Why the Choice of Risk Measure Matters
A capital decision depends not only on the expected aggregate loss but also on how losses can vary around that expectation and how severe they may be in adverse scenarios. Different risk measures answer different questions. Variance describes squared dispersion around the mean; value-at-risk (VaR) identifies a loss threshold at a specified probability level; and tail value-at-risk (TVaR) averages losses in the upper tail beyond that level.
These measures need not rank two hypothetical portfolios in the same order. In particular, VaR focuses on a threshold and does not describe how large losses may be beyond it. TVaR incorporates more of that tail severity, while variance summarizes deviations on both sides of the mean. A capital assessment should therefore begin by identifying which feature of loss is relevant to the decision.
The confidence level \(p\) is a probability, not a dollar amount. For example, \(\operatorname{VaR}_{0.95}(X)\) is a loss threshold associated with the 95th percentile. In a discrete distribution, the cumulative probability can jump across \(p\). The infimum definition selects the smallest loss whose cumulative probability is at least \(p\); it does not require the cumulative probability to equal \(p\).
Variance has units of dollars squared. Its square root, the standard deviation, has units of dollars, but it is not itself a quantile or an upper-tail average. VaR and TVaR are expressed in the same units as the loss. Keeping these units and interpretations distinct is essential when comparing capital indications.
How VaR and TVaR Treat a Discrete Tail
For a loss distribution with possible atoms, the quantile definition gives a precise way to locate the VaR threshold. TVaR then averages exactly the worst \(1-p\) fraction of probability, taking only the required fraction of probability mass at the threshold if the threshold lies inside an atom.
This is not generally the same as \(E[X\mid X\geq v]\). That conditional mean includes all probability at the VaR point, even when only part of that mass belongs in the worst \(1-p\) fraction. The quantile-integral definition handles that boundary consistently.
Worked Comparisons
Worked Example: Different Tail Risk Despite Low VaR
Two hypothetical portfolios have the following aggregate annual losses, in dollars. Portfolio A has losses \(0\), \(100\), and \(500\), with probabilities \(0.90\), \(0.08\), and \(0.02\). Portfolio B has losses \(0\), \(400\), and \(1{,}000\), with probabilities \(0.96\), \(0.03\), and \(0.01\). At \(p=0.95\), calculate each portfolio's mean, variance, VaR, and TVaR.
Method and threshold. First calculate moments from the stated probabilities, then use the cumulative distribution to find VaR. For A, \(F_A(0)=0.90\) and \(F_A(100)=0.98\), so \(\operatorname{VaR}_{0.95}(A)=100\). For B, \(F_B(0)=0.96\), so \(\operatorname{VaR}_{0.95}(B)=0\). The latter illustrates why a 95% VaR can be zero even when the distribution has substantial losses in its upper tail.
Portfolio A moments. Its mean is \(0(0.90)+100(0.08)+500(0.02)=8+10=18\). Its second moment is \(0^2(0.90)+100^2(0.08)+500^2(0.02)=800+5{,}000=5{,}800\). Therefore, \(\operatorname{Var}(A)=5{,}800-18^2=5{,}800-324=5{,}476\) dollars squared.
Portfolio A TVaR. The worst 5% of outcomes contains all of the 2% probability at \(500\) and 3% of the 8% probability at \(100\). Its average loss is \(\operatorname{TVaR}_{0.95}(A)=\frac{0.02(500)+0.03(100)}{0.05}=\frac{10+3}{0.05}=260\).
Portfolio B moments. Its mean is \(0(0.96)+400(0.03)+1{,}000(0.01)=12+10=22\). Its second moment is \(400^2(0.03)+1{,}000^2(0.01)=4{,}800+10{,}000=14{,}800\). Thus, \(\operatorname{Var}(B)=14{,}800-22^2=14{,}800-484=14{,}316\) dollars squared.
Portfolio B TVaR. The worst 5% consists of all 1% at \(1{,}000\), all 3% at \(400\), and 1% of the probability at zero. Thus, \(\operatorname{TVaR}_{0.95}(B)=\frac{0.01(1{,}000)+0.03(400)+0.01(0)}{0.05}=\frac{22}{0.05}=440\).
Check and interpretation. B has the higher mean, variance, and TVaR, while its VaR is zero rather than A's \(100\). This is consistent with the cumulative probabilities: B's zero-loss probability already exceeds 95%, but the worst 5% still contains losses averaging \(440\). The common mistake is to report B's VaR as \(400\) because it is the first positive loss. The correction is to apply the quantile definition: \(F_B(0)=0.96\geq0.95\), so the VaR is \(0\).
Worked Example: VaR and TVaR Can Rank Portfolios Differently
Consider two hypothetical aggregate-loss distributions, in dollars, at confidence level \(p=0.90\). Portfolio C has losses \(0\), \(200\), and \(300\), with probabilities \(0.90\), \(0.09\), and \(0.01\). Portfolio D has losses \(0\), \(100\), and \(1{,}000\), with probabilities \(0.89\), \(0.10\), and \(0.01\). Calculate the measures and compare their rankings.
