From a Loss Quantile to Economic Capital
A selected loss quantile gives a threshold that losses do not exceed with at least a specified probability. It does not, by itself, measure the additional amount needed above expected loss. Quantile-based economic capital makes that comparison explicit: it subtracts the expected loss from the selected quantile, with both quantities measured on the same basis.
The distinction matters in practical calculations. A percentile is a level of total loss; economic capital under this definition is the difference between that level and the mean loss. For a discrete distribution, finding the percentile requires care because the cumulative probability can jump over the selected confidence level or land exactly on it.
This tutorial uses the generalized-inverse definition throughout. For a discrete distribution, it means selecting the smallest loss value whose cumulative probability is at least \(\alpha\). It does not mean choosing the first value whose cumulative probability is strictly greater than \(\alpha\), nor does it require interpolation between possible loss values.
Why the Expected Loss Is Subtracted
Suppose expected loss is already provided for in the financial plan, and the selected quantile is the target loss level to be supported. The total amount at that target is \(q_\alpha\). The amount remaining to provide above the expected-loss provision is therefore the difference between the target and the provision.
Writing the expected-loss provision as \(E[X]\) and the additional amount as \(EC_\alpha\), this relationship gives \(E[X]+EC_\alpha=q_\alpha\). Rearranging yields the definition. The subtraction is not an arbitrary adjustment: it isolates the part of the selected loss threshold that lies above expected loss.
For a discrete loss \(X\) that takes values \(x_1<x_2<\cdots<x_n\), with probabilities \(p_1,p_2,\ldots,p_n\), the probability-weighted mean and cumulative probabilities are
The probabilities must be nonnegative and sum to one. Once the cumulative probabilities have been calculated in order, \(q_\alpha\) is the first \(x_k\) for which \(F_X(x_k)\geq\alpha\). The calculation then uses that value and the probability-weighted mean; the quantile is not substituted for the mean.
Calculating from a Discrete Distribution
The following process keeps the quantile selection and the capital calculation separate. It is especially useful when a confidence level coincides with a cumulative probability or when the answer is negative.
Confirm that the listed scenarios cover the stated outcomes and that their probabilities sum to one.
Multiply each loss by its probability and add the products.
Order losses from smallest to largest and accumulate their probabilities.
Choose the first loss with cumulative probability at least the stated confidence level.
Calculate \(q_\alpha-E[X]\), retaining the sign unless a different capital rule has been specified.
Worked Example: A Quantile at an Exact Cumulative Probability
A hypothetical insurer models a one-year net loss, in dollars, using the distribution below. There is no discounting, so the interest rate is 0%. No mortality assumption is used; the table directly specifies the one-year loss distribution. Calculate economic capital at \(\alpha=0.90\).
| Loss \(x\) | Probability \(P(X=x)\) | Cumulative probability \(F_X(x)\) |
|---|---|---|
| $0 | 0.65 | 0.65 |
| $120 | 0.25 | 0.90 |
| $700 | 0.10 | 1.00 |
Method. First calculate the probability-weighted mean, then apply the generalized-inverse rule to the cumulative probabilities, and finally take the difference.
1. Check the probabilities. Their sum is \(0.65+0.25+0.10=1.00\), so the three scenarios form a complete distribution.
2. Calculate expected loss.
3. Find the 90th-percentile loss. The cumulative probability at $0 is 0.65, which is below 0.90. At $120, it is exactly 0.90. Therefore, the smallest loss with cumulative probability at least 0.90 is \(q_{0.90}=\$120\).
4. Calculate capital.
The result is checked against the definition: adding the $20 capital difference to the $100 expected-loss provision gives the $120 selected threshold. A common mistake this method avoids is using the first loss whose cumulative probability is strictly greater than 0.90. That would select $700 and produce $600, but it contradicts the stated generalized-inverse convention because the cumulative probability has already reached 0.90 at $120.
When the Difference Is Negative
The formula is an algebraic difference and does not guarantee a positive result. If the selected quantile is below the mean, then \(q_\alpha-E[X]<0\). This can occur, for example, when the selected confidence level is low relative to the shape of the loss distribution. A negative value does not mean the quantile was calculated incorrectly.
Do not silently replace a negative result with zero unless a separate rule explicitly requires a nonnegative capital amount. The quantile-based definition itself gives the signed difference. In all cases, check that the quantile was selected using the stated confidence level and that the subtraction is quantile minus expected loss.
Worked Example: A Quantile with a Negative Capital Difference
A hypothetical specialty insurer models a one-year net loss in dollars. The interest rate is 0%, so no discounting is applied. No mortality assumption is involved; the loss distribution is specified directly. Calculate economic capital at \(\alpha=0.70\) and interpret the sign.
| Loss \(x\) | Probability \(P(X=x)\) | Cumulative probability \(F_X(x)\) |
|---|---|---|
| $0 | 0.40 | 0.40 |
| $60 | 0.35 | 0.75 |
| $240 | 0.25 | 1.00 |
Method. Calculate the mean from all scenarios, independently identify the first cumulative probability that reaches 0.70, and subtract in the defined order.
