpha}$. The GPD model gives the conditional mean excess above any point $x\geq u$, provided $\xi<1$. This restriction matters: if $\xi\geq 1$, the fitted conditional tail mean is infinite.

\operatorname{TVaR}_{\alpha} = \operatorname{VaR}_{\alpha} + \frac{\beta+\xi(\operatorname{VaR}_{\alpha}-u)}{1-\xi}, \qquad \xi<1.

In an application, use the fitted parameters and the fitted VaR in this formula. The quantity $\beta+\xi(\operatorname{VaR}_{\alpha}-u)$ is the GPD scale at the VaR level. For $\xi=0$, the formula becomes $\operatorname{VaR}_{\alpha}+\beta$.

Why the TVaR Formula Holds

The derivation uses the GPD survival function conditional on exceeding the threshold. Write $v=\operatorname{VaR}_{\alpha}$, and assume $v\geq u$. For $z\geq 0$, the probability of exceeding $v+z$, conditional on exceeding $v$, is the ratio of the two fitted tail probabilities. Simplifying that ratio gives another GPD survival function, with scale $\beta+\xi(v-u)$:

P(X>v+z\mid X>v) = \left( 1+\frac{\xi z}{\beta+\xi(v-u)} \right)^{-1/\xi}.

For a nonnegative random variable, its mean is the integral of its survival function. Therefore, when $\xi<1$, the mean excess above $v$ is

E[X-v\mid X>v] = \int_0^\infty \left( 1+\frac{\xi z}{\beta+\xi(v-u)} \right)^{-1/\xi}\,dz = \frac{\beta+\xi(v-u)}{1-\xi}.

The integral is finite only for $\xi<1$; for $\xi=0$, it is the integral of an exponential survival function and equals $\beta$. Adding this conditional mean excess to $v$ yields the TVaR formula. This justification also shows why a high VaR does not alone determine TVaR: the fitted shape controls how much expected loss remains beyond that VaR.

Worked Examples

Worked Example: Extrapolating from a Fitted Tail

A hypothetical portfolio has 1,000 annual losses, of which 50 exceed a threshold of $u=\$1{,}000$ thousand. A GPD is fitted to the exceedances with $\hat{\xi}=0.25$ and $\hat{\beta}=\$200$ thousand. Estimate the 99.5% VaR and TVaR. Use the fitted model beyond the observed data range.

Method and threshold probability. The observed threshold exceedance fraction is $\hat{p}_u=50/1{,}000=0.05$. The target tail probability is $1-0.995=0.005$, so the ratio in the quantile formula is $0.05/0.005=10$. Since 0.005 is less than 0.05, the target lies above the threshold.

VaR. The power term is $10^{0.25}=1.778279410$. Thus

\widehat{\operatorname{VaR}}_{0.995} = 1{,}000+\frac{200}{0.25}(1.778279410-1) = 1{,}622.623528

The estimated VaR is $\$1{,}622.62$ thousand.

TVaR. The scale at the VaR is $200+0.25(1{,}622.623528-1{,}000)=355.655882$ thousand. Since $0.25<1$, the mean excess exists:

\widehat{\operatorname{TVaR}}_{0.995} = 1{,}622.623528+\frac{355.655882}{1-0.25} = 2{,}096.831371

The estimated TVaR is $\$2{,}096.83$ thousand. The computation is checked by confirming that the fitted VaR exceeds the threshold and that TVaR exceeds VaR. The method avoids treating the largest observed loss as if it were a reliable estimate of a more extreme quantile.

Worked Example: Moment-Fitting a GPD to Exceedances

A hypothetical insurer records four positive exceedances over a threshold of $\$500$ thousand: 5, 5, 5, and 105 thousand dollars. Estimate GPD parameters by matching the empirical mean and variance to the GPD mean and variance, using divisor $n$ for the empirical variance. There are 100 losses in total. Then estimate the 99% VaR and TVaR.

Fit the parameters. The mean exceedance is $(5+5+5+105)/4=30$ thousand. Deviations from the mean are $-25,-25,-25,75$, so the empirical variance is $(625+625+625+5{,}625)/4=1{,}875$. The squared coefficient of variation is $1{,}875/30^2=25/12$.

For a GPD with $\xi<1/2$, its mean is $\beta/(1-\xi)$ and its variance is $\beta^2/[(1-\xi)^2(1-2\xi)]$. Consequently, the squared coefficient of variation is $1/(1-2\xi)$. Matching it to $25/12$ gives $\hat{\xi}=(1-12/25)/2=0.26$. Matching the mean gives $\hat{\beta}=30(1-0.26)=22.2$ thousand.

Estimate VaR. The threshold probability estimate is $4/100=0.04$, and the 99% target tail probability is 0.01. The ratio is $0.04/0.01=4$, and $4^{0.26}=1.433954$. Therefore,

\widehat{\operatorname{VaR}}_{0.99} = 500+\frac{22.2}{0.26}(1.433954-1) \approx 537.05

All amounts here are in thousands of dollars. The scale at the estimated VaR is $22.2(1.433954)=31.8338$ thousand. Since $0.26<1$, the estimated TVaR is

\widehat{\operatorname{TVaR}}_{0.99} = 537.052995+\frac{31.833779}{0.74} \approx 580.07

The estimates are $\$537.05$ thousand for VaR and $\$580.07$ thousand for TVaR. The calculation is internally consistent: the VaR is above the $\$500$ thousand threshold, and the finite mean tail produces TVaR above VaR. The method avoids fitting directly to the original losses when the stated model applies to threshold exceedances.

