Selecting a Risk Measure for Capital Decisions

Start With the Capital Objective

A capital measure is useful only if it answers the decision being made. A portfolio manager concerned with ordinary fluctuations may want a measure of overall dispersion. A capital committee asking how much money is needed to cover a specified fraction of annual losses needs a tail measure. Those objectives are different, so the numerical measures should not be treated as interchangeable.

This tutorial compares variance, value-at-risk (VaR), and tail value-at-risk (TVaR) for loss $X$. It focuses on choosing a measure for a stated purpose, not on estimating a full aggregate loss distribution. The distinctions below are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: For confidence level $\alpha$, where $0<\alpha<1$, define $\operatorname{VaR}_{\alpha}(X)$ as the $\alpha$-quantile of the loss distribution: the smallest $x$ for which $F_X(x)\geq\alpha$. Define $\operatorname{TVaR}_{\alpha}(X)$ as the average of the loss quantiles above $\alpha$: $$ \operatorname{TVaR}_{\alpha}(X)=\frac{1}{1-\alpha}\int_{\alpha}^{1}\operatorname{VaR}_{u}(X)\,du. $$ For a continuous loss distribution, this is $E[X\mid X>\operatorname{VaR}_{\alpha}(X)]$. The quantile-integral definition also handles discrete distributions, where a simple conditional expectation above VaR can omit losses at the VaR boundary.

Variance, $\operatorname{Var}(X)$, measures squared deviations from the mean. It is not itself a dollar capital amount: its units are squared dollars. The standard deviation, $\sqrt{\operatorname{Var}(X)}$, is in dollars, but it still describes overall spread rather than a particular tail percentile. VaR is a loss threshold at a specified confidence level. TVaR describes the average of the worst $1-\alpha$ fraction of losses, accounting for the severity of that tail.

Match the Measure to the Question

Measure	What it describes	Useful when	Important limitation
Variance	Average squared distance from expected loss	Comparing dispersion or applying a variance-based pricing rule	Does not isolate the upper tail; its units are squared
VaR at level $\alpha$	A loss threshold exceeded with probability at most about $1-\alpha$	The decision specifies a percentile or threshold to cover	Does not describe how severe losses are beyond the threshold; may fail subadditivity
TVaR at level $\alpha$	Average loss in the worst $1-\alpha$ fraction of outcomes	The decision is sensitive to severity beyond a percentile	Tail estimates can be sensitive to model assumptions and limited data

A capital convention must also be stated. VaR and TVaR above are measures of total loss. If a decision instead defines capital as an amount above expected loss, the corresponding quantities are $\operatorname{VaR}_{\alpha}(X)-E[X]$ and $\operatorname{TVaR}_{\alpha}(X)-E[X]$. This subtraction changes the reported capital, not which tail losses the measures reflect. A variance-based loading likewise needs a specified rule, such as a multiple of standard deviation; variance alone cannot be added to expected loss in dollars.

Worked Example: When the Tail Average Changes the Decision

A hypothetical insurer models annual loss $X$ in dollars using this distribution: loss is $0$ with probability $0.90$, $100$ with probability $0.05$, and $400$ with probability $0.05$. At $\alpha=0.90$, calculate variance, VaR, and TVaR, then assess which measure answers a question about average loss in the worst 10% of outcomes.

Method. Compute the moments for variance, use the smallest loss whose cumulative probability reaches $0.90$ for VaR, and average the worst 10% of outcomes for TVaR. This example avoids the mistake of treating VaR as the average loss beyond the quantile.

Step 1: Mean and variance. The expected loss is $$ E[X]=(0)(0.90)+(100)(0.05)+(400)(0.05)=25. $$ The second moment is $$ E[X^2]=(0^2)(0.90)+(100^2)(0.05)+(400^2)(0.05)=500+8{,}000=8{,}500. $$ Therefore, $$ \operatorname{Var}(X)=E[X^2]-(E[X])^2=8{,}500-25^2=7{,}875. $$ Variance is $7{,}875$ dollars squared, not $7{,}875$ dollars of capital.

Step 2: VaR. The cumulative probability at loss $0$ is $0.90$. Hence $\operatorname{VaR}_{0.90}(X)=0$.

Step 3: TVaR. The worst 10% consists of the 5% of outcomes with loss $400$ and the 5% with loss $100$. Thus $$ \operatorname{TVaR}_{0.90}(X)=\frac{(0.05)(400)+(0.05)(100)}{0.10} =\frac{25}{0.10}=250. $$

Check and interpretation. VaR is $0$, while the average loss in the worst 10% is $250$. That is consistent with the specified distribution: the tail includes equal probability masses at $100$ and $400$. TVaR answers the stated tail-severity question; VaR does not. The variance summarizes dispersion across all outcomes and cannot replace that tail average.

