Sensitivity Of TVaR To Tail Shape Assumptions

Why Tail-Shape Sensitivity Matters

A high-confidence TVaR estimate can depend heavily on the assumed behavior of losses beyond the observations that informed a fitted tail. Two models may use the same threshold and the same estimated frequency of threshold exceedance, yet imply very different losses at an extreme confidence level. The difference is not necessarily a calculation error: it can be a consequence of different assumptions about how quickly the tail decays.

This tutorial isolates that effect. It uses a generalized Pareto distribution (GPD) for losses above a threshold, then compares its estimates under alternative shape parameters and against an exponential excess model. These are hypothetical modeling examples, not estimates for an actual insurer. Sensitivity analysis helps show which assumptions drive an estimate; it does not, by itself, determine which model is best.

Definition: Let $X$ be a continuous loss with threshold $u$, and let $r=P(X>u)$. Model the excess $Y=X-u$, conditional on $X>u$, by a generalized Pareto distribution with scale $\beta>0$ and shape $\xi$. For $\xi\ne 0$, its conditional survival function is $P(Y>y\mid X>u)=(1+\xi y/\beta)^{-1/\xi}$, on values $y\ge 0$ for which $1+\xi y/\beta>0$. For $\xi=0$, the limiting model is exponential with mean $\beta$.

From a Tail Fit to a TVaR Estimate

Suppose the target confidence level is $p$, with $p>1-r$, so the target quantile lies above the threshold. Define $t$ as the probability of exceeding the target quantile, conditional on already exceeding $u$. The unconditional probability of exceeding the target is $1-p$; conditioning on threshold exceedance gives $t=(1-p)/r$. Thus $0<t<1$. The distinction between $t$ and $1-p$ matters whenever $r\ne 1$.

For a continuous loss, the target quantile is $q_p=u+y_p$, where the GPD survival probability of the excess $y_p$ equals $t$. Inverting the survival function gives the quantile formula below. If $\xi<0$, the model has a finite upper endpoint, so the target must lie within its support. For finite TVaR under this model, require $\xi<1$.

t=\frac{1-p}{r}, \qquad y_p= \begin{cases} \displaystyle \frac{\beta}{\xi}\left(t^{-\xi}-1\right), & \xi\ne 0,\\[6pt] \displaystyle \beta\log(1/t), & \xi=0, \end{cases} \qquad q_p=u+y_p .

For $\xi<1$, the conditional mean excess above $q_p$ is $(\beta+\xi y_p)/(1-\xi)$. Therefore, for this continuous model, $\operatorname{TVaR}_p(X)=q_p+(\beta+\xi y_p)/(1-\xi)$. When $\xi=0$, this reduces to $q_p+\beta$. A GPD with $\xi\ge 1$ has infinite mean excess, so its TVaR is not finite.

Why the Remaining Expected Excess Has This Form

The TVaR expression is not just the quantile plus an arbitrary adjustment. It adds the expected amount by which a loss, already known to exceed the quantile, exceeds that quantile. To see the adjustment, let $z\ge0$ be an additional excess beyond $y_p$. Conditional survival above $y_p$ is $(1+\xi z/(\beta+\xi y_p))^{-1/\xi}$. This follows by dividing the GPD survival at $y_p+z$ by the survival at $y_p$:

\begin{aligned} P(Y-y_p>z\mid Y>y_p) &=\frac{(1+\xi(y_p+z)/\beta)^{-1/\xi}} {(1+\xi y_p/\beta)^{-1/\xi}}\\ &=\left(1+\frac{\xi z}{\beta+\xi y_p}\right)^{-1/\xi}. \end{aligned}

For a nonnegative random variable, its expectation is the integral of its survival function. Integrating this conditional survival over $z\ge0$ gives $(\beta+\xi y_p)/(1-\xi)$ when $\xi<1$. In the exponential limit $\xi=0$, the survival is $\exp(-z/\beta)$, whose integral is $\beta$. This also explains the finiteness condition: when $\xi\ge1$, the GPD survival integral does not converge.

Key result: For a continuous threshold-tail GPD with $r=P(X>u)$, $p>1-r$, and $\xi<1$, calculate $t=(1-p)/r$, find $y_p$ from the conditional excess survival function, and then add the conditional mean excess above $y_p$ to $q_p=u+y_p$. To isolate sensitivity to $\xi$, hold $u$, $r$, $\beta$, and $p$ fixed across the scenarios being compared.

Worked Examples: Isolating Shape and Model Effects

Worked Example: Comparing Three Shape Assumptions

A hypothetical portfolio has threshold $u=\$100{,}000$, threshold exceedance probability $r=0.0400$, and fitted GPD scale $\beta=\$10{,}000$. Compare 99.75% TVaR under $\xi=0.50$, $\xi=0$, and $\xi=-0.50$. Assume the same $u$, $r$, and $\beta$ in all three fits.

