Choosing VaR or TVaR for Capital

Why the Choice of Capital Measure Matters

A capital measure should reflect what the organization needs capital to protect against. A threshold-based measure answers how large losses can become before a specified proportion of outcomes is exceeded. A tail-average measure also considers how severe losses are after that threshold is crossed. Those questions are related, but they do not give the same capital amount.

This tutorial compares Value at Risk (VaR) and Tail Value at Risk (TVaR) for the same loss distribution and confidence level. The calculations use unexpected-loss capital: the selected risk measure less expected loss. They assume the losses are measured on the same basis and over the same period. When a normal distribution is used, losses are allowed to be signed; that model is an approximation and is not constrained to produce nonnegative losses.

Definition: For a loss $X$ and confidence level $\alpha$, where $0<\alpha<1$, $\operatorname{VaR}_{\alpha}(X)$ is the $\alpha$-quantile of the loss distribution. Using the generalized-inverse convention, $\operatorname{VaR}_{\alpha}(X)=\inf\{x:F_X(x)\geq\alpha\}$. For an integrable loss, $\operatorname{TVaR}_{\alpha}(X)$ is the average of the loss quantiles from $\alpha$ to 1. For a continuous distribution, it is the expected loss conditional on exceeding the VaR threshold. When there is probability mass at that threshold, the quantile-average definition includes only the fraction of that mass needed to make up the upper $1-\alpha$ tail.

The confidence level determines how much of the distribution is treated as the tail. At a 95% confidence level, VaR is the 95th percentile: about 5% of outcomes lie above it, subject to the convention used when losses have atoms. TVaR averages the worst 5% of outcomes. In a discrete distribution, “the worst 5%” may require taking only part of the probability at the VaR value.

For a loss model with finite expected loss, the corresponding unexpected-loss capital amounts are

K_{\operatorname{VaR}}=\operatorname{VaR}_{\alpha}(X)-E[X], \qquad K_{\operatorname{TVaR}}=\operatorname{TVaR}_{\alpha}(X)-E[X].

These amounts measure the selected tail measure above the mean. They are not interchangeable with the full risk measures: if the question asks for VaR or TVaR, report that measure; if it asks for unexpected-loss capital, subtract expected loss using the same loss basis.

How to Interpret and Calculate the Tail Average

For a continuous loss distribution with quantile function $F_X^{-1}$, TVaR is the mean of the quantiles in the upper tail:

\operatorname{TVaR}_{\alpha}(X) = \frac{1}{1-\alpha}\int_{\alpha}^{1}F_X^{-1}(u)\,du.

This formula explains why TVaR measures severity beyond a threshold rather than merely locating the threshold. As $u$ ranges from $\alpha$ to 1, the quantile function traces the losses from the VaR level through increasingly extreme outcomes. Dividing by $1-\alpha$ converts the total quantile area into the average loss in that upper tail.

For a continuous distribution, the probability of a loss exactly equal to the VaR threshold is zero. The upper $1-\alpha$ fraction is therefore the event $X>\operatorname{VaR}_{\alpha}(X)$, and TVaR can be calculated as the conditional mean on that event. For a discrete distribution, the event $X>\operatorname{VaR}_{\alpha}(X)$ may have probability less than $1-\alpha$. In that case, simply conditioning on a strict exceedance excludes part of the required tail and generally does not give TVaR.

Find the VaR threshold.
Use the cumulative distribution and the stated generalized-inverse convention to find the smallest loss whose cumulative probability is at least $\alpha$.

Identify the upper-tail probability.
The tail must contain probability $1-\alpha$. If losses are discrete, determine whether the probability strictly above VaR is less than that amount.

Average the required tail losses.
Include all probability above the threshold and, when necessary, just enough probability at the threshold to reach $1-\alpha$. Divide the tail-weighted loss by $1-\alpha$.

Convert to unexpected-loss capital if requested.
Subtract $E[X]$ from the chosen risk measure. Do not subtract expected loss when the question asks only for VaR or TVaR.

Worked Comparisons

Worked Example: A Discrete Loss With Probability at the Threshold

A hypothetical portfolio has a one-period loss $X$ with the following distribution, in dollars: loss 0 with probability 0.92, loss 50 with probability 0.04, loss 300 with probability 0.03, and loss 900 with probability 0.01. At the 95% confidence level, calculate VaR, TVaR, expected loss, and both unexpected-loss capital amounts.