Why Tail Sensitivity and Diversification Matter

Two portfolios can have the same VaR but very different losses beyond that threshold. A decision that only sets a percentile threshold may therefore be served by VaR, while one that needs to reflect the severity of bad outcomes may call for TVaR. Conversely, a tail measure can be difficult to estimate reliably when observations in the relevant tail are scarce. The measure should reflect both the objective and what the data and model can support.

Diversification is another important distinction. For losses $X$ and $Y$, variance satisfies $$ \operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y). $$ Thus, the variance of the portfolio depends explicitly on the covariance between its components. VaR does not always satisfy subadditivity: it is possible for portfolio VaR to exceed the sum of standalone VaRs. TVaR, defined using the quantile integral, is subadditive under the usual conditions for finite expected losses. In practical terms, adding positions can produce diversification benefits in TVaR, although the size of those benefits still depends on dependence between losses.

Justification: The variance identity follows by expanding around the means. Since $E[X+Y]=E[X]+E[Y]$, $$ \begin{aligned} \operatorname{Var}(X+Y) &=E\left[\big((X-E[X])+(Y-E[Y])\big)^2\right]\\ &=E[(X-E[X])^2]+E[(Y-E[Y])^2]\\ &\quad+2E[(X-E[X])(Y-E[Y])]\\ &=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y). \end{aligned} $$ The cross-product is the covariance term, so ignoring dependence gives an incorrect portfolio variance unless that term is zero.

Worked Example: A VaR Diversification Trap

Two hypothetical insurance covers each have a loss of $100$ dollars when a distinct event occurs, and otherwise have loss $0$. The events are mutually exclusive, each has probability $0.04$, and neither occurs with probability $0.92$. At $\alpha=0.95$, calculate each standalone VaR and TVaR, then calculate those measures for the combined loss. Also compare the variance of the combined loss with the sum of the standalone variances.

Method. Derive the marginal and combined distributions from the event probabilities, then apply the same quantile and tail-fraction definitions. This example avoids assuming that portfolio VaR is automatically no greater than the sum of standalone VaRs.

Step 1: Standalone VaR and TVaR. For either cover, the probability of loss $0$ is $0.96$. Since $0.96\geq0.95$, each standalone $\operatorname{VaR}_{0.95}$ is $0$. The worst 5% consists of the 4% with loss $100$ and 1% with loss $0$, so $$ \operatorname{TVaR}_{0.95}=\frac{(0.04)(100)+(0.01)(0)}{0.05}=80. $$

Step 2: Combined VaR and TVaR. Because the events are mutually exclusive, combined loss is $100$ with probability $0.04+0.04=0.08$, and $0$ with probability $0.92$. The cumulative probability at zero is below $0.95$, so the combined $\operatorname{VaR}_{0.95}$ is $100$. The worst 5% all have loss $100$, giving combined $\operatorname{TVaR}_{0.95}=100$.

Step 3: Variances. Each standalone loss has mean $(0.04)(100)=4$ and second moment $(0.04)(100^2)=400$, so its variance is $400-4^2=384$. The sum of the standalone variances is $768$. Combined loss has mean $(0.08)(100)=8$ and second moment $(0.08)(100^2)=800$, so its variance is $$ 800-8^2=736. $$ The combined variance is below the sum of standalone variances.

Check and interpretation. Standalone VaRs sum to $0$, but combined VaR is $100$, so VaR is not subadditive in this example. Standalone TVaRs sum to $160$, compared with combined TVaR $100$. The dependence is mutually exclusive: the covariance is $0-(4)(4)=-16$, and $384+384+2(-16)=736$, confirming the combined variance calculation.

Selection Requires Judgment About the Model

A stated objective is the starting point, not the only consideration. A high-confidence VaR may be easy to explain as a threshold, but it gives no direct information about losses above that threshold. TVaR uses the severity of those losses; its tail average can be more useful for an objective that explicitly concerns the cost of severe outcomes. Variance can be suitable for a rule based on overall volatility, but it weights squared deviations and may be strongly affected by observations far from the mean on either side.

All three measures depend on the loss distribution. Tail measures can be particularly uncertain if the data contain few observations in the relevant tail. Model choice, parameter uncertainty, changes in exposure, and dependence assumptions can materially affect a high-level VaR or TVaR. A precise numerical result is not proof that its inputs are reliable. For capital decisions, report the confidence level, whether the result is total loss or capital above expected loss, the distributional assumptions, and the material limitations of the data.

Worked Example: Choosing Between a Percentile and Tail Average

A hypothetical portfolio has annual net loss $X$ approximated by a normal distribution $N(100,20^2)$, with amounts in thousands of dollars. The normal approximation assigns a very small probability to negative loss, so it is being used as an approximation rather than a literal model of nonnegative claim amounts. At $\alpha=0.975$, compare VaR and TVaR with a capital threshold of $145$ thousand dollars. Which measure should govern if the objective is to cover the average loss in the worst 2.5% of outcomes?

