TVaR-Based Economic Capital Calculations

Why TVaR Needs Care for Discrete Losses

A quantile-based capital calculation identifies a loss threshold at a chosen confidence level. A tail-value-at-risk calculation instead averages losses in the upper tail beyond that confidence level. For a discrete loss distribution, the threshold itself may have positive probability. The tail can therefore include only part of the probability assigned to that threshold.

That distinction matters when translating a loss distribution into capital. The quantile is not generally the average of tail losses, and the conditional average given that loss is at least the quantile is not generally the standard discrete TVaR either. The calculation must include exactly the fraction of the quantile atom needed to make the tail probability equal to one minus the confidence level.

Definition: Let $X$ be a loss over a specified measurement horizon, with cumulative distribution function $F_X(x)=P(X\leq x)$. For confidence level $\alpha$, where $0<\alpha<1$, define $$ \operatorname{VaR}_{\alpha}(X)=\inf\{x:F_X(x)\geq\alpha\}. $$ Define TVaR by the upper-quantile average $$ \operatorname{TVaR}_{\alpha}(X)=\frac{1}{1-\alpha}\int_{\alpha}^{1}\operatorname{VaR}_{p}(X)\,dp. $$ TVaR is an average loss over an upper tail of probability $1-\alpha$, with a fractional contribution from the VaR loss when its probability mass crosses the confidence-level boundary.

For a discrete loss distribution, the VaR is the smallest loss value whose cumulative probability is at least $\alpha$. If $q=\operatorname{VaR}_{\alpha}(X)$, then $F_X(q-)<\alpha\leq F_X(q)$, where $F_X(q-)=P(X<q)$. The difference between $\alpha$ and $F_X(q-)$ determines how much of the probability at $q$ lies below the upper tail.

In economic-capital applications, be explicit about the capital basis. The tail loss measure is $\operatorname{TVaR}_{\alpha}(X)$. If economic capital is defined as unexpected loss above the expected loss, then the TVaR-based amount is $\operatorname{TVaR}_{\alpha}(X)-E[X]$. A quantile-based unexpected-loss amount is instead $\operatorname{VaR}_{\alpha}(X)-E[X]$. These quantities answer different questions and should not be interchanged.

Calculating TVaR from a Discrete Distribution

Suppose the possible losses are $x_1,\ldots,x_n$, with probabilities $p_1,\ldots,p_n$, where the probabilities are nonnegative and sum to 1. Order the losses from smallest to largest and find the first loss $q$ for which the cumulative probability reaches or exceeds $\alpha$. Then calculate the probability-weighted loss strictly above $q$, and add the needed fraction of the probability at $q$.

\operatorname{TVaR}_{\alpha}(X) = \frac{E[X\,\mathbf{1}_{\{X>q\}}]+q\bigl(F_X(q)-\alpha\bigr)}{1-\alpha}, \qquad q=\operatorname{VaR}_{\alpha}(X).

Here $\mathbf{1}_{\{X>q\}}$ is 1 when $X>q$ and 0 otherwise. Thus $E[X\,\mathbf{1}_{\{X>q\}}]$ is the sum of each loss strictly greater than $q$ multiplied by its probability. The quantity $F_X(q)-\alpha$ is the probability at the quantile that must be included in the upper tail. If $\alpha=F_X(q)$, none of the mass at $q$ is needed; if $\alpha$ falls inside the probability mass at $q$, only the portion above $\alpha$ is included.

Model condition: The formula applies to a specified loss distribution and confidence level using the quantile definition above. The discrete scenarios must represent mutually exclusive and exhaustive outcomes with probabilities summing to 1. It calculates the upper-quantile average, not an average over the whole distribution and not, in general, $E[X\mid X\geq q]$.

Why the Fractional-Atom Formula Holds

The integral definition of TVaR averages the quantile function over percentile levels from $\alpha$ to 1. In a discrete distribution, the quantile function is constant at each loss value across a percentile interval whose length is the probability of that loss. The upper-tail integral therefore collects full contributions from losses above $q$, but only the part of the interval at $q$ that lies above $\alpha$.

Locate the quantile interval.
Because $F_X(q-)<\alpha\leq F_X(q)$, the quantile is $q$ over the percentile interval from $F_X(q-)$ to $F_X(q)$. Only the interval from $\alpha$ to $F_X(q)$ contributes at $q$, with length $F_X(q)-\alpha$.

Add the higher losses.
For every loss value $x>q$, the quantile function contributes $x$ across a percentile interval of length $P(X=x)$. The total contribution from those intervals is $E[X\,\mathbf{1}_{\{X>q\}}]$.