Method. Find the 95th percentile, then average exactly the upper 5% of probability. This method avoids treating all losses strictly greater than VaR as the entire TVaR tail.

Step 1: Find VaR. The cumulative probability is 0.92 at loss 0 and 0.96 at loss 50. Since 0.92<0.95 and 0.96≥0.95, $\operatorname{VaR}_{0.95}(X)=\$50$.

Step 2: Form the upper 5%. Losses strictly above 50 have probability $0.03+0.01=0.04$, or 4%. The upper tail needs 5%, so it includes an additional 1% probability at loss 50. Its probability-weighted loss is

0.01(\$50)+0.03(\$300)+0.01(\$900) =\$0.50+\$9.00+\$9.00 =\$18.50.

Therefore, $\operatorname{TVaR}_{0.95}(X)=\$18.50/0.05=\$370.00$. Check: the included probabilities sum to $0.01+0.03+0.01=0.05$, as required.

Step 3: Find expected loss.

E[X]=0.92(\$0)+0.04(\$50)+0.03(\$300)+0.01(\$900) =\$2+\$9+\$9 =\$20.

Step 4: Calculate capital. VaR-based unexpected-loss capital is $\$50-\$20=\$30$. TVaR-based unexpected-loss capital is $\$370-\$20=\$350$. All amounts are in dollars.

Common mistake and correction. The strict exceedance conditional mean is $(0.03(\$300)+0.01(\$900))/0.04=\$450$. That is not TVaR here: it averages only the 4% strictly above \$50, while TVaR averages the upper 5% and includes 1% at \$50. The correct TVaR is \$370.00.

Worked Example: Comparing Measures Under a Normal Loss Model

For a hypothetical one-period portfolio, suppose the signed loss approximation is $X\sim N(200,30^2)$, with amounts in thousands of dollars. The mean is 200 and the standard deviation is 30. At 95%, calculate VaR, TVaR, and each unexpected-loss capital amount. Use the unrounded standard-normal values $z_{0.95}=1.6448536$ and $\phi(z_{0.95})=0.103135$, then round final amounts to two decimals in thousands.

Model and method. A normal distribution has support over all real numbers, so this approximation allows negative losses as well as positive ones. For $X\sim N(\mu,\sigma^2)$, the quantile is $\mu+\sigma z_\alpha$. For a continuous normal loss, the upper-tail mean is $\mu+\sigma\phi(z_\alpha)/(1-\alpha)$.

Step 1: Calculate VaR.

\operatorname{VaR}_{0.95}(X) =200+30(1.6448536) =200+49.345608 =249.345608.

Thus VaR is 249.35 thousand dollars.

Step 2: Calculate TVaR.

\operatorname{TVaR}_{0.95}(X) =200+30\left(\frac{0.103135}{0.05}\right) =200+30(2.0627) =200+61.881 =261.881.

Thus TVaR is 261.88 thousand dollars. The calculation uses the density value specified in the problem, not a different rounded value.

Step 3: Calculate unexpected-loss capital. Expected loss is $E[X]=200$ thousand dollars. VaR-based capital is $249.345608-200=49.345608$, or 49.35 thousand dollars. TVaR-based capital is $261.881-200=61.881$, or 61.88 thousand dollars.

Check. TVaR exceeds VaR by $261.881-249.345608=12.535392$ thousand dollars, reflecting the average severity of losses above the 95th-percentile threshold. The capital difference is the same because the same expected loss is subtracted from each measure.

Worked Example: An Exponential Loss and Its Tail Mean

A hypothetical product has a nonnegative loss $X$ with an exponential distribution and mean $\theta=\$1{,}000$. At a 99% confidence level, find VaR, TVaR, expected loss, and both unexpected-loss capital amounts. Round final dollar amounts to the nearest cent.

Model and method. For an exponential loss with mean $\theta$, the survival function is $S_X(x)=e^{-x/\theta}$ for $x\geq0$. Solving $S_X(x)=1-\alpha$ gives the VaR. The exponential distribution is memoryless: conditional on exceeding a threshold, the expected additional loss beyond it is $\theta$. Consequently, its upper-tail mean is VaR plus $\theta$.