Portfolio C. The mean is \(0.09(200)+0.01(300)=18+3=21\). The second moment is \(0.09(200^2)+0.01(300^2)=3{,}600+900=4{,}500\), so the variance is \(4{,}500-21^2=4{,}500-441=4{,}059\) dollars squared. Since \(F_C(0)=0.90\), \(\operatorname{VaR}_{0.90}(C)=0\). The worst 10% includes all 1% at \(300\) and 9% at \(200\), giving \(\operatorname{TVaR}_{0.90}(C)=\frac{0.01(300)+0.09(200)}{0.10}=\frac{21}{0.10}=210\).
Portfolio D. The mean is \(0.10(100)+0.01(1{,}000)=10+10=20\). The second moment is \(0.10(100^2)+0.01(1{,}000^2)=1{,}000+10{,}000=11{,}000\), so the variance is \(11{,}000-20^2=10{,}600\) dollars squared. Here \(F_D(0)=0.89\) and \(F_D(100)=0.99\), so \(\operatorname{VaR}_{0.90}(D)=100\). The worst 10% contains 1% at \(1{,}000\) and 9% at \(100\), giving \(\operatorname{TVaR}_{0.90}(D)=\frac{0.01(1{,}000)+0.09(100)}{0.10}=\frac{19}{0.10}=190\).
Check and interpretation. VaR ranks C as lower at this confidence level (\(0\) versus \(100\)), but TVaR is higher for C (\(210\) versus \(190\)). Variance is lower for C (\(4{,}059\) versus \(10{,}600\) dollars squared). The difference between the VaR and TVaR rankings arises because VaR reports a threshold, while TVaR averages the full upper 10% of probability. The calculations also verify that the respective TVaR averages use probability mass totaling exactly \(0.10\).
Worked Example: Equal VaR, Different Tail Severity
Two hypothetical aggregate-loss distributions have losses in dollars. Portfolio E has losses \(0\), \(100\), and \(200\), with probabilities \(0.94\), \(0.05\), and \(0.01\). Portfolio F has losses \(0\), \(100\), and \(1{,}000\), with the same probabilities. At \(p=0.95\), calculate and compare variance, VaR, and TVaR.
VaR. For both portfolios, the cumulative probability at zero is \(0.94\), and at \(100\) it is \(0.99\). Therefore, \(\operatorname{VaR}_{0.95}(E)=\operatorname{VaR}_{0.95}(F)=100\).
Portfolio E. Its mean is \(0.05(100)+0.01(200)=5+2=7\). Its second moment is \(0.05(100^2)+0.01(200^2)=500+400=900\), and its variance is \(900-7^2=851\) dollars squared. The worst 5% comprises 1% at \(200\) and 4% at \(100\). Hence, \(\operatorname{TVaR}_{0.95}(E)=\frac{0.01(200)+0.04(100)}{0.05}=\frac{6}{0.05}=120\).
Portfolio F. Its mean is \(0.05(100)+0.01(1{,}000)=5+10=15\). Its second moment is \(0.05(100^2)+0.01(1{,}000^2)=500+10{,}000=10{,}500\), and its variance is \(10{,}500-15^2=10{,}275\) dollars squared. The worst 5% comprises 1% at \(1{,}000\) and 4% at \(100\). Hence, \(\operatorname{TVaR}_{0.95}(F)=\frac{0.01(1{,}000)+0.04(100)}{0.05}=\frac{14}{0.05}=280\).
Check and interpretation. Both portfolios have VaR \(100\), but F's TVaR is \(280\), compared with E's \(120\), and F's variance is also larger. The shared VaR does not mean their tail risks are alike: it records the same boundary while ignoring the difference between losses of \(200\) and \(1{,}000\) beyond that boundary. The common mistake is to conclude that equal VaR implies equal capital risk; the TVaR and variance calculations show why that conclusion does not follow.
What Each Measure Contributes to a Capital Decision
The variance, VaR, and TVaR calculations answer distinct questions. Their results should be interpreted in the context of the decision rather than treated as interchangeable measures of one underlying quantity.
| Measure | What it summarizes | Important limitation |
|---|---|---|
| Variance | Expected squared deviation from the mean | Measured in squared loss units; it does not directly identify a loss threshold or average tail loss |
| VaR at level \(p\) | The quantile threshold at cumulative probability \(p\) | Does not distinguish severity beyond the threshold and can be affected by discrete probability jumps |
| TVaR at level \(p\) | The average of losses in the worst \(1-p\) fraction of outcomes | Depends on the chosen confidence level and the modeled tail of the distribution |
A variance-based measure can be useful when the decision explicitly values dispersion around the mean, but variance alone is not a dollar capital threshold. VaR answers a percentile question, which can be suitable when a decision is framed around a specified probability of not exceeding a loss level. TVaR gives additional information when the severity of outcomes beyond that level matters. None of the three measures determines the appropriate capital target by itself.