Normalize by the tail probability.
The percentile interval from $\alpha$ to 1 has length $1-\alpha$. Dividing the total integral by that length gives the stated TVaR formula.

\int_{\alpha}^{1}\operatorname{VaR}_{p}(X)\,dp = q\bigl(F_X(q)-\alpha\bigr)+E[X\,\mathbf{1}_{\{X>q\}}], \qquad \operatorname{TVaR}_{\alpha}(X) = \frac{\int_{\alpha}^{1}\operatorname{VaR}_{p}(X)\,dp}{1-\alpha}.

Worked Calculations

Worked Example: A Confidence Level Inside a Quantile Atom

A hypothetical insurer models its one-year loss $X$, in thousands of dollars, as 0 with probability 0.70, 100 with probability 0.20, and 500 with probability 0.10. These mutually exclusive scenarios are exhaustive, the probabilities sum to 1, and losses are measured at year-end without discounting. At $\alpha=0.80$, calculate VaR, TVaR, TVaR-based unexpected-loss capital, and the corresponding VaR-based unexpected-loss amount.

Method. Order the losses and accumulate their probabilities: at 0, the cumulative probability is 0.70; at 100, it is 0.90; at 500, it is 1.00. The first cumulative probability at least 0.80 occurs at 100, so $\operatorname{VaR}_{0.80}(X)=100$.

The upper tail has probability $1-0.80=0.20$. The loss strictly above the quantile contributes $500(0.10)=50$. At the quantile, the probability to include is $F_X(100)-\alpha=0.90-0.80=0.10$, contributing $100(0.10)=10$. Therefore,

\operatorname{TVaR}_{0.80}(X)=\frac{50+10}{0.20}=300 \quad\text{thousand dollars}.

The expected loss is $E[X]=0(0.70)+100(0.20)+500(0.10)=0+20+50=70$ thousand dollars. TVaR-based unexpected-loss capital is $300-70=230$ thousand dollars. VaR-based unexpected-loss capital is $100-70=30$ thousand dollars. The check is that the TVaR tail contains probability $0.10+0.10=0.20$, as required, and its average loss is $(50+10)/0.20=300$.

Common mistake avoided: Using $E[X\mid X\geq100]$ includes all probability at 100, giving $(20+50)/0.30=233.33$ thousand dollars. That conditional average uses a tail probability of 0.30, not the required 0.20, so it is not $\operatorname{TVaR}_{0.80}(X)$.

Worked Example: A Higher Quantile with a Partial Atom

A hypothetical specialty insurer models a one-year loss in thousands of dollars as 0 with probability 0.80, 200 with probability 0.15, and 1,000 with probability 0.05. The outcomes are mutually exclusive and exhaustive, with no discounting. At $\alpha=0.90$, calculate the quantile, TVaR, and TVaR-based unexpected-loss capital.

The cumulative probabilities at the ordered losses are 0.80 at 0, 0.95 at 200, and 1.00 at 1,000. Thus $\operatorname{VaR}_{0.90}(X)=200$. The upper tail probability is 0.10. Losses strictly above 200 contribute $1{,}000(0.05)=50$. Since $F_X(200)-\alpha=0.95-0.90=0.05$, the quantile contributes $200(0.05)=10$.

\operatorname{TVaR}_{0.90}(X)=\frac{50+10}{0.10}=600 \quad\text{thousand dollars}.

Expected loss is $E[X]=0(0.80)+200(0.15)+1{,}000(0.05)=0+30+50=80$ thousand dollars. Therefore TVaR-based unexpected-loss capital is $600-80=520$ thousand dollars. VaR-based unexpected-loss capital would be $200-80=120$ thousand dollars. As a check, the tail includes 0.05 probability at 1,000 and 0.05 at 200, totaling 0.10; their probability-weighted loss is 60, so the tail average is 600.

Method and trap avoided: The calculation uses the upper-quantile integral, not the conditional average given $X\geq200$. Including all 0.15 probability at 200 would make the denominator $0.20$, which is the wrong tail probability.

Worked Example: When the VaR Atom Is Not Part of the Tail

A hypothetical portfolio has one-year losses, in thousands of dollars, of 10 with probability 0.40, 50 with probability 0.30, 150 with probability 0.20, and 400 with probability 0.10. The scenarios are exhaustive and mutually exclusive, and losses are measured without discounting. At $\alpha=0.75$, find VaR, TVaR, and unexpected-loss capital on both the VaR and TVaR bases.

