What a TVaR Number Describes
A reported tail value at risk (TVaR) combines a confidence level with a loss amount. The confidence level specifies how far into the loss distribution the tail begins; TVaR then describes the average loss across that upper tail. It is therefore a monetary measure of tail severity, not a probability and not the loss threshold itself.
For example, a hypothetical annual portfolio loss with a 99% TVaR of $13 million has an average loss of $13 million in the worst 1% of outcomes, under the applicable TVaR convention. It does not mean that losses exceed $13 million with probability 1%, nor does it mean that $13 million is the 99th-percentile cutoff. Those statements concern different quantities.
The quantile-integral definition is useful even when the loss distribution has discrete outcomes or a probability mass at the VaR threshold. When the distribution is continuous at the threshold and has no probability mass there, TVaR also equals the conditional mean \(E[X\mid X>\operatorname{VaR}_{\alpha}(X)]\). With an atom at the threshold, that conditional mean can differ from TVaR: the upper tail may include only part of the probability mass at the threshold.
VaR, TVaR, Exceedance Probability, and Expected Loss
These four quantities answer different questions. Keeping their roles separate is essential when communicating a reported risk measure.
| Quantity | What it describes | Units |
|---|---|---|
| \(\operatorname{VaR}_{\alpha}(X)\) | The loss threshold at the \(\alpha\)-quantile | Loss units |
| \(\operatorname{TVaR}_{\alpha}(X)\) | The average loss across the upper \(1-\alpha\) fraction of outcomes | Loss units |
| \(P(X>\operatorname{VaR}_{\alpha}(X))\) | The probability of strictly exceeding the VaR threshold | Probability |
| \(E[X]\) | The unconditional average loss across all outcomes | Loss units |
If there is no atom at the VaR threshold, the probability of exceeding it is \(1-\alpha\). If there is an atom, the probability of strictly exceeding the threshold can be less than \(1-\alpha\). In that case, some of the upper-tail probability is supplied by outcomes equal to the threshold. TVaR still averages the upper \(1-\alpha\) fraction, including the necessary portion of that threshold mass.
The unconditional expected loss averages over the entire distribution, not just the upper tail. It can be below TVaR, but the size of the difference depends on the distribution and the selected confidence level. Neither the confidence level nor the TVaR alone specifies the expected loss.
Why the VaR-and-Excess Formula Works
Let \(q=\operatorname{VaR}_{\alpha}(X)\). Under the quantile-integral convention, the TVaR can also be written as
This form explains the relationship between the tail average and the threshold. It says to start at the VaR cutoff and add the average excess above that cutoff, allocated across a tail of probability \(1-\alpha\). In particular, a reported TVaR and VaR determine the unconditional expected excess above the VaR threshold:
Let \(q\) be the \(\alpha\)-quantile. Losses above \(q\) contribute their excess \(X-q\); losses at or below \(q\) contribute no positive excess.
The upper \(1-\alpha\) fraction contains all outcomes above \(q\), together with enough probability mass at \(q\), if needed, to make the tail fraction exactly \(1-\alpha\).
Every tail outcome starts with the baseline \(q\). The total amount above that baseline is \(E[(X-q)^+]\). Dividing this excess by the tail probability \(1-\alpha\), then adding \(q\), gives the tail average.
The formula does not say that the TVaR is the average loss conditional on strict exceedance in every distribution. That conditional average excludes outcomes equal to \(q\), whereas the quantile-integral definition can include part of the probability mass at \(q\). When there is no mass at \(q\), the upper-tail probability is \(1-\alpha\), and the two interpretations agree.
Worked Interpretations
Worked Example: Reading TVaR for a Continuous Loss
A hypothetical annual loss \(X\), measured in thousands of dollars, is uniformly distributed from 0 to 100. Find its 90% VaR and TVaR, the probability of exceeding that VaR, and the unconditional expected loss. Interpret the reported TVaR.
Method. Use the uniform distribution’s quantile function \( \operatorname{VaR}_{u}(X)=100u \), then average its quantiles over the upper 10% of the distribution.
VaR threshold. The 90% quantile is
Thus the 90% VaR is $90,000. For a continuous uniform distribution, the probability of a loss above 90 is \(1-0.90=0.10\), or 10%.
TVaR. Apply the quantile integral, keeping the loss in thousands of dollars:
The 90% TVaR is $95,000. It is the average loss in the worst 10% of outcomes, not the threshold of $90,000. The unconditional expected loss is \((0+100)/2=50\) thousand dollars, or $50,000.
Check. The losses in the upper 10% range uniformly from 90 to 100, whose average is \((90+100)/2=95\). This confirms the quantile-integral calculation. A common mistake would be to report $90,000 as the TVaR: that is the VaR cutoff, not the tail average.
Worked Example: Interpreting TVaR When the VaR Is an Atom
A hypothetical annual loss, in thousands of dollars, is 0 with probability 0.80, 100 with probability 0.15, and 500 with probability 0.05. Find the 90% VaR, 90% TVaR, strict exceedance probability at the VaR threshold, and expected loss.
