VaR's Blind Spot Beyond the Quantile

What VaR Does—and Does Not—Measure

Value at Risk (VaR) gives a loss threshold at a selected confidence level. It is useful for describing a point in a loss distribution, but that point does not summarize the entire distribution. In particular, two portfolios can have the same VaR and very different losses beyond it. This distinction matters when a risk decision depends not only on whether a threshold is crossed, but also on how severe a crossing could be.

Let $X$ be a loss over a specified horizon, with cumulative distribution function $F_X(x)=P(X\le x)$. The confidence level $\alpha$ is between 0 and 1. The examples below use hypothetical loss distributions; their values are not claims about actual insurers or markets.

Definition: For a loss $X$ and confidence level $\alpha$, define $\operatorname{VaR}_{\alpha}(X)$ as the lower $\alpha$-quantile: $$ \operatorname{VaR}_{\alpha}(X)=\inf\{x:F_X(x)\ge \alpha\}. $$ For a continuous distribution that increases through the quantile, this is the loss $x$ for which $F_X(x)=\alpha$. For a discrete distribution, the definition selects the smallest loss at which the cumulative probability reaches or exceeds $\alpha$; it does not interpolate between possible losses.

Under this definition, at $q=\operatorname{VaR}_{\alpha}(X)$, the cumulative probability through $q$ is at least $\alpha$, while the cumulative probability at any smaller loss is below $\alpha$. In a continuous model, this usually means $P(X\le q)=\alpha$. With a discrete model, probability can accumulate at $q$, so the cumulative probability may jump past $\alpha$. Neither case tells us the sizes or detailed probabilities of losses above $q$.

Why Equal VaR Can Hide Different Tails

The quantile definition depends on where the cumulative probability first reaches the selected confidence level. Once it has reached that level, probabilities assigned to still larger losses do not change the location of that first crossing. This gives a direct reason two distributions can share a VaR while having different severe outcomes.

F_A(x)<\alpha \text{ for } x<q,\quad F_A(q)\ge\alpha \qquad\text{and}\qquad F_B(x)<\alpha \text{ for } x<q,\quad F_B(q)\ge\alpha \ \Longrightarrow\ \operatorname{VaR}_{\alpha}(A)=\operatorname{VaR}_{\alpha}(B)=q.

The conditions establish the same first crossing for both distributions. They impose no requirement that $F_A(x)=F_B(x)$ for $x>q$. Therefore, losses beyond $q$, the probabilities of particular high losses, and quantiles at higher confidence levels may differ substantially. This is the blind spot: VaR reports the threshold but does not describe the severity beyond it.

Worked Example: Same 99% VaR, Different Extreme Losses

Two hypothetical portfolios have one-year losses measured in millions of dollars. Portfolio A has loss $0$ with probability $0.98$, loss $10$ with probability $0.01$, and loss $40$ with probability $0.01$. Portfolio B has loss $0$ with probability $0.98$, loss $10$ with probability $0.01$, and loss $100$ with probability $0.01$. Compare their 99% and 99.5% VaR values.

Method. Add probabilities from the smallest loss upward and find the first loss at which the cumulative probability reaches the confidence level.

For both portfolios, the cumulative probability at loss $0$ is $0.98$, which is below $0.99$. At loss $10$, it is $0.98+0.01=0.99$. Therefore,

\operatorname{VaR}_{0.99}(A)=10,\qquad \operatorname{VaR}_{0.99}(B)=10.

At the 99.5% level, the cumulative probability through loss $10$ is still $0.99$, below $0.995$. The next possible loss is $40$ for A and $100$ for B; in each case the cumulative probability then reaches $1.00$. Thus,

\operatorname{VaR}_{0.995}(A)=40,\qquad \operatorname{VaR}_{0.995}(B)=100.

Both portfolios have a 99% VaR of $10 million, but their 99.5% VaR values differ by $60 million. Also, each has a 1% probability of a loss strictly greater than $10 million; the loss in that event is $40 million for A and $100 million for B. The shared 99% threshold does not make those outcomes alike.

Check. For each portfolio, the cumulative probability at $10 million is exactly $0.99$, confirming the 99% result. The cumulative probability at the next loss is $1.00$, confirming the 99.5% result. The common mistake this method avoids is inferring that equal VaR implies equal losses above the VaR threshold.

Comparing Losses Beyond the Threshold

A useful comparison begins by keeping the VaR level fixed and then asking a separate question about what lies beyond that level. For example, a risk report can show a higher-confidence quantile, the possible loss amounts above the threshold, or the probability of exceeding a specified larger loss. Those are additional facts about the distribution; they are not contained in the VaR value alone.

Worked Example: Equal 95% VaR in Two Continuous Models

Consider two hypothetical one-year loss models, in millions of dollars. In each model, 95% of the probability is uniformly distributed between $0 million and $10 million. The remaining 5% is uniformly distributed between $10 million and $30 million in Model A, and between $10 million and $110 million in Model B. Find each model’s 95% and 99% VaR.

Method. Locate the quantile first in the cumulative probability. The initial part of each model contributes 0.95 probability by $10 million, so the 95th percentile is at that boundary. For a higher quantile, calculate how far through the remaining 5% of probability the target lies.

For either model, all losses below $10 million come from the first component, which has total probability $0.95$. At $10 million, the cumulative probability is $0.95$, so

\operatorname{VaR}_{0.95}(A)=\operatorname{VaR}_{0.95}(B)=10.

For the 99th percentile, the cumulative probability must increase from $0.95$ to $0.99$. The required $0.04$ probability is $0.04/0.05=0.80$, or 80%, of the second component. In Model A, that component spans $20 million, so the quantile is 80% of the way from $10 million to $30 million:

\operatorname{VaR}_{0.99}(A) =10+0.80(30-10) =10+16 =26.