Step 1: Find VaR.

e^{-x/\theta}=1-\alpha \quad\Longrightarrow\quad \operatorname{VaR}_{0.99}(X) =-\theta\ln(0.01) =-\$1{,}000\ln(0.01) =\$4{,}605.170186\ldots

Step 2: Find TVaR. By the memoryless property, the expected loss conditional on exceeding this continuous threshold is the threshold plus the mean excess:

\operatorname{TVaR}_{0.99}(X) =\operatorname{VaR}_{0.99}(X)+\theta =\$4{,}605.170186\ldots+\$1{,}000 =\$5{,}605.170186\ldots

Step 3: Calculate expected loss and capital. The exponential mean gives $E[X]=\$1{,}000$. VaR-based capital is $\$4{,}605.170186\ldots-\$1{,}000=\$3{,}605.170186\ldots$. TVaR-based capital is $\$5{,}605.170186\ldots-\$1{,}000=\$4{,}605.170186\ldots$.

Rounded to the nearest cent, VaR is \$4,605.17, TVaR is \$5,605.17, VaR-based capital is \$3,605.17, and TVaR-based capital is \$4,605.17. Check: TVaR exceeds VaR by exactly the exponential mean, \$1,000, before rounding.

Choosing Between VaR and TVaR

VaR is useful when the intended question is explicitly about a loss threshold at a stated confidence level. It communicates a percentile: under the modeled distribution, losses exceed that amount with probability approximately $1-\alpha$, with details depending on whether the distribution is continuous or discrete. VaR does not, by itself, indicate how large those exceedance losses may be.

TVaR answers a different question by averaging the upper tail. It distinguishes two distributions that can have the same VaR but substantially different loss severity beyond that point. This makes TVaR informative when the purpose of the capital measure is to represent the scale of severe outcomes, not just the boundary separating most outcomes from the tail.

The choice is not simply a matter of selecting the larger number. TVaR can be more sensitive to assumptions about extreme losses and to limited data in that part of the distribution. VaR can be less informative about the magnitude of losses after its threshold is crossed. A sound comparison therefore identifies the confidence level, loss basis, model assumptions, and intended interpretation. The formulas also require a finite expected loss if unexpected-loss capital is defined by subtracting $E[X]$.

Key takeaway: VaR locates a loss quantile; TVaR averages the upper tail. At a common confidence level and on a common loss basis, TVaR is generally at least as large as VaR. For discrete losses, construct the required tail probability carefully: the tail average may include only part of the probability mass at the VaR threshold.

Check Your Understanding

For each problem, use the stated confidence level and distinguish the risk measure from unexpected-loss capital.

A hypothetical loss is 0 with probability 0.90, \$100 with probability 0.06, and \$500 with probability 0.04. At 95%, find VaR, TVaR, expected loss, and both unexpected-loss capital amounts. (Answer: VaR = \$100; TVaR = \$420; expected loss = \$26; VaR capital = \$74; TVaR capital = \$394.)
A hypothetical exponential loss has mean \$200. At 90%, find VaR and TVaR, then calculate each unexpected-loss capital amount. Round to cents. (Answer: VaR = \$460.52; TVaR = \$660.52; VaR capital = \$260.52; TVaR capital = \$460.52.)
A hypothetical loss is 0 with probability 0.94, \$200 with probability 0.04, and \$1,000 with probability 0.02. At 95%, calculate VaR, TVaR, expected loss, and both capital amounts. (Answer: VaR = \$200; TVaR = \$520; expected loss = \$28; VaR capital = \$172; TVaR capital = \$492.)
For a hypothetical signed loss approximation $X\sim N(80,10^2)$, use $\alpha=0.90$, $z_{0.90}=1.2816$, and $\phi(z_{0.90})=0.1755$. Calculate VaR, TVaR, and both unexpected-loss capital amounts, in the same units as the distribution. Round to two decimals. (Answer: VaR = 92.82; TVaR = 97.55; VaR capital = 12.82; TVaR capital = 17.55.)
Two continuous loss models have the same 99% VaR of \$2 million, but their 99% TVaRs are \$2.4 million and \$3.1 million. Which model has the larger TVaR-based unexpected-loss capital if both have expected loss of \$0.5 million, and by how much? (Answer: The model with TVaR \$3.1 million; its capital is \$2.6 million versus \$1.9 million, a difference of \$0.7 million.)

What You'll Learn