Worked Example: A Negative Shape and a Finite Endpoint

For a hypothetical loss portfolio, losses above $u=\$2{,}000$ thousand are modeled with a GPD having $\xi=-0.20$ and $\beta=\$300$ thousand. The estimated probability of exceeding the threshold is 0.10. Find the 99% VaR and TVaR, and check the model’s upper endpoint.

VaR. The target tail probability is 0.01, so the ratio is $0.10/0.01=10$. The power term is $10^{-0.20}=0.630957345$. Applying the quantile formula,

\widehat{\operatorname{VaR}}_{0.99} = 2{,}000+\frac{300}{-0.20}(0.630957345-1) = 2{,}553.563983

The estimated VaR is $\$2{,}553.56$ thousand. The finite upper endpoint for the loss is $u-\beta/\xi=2{,}000-300/(-0.20)=\$3{,}500$ thousand, so the estimate is within the fitted support.

TVaR. The scale at the VaR is $300(0.630957345)=189.287203$ thousand. The mean excess is $189.287203/(1-(-0.20))=157.739336$ thousand. Thus

\widehat{\operatorname{TVaR}}_{0.99} = 2{,}553.563983+157.739336 = 2{,}711.303319

The estimated TVaR is $\$2{,}711.30$ thousand, below the $\$3{,}500$ thousand endpoint as it must be. This check helps catch an invalid extrapolated quantile outside a bounded fitted tail.

Assumptions and Common Exam Traps

A GPD tail fit requires more than substituting values into formulas. The threshold exceedances are assumed to be adequately represented by a GPD, and the estimated probability of exceeding the threshold must represent the loss population of interest. In using the simple fraction $k/n$, the observations are treated as representative; dependence, changing exposure, and other data issues can make that estimate inappropriate. The formulas extrapolate the fitted model and do not establish that the model is appropriate.

Parameter fitting also has conditions. The moment formulas used above require $\xi<1/2$ for a finite variance. The TVaR formula requires $\xi<1$ for a finite conditional mean. A fitted $\xi\geq1$ still permits a finite VaR at a probability within the model’s support, but its TVaR is infinite under the unbounded GPD tail.

A frequent calculation error is to reverse the probability ratio. In Worked Example 1, using $0.005/0.05=0.1$ in place of $0.05/0.005=10$ gives $10^{0.25}-1$ with the wrong ratio, producing a negative excess over the threshold under the stated formula. That is inconsistent with the target being more extreme than the threshold. The correction is to set the fitted survival probability equal to $1-\alpha$: $\hat{p}_u$ times the conditional exceedance probability must equal $1-\alpha$. The resulting ratio is $\hat{p}_u/(1-\alpha)$.

Key takeaway: Estimate the threshold exceedance probability, fit the GPD to the excesses, and use the ratio of threshold tail probability to target tail probability for extrapolation. Before reporting TVaR, check that the fitted shape is less than 1, the VaR lies within the fitted support, and TVaR is at least as large as VaR.

Check Your Understanding

For each problem, work in the stated units and check that the target tail probability is below the threshold exceedance probability before using the extrapolation formula.

A fitted GPD has $u=800$, $\beta=120$, $\xi=0.20$, and threshold exceedance probability 0.08. Estimate the 99% VaR and TVaR. (Answer: VaR $\approx 1{,}109.43$; TVaR $\approx 1{,}336.79$, in the same units as $u$.)
A fitted exponential tail has $u=1{,}000$, $\beta=250$, and threshold exceedance probability 0.10. Estimate the 99.5% VaR and TVaR. (Answer: VaR $\approx 1{,}748.93$; TVaR $\approx 1{,}998.93$.)
A fitted GPD has $u=1{,}000$, $\beta=300$, $\xi=0.40$, and threshold exceedance probability 0.02. Estimate the 99% VaR and TVaR. (Answer: VaR $\approx 1{,}239.63$; TVaR $\approx 1{,}899.38$.)
A fitted GPD has $u=400$, $\beta=100$, $\xi=0.70$, and threshold exceedance probability 0.05. At the 99% level, determine the VaR and whether TVaR is finite. (Answer: VaR $\approx 697.88$; TVaR is infinite under this fitted model because $\xi\geq1$ is false? The correct condition is $\xi<1$, so with $\xi=0.70$, TVaR is finite.)
A fitted GPD has $u=600$, $\beta=150$, $\xi=-0.25$, and threshold exceedance probability 0.04. Estimate the 99% VaR and TVaR, and state the endpoint. (Answer: VaR $\approx 775.74$; TVaR $\approx 860.59$; endpoint $=1{,}200$, in the same units.)

What You'll Learn