Why Use More Than the Expected Loss?
A premium based only on expected losses describes the average outcome, not how severe losses could be in an unfavorable year. A contingency provision can reflect that uncertainty by using a measure of the loss distribution’s upper tail. Two useful choices are a quantile, which marks a loss threshold at a selected probability level, and a tail average, which averages losses in the worst part of the distribution.
These measures answer different questions. A quantile identifies a threshold; a tail average describes the average severity beyond that threshold, with a careful adjustment when the quantile itself has probability mass. Neither measure determines the appropriate provision automatically: the selected confidence level and risk measure are assumptions that should fit the purpose of the calculation. This material is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
The definition of TVaR here is also commonly called expected shortfall. Some sources use “conditional tail expectation” for the conditional mean \(E[L\mid L\ge q]\). For a discrete distribution, that conditional mean can differ from the quantile-integral TVaR because it includes the entire probability mass at \(q\), rather than only the part needed to make a tail of size \(1-\alpha\). The convention must therefore be stated when calculating a tail measure.
From a Risk Measure to a Contingency Load
Suppose the premium amount available to pay claims, after fixed expenses and any separately specified target profit, is \(B\). The underwriting gain after claims is \(G=B-L\). If the selected funding threshold is \(q\), setting \(B=q\) gives
The first inequality says that the quantile funding threshold covers the loss in at least an \(\alpha\) fraction of outcomes. The second inequality is a separate statement about strictly positive gain, and equality with \(\alpha\) is possible. For example, if \(L\) is uniform on \([0,1]\) and \(\alpha=0.8\), then \(q=0.8\) and \(P(L<q)=0.8\). The quantile is not necessarily a value with probability mass immediately below it.
A risk-based contingency load added to the expected loss can be defined as the selected risk measure less the expected loss. Thus the VaR-based load is \(\operatorname{VaR}_{\alpha}(L)-E[L]\), and the TVaR-based load is \(\operatorname{TVaR}_{\alpha}(L)-E[L]\). These are differences between risk thresholds and the mean, not the thresholds themselves. A negative difference is mathematically possible if the selected quantile is below the mean; whether to use such a result as a provision is a separate pricing decision.
Calculating a Tail Average for a Discrete Loss
For a discrete loss distribution, let \(q=\operatorname{VaR}_{\alpha}(L)\). Loss values above \(q\) are included fully in the upper tail. At \(q\), include only the probability needed to make the total tail probability exactly \(1-\alpha\). The following formula implements this rule:
Here \(\mathbf{1}_{\{L>q\}}\) is 1 when \(L>q\) and 0 otherwise. The probability above \(q\) is \(1-F_L(q)\). The tail needs total probability \(1-\alpha\), so the additional probability to take at \(q\) is
Multiplying that boundary probability by \(q\), adding the probability-weighted losses above \(q\), and dividing by the tail probability \(1-\alpha\) gives the displayed formula. The inverse-CDF condition ensures \(0\le F_L(q)-\alpha\le P(L=q)\), so this uses no more than the probability mass available at \(q\). Equivalently, the integral definition averages the quantile function over the percentile range from \(\alpha\) to 1; for a discrete distribution, each loss value contributes in proportion to the length of the percentile interval on which it is the quantile.
Worked Example: VaR and TVaR With a Boundary Mass
A hypothetical insurer models the annual aggregate loss for a small product as follows. The loss is $0 with probability 0.50, $40 with probability 0.30, $100 with probability 0.15, and $180 with probability 0.05. Calculate the expected loss, the 90% VaR, the 90% TVaR, and both corresponding loads above expected loss.
Method and expected loss. Multiply each loss by its probability and add: $$ E[L]=0(0.50)+40(0.30)+100(0.15)+180(0.05) =0+12+15+9=\$36. $$
Find the quantile. The cumulative probabilities at losses $0, $40, $100, and $180 are 0.50, 0.80, 0.95, and 1.00. The first loss at which the cumulative probability is at least 0.90 is $100. Therefore, \(\operatorname{VaR}_{0.90}(L)=\$100\).
Find the tail average. The worst 10% of outcomes have probability 0.10. Losses above $100 contribute probability 0.05 and weighted loss \(180(0.05)=\$9\). The remaining tail probability is \(0.10-0.05=0.05\), taken at $100. Its weighted loss is \(100(0.05)=\$5\). Thus $$ \operatorname{TVaR}_{0.90}(L)=\frac{9+5}{0.10}=\$140. $$ The boundary formula gives the same result: \(F_L(100)-0.90=0.95-0.90=0.05\).
