Stress Testing a Risk-Measure Choice

Why Stress-Test a Risk-Measure Choice?

A risk measure can support a decision only to the extent that its result is informative under plausible changes in assumptions. A ranking between two portfolios may reverse when the confidence level changes, when a rare loss becomes more severe, or when dependence assumptions change. A large change in the reported risk amount is not necessarily a decision problem; a ranking reversal often is.

Stress testing here means recalculating candidate risk measures after changing one specified input while holding the others fixed. This is a sensitivity analysis, not a claim that the stressed input is more likely. It helps separate a robust choice, which remains preferred across a reasonable set of stresses, from a choice that depends on one particular calibration.

Definition: For a loss $X$ and confidence level $\alpha$, where $0<\alpha<1$, define $\operatorname{VaR}_{\alpha}(X)=\inf\{x:F_X(x)\geq\alpha\}$. Define $\operatorname{TVaR}_{\alpha}(X)$ as the average of the quantiles above level $\alpha$: $\operatorname{TVaR}_{\alpha}(X)=\frac{1}{1-\alpha}\int_{\alpha}^{1}\operatorname{VaR}_{p}(X)\,dp$, for an integrable loss. This quantile-integral definition applies to continuous and discrete losses, including losses with atoms.

For a discrete loss, TVaR averages exactly the upper $1-\alpha$ fraction of probability, using only the required fraction of probability at the boundary quantile. Thus, it need not equal the conditional mean given $X\geq\operatorname{VaR}_{\alpha}(X)$: that conditional mean can include more than the upper $1-\alpha$ probability when there is an atom at the quantile. Unless stated otherwise, examples below are one-period losses in dollars, with no discounting.

Stress Confidence Levels Before Trusting a Ranking

The confidence level controls how far into the loss distribution a quantile measure looks. A small change can have a substantial effect when the loss distribution has a probability jump or a thin but severe tail. Comparing more than one level can show whether a decision reflects broad risk behavior or a particular cutoff.

Worked Example: A Ranking That Depends on the Confidence Level

A hypothetical insurer compares portfolio A, which has loss $0$ with probability $0.90$ and loss $\$100$ with probability $0.10$, with portfolio B, which always has loss $\$15$. Calculate VaR and TVaR for each portfolio at confidence levels $0.90$ and $0.95$, then assess the ranking.

Method. First locate each quantile using the cumulative distribution function. Then average the upper $1-\alpha$ probability to obtain TVaR.

For A, $F_A(0)=0.90$ and $F_A(100)=1$. Therefore, $\operatorname{VaR}_{0.90}(A)=0$ and $\operatorname{VaR}_{0.95}(A)=100$. At level $0.90$, the upper $10\%$ of losses are all $\$100$, so $\operatorname{TVaR}_{0.90}(A)=\$100$. At level $0.95$, the upper $5\%$ are also all $\$100$, so $\operatorname{TVaR}_{0.95}(A)=\$100$.

B is constant at $\$15$, so its quantile at either level is $\$15$, and its TVaR at either level is also $\$15$. The results are:

Portfolio	VaR at 90%	TVaR at 90%	VaR at 95%	TVaR at 95%
A	$0	$100	$100	$100
B	$15	$15	$15	$15

At 90%, VaR ranks A below B, while TVaR ranks A above B. At 95%, both measures rank A above B. This choice is not robust to either the selected measure or, for VaR, the confidence level.

Check. At 90%, exactly $10\%$ of A's outcomes are $\$100$, so the upper $10\%$ average is $\$100$, consistent with the TVaR calculation. A common mistake is to infer from $\operatorname{VaR}_{0.90}(A)=0$ that most of A's losses are small enough to make it the safer portfolio under every measure. VaR at that level does not summarize the severity of losses beyond the quantile.

Stress Tail Severity, Not Just the Quantile

A severity stress changes the size of a loss while retaining its probability, unless the scenario explicitly changes both. This distinction matters: a quantile can stay fixed even as the losses beyond it grow. A tail-sensitive measure can reveal that change, while a quantile measure may not.

Worked Example: A Tail-Severity Stress Reverses a Choice

A hypothetical insurer compares portfolio A, which pays $\$350$ with probability $0.04$ and zero otherwise, with portfolio B, which pays $\$300$ with probability $0.10$ and zero otherwise. At the 95% level, compare VaR and TVaR. Then stress A's severe loss from $\$350$ to $\$400$, leaving its probability unchanged.