Method. First identify the quantile cutoff from the cumulative probabilities. Then construct the worst 10% of outcomes, taking account of the probability mass at the cutoff.
VaR threshold. The cumulative probability through loss 0 is 0.80, and through loss 100 it is \(0.80+0.15=0.95\). Therefore the 90th percentile is 100, so the 90% VaR is $100,000.
Tail average. The worst 10% consists of all 5% of outcomes at 500 and 5% of the outcomes at 100. The latter amount is needed because the outcomes strictly above 100 account for only 5% of probability. Hence
The 90% TVaR is $300,000. The strict probability of exceeding the $100,000 VaR threshold is only 0.05, or 5%, not 10%.
Expected loss.
The unconditional expected loss is $40,000.
Common mistake and correction. A tempting calculation is \(E[X\mid X>100]=500\), since every loss strictly above 100 is 500. That is not the 90% TVaR: it drops the 5% of probability at the VaR threshold that is needed to complete the worst 10%. Including that portion gives 300, as calculated above.
Check using the excess formula. The expected excess over 100 is \(0.05(500-100)=20\) thousand dollars. Therefore \(100+20/0.10=300\) thousand dollars, agreeing with the direct tail average.
Worked Example: Separating a Reported TVaR from Expected Loss
A hypothetical insurer reports a 99% VaR of $8 million and a 99% TVaR of $13 million for annual aggregate losses. Its separately calculated expected annual loss is $2.4 million. Interpret the figures and calculate the expected excess above the VaR threshold.
Method. Read the 99% TVaR as the average in the worst \(1-0.99=0.01\), or 1%, of outcomes. Use the VaR-and-excess formula to calculate the unconditional expected excess above $8 million.
The $8 million figure is the 99th-percentile threshold. The $13 million TVaR is the average loss in the upper 1% of outcomes under the quantile-integral convention. It is not a statement that losses exceed $13 million with probability 1%.
The excess of the TVaR over the VaR is
Consequently, the unconditional expected excess above the VaR threshold is
The $2.4 million expected annual loss is an average over all outcomes; it is not the upper-tail average. These reported quantities describe different features of the loss distribution and are not interchangeable. Without a continuity assumption at the $8 million threshold, the figures also do not establish that the strict exceedance probability is exactly 1%.
Check. The expected excess calculation is $50,000, while the excess of the tail average over the threshold is $5 million. The former is an unconditional expected amount; the latter is a difference between two tail loss amounts. Confusing them would omit the factor \(1-\alpha=0.01\).
Common Interpretation Traps
- Calling TVaR a cutoff. The cutoff is VaR. TVaR is the average over the specified upper tail and is generally different from the cutoff.
- Reading the confidence level as an exceedance probability for TVaR. A 99% TVaR uses the upper 1% of the loss distribution; it does not assert that losses exceed the TVaR amount 1% of the time.
- Assuming strict exceedance probability always equals the tail probability. With probability mass at the VaR threshold, some of the selected tail can consist of outcomes equal to the threshold.
- Substituting expected loss for TVaR. Expected loss averages all outcomes, while TVaR averages only the specified upper tail.
- Dropping the loss units. If losses are reported in thousands of dollars, the TVaR is also in thousands of dollars; converting to dollars requires multiplying by 1,000.
Check Your Understanding
For each problem, distinguish the tail average from the cutoff and from the unconditional mean. Loss amounts are stated in the units given.
- A hypothetical loss is uniform from 0 to 200 thousand dollars. Find the 95% VaR, 95% TVaR, probability of exceeding VaR, and expected loss. (Answer: VaR = $190,000; TVaR = $195,000; exceedance probability = 0.0500; expected loss = $100,000.)
- A loss in thousands of dollars is 0 with probability 0.70, 40 with probability 0.20, and 240 with probability 0.10. Find the 85% VaR, 85% TVaR, strict exceedance probability, and expected loss. (Answer: VaR = $40,000; TVaR = $173,333.33; exceedance probability = 0.1000; expected loss = $32,000.)
- A report gives a 97.5% VaR of $2.0 million and a 97.5% TVaR of $2.8 million. Calculate the unconditional expected excess above the VaR threshold. (Answer: $20,000.)
- A loss is $1 million with probability 0.90 and $9 million with probability 0.10. Find its 95% VaR, 95% TVaR, strict exceedance probability at VaR, and expected loss. (Answer: VaR = $9 million; TVaR = $9 million; exceedance probability = 0.0000; expected loss = $1.8 million.)
- A hypothetical loss has a 99% VaR of $5 million and a 99% TVaR of $9 million. What is the average upper-tail loss, and what unconditional expected excess above $5 million follows from these figures? Can the strict probability of exceeding $5 million be inferred without an assumption about the distribution at the threshold? (Answer: The average loss in the worst 1% is $9 million; the unconditional expected excess is $40,000; no, the strict exceedance probability cannot be inferred without additional information about probability mass at the threshold.)