Why TVaR Requires Care at the Tail Boundary
An aggregate loss distribution may assign positive probability to a single loss amount. That is common when losses are discrete, scenario-based, or represented by a finite simulation sample. In such cases, the confidence level may cut through the probability at the VaR threshold. Calculating TVaR then requires using only the right fraction of that threshold probability—not including the entire mass automatically.
The method applies to the distribution of the aggregate loss being measured. If a portfolio has component losses, first define the aggregate loss for each outcome as the sum of the relevant component losses on a consistent basis. The calculations below take that aggregate distribution as given or show the component sums explicitly.
The quantile convention matters. With this definition, VaR is the smallest loss whose cumulative probability is at least \(\alpha\). For a discrete distribution, the probability strictly above VaR may be less than \(1-\alpha\); the shortfall is supplied by taking just enough mass at VaR to make the tail probability exactly \(1-\alpha\).
Discrete TVaR Formula and Its Justification
Let \(q=\operatorname{VaR}_{\alpha}(X)\). Since \(F_X(q)\geq\alpha\), the probability at or below \(q\) is at least \(\alpha\). The probability strictly above \(q\) is \(1-F_X(q)\). To form an upper tail with total probability \(1-\alpha\), include the excess-loss outcomes in full and then add probability \(F_X(q)-\alpha\) at \(q\).
Here \(\mathbf{1}_{\{X>q\}}\) is 1 when \(X>q\) and 0 otherwise. The probability used in the numerator is exactly \(P(X>q)+F_X(q)-\alpha=1-F_X(q)+F_X(q)-\alpha=1-\alpha\). Dividing the corresponding probability-weighted losses by that probability gives the average loss in the required tail. This also explains why the formula uses only a fraction of any atom at \(q\).
For a finite distribution with loss values \(x_j\) and probabilities \(p_j\), calculate \(q\), then evaluate \( \sum_{x_j>q}x_jp_j+q(F_X(q)-\alpha) \), and divide by \(1-\alpha\). For \(n\) equally weighted simulation outcomes, each observation has probability \(1/n\). If the desired tail contains a non-integer number \(n(1-\alpha)\) of observations, the boundary observation is included fractionally.
Use aggregate losses on a consistent basis, together with their probabilities or simulation weights.
Order losses and identify the smallest \(q\) for which cumulative probability reaches or exceeds \(\alpha\).
Include all probability strictly above \(q\), then add probability \(F_X(q)-\alpha\) at \(q\).
Divide the selected probability-weighted loss total by \(1-\alpha\).
Worked Example: A Discrete Aggregate Distribution
A hypothetical insurer models annual aggregate losses, in thousands of dollars, as follows: loss 0 with probability 0.50, loss 40 with probability 0.25, loss 100 with probability 0.15, and loss 200 with probability 0.10. Calculate VaR and TVaR at \(\alpha=0.80\). State the monetary results in thousands of dollars.
Method. Build the cumulative distribution, locate the 80th percentile, then use the fractional-threshold formula. This avoids treating all probability at the VaR amount as if it necessarily belonged in the tail.
The cumulative probabilities are \(F_X(0)=0.50\), \(F_X(40)=0.75\), \(F_X(100)=0.90\), and \(F_X(200)=1.00\). The first cumulative probability at least 0.80 occurs at 100, so \(q=\operatorname{VaR}_{0.80}(X)=100\).
The required tail probability is \(1-0.80=0.20\). Losses strictly above 100 contribute probability 0.10 and weighted loss \(200(0.10)=20\). The amount to include at the threshold is \(F_X(100)-0.80=0.90-0.80=0.10\), contributing weighted loss \(100(0.10)=10\).
Thus VaR is 100 thousand dollars and TVaR is 150 thousand dollars. The selected tail consists of all the 200-loss probability and 0.10 probability at loss 100; these probabilities sum to 0.20. The weighted loss total is 30, and \(30/0.20=150\), which checks the calculation.
Using Equally Weighted Simulation Results
A finite simulation sample can be treated as an empirical distribution: each simulated outcome receives probability \(1/n\), unless unequal weights are specified. Sort the aggregate losses from smallest to largest. The VaR quantile identifies the boundary, but the TVaR calculation may require a fractional number of observations.
For a sample of \(n\) equally weighted outcomes, a tail probability of \(1-\alpha\) corresponds to \(n(1-\alpha)\) observation-units. If this is not an integer, include the largest losses in full and the next loss only to the fraction needed. This is the empirical version of the discrete formula, not a different definition.
Worked Example: Fractional Observation at the VaR Threshold
A hypothetical portfolio has ten simulated aggregate losses, in thousands of dollars, sorted as \(0,10,20,30,40,50,60,100,100,160\). Calculate VaR and TVaR at \(\alpha=0.75\). Round the final monetary results to two decimal places.
Method. Give each result probability \(0.10\), construct the empirical cumulative distribution, and select a tail of \(1-0.75=0.25\), or \(10(0.25)=2.5\) observations.
The first seven observations have cumulative probability 0.70, while the first nine have cumulative probability 0.90. The smallest loss with cumulative probability at least 0.75 is 100, so \(q=100\) and \(F_X(q)=0.90\).
There is one observation strictly above 100, namely 160. Its probability-weighted loss is \(160(0.10)=16\). The probability to include at the threshold is \(0.90-0.75=0.15\), corresponding to 1.5 of the ten equally weighted observations. Its weighted loss is \(100(0.15)=15\).