Baseline for A. The probability of zero loss is $0.96$, so $F_A(0)=0.96\geq0.95$ and $\operatorname{VaR}_{0.95}(A)=0$. The upper $5\%$ contains all $4\%$ of the $\$350$ outcomes and $1\%$ of zero-loss outcomes. Therefore,

\operatorname{TVaR}_{0.95}(A) =\frac{0.04(\$350)+0.01(\$0)}{0.05} =\$280.

For B, $F_B(0)=0.90<0.95$, so $\operatorname{VaR}_{0.95}(B)=\$300$. The upper $5\%$ lies entirely among the $\$300$ outcomes, giving $\operatorname{TVaR}_{0.95}(B)=\$300$. Baseline VaR favors A, but TVaR favors A as well because $\$280<\$300$.

Severity stress for A. Its probability remains $0.04$, but the severe outcome becomes $\$400$. The zero-loss probability remains $0.96$, so VaR is still zero. TVaR becomes

\operatorname{TVaR}_{0.95}(A) =\frac{0.04(\$400)+0.01(\$0)}{0.05} =\$320.

Now TVaR favors B, since $\$300<\$320$, while VaR still favors A. The threshold at which A's TVaR matches B's is obtained from $0.04s/0.05=\$300$, giving $s=\$375$. Thus a severity above $\$375$ reverses the TVaR ranking.

Check. The upper $5\%$ probability used in each TVaR calculation is $0.04+0.01=0.05$. A common mistake is to average only the $4\%$ of positive outcomes and report $\$350$ or $\$400$ as TVaR. The definition averages the upper $5\%$, including the required $1\%$ at zero.

Stress Dependence While Holding Marginals Fixed

When combining risks, changing dependence while preserving each component's marginal distribution isolates the effect of aggregation assumptions. The example below compares independence with perfect positive dependence, also called comonotonic dependence: the two losses are high together and low together. The individual risks are unchanged; only their joint behavior differs.

Worked Example: Aggregate Tail Risk Under Two Dependence Assumptions

Two hypothetical coverages each produce a loss of $\$100$ with probability $0.05$, and zero otherwise. Compare the aggregate $S=X+Y$ at the 95% level when the losses are independent and when they are perfectly positively dependent.

Independent case. The aggregate is zero with probability $0.95^2=0.9025$. It is $\$100$ when exactly one loss occurs, with probability $2(0.05)(0.95)=0.0950$, and $\$200$ when both occur, with probability $0.05^2=0.0025$. The probabilities sum to $0.9025+0.0950+0.0025=1.0000$. Since $F_S(0)=0.9025<0.95$, and $F_S(100)=0.9975\geq0.95$, the aggregate VaR is $\$100$. The upper $5\%$ consists of all $0.25\%$ at $\$200$ and $4.75\%$ at $\$100$. Hence,

\operatorname{TVaR}_{0.95}(S) =\frac{0.0025(\$200)+0.0475(\$100)}{0.05} =\$105.

Perfect positive dependence. Both losses are zero together with probability $0.95$, and both are $\$100$ together with probability $0.05$. Thus the aggregate is zero with probability $0.95$ and $\$200$ with probability $0.05$. Under the stated lower-quantile convention, $\operatorname{VaR}_{0.95}(S)=0$, because $F_S(0)=0.95$. The entire upper $5\%$ is at $\$200$, so $\operatorname{TVaR}_{0.95}(S)=\$200$.

The comparison shows that aggregate TVaR rises from $\$105$ under independence to $\$200$ under perfect positive dependence. VaR does not move in the same direction in this example: it changes from $\$100$ to zero because the perfect-dependence distribution has an atom exactly at the confidence boundary. A common mistake is to use a strict inequality at that boundary and assign VaR of $\$200$ in the comonotonic case. With $\inf\{x:F_S(x)\geq0.95\}$, the quantile is zero.

Check What Changes When Losses Are Rescaled

A common positive scaling of every loss changes the monetary amount of a risk measure but not the ranking of candidates under VaR or TVaR. This is different from scaling only one portfolio or one component: that is a change to the comparison itself and can change the ranking.

Justification: For a positive constant $c$, $F_{cX}(x)=F_X(x/c)$. Therefore the lower quantile satisfies $\operatorname{VaR}_{p}(cX)=c\operatorname{VaR}_{p}(X)$. Substituting this into the quantile-integral definition gives $\operatorname{TVaR}_{\alpha}(cX)=c\operatorname{TVaR}_{\alpha}(X)$. Thus a common positive scaling multiplies both candidates' VaR and TVaR by the same factor and preserves their ranking.