Method and conditional probability. Here $p=0.9975$, so the unconditional exceedance probability is $1-p=0.0025$. Conditional on exceeding the threshold, $t=0.0025/0.0400=0.0625$. The common threshold and scale make the effect of changing $\xi$ visible.

Shape $\xi=0.50$. Since $t^{-0.50}=4$, $y_p=(\$10{,}000/0.50)(4-1)=\$60{,}000$, and $q_p=\$100{,}000+\$60{,}000=\$160{,}000$. The conditional mean excess above $q_p$ is $(\$10{,}000+0.50(\$60{,}000))/(1-0.50)=\$40{,}000/0.50=\$80{,}000$. Thus TVaR is $\$160{,}000+\$80{,}000=\$240{,}000$.

Shape $\xi=0$. The exponential-limit quantile excess is $y_p=\$10{,}000\log(16)=\$27{,}725.8872\ldots$. Hence $q_p=\$127{,}725.8872\ldots$, and TVaR is $q_p+\$10{,}000=\$137{,}725.8872\ldots$, or \$137,725.89 to the nearest cent.

Shape $\xi=-0.50$. Now $t^{0.50}=0.25$, so $y_p=(\$10{,}000/-0.50)(0.25-1)=\$15{,}000$, and $q_p=\$115{,}000$. The mean excess above the quantile is $(\$10{,}000-0.50(\$15{,}000))/(1.50)=\$2{,}500/1.50=\$1{,}666.6667\ldots$. TVaR is \$116,666.67 to the nearest cent. The finite endpoint is $u-\beta/\xi=\$100{,}000+\$20{,}000=\$120{,}000$, consistent with the estimated quantile and TVaR being below that endpoint.

Check and lesson. Each calculation uses the same conditional tail probability, threshold, and scale; only the shape changes. The resulting TVaRs range from \$116,666.67 to \$240,000. The negative-shape fit is bounded, the zero-shape fit has an exponential excess tail, and the positive-shape fit has a heavier tail. The common mistake this comparison avoids is changing several inputs at once and then attributing the entire difference to shape.

Worked Example: A Modest Shape Change at a More Extreme Level

For a hypothetical tail fit, let $u=\$80{,}000$, $r=0.0500$, and $\beta=\$20{,}000$. Estimate 99.95% TVaR once with $\xi=0.10$ and once with $\xi=0.20$, holding the other inputs fixed.

Set the conditional tail probability. Here $1-p=0.0005$, so $t=0.0005/0.0500=0.01$. For $\xi=0.10$, $t^{-\xi}=0.01^{-0.10}=10^{0.20}=1.584893192461\ldots$.

Calculate the $\xi=0.10$ estimate. The quantile excess is $y_p=(\$20{,}000/0.10)(1.584893192461\ldots-1)=\$116{,}978.638492\ldots$. Thus $q_p=\$196{,}978.638492\ldots$. The scale at the target quantile is $\beta+\xi y_p=\$20{,}000(1.584893192461\ldots)=\$31{,}697.863849\ldots$. The conditional mean excess is this amount divided by $1-0.10=0.90$, or $\$35{,}219.848721\ldots$. TVaR is \$232,198.49 to the nearest cent.

Calculate the $\xi=0.20$ estimate. Now $t^{-\xi}=0.01^{-0.20}=10^{0.40}=2.511886431510\ldots$. Therefore, $y_p=(\$20{,}000/0.20)(2.511886431510\ldots-1)=\$151{,}188.643151\ldots$, and $q_p=\$231{,}188.643151\ldots$. The scale at the quantile is $\$20{,}000(2.511886431510\ldots)=\$50{,}237.728630\ldots$; dividing by $0.80$ gives a conditional mean excess of $\$62{,}797.160788\ldots$. TVaR is \$293,985.80 to the nearest cent.

Check and lesson. The change in TVaR is $\$293{,}985.80-\$232{,}198.49=\$61{,}787.31$. The shape increase affects both the target quantile and the expected excess beyond it. The common mistake is to compare only the quantiles and overlook the shape-dependent mean excess included in TVaR.

Worked Example: Comparing Exponential and GPD Excess Fits

Two hypothetical models are fitted above the same threshold $u=\$50{,}000$, with $r=0.0500$, and are used to estimate 99.75% TVaR. The exponential excess fit has mean $\beta=\$15{,}000$. The alternative GPD excess fit has $\beta=\$10{,}000$ and $\xi=0.25$. Compare the estimates.

Find the conditional tail probability. Since $1-p=0.0025$, $t=0.0025/0.0500=0.05$ for both fits.

Exponential fit. The excess quantile is $y_p=\$15{,}000\log(20)=\$44{,}935.984103\ldots$. The loss quantile is $q_p=\$50{,}000+\$44{,}935.984103\ldots=\$94{,}935.984103\ldots$. By the exponential memoryless property, the expected excess beyond this quantile is $\$15{,}000$. Thus TVaR is $\$109{,}935.984103\ldots$, or \$109,935.98 to the nearest cent.