The cumulative probabilities are 0.40 at 10, 0.70 at 50, 0.90 at 150, and 1.00 at 400. The first cumulative probability to reach 0.75 is 0.90, so $\operatorname{VaR}_{0.75}(X)=150$. The tail probability is 0.25. Losses strictly above 150 contribute $400(0.10)=40$. The quantile contribution uses $F_X(150)-\alpha=0.90-0.75=0.15$, or $150(0.15)=22.5$.

\operatorname{TVaR}_{0.75}(X)=\frac{40+22.5}{0.25}=250 \quad\text{thousand dollars}.

The expected loss is $E[X]=10(0.40)+50(0.30)+150(0.20)+400(0.10)=4+15+30+40=89$ thousand dollars. The TVaR-based unexpected-loss capital is $250-89=161$ thousand dollars; the VaR-based amount is $150-89=61$ thousand dollars. The tail probability check is $0.10+0.15=0.25$, and the tail loss total is $40+22.5=62.5$, yielding an average of 250.

Common mistake avoided: Averaging 150 and 400 with their full probabilities would include 0.30 probability, although the specified upper tail has probability 0.25. Here only 0.15 of the 0.20 probability at 150 belongs in the TVaR tail.

Interpretation and Exam Traps

TVaR and VaR are both evaluated at a stated confidence level, but their outputs have different meanings. VaR is a loss threshold. TVaR is an average over the upper tail of specified probability. When the distribution has no probability mass at the quantile, the tail boundary does not cut through an atom; with a discrete distribution, it often does. Applying the continuous-distribution shortcut $E[X\mid X\geq q]$ without checking the atom can therefore change the tail probability and the result.

A reliable calculation keeps three quantities separate:

VaR: the loss threshold at confidence level $\alpha$.
TVaR: the average loss over the upper tail of probability $1-\alpha$, including only the required part of the VaR atom.
Unexpected-loss capital: the chosen risk measure minus $E[X]$, when capital is defined on an unexpected-loss basis.

Before finalizing a discrete TVaR calculation, check that the probability included in the tail is exactly $1-\alpha$, that each included loss is multiplied by its included probability, and that the numerator is divided by $1-\alpha$. If the question requests a tail loss measure, report TVaR itself; subtract expected loss only when the requested capital basis is unexpected loss above the mean. This distinction is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Key takeaway: For discrete losses, TVaR is not generally the conditional mean of all losses at least as large as VaR. Find the VaR atom, include only enough of its probability to create an upper tail of size $1-\alpha$, then average that tail. Subtract expected loss only when the requested economic-capital measure is defined as unexpected loss.

Check Your Understanding

For each hypothetical one-year loss distribution, outcomes are mutually exclusive and exhaustive, losses are in thousands of dollars, and no discounting applies. Calculate as requested, checking the tail probability as well as the loss average.

$X$ is 0 with probability 0.60, 80 with probability 0.25, and 300 with probability 0.15. At $\alpha=0.85$, find VaR, TVaR, and TVaR-based unexpected-loss capital. (Answer: VaR $=80$; TVaR $=300$; expected loss $=65$; TVaR-based capital $=235$, all in thousands of dollars.)
$X$ is 20 with probability 0.50, 100 with probability 0.30, and 500 with probability 0.20. At $\alpha=0.80$, find VaR, TVaR, and TVaR-based unexpected-loss capital. (Answer: VaR $=100$; TVaR $=500$; expected loss $=140$; TVaR-based capital $=360$, all in thousands of dollars.)
$X$ is 0 with probability 0.70, 200 with probability 0.20, and 800 with probability 0.10. At $\alpha=0.75$, calculate TVaR and TVaR-based unexpected-loss capital. (Answer: VaR $=200$; TVaR $=440$; expected loss $=120$; TVaR-based capital $=320$, all in thousands of dollars.)
$X$ is 50 with probability 0.40, 150 with probability 0.35, and 600 with probability 0.25. At $\alpha=0.90$, calculate TVaR and TVaR-based unexpected-loss capital. (Answer: VaR $=600$; TVaR $=600$; expected loss $=222.5$; TVaR-based capital $=377.5$, all in thousands of dollars.)
$X$ is 0 with probability 0.82, 100 with probability 0.10, and 400 with probability 0.08. At $\alpha=0.85$, calculate TVaR, VaR-based unexpected-loss capital, and TVaR-based unexpected-loss capital. (Answer: VaR $=100$; TVaR $=260$; expected loss $=42$; VaR-based capital $=58$; TVaR-based capital $=218$, all in thousands of dollars.)

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