Worked Example: Common Scaling Versus Portfolio-Specific Scaling

Two hypothetical portfolios have losses measured in thousands of dollars. Portfolio C has loss $0$ with probability $0.95$ and loss $200$ with probability $0.05$. Portfolio D always has loss $110$. Compare 95% VaR and TVaR before and after a common 1.2 scaling; then consider a 0.5 scaling applied to C alone.

Initially, $F_C(0)=0.95$, so $\operatorname{VaR}_{0.95}(C)=0$. Its upper $5\%$ consists entirely of losses of 200, so $\operatorname{TVaR}_{0.95}(C)=200$. D is constant, giving $\operatorname{VaR}_{0.95}(D)=\operatorname{TVaR}_{0.95}(D)=110$. VaR favors C, while TVaR favors D.

Under a common 1.2 scaling, C's outcomes become $0$ and $1.2(200)=240$, and D's constant loss becomes $1.2(110)=132$. The scaled results are: C has VaR $0$ and TVaR $240$; D has VaR $132$ and TVaR $132$. Both measure rankings are unchanged from their respective initial rankings.

If instead only C is scaled by 0.5, its outcomes are $0$ and $100$. Its 95% VaR remains zero, and its 95% TVaR becomes $100$. D remains at $110$. TVaR now favors C rather than D, while VaR continues to favor C. The common-scaling result does not apply to a portfolio-specific change.

Check. Each reported amount is in thousands of dollars. A common mistake is to multiply only one candidate's risk amount by the common scaling factor; if the loss basis changes uniformly, both candidates must be rescaled.

Turning Sensitivity Results into a Decision

A useful stress test changes one assumption at a time, records the resulting measure and ranking, and identifies the point at which the preferred option changes. Then consider whether that point is plausible relative to the decision's uncertainty. For example, the severity threshold in the second example makes the TVaR decision's dependence on tail size explicit; the dependence example shows that an atom can make a VaR comparison particularly sensitive to a confidence boundary.

A decision is more robust when its ranking persists across reasonable confidence levels, severity stresses, dependence assumptions, and loss bases. If rankings change, report the change rather than selecting a single favorable scenario. Also distinguish monetary sensitivity from decision sensitivity: if every candidate's measure increases by the same positive factor, the risk amounts change, but the ranking does not.

State the loss definition, confidence level, and dependence assumption used for each comparison.
Vary one assumption at a time before combining stresses, so the source of a change is identifiable.
For discrete losses, check the cumulative probability at the quantile and the exact upper-tail probability included in TVaR.
Report ranking reversals and boundary effects, not only changes in the numerical risk amounts.

Key takeaway: A risk-measure choice is robust only if the decision it supports survives reasonable changes to the inputs that matter. Stress the confidence level, tail severity, dependence, and scale, and distinguish a changed risk estimate from a changed decision.

Check Your Understanding

For each problem, use the lower-quantile convention and the quantile-integral definition of TVaR. Monetary amounts are in dollars unless noted.

A loss is $0$ with probability $0.92$ and $\$80$ with probability $0.08$. At the 90% level, calculate VaR and TVaR. (Answer: VaR = $0; TVaR = $64.)
Portfolio A has loss $\$600$ with probability $0.03$ and zero otherwise. Portfolio B is constant at $\$350$. At the 95% level, calculate both measures for A and state which portfolio each measure ranks lower. (Answer: A has VaR = $0 and TVaR = $360; B has VaR = TVaR = $350; VaR ranks A lower, while TVaR ranks B lower.)
Two independent risks each pay $\$40$ with probability $0.10$, otherwise zero. Find the 95% VaR and TVaR of their sum. (Answer: VaR = $40; TVaR = $48.)
A loss is $0$ with probability $0.90$ and $\$50$ with probability $0.10$. At the 95% level, find VaR and TVaR before and after multiplying every loss by 1.5. (Answer: Before, VaR = $50 and TVaR = $50; after, VaR = $75 and TVaR = $75.)
Portfolio A has loss $s$ with probability $0.04$ and zero otherwise. Portfolio B has loss $\$250$ with probability $0.10$ and zero otherwise. At the 95% level, find the value of $s$ at which their TVaRs are equal, and state which has lower TVaR when $s=\$350$. (Answer: Equality occurs at $s=$312.50$; when $s=$350$, B has lower TVaR.)

What You'll Learn