The same result follows by counting observation-units: include the 160 loss, one full 100 loss, and half of the other 100 loss. Their weighted total in observation-units is \(160+100+0.5(100)=310\); dividing by 2.5 gives 124.00 thousand dollars.
Common mistake avoided. Averaging all three largest losses gives \((100+100+160)/3=120\) thousand dollars. That uses a tail of 3 out of 10 observations, or 30%, not the required 25%. The correct tail has 2.5 observation-units, so the boundary loss is only partly included and the answer is 124.00 thousand dollars. The arithmetic check is \(16+15=31\) in probability-weighted thousands, and \(31/0.25=124\).
Aggregating Components Before Measuring Tail Risk
TVaR must be calculated from the distribution of the quantity being measured. When each scenario specifies several component losses, first sum the components within each scenario. Then attach that scenario's probability to the resulting aggregate loss. Averaging component TVaRs or selecting different component tail scenarios does not, in general, calculate the TVaR of the aggregate.
Worked Example: Aggregate Scenario Losses With Threshold Mass
A hypothetical insurer considers four mutually exclusive annual scenarios for two lines of business. Scenario A has probability 0.50 and component losses 10 and 20 thousand dollars. Scenario B has probability 0.20 and losses 40 and 50. Scenario C has probability 0.20 and losses 70 and 80. Scenario D has probability 0.10 and losses 120 and 180. Calculate aggregate VaR and TVaR at \(\alpha=0.80\), in thousands of dollars.
Method. Sum the two component losses in each scenario, retain the scenario probabilities, and apply the quantile-tail calculation to that aggregate distribution.
| Scenario | Probability | Component loss 1 | Component loss 2 | Aggregate loss |
|---|---|---|---|---|
| A | 0.50 | 10 | 20 | 30 |
| B | 0.20 | 40 | 50 | 90 |
| C | 0.20 | 70 | 80 | 150 |
| D | 0.10 | 120 | 180 | 300 |
The aggregate losses are 30, 90, 150, and 300. Their cumulative probabilities are 0.50, 0.70, 0.90, and 1.00. Thus \(q=\operatorname{VaR}_{0.80}(X)=150\), and \(F_X(q)=0.90\). The required tail probability is 0.20. The loss strictly above 150 contributes \(300(0.10)=30\); the threshold probability to include is \(0.90-0.80=0.10\), contributing \(150(0.10)=15\).
The answer is aggregate VaR of 150 thousand dollars and aggregate TVaR of 225 thousand dollars. The selected tail comprises all of Scenario D's 0.10 probability and 0.10 probability from Scenario C, for a total of 0.20. The weighted loss total is 45, and \(45/0.20=225\), confirming the result.
Interpretation and Common Traps
TVaR is the average loss in the worst \(1-\alpha\) fraction of the distribution, with fractional probability allowed at the boundary. If the loss distribution is continuous at VaR, there is no positive probability mass at \(q\), and the formula reduces to the conditional mean of losses above \(q\), when that conditional mean exists. For a discrete distribution with positive mass at \(q\), simply computing \(E[X\mid X\geq q]\) usually includes too much threshold probability. It is not generally the same as TVaR.
The definition also clarifies why an empirical tail average is sensitive to how many observations are selected. If the requested tail contains 2.5 equally weighted observations, averaging the three largest observations changes the tail probability. The boundary observation must instead be assigned the fraction required to reach exactly 2.5.
- Using \(>\alpha\) instead of reaching \(\alpha\). VaR is the smallest value whose cumulative probability is at least \(\alpha\), not necessarily the first value where it exceeds \(\alpha\).
- Including all mass at the threshold. Include \(F_X(q)-\alpha\) probability at \(q\), not automatically all of \(P(X=q)\).
- Using the wrong tail size. Divide by \(1-\alpha\), or by \(n(1-\alpha)\) observation-units in an equally weighted sample—not by a rounded-up count of full observations.
- Aggregating after tail selection. Sum component losses within each outcome first, then find the aggregate distribution and its tail.
These calculations are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. In applied work, the chosen confidence level, distribution or simulation weights, loss basis, and treatment of ties should be stated so another analyst can reproduce the tail calculation.
Check Your Understanding
For each problem, use the smallest-loss quantile convention and show how much probability is taken at the VaR threshold.
- A loss is 0 with probability 0.60, 20 with probability 0.20, 60 with probability 0.15, and 140 with probability 0.05. Find VaR and TVaR at \(\alpha=0.80\). (Answer: VaR = 20; TVaR = 80.)
- Eight equally weighted simulated aggregate losses, already sorted, are 5, 10, 15, 25, 40, 40, 80, and 120. Find VaR and TVaR at \(\alpha=0.75\). (Answer: VaR = 40; TVaR = 100.)
- A loss distribution has values 10, 30, 70, and 200 with probabilities 0.50, 0.20, 0.20, and 0.10, respectively. Find VaR and TVaR at \(\alpha=0.85\). (Answer: VaR = 70; TVaR = 156.6667.)
- Twenty equally weighted simulation results are sorted. The 18th loss is 100, the 19th loss is 200, and the 20th loss is 300. Find TVaR at \(\alpha=0.925\). (Answer: 266.6667, using 1.5 observation-units in the tail.)
- A hypothetical aggregate loss is 0 with probability 0.40, 50 with probability 0.30, 100 with probability 0.20, and 400 with probability 0.10. Find VaR and TVaR at \(\alpha=0.80\). (Answer: VaR = 100; TVaR = 250.)