1. Check the probabilities. \(0.40+0.35+0.25=1.00\).
2. Calculate expected loss.
3. Select the quantile. At $0, the cumulative probability is 0.40, which is less than 0.70. At $60, it is 0.75, which is at least 0.70. Thus \(q_{0.70}=\$60\).
4. Subtract expected loss.
The check is \(\$81+(-\$21)=\$60\), exactly the selected quantile. The result is negative because the selected quantile is below the mean. A common mistake this method avoids is reversing the subtraction and reporting \(\$81-\$60=\$21\). That positive number is not the defined quantile-minus-expected-loss difference.
Combining Probability Mass at the Quantile
A quantile identifies a loss threshold, not necessarily an outcome that has exactly the stated probability of occurring. With discrete losses, probability accumulates in jumps. The selected threshold satisfies \(F_X(q_\alpha)\geq\alpha\), while the cumulative probability immediately below the threshold is less than \(\alpha\), when such a lower outcome exists. Therefore, \(F_X(q_\alpha)\) may be greater than \(\alpha\).
This is why the cumulative-probability table is more reliable than trying to infer a quantile from a verbal description such as “the loss at 90%.” A confidence level of 0.90 means find the first threshold whose cumulative probability reaches or passes 0.90; it does not mean that the probability at that exact loss must be 0.90.
Worked Example: A Quantile That Jumps Past the Confidence Level
A hypothetical equipment insurer models a one-year net loss, in dollars, with the distribution shown below. The interest rate is 0%; no discounting is made. No mortality assumption is used. Calculate economic capital at \(\alpha=0.85\), rounding only the final amount to the nearest cent.
| Loss \(x\) | Probability \(P(X=x)\) | Cumulative probability \(F_X(x)\) |
|---|---|---|
| $0 | 0.50 | 0.50 |
| $40 | 0.25 | 0.75 |
| $150 | 0.20 | 0.95 |
| $900 | 0.05 | 1.00 |
Method. Use the discrete generalized-inverse quantile. Do not interpolate between the $40 and $150 outcomes.
1. Check the probabilities. \(0.50+0.25+0.20+0.05=1.00\).
2. Calculate expected loss.
3. Find the 85th-percentile loss. The cumulative probability is 0.75 at $40, below 0.85, and 0.95 at $150, at least 0.85. Hence \(q_{0.85}=\$150\). The cumulative probability jumps over 0.85; the definition does not call for interpolation.
4. Calculate and check capital.
A common mistake this method avoids is interpolating between $40 and $150 to manufacture an outcome with cumulative probability 0.85. Under the stated discrete generalized-inverse convention, the selected loss must be one of the possible loss values; here it is $150.
Common Calculation Checks
A correct-looking final subtraction can still be based on an incorrect quantile or mean. Before reporting the result, use these checks:
- Probability check: confirm that all scenario probabilities sum to one.
- Mean check: include every scenario in the probability-weighted sum, including zero-loss outcomes.
- Quantile check: verify that the cumulative probability at the selected outcome is at least \(\alpha\), and that the previous cumulative probability is below \(\alpha\).
- Basis check: confirm that the quantile and expected loss use the same time horizon, net or gross basis, and currency units.
- Sign check: compute quantile minus expected loss, then verify \(E[X]+EC_\alpha=q_\alpha\).
The reported quantity is sensitive to the selected confidence level and the loss distribution. That sensitivity is a feature of the definition, not a reason to change the quantile rule midway through a calculation. The method here is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
Check Your Understanding
For each problem, use the generalized-inverse quantile definition. Unless stated otherwise, losses are one-year dollar amounts with no discounting (interest rate 0%) and no mortality assumption. Calculate the expected loss, selected quantile, and economic capital.
- A hypothetical service insurer has losses of $0, $50, and $200 with probabilities 0.70, 0.20, and 0.10, respectively. Find economic capital at \(\alpha=0.80\). (Answer: expected loss $30; quantile $50; economic capital $20.)
- A hypothetical cyber portfolio has losses of $10, $40, and $160 with probabilities 0.20, 0.50, and 0.30. Find economic capital at \(\alpha=0.60\). (Answer: expected loss $70; quantile $40; economic capital \(-\$30\).)
- A hypothetical municipal risk pool has losses of $0, $100, and $400 with probabilities 0.55, 0.25, and 0.20. Find economic capital at \(\alpha=0.80\), noting what happens when cumulative probability equals the confidence level. (Answer: expected loss $105; quantile $100; economic capital \(-\$5\).)
- A hypothetical machinery insurer has losses of $0, $20, $100, and $500 with probabilities 0.30, 0.40, 0.20, and 0.10. Find economic capital at \(\alpha=0.90\). (Answer: expected loss $78; quantile $100; economic capital $22.)
- A hypothetical catastrophe pool has losses of $25, $75, $250, and $1,000 with probabilities 0.15, 0.45, 0.30, and 0.10. Find economic capital at \(\alpha=0.90\) and verify the subtraction by adding expected loss back to the capital result. (Answer: expected loss $212.50; quantile $250; economic capital $37.50; check: $212.50+$37.50=$250.)