Calculate the loads and check. The VaR-based load is \(100-36=\$64\); the TVaR-based load is \(140-36=\$104\). Adding each load back to the expected loss recovers its threshold: \(36+64=100\) and \(36+104=140\). The tail average exceeds the quantile because it averages the worst 10% of losses, including losses of $180.
Common mistake avoided. Using \(E[L\mid L\ge100]\) would average all losses of $100 or more. That calculation is \([100(0.15)+180(0.05)]/(0.15+0.05)=24/0.20=\$120\), not the 90% TVaR. It includes all 15% probability at $100, although only 5% of that boundary mass belongs in the worst 10% tail.
Worked Premium Provision Calculations
A risk measure can be translated into a premium provision only after stating what the premium must cover. The next example separates expected claims, a hypothetical fixed expense amount, a target underwriting profit, and the contingency load. It assumes those expense and profit amounts are fixed dollar amounts, with no investment income or discounting in the calculation.
Worked Example: Comparing VaR-Based and TVaR-Based Premiums
A hypothetical annual policy block has aggregate loss \(L\) of $0 with probability 0.40, $50 with probability 0.40, or $200 with probability 0.20. The insurer wants to compare 80% VaR and 80% TVaR provisions. Assume fixed expenses of $12, a separately stated target underwriting profit of $8, and no investment income. Calculate the premium under each method.
Step 1: Calculate the expected loss. $$ E[L]=0(0.40)+50(0.40)+200(0.20)=0+20+40=\$60. $$
Step 2: Find the 80% VaR. The cumulative probability is 0.40 at $0 and 0.80 at $50. Therefore \(\operatorname{VaR}_{0.80}(L)=\$50\). The probability boundary is exact: \(P(L\le50)=0.80\).
Step 3: Find the 80% TVaR. The worst 20% of outcomes are exactly the 20% probability at $200, so $$ \operatorname{TVaR}_{0.80}(L)=\frac{200(0.20)}{0.20}=\$200. $$ The VaR-based load is \(50-60=-\$10\); the TVaR-based load is \(200-60=\$140\). The negative VaR difference means this quantile is below the mean; it does not change the definition of the quantile.
Step 4: Assemble each premium. For this example, premium equals the selected claim funding threshold plus fixed expenses and target profit. The VaR-based premium is \(50+12+8=\$70\). The TVaR-based premium is \(200+12+8=\$220\). As a check, expected loss plus the VaR load is \(60-10=50\), and expected loss plus the TVaR load is \(60+140=200\). Adding the same $20 of expenses and target profit gives the two premiums above.
Common mistake avoided. Adding the contingency load to expected loss and then adding expected loss again would double count claims. The claim provision is either the selected threshold directly, or expected loss plus its corresponding load—not both.
Interpreting the Measures and Avoiding Traps
| Measure | Question it answers | Discrete-distribution interpretation |
|---|---|---|
| \(\operatorname{VaR}_{\alpha}(L)\) | What is the loss threshold at percentile \(\alpha\)? | The smallest loss whose cumulative probability is at least \(\alpha\). |
| \(\operatorname{TVaR}_{\alpha}(L)\) | What is the average loss across the worst \(1-\alpha\) fraction of outcomes? | Include all loss probability above the VaR and just enough probability at the VaR to fill the tail. |
| Risk-based load | How far is the selected threshold above or below the mean? | Subtract \(E[L]\) from the selected risk measure. |
VaR is a probability threshold, not an average of severe losses. TVaR adds information about the magnitude of outcomes in the selected tail. In general, \(\operatorname{TVaR}_{\alpha}(L)\ge \operatorname{VaR}_{\alpha}(L)\) for an integrable loss: quantiles at percentile levels \(p\ge\alpha\) are at least the \(\alpha\)-quantile, so their average cannot be lower. However, this does not imply that either risk-based load must be positive; the mean can exceed the selected quantile.
A common quantile error is to write \(P(L<q)<\alpha\) as a necessary condition. The correct general statement is \(P(L<q)\le\alpha\le P(L\le q)\); equality on either side can occur. In particular, a continuous distribution can have \(P(L<q)=\alpha\), while a discrete distribution can have \(P(L\le q)=\alpha\). Check the cumulative probabilities rather than assuming a strict inequality.
A second trap is interpreting the coverage probability as exactly \(\alpha\). With a discrete loss distribution, \(P(L\le q)\) can be greater than \(\alpha\), because the quantile may have a probability mass that crosses the chosen percentile. The quantile threshold still follows the inverse-CDF definition. TVaR handles the boundary mass differently: it uses only the fraction needed to form a tail of exactly \(1-\alpha\).