In Model B, the second component spans $100 million. Its 80%-point is:

\operatorname{VaR}_{0.99}(B) =10+0.80(110-10) =10+80 =90.

The two models have the same 95% VaR, $10 million, but their 99% VaR values are $26 million and $90 million. The calculation also checks the component probabilities: the 99th percentile lies 80% through the last 5%, adding $0.80(0.05)=0.04$ to the initial $0.95$, for a cumulative probability of $0.99$.

Common mistake. It would be wrong to assume that because 95% of probability lies at or below $10 million, the remaining 5% must behave similarly in both models. That assumption would give no basis for the different upper ranges specified in the problem. The correct calculation uses each model’s own tail interval.

Probability Mass at VaR Matters

Discrete loss models make another limitation especially visible. The VaR definition guarantees that the cumulative probability reaches the confidence level at the selected loss; it does not guarantee that the probability of exceeding that loss is exactly $1-\alpha$. If there is probability mass at VaR, the probability of a loss strictly greater than VaR can be smaller than $1-\alpha$. Two models can even have the same VaR while that exceedance probability differs.

Worked Example: Same 97.5% VaR, Different Exceedance Outcomes

Two hypothetical portfolios have one-year losses in millions of dollars. Portfolio C has loss $0$ with probability $0.970$, loss $4$ with probability $0.010$, and loss $8$ with probability $0.020$. Portfolio D has loss $0$ with probability $0.970$, loss $4$ with probability $0.005$, and loss $50$ with probability $0.025$. Compare their 97.5% VaR, probabilities of losses strictly greater than that VaR, and 99% VaR.

Method. First identify the cumulative probability at each possible loss. Then add probabilities strictly above the VaR separately; this avoids treating all probability at the VaR threshold as an exceedance.

For Portfolio C, the cumulative probability at $0$ is $0.970$. At $4$, it is $0.970+0.010=0.980$, which reaches 0.975. For Portfolio D, the cumulative probability at $0$ is $0.970$, and at $4$ it is $0.970+0.005=0.975$. Thus both have 97.5% VaR of $4 million.

\operatorname{VaR}_{0.975}(C)=\operatorname{VaR}_{0.975}(D)=4.

For C, only the loss of $8 million is strictly greater than $4 million, so the exceedance probability is $0.020$. For D, only the loss of $50 million is strictly greater than $4 million, so the exceedance probability is $0.025$. Their 99% VaR values are also different: C has cumulative probability $0.980$ through $4 million and reaches $1.000$ at $8 million; D has cumulative probability $0.975$ through $4 million and reaches $1.000$ at $50 million. Therefore, the 99% VaR values are $8 million for C and $50 million for D.

Check. Each probability distribution sums to 1.000. For C, $0.970+0.010+0.020=1.000$; for D, $0.970+0.005+0.025=1.000$. The example avoids the mistake of equating a 97.5% VaR with a guaranteed 2.5% exceedance probability: that probability is 2.0% for C and 2.5% for D.

Common Interpretation Trap

A frequent error is to read “99% VaR is $10 million” as meaning that all losses are at most $10 million, or that the worst 1% of losses has a known size. The statement instead identifies a quantile under the specified loss model and horizon. It leaves the shape and severity of the larger losses unspecified.

The discrete examples also show why “the loss exceeds VaR with probability exactly 1%” needs qualification. That equality holds for a continuous distribution at the stated quantile, but an atom at the threshold can change the probability of a strictly greater loss. The quantile definition remains applicable in either setting; the interpretation of the tail must account for the distribution’s probability mass.

VaR remains useful as a standardized threshold for comparing risk at a chosen confidence level. Its limitation is not that the threshold is meaningless, but that it is only one feature of the loss distribution. A sound assessment should use other relevant distributional information—such as higher quantiles or explicit severe-loss scenarios—when decisions depend on what may happen beyond the reported VaR. This point is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Key takeaway: Equal VaR means that two loss distributions reach the selected cumulative probability at the same loss threshold. It does not mean their probabilities, quantiles, or loss amounts beyond that threshold are equal.

Check Your Understanding

For each hypothetical loss model, use the lower-quantile definition and distinguish the VaR threshold from losses strictly above it.

A portfolio loses $0 million with probability 0.94, $5 million with probability 0.02, and $30 million with probability 0.04. What is its 95% VaR? (Answer: $5 million.)
A model assigns probability 0.90 uniformly over losses from $0 million to $8 million, and probability 0.10 uniformly over losses from $8 million to $48 million. Find its 90% and 95% VaR values. (Answer: 90% VaR = $8 million; 95% VaR = $28 million.)
A portfolio loses $0 million with probability 0.97, $6 million with probability 0.02, and $60 million with probability 0.01. Find its 98% VaR and the probability of a loss strictly greater than that VaR. (Answer: 98% VaR = $6 million; exceedance probability = 0.01.)
Portfolio E has losses of $0 million, $12 million, and $25 million with probabilities 0.98, 0.01, and 0.01. Portfolio F has losses of $0 million, $12 million, and $90 million with probabilities 0.98, 0.01, and 0.01. Find each 99% VaR and each 99.5% VaR. (Answer: Both 99% VaR values = $12 million; 99.5% VaR values are $25 million for E and $90 million for F.)
Model G has losses of $0 million, $5 million, and $10 million with probabilities 0.94, 0.02, and 0.04. Model H has losses of $0 million, $5 million, and $100 million with probabilities 0.94, 0.02, and 0.04. Find each 95% VaR and each expected loss. (Answer: Both 95% VaR values = $5 million; expected loss is $0.50 million for G and $4.10 million for H.)

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