Estimating TVaR From Different Sources of Loss Information
A TVaR estimate can be based on observed losses, a fitted probability distribution, or simulated losses. Each approach answers a related question, but they do not have the same information behind them. An empirical estimate depends on the observations and the convention used at the tail boundary; a parametric estimate depends on the fitted distribution; and a simulation estimate depends on both the underlying model and the number of simulated outcomes.
A useful comparison therefore begins by checking that estimates use the same confidence level and loss definition. Then check how each method handles the tail. In particular, a confidence level can cut through a group of equal losses or through an observation in a finite simulated sample, so the entire boundary value should not automatically be included in the tail.
Empirical TVaR and the Boundary Observation
For observations \(x_1,\ldots,x_n\), the empirical distribution assigns probability \(1/n\) to each observation. Sort the values from smallest to largest. At confidence level \(\alpha\), the empirical VaR is the observation in position \(\lceil n\alpha\rceil\), using the lower-quantile convention above. The tail contains \(n(1-\alpha)\) observation-equivalents, which need not be a whole number. If it is fractional, use the required fraction of the boundary observation.
Equivalently, if \(q\) is the empirical VaR and \(F_n(q)\) is the empirical proportion at or below \(q\), then
The first term in the numerator is the probability-weighted contribution from observations strictly above \(q\). The second adds just enough of the probability mass at \(q\) to complete a tail of size \(1-\alpha\).
Worked Example: Empirical TVaR With a Fractional Boundary
A hypothetical insurer records ten annual losses, in thousands of dollars: four losses of 0, three of 10, two of 20, and one of 50. Calculate empirical VaR and TVaR at \(\alpha=0.75\). Use the empirical lower-quantile convention and report amounts in thousands of dollars to three decimals.
Method. Sort the observations and identify the required tail size. The ordered sample has positions 1–4 equal to 0, positions 5–7 equal to 10, positions 8–9 equal to 20, and position 10 equal to 50. Since \(\lceil 10(0.75)\rceil=8\), the empirical VaR is \(q=20\).
The tail contains \(10(1-0.75)=2.5\) observation-equivalents. There is one observation above 20, namely 50. The remaining 1.5 observation-equivalents must come from the two observations equal to 20. Thus,
The empirical proportion at or below 20 is \(F_n(20)=9/10=0.9\). The boundary-mass formula gives the same result: the strict-above contribution is \(50/10=5\), and the boundary contribution is \(20(0.9-0.75)=3\). Dividing their sum by \(1-0.75=0.25\) gives \((5+3)/0.25=32\). The answer is 32.000 thousand dollars.
Check. The tail uses 2.5 observations, and its weighted loss total is \(50+1.5(20)=80\); \(80/2.5=32\). A common mistake is to average all observations greater than or equal to VaR, giving \((50+20+20)/3=30\). That includes too much boundary mass: the prescribed tail has 2.5 observation-equivalents, not 3.
TVaR From a Fitted Continuous Distribution
A parametric estimate substitutes a fitted distribution for the empirical distribution. The fitted model can provide a smooth tail estimate, but its result depends on whether the chosen distribution describes the relevant loss tail adequately. Parameter estimation uncertainty is also distinct from the uncertainty represented by the fitted model itself.
For example, suppose severity \(X\) is modeled as exponential with scale parameter \(\theta>0\), so \(S_X(x)=e^{-x/\theta}\) for \(x\geq0\). If independent observations are used to fit this model by maximum likelihood, the estimate of \(\theta\) is the sample mean. The model gives the VaR and TVaR formulas below.
To justify the TVaR formula, condition on exceeding the VaR threshold \(q\). For \(y\geq0\), the exponential survival function gives
Thus the excess \(X-q\), conditional on \(X>q\), has the same exponential distribution and mean \(\theta\). Since this model is continuous, TVaR is the conditional mean above \(q\), so it equals \(q+\theta\). This derivation relies on the exponential model and does not establish the same formula for other severity distributions.
Worked Example: Fitting an Exponential Severity Model
For a hypothetical portfolio, 15 observed claim severities total $1,200,000. Assume the observations are independent and identically distributed exponential severities, and fit the scale parameter by maximum likelihood. Estimate the 90% VaR and TVaR. Report dollars to the cent.
Method. Under the stated exponential model, the maximum-likelihood estimate of the scale is the sample mean:
At \(\alpha=0.90\), \(1-\alpha=0.10\), so \(-\ln(1-\alpha)=-\ln(0.10)=\ln(10)\). Therefore,
TVaR is the VaR plus the fitted scale:
The result is estimated 90% VaR of $184,206.81 and estimated 90% TVaR of $264,206.81, rounded to the nearest cent. The arithmetic is consistent with the model relation \(\operatorname{TVaR}_{0.90}-\operatorname{VaR}_{0.90}=\widehat{\theta}\). The model assumption is substantive: a different fitted tail shape could produce a different TVaR, even if it gave a similar VaR.
TVaR From Simulated Losses
Monte Carlo estimation generates losses from a specified model and estimates tail quantities from the resulting sample. Once the simulated losses are sorted, the empirical tail calculation applies. A simulation does not remove model risk: it approximates the consequences of the model used to generate the losses. Nor does a large number of simulations guarantee a precise extreme-tail estimate if relatively few simulations lie in the tail of interest.