The formulas also depend on the loss distribution being used. In these examples, each distribution is stated directly, so the calculations require no additional assumptions about frequency, severity, or dependence. In an application, an aggregate-loss model and its assumptions must be appropriate for the portfolio and decision. A precise calculation from an unsuitable model does not make the resulting capital indication reliable.
Derivation of the Discrete TVaR Calculation
The discrete TVaR formula follows from its definition as an average of quantiles, not from conditioning on every outcome at or above the VaR. Let \(v=\operatorname{VaR}_p(X)\), and suppose \(X\) has a discrete distribution. By the definition of the quantile, \(F_X(v-)\leq p\leq F_X(v)\), where \(F_X(v-)=P(X<v)\). The quantile equals \(v\) for probability levels between \(F_X(v-)\) and \(F_X(v)\).
All outcomes with \(X>v\) lie in the upper tail. Their contribution to the integral of the quantile is \(E[X\mathbf{1}_{\{X>v\}}]\).
The worst \(1-p\) fraction contains probability \(1-F_X(v)\) above \(v\). The remaining probability is \((1-p)-(1-F_X(v))=F_X(v)-p\), taken from the atom at \(v\). Its loss contribution is \(v(F_X(v)-p)\).
The combined probability is \(1-p\), so dividing the total loss contribution by \(1-p\) gives \(\operatorname{TVaR}_p(X)=\frac{E[X\mathbf{1}_{\{X>v\}}]+v(F_X(v)-p)}{1-p}\).
This derivation also explains why using \(E[X\mid X\geq v]\) can be wrong for a discrete distribution. That conditional expectation includes the entire atom at \(v\). The quantile-integral definition includes only the amount \(F_X(v)-p\) needed to complete the worst \(1-p\) fraction.
Common Exam Traps
- Using the first positive loss as VaR. Check the cumulative probability at every candidate loss. If \(F_X(0)\geq p\), then \(\operatorname{VaR}_p(X)=0\).
- Including all mass at the VaR point in TVaR. Include only the portion needed to make the selected tail probability equal to \(1-p\).
- Equating VaR with tail severity. Two distributions can have the same VaR and very different TVaR because losses beyond the threshold may differ greatly.
- Comparing variance directly with dollar amounts. Variance is in squared units. A dollar-valued capital interpretation requires a clearly specified measure and decision framework.
- Assuming one measure must always be largest or most appropriate. Rankings depend on the loss distributions and on which feature the capital decision is intended to address.
Check Your Understanding
For each hypothetical loss distribution, use the stated confidence level and distinguish the quantile threshold from the upper-tail average.
- A portfolio has aggregate loss \(0\) with probability \(0.80\) and \(50\) with probability \(0.20\). Calculate its mean, variance, \(\operatorname{VaR}_{0.80}\), and \(\operatorname{TVaR}_{0.80}\). (Answer: Mean \(10\); variance \(400\) dollars squared; VaR \(0\); TVaR \(50\).)
- A portfolio has aggregate losses \(0\), \(40\), and \(140\), with probabilities \(0.85\), \(0.10\), and \(0.05\). Calculate its mean, variance, \(\operatorname{VaR}_{0.90}\), and \(\operatorname{TVaR}_{0.90}\). (Answer: Mean \(11\); variance \(1{,}019\) dollars squared; VaR \(40\); TVaR \(90\).)
- A portfolio has aggregate loss \(0\) with probability \(0.97\) and \(500\) with probability \(0.03\). At \(p=0.95\), calculate VaR and TVaR, and explain briefly why the VaR is not the positive loss. (Answer: VaR \(0\); TVaR \(300\); \(F_X(0)=0.97\geq0.95\).)
- A portfolio has aggregate losses \(10\), \(30\), and \(80\), with probabilities \(0.70\), \(0.20\), and \(0.10\). Calculate the mean, variance, \(\operatorname{VaR}_{0.90}\), and \(\operatorname{TVaR}_{0.90}\). (Answer: Mean \(21\); variance \(449\) dollars squared; VaR \(30\); TVaR \(80\).)
- Portfolio G has losses \(0\), \(100\), and \(300\), with probabilities \(0.90\), \(0.08\), and \(0.02\). Portfolio H has losses \(0\), \(100\), and \(800\), with probabilities \(0.90\), \(0.08\), and \(0.02\). At \(p=0.95\), calculate each VaR and TVaR and identify which portfolio has the larger tail average. (Answer: Both VaRs are \(100\); \(\operatorname{TVaR}_{0.95}(G)=180\), \(\operatorname{TVaR}_{0.95}(H)=380\); H has the larger tail average.)