GPD fit. Since $0.05^{-0.25}=20^{0.25}=2.114742526881\ldots$, $y_p=(\$10{,}000/0.25)(2.114742526881\ldots-1)=\$44{,}589.701075\ldots$. The quantile is $q_p=\$94{,}589.701075\ldots$. The scale at that quantile is $\$10{,}000(2.114742526881\ldots)=\$21{,}147.425269\ldots$, so the mean excess is $\$21{,}147.425269\ldots/0.75=\$28{,}196.567025\ldots$. The GPD TVaR is \$122,786.27 to the nearest cent.

Check and lesson. The GPD estimate is $\$12{,}850.29$ higher. This comparison changes both the family and the fitted scale, so the difference cannot be assigned to shape alone. The common mistake is to treat a family comparison as a controlled shape-sensitivity test; for that purpose, hold the other parameters fixed, as in the first two examples.

Using Sensitivity Results Responsibly

Tail-shape sensitivity is useful because extreme quantiles and TVaR often depend on extrapolation beyond the central body of a loss distribution. A small change in the fitted shape can therefore have a material effect, especially when the target probability is far into the tail. A sensitivity table can make this dependence explicit:

Model change	What it isolates	Interpretation to keep in view
Change $\xi$, holding $u$, $r$, $\beta$, and $p$ fixed	Effect of the GPD shape assumption	Both the quantile and remaining mean excess can change
Change the fitted family and its parameters	Combined effect of family and refitting	Differences cannot automatically be attributed to one parameter
Change $r$ while holding $p$ fixed	Effect of the estimated frequency above the threshold	The conditional probability $t=(1-p)/r$ changes too

A sensitivity calculation is not a forecast that every model is equally plausible. It is a way to reveal how much the reported estimate relies on tail assumptions. Interpret the result alongside the threshold choice, data available for fitting, parameter uncertainty, and whether the model’s support and moment conditions are appropriate. A negative shape imposes a finite endpoint; a shape at or above 1 implies infinite model TVaR. Neither feature should be ignored just because the fitted formula can be evaluated.

Common Exam Trap: Using the Wrong Tail Probability

Suppose $r=0.0400$ and the target level is $p=0.9975$. The unconditional tail probability is $1-p=0.0025$, but the GPD is conditional on exceeding the threshold. The correct probability to put into its survival function is $t=0.0025/0.0400=0.0625$. Using $0.0025$ instead asks the fitted excess distribution to reach a much more extreme conditional quantile than the target requires.

Another trap is to report $q_p$ as TVaR. For a continuous tail, TVaR includes the expected excess beyond $q_p$, which is positive. Use the full formula, and verify that the quantile lies inside the fitted support and that $\xi<1$ for finite TVaR. The discussion here is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each problem, use the conditional excess-tail probability and round money to the nearest cent only at the final step.

A hypothetical GPD tail has $u=\$40{,}000$, $r=0.0400$, $\beta=\$8{,}000$, and $\xi=0.50$. Find TVaR at $p=0.9975$. (Answer: \$152,000.00)
An exponential excess tail has threshold $u=\$30{,}000$, scale $\beta=\$6{,}000$, and threshold exceedance probability $r=0.0200$. Find TVaR at $p=0.999$. (Answer: \$53,974.39)
A GPD tail has $u=\$100{,}000$, $\beta=\$12{,}000$, $\xi=-0.50$, and $r=0.0400$. Find 99% TVaR and the model endpoint. (Answer: TVaR \$116,000.00; endpoint \$124,000)
A GPD excess tail has $u=\$60{,}000$, $\beta=\$10{,}000$, $\xi=0.25$, and $r=0.0400$. Find TVaR at $p=0.9975$. (Answer: \$126,666.67)
An exponential excess model has $u=\$25{,}000$, $\beta=\$4{,}000$, and $r=0.1000$. For $p=0.999$, calculate the correct conditional tail probability and TVaR. Explain why using $1-p$ directly in the exponential quantile formula would be wrong. (Answer: $t=0.0100$; TVaR \$47,420.68; $1-p$ is unconditional, whereas the fitted excess model is conditional on exceeding the threshold.)

Model change	What it isolates	Interpretation to keep in view
Change \(\xi\), holding \(u\), \(r\), \(\beta\), and \(p\) fixed	Effect of the GPD shape assumption	Both the quantile and remaining mean excess can change
Change the fitted family and its parameters	Combined effect of family and refitting	Differences cannot automatically be attributed to one parameter
Change \(r\) while holding \(p\) fixed	Effect of the estimated frequency above the threshold	The conditional probability \(t=(1-p)/r\) changes too

What You'll Learn