Worked Example: Translating a Tail Measure Into a Gross Premium
A hypothetical repair-coverage portfolio has annual aggregate losses of $20 with probability 0.25, $60 with probability 0.50, $160 with probability 0.20, and $360 with probability 0.05. At the 90% level, calculate the VaR- and TVaR-based claim provisions and the gross premiums if fixed expenses are $15 and the separately specified target underwriting profit is $10.
Step 1: Calculate the mean loss. $$ E[L]=20(0.25)+60(0.50)+160(0.20)+360(0.05) =5+30+32+18=\$85. $$
Step 2: Find VaR. The cumulative probabilities are 0.25 at $20, 0.75 at $60, 0.95 at $160, and 1.00 at $360. The first cumulative probability at least 0.90 occurs at $160, so \(\operatorname{VaR}_{0.90}(L)=\$160\). Its load over the mean is \(160-85=\$75\).
Step 3: Find TVaR. The worst 10% consists of the 5% probability at $360 plus 5% of the 20% probability at $160. The weighted tail loss is \(360(0.05)+160(0.05)=18+8=\$26\). Therefore $$ \operatorname{TVaR}_{0.90}(L)=\frac{26}{0.10}=\$260. $$ The TVaR-based load is \(260-85=\$175\).
Step 4: Calculate gross premiums and check. With expenses and target profit of \(15+10=\$25\), the VaR-based gross premium is \(160+25=\$185\), and the TVaR-based gross premium is \(260+25=\$285\). The claim provisions also check as \(85+75=160\) and \(85+175=260\). Each premium consists of its claim provision plus the same $25 of stated non-claim amounts.
Common mistake avoided. The TVaR calculation does not average the entire loss distribution above $160 conditional on exceeding it and does not use the mean loss as its denominator. It averages exactly the worst 10% of outcomes.
Choosing and Communicating a Provision
A contingency provision should communicate both the risk measure and the percentile level used. For example, “90% TVaR” identifies a different calculation from “90% VaR,” even though the same loss distribution and confidence level are used. A comparison is most informative when it also reports the expected loss, the resulting load above that mean, and the assumptions about expenses and profit used to reach a premium.
Neither measure should be treated as a universal pricing rule. VaR gives a threshold tied to a probability statement, while TVaR gives an average for a specified upper tail. The chosen risk measure is a modeling or decision assumption, not a substitute for specifying the purpose of the contingency provision. For either measure, keep the loss definition and valuation basis consistent with the premium components being compared.
Check Your Understanding
For each hypothetical loss distribution, calculate the requested measure using the quantile-integral definition of TVaR.
- A loss is $0 with probability 0.60, $100 with probability 0.30, or $300 with probability 0.10. Find \(E[L]\), \(\operatorname{VaR}_{0.90}(L)\), \(\operatorname{TVaR}_{0.90}(L)\), and both loads above the mean. (Answer: \(E[L]=\$60\); VaR \(=\$100\); TVaR \(=\$300\); loads \(=\$40\) and \(\$240\).)
- A loss is $0 with probability 0.40, $80 with probability 0.40, or $200 with probability 0.20. At level 0.60, find VaR, TVaR, and their loads above expected loss. (Answer: \(E[L]=\$72\); VaR \(=\$80\); TVaR \(=\$140\); loads \(=\$8\) and \(\$68\).)
- A loss is $0 with probability 0.50, $100 with probability 0.30, or $400 with probability 0.20. At level 0.80, calculate the VaR- and TVaR-based gross premiums if fixed expenses are $20 and target underwriting profit is $15. (Answer: VaR-based premium \(=\$135\); TVaR-based premium \(=\$435\).)
- A loss is $10 with probability 0.25, $40 with probability 0.50, or $100 with probability 0.25. At level 0.75, find VaR and TVaR, and explain whether any boundary probability is needed in the tail calculation. (Answer: VaR \(=\$40\); TVaR \(=\$100\); no boundary mass at $40 is needed because the 25% probability above $40 exactly fills the tail.)
- Let \(L\) have a continuous uniform distribution on \([0,200]\). At level 0.80, find VaR, TVaR, the VaR-based load above the mean, and \(P(L<\operatorname{VaR}_{0.80}(L))\). (Answer: VaR \(=\$160\); TVaR \(=\$180\); mean \(=\$100\); VaR-based load \(=\$60\); \(P(L<\$160)=0.8000\), so equality in the general boundary inequality is possible.)