Method. Use the normal quantile for VaR and the normal upper-tail conditional mean for TVaR. The calculation avoids the mistake of using the VaR threshold as though it were the tail average.

Step 1: VaR. For a standard normal variable, the 97.5th percentile is approximately $1.96$. With mean $100$ and standard deviation $20$, in thousands of dollars, $$ \operatorname{VaR}_{0.975}(X)=100+(1.96)(20)=139.20. $$

Step 2: TVaR. For a normal loss with mean $\mu$, standard deviation $\sigma$, and standardized quantile $z_{\alpha}$, $$ \operatorname{TVaR}_{\alpha}(X)=\mu+\sigma\frac{\phi(z_{\alpha})}{1-\alpha}, $$ where $\phi$ is the standard normal density. Here $\phi(1.96)\approx0.05844$, so $$ \operatorname{TVaR}_{0.975}(X) =100+20\left(\frac{0.05844}{0.025}\right) \approx100+46.75=146.75. $$

Step 3: Apply the objective. VaR is below the $145$-thousand-dollar threshold; TVaR is above it. Since the stated objective is the average loss in the worst 2.5% of outcomes, TVaR is the measure that matches the objective. A decision based only on the 97.5th percentile would answer a different question.

Check and interpretation. The TVaR exceeds VaR, as expected for this continuous upper-tail distribution. The conclusion depends on the normal approximation: the calculation is internally consistent, but the model and the tail data should be reviewed before treating $146.75$ thousand dollars as a reliable capital estimate.

A Practical Selection Sequence

State the decision.
Decide whether it concerns overall fluctuation, a loss threshold at a specified percentile, or average severity in the worst fraction of outcomes.

Match the measure.
Use variance for squared dispersion, VaR for a percentile threshold, or TVaR when severity beyond the percentile is part of the objective.

Check the portfolio effect.
For variance, include covariance. For VaR, do not assume subadditivity. Consider whether a subadditive tail measure better reflects the portfolio decision.

Check the evidence and convention.
State the confidence level, model assumptions, data limitations, and whether reported capital is total loss or an amount above expected loss.

Common Exam Traps

Comparing unlike units. Variance is in squared dollars; VaR and TVaR are in dollars. Do not rank them numerically as though they were the same kind of amount.
Calling VaR a tail average. VaR is a threshold. TVaR reflects the average of the worst fraction of losses.
Assuming diversification always lowers VaR. Portfolio VaR can exceed the sum of standalone VaRs, as the mutually exclusive-event example demonstrates.
Ignoring the definition at an atom. In a discrete distribution, conditioning only on losses strictly above VaR may not select exactly the worst $1-\alpha$ fraction. Use the quantile-integral definition or explicitly include the needed fraction at the boundary.
Reporting a precise tail number without discussing the model. A high-confidence estimate can depend heavily on distribution and dependence assumptions, especially when observations in the tail are limited.

Key takeaway: Select a measure by the objective: variance describes overall squared dispersion, VaR describes a percentile threshold, and TVaR describes average severity in the upper tail. Then check diversification behavior, units, model assumptions, and the strength of the available data.

Check Your Understanding

For each problem, identify the relevant calculation and check that the measure matches the stated objective.

A hypothetical loss is $0$ with probability $0.80$, $50$ with probability $0.15$, and $200$ with probability $0.05$. Calculate $\operatorname{VaR}_{0.90}$ and $\operatorname{TVaR}_{0.90}$. (Answer: VaR $=50$; TVaR $=125$.)
A hypothetical loss is approximated by $N(60,10^2)$, in thousands of dollars. Use $z_{0.95}=1.645$ and $\phi(1.645)=0.1030$ to calculate VaR and TVaR at 95%. (Answer: VaR $\approx76.45$ thousand dollars; TVaR $\approx80.60$ thousand dollars.)
Two covers each pay $100$ dollars on distinct mutually exclusive events, each with probability $0.03$, and neither event occurs with probability $0.94$. At the 95% level, find each standalone VaR and the combined VaR. (Answer: Each standalone VaR is $0$; combined VaR is $100$ dollars.)
Two losses have variances $25$ and $36$, and covariance $-8$, in squared-dollar units. Find the variance of their sum and compare it with the sum of standalone variances. (Answer: Portfolio variance $=25+36+2(-8)=45$, below the standalone sum of $61$.)
A capital decision is explicitly about the average loss in the worst 1% of outcomes. The available annual data contain few observations in that tail, and the fitted model has substantial tail uncertainty. Which measure matches the objective, and what limitation must be reported? (Answer: TVaR at the 99% level matches the objective; report that the tail estimate is uncertain because tail data are sparse and model-dependent.)

What You'll Learn