When reporting a simulated estimate, state the confidence level, sample size, and boundary convention. For a sample of \(n\) simulated values, the empirical quantile-integral convention uses \(n(1-\alpha)\) observation-equivalents in the upper tail, including a fractional boundary value where needed.
Worked Example: TVaR From a Finite Simulation Sample
A hypothetical model produces 12 simulated annual losses, sorted and expressed in thousands of dollars: 40, 45, 50, 55, 60, 65, 70, 75, 80, 90, 120, and 200. Estimate VaR and TVaR at \(\alpha=0.90\), using the empirical quantile-integral convention. Report amounts in thousands to three decimals.
Method. The empirical VaR is at position \(\lceil12(0.90)\rceil=\lceil10.8\rceil=11\), so \(q=120\). The tail has \(12(0.10)=1.2\) observation-equivalents. The single observation above 120 is 200; the remaining 0.2 observation-equivalents come from the boundary value 120.
The corresponding boundary-formula check uses \(F_n(120)=11/12\). The strict-above contribution is \(200/12\), and the boundary contribution is \(120(11/12-0.90)\). The latter equals \(120(1/60)=2\). Thus,
The estimates are VaR of 120.000 thousand dollars and TVaR of 186.667 thousand dollars, rounded to three decimals. The check confirms both the tail weighting and the result. With only 12 simulations, the estimate is especially sensitive to the largest simulated losses; this calculation alone does not quantify its sampling uncertainty.
Comparing Estimates and Their Assumptions
Estimates should be compared only after aligning their confidence levels, loss units, and definitions. If one estimate uses a 90% tail and another a 95% tail, the difference combines method effects with a change in the quantity being estimated. Even at a common confidence level, an empirical estimate and a parametric estimate may differ because the fitted distribution smooths or extends beyond the observed losses.
The following comparison summarizes what each method uses and what needs attention:
| Approach | Information used | Important assumption or check |
|---|---|---|
| Observed-loss data | Recorded values and their empirical probabilities | Use a stated boundary convention when the tail cuts through a tied value or observation. |
| Fitted continuous model | A distribution and estimated parameters | Check whether the chosen distribution and fitted tail are plausible for the intended use. |
| Simulated losses | Model-generated outcomes | Check both the generating model and whether the simulation contains enough tail observations. |
Worked Example: Comparing Parametric and Simulation Estimates
At the same 90% confidence level, a hypothetical exponential fit gives estimated TVaR of $264,206.81, while a simulation-based estimate gives $186,666.67. Both estimates are for annual losses in dollars. Calculate the dollar difference and the percentage by which the simulation estimate is below the parametric estimate. Then state one reason the estimates could differ.
Method. Subtract the smaller estimate from the larger:
Use the parametric estimate as the denominator because the question asks how far below that estimate the simulation result is:
The simulation estimate is $77,540.14, or 29.35%, below the parametric estimate. The estimates could differ because the exponential fit imposes an exponential tail, whereas a finite simulation has sampling variability and may produce few very large losses. The comparison does not by itself establish which estimate is more appropriate; that requires assessing the model and the simulation sample for the intended application.
Common Estimation Traps
- Including the whole boundary atom. If only part of the probability mass at VaR is needed to make a tail of size \(1-\alpha\), include only that part. Averaging every observation equal to or above VaR can change the tail probability.
- Changing the confidence level in a comparison. A difference between estimates at different confidence levels is not a like-for-like method comparison.
- Treating fitted-model output as observed fact. A fitted TVaR is conditional on the distributional form and parameter estimates. It is not a direct measurement of losses beyond the data.
- Assuming simulation eliminates uncertainty. Simulated outcomes vary from run to run, and their tail behavior also depends on the generating model. A small number of simulated tail outcomes can make the estimate unstable.
- Rounding too early. Keep full precision through the quantile and tail calculations, then round the final reported amount. Early rounding can alter a fractional boundary contribution.
Check Your Understanding
For each problem, apply the stated empirical or model convention and keep units consistent.
- A hypothetical loss distribution has probability 0.60 at 0, 0.25 at 5, and 0.15 at 25, with amounts in thousands of dollars. Find VaR and TVaR at \(\alpha=0.80\), using the quantile-integral definition. (Answer: VaR = 5 thousand dollars; TVaR = 20 thousand dollars.)
- Assume a fitted exponential severity model has scale \(\theta=\$50{,}000\). Calculate VaR and TVaR at \(\alpha=0.80\), rounding to the nearest cent. (Answer: VaR = $80,471.90; TVaR = $130,471.90.)
- Eight sorted simulated losses, in thousands of dollars, are 10, 20, 30, 40, 50, 60, 90, and 150. Using the empirical quantile-integral convention, find TVaR at \(\alpha=0.875\). (Answer: 150 thousand dollars.)
- Ten sorted observed losses, in thousands of dollars, are 5, 8, 12, 15, 20, 25, 30, 40, 60, and 100. Find empirical VaR and TVaR at \(\alpha=0.85\), using fractional boundary weighting. (Answer: VaR = 60 thousand dollars; TVaR = 86.667 thousand dollars.)
- At the same confidence level, two hypothetical methods estimate TVaR as $240,000 and $210,000. By how much, in dollars and as a percentage of the first estimate, is the second estimate lower? (Answer: $30,000, or 12.5% lower.)