From a Loss Distribution to an Economic Capital Assessment
An economic capital result is meaningful only when the calculation follows a consistent set of choices. The assessor must identify the loss measure, confidence level, risk horizon, loss basis, treatment of expected losses, and tax basis. A correct quantile applied to the wrong horizon or compared with an expected loss on a different basis can produce a precise-looking but invalid result.
This tutorial uses hypothetical loss distributions supplied as assessment inputs. Unless an example says otherwise, losses and capital are stated in thousands of dollars, are undiscounted, and refer to the stated horizon. No interest-rate or mortality assumption is used: neither is needed to calculate a quantile or expectation from a supplied aggregate loss distribution.
The loss basis matters as much as the measure. For example, a supplied distribution may represent gross losses or losses net of reinsurance. The expected loss must use the same basis. Likewise, a one-year distribution cannot be substituted for a multi-year distribution without a justified method of conversion. When the required horizon distribution is supplied directly, use it directly rather than applying an additional time-scaling adjustment.
A Consistent Calculation Sequence
Use the following sequence to keep the assessment auditable. The order prevents a common error: performing a familiar calculation before checking what the requested capital number represents.
Record the loss perimeter, whether losses are gross or net, the horizon, and whether amounts are pre-tax or after-tax.
Use the stated VaR or TVaR confidence level. Do not substitute one measure for the other.
For a discrete distribution, accumulate probability to identify VaR and sum probability times loss to calculate \(E[X]\).
If expected losses are funded or otherwise provided for separately, subtract the matching expected loss. If not, do not subtract it.
Apply the stated tax adjustment to unexpected-loss capital only if the assumed tax relief is available and applies on that basis.
Why Expected Loss Is Subtracted
Suppose the selected risk measure is \(\rho(X)\), such as \(\operatorname{VaR}_p(X)\). If the insurer has provided separately for the expected loss \(E[X]\), the capital meant to cover adverse deviation above that provision is the excess of the risk measure over the mean. This gives \(\rho(X)-E[X]\). The subtraction is not a general property of VaR; it follows from the stated purpose of measuring unexpected loss after expected loss is separately recognized.
The same logic applies when \(\rho\) is TVaR, provided the assessment defines capital as the risk measure less separately provided expected loss. Conversely, if the question asks for the total amount at risk at the selected confidence level, the requested quantity may be \(\rho(X)\) itself. Read the treatment of expected losses as a substantive instruction, not as an automatic adjustment.
Worked Assessments
Worked Example: One-Year VaR With Expected Loss Provided Separately
A hypothetical insurer assesses one-year, pre-tax aggregate losses using the following supplied distribution. Loss amounts are in $000s.
| One-year loss | Probability |
|---|---|
| 0 | 0.9000 |
| 4,000 | 0.0700 |
| 12,000 | 0.0300 |
The required measure is 95% VaR. Expected losses are provided for separately. The assessment then assumes a 25% tax offset on unexpected-loss capital, fully realizable. There is no discounting or mortality component; all values are for the stated one-year horizon.
Step 1: Find the 95% VaR. The cumulative probabilities are \(F_X(0)=0.9000\), \(F_X(4{,}000)=0.9700\), and \(F_X(12{,}000)=1.0000\). The first cumulative probability at least 0.95 occurs at a loss of 4,000. Thus \(\operatorname{VaR}_{0.95}(X)=4{,}000\).
Step 2: Calculate expected loss.
Step 3: Calculate pre-tax unexpected-loss capital. Because expected loss is provided separately, subtract 640 from the VaR: \(4{,}000-640=3{,}360\) ($000s).
Step 4: Apply the stated tax convention. After-tax capital is \((1-0.25)(3{,}360)=0.75(3{,}360)=2{,}520\) ($000s).
Answer and check. The assessment produces pre-tax economic capital of $3,360,000 and after-tax economic capital of $2,520,000. The result is below the 95% loss quantile because the separately provided expected loss and the stated tax offset have both been applied. The common mistake avoided is reporting the $4,000,000 VaR as economic capital without following either adjustment.
Worked Example: A Supplied Two-Year Distribution
A hypothetical insurer is assessing aggregate losses over a two-year horizon. The supplied distribution is already the two-year distribution, is pre-tax, and is stated in $000s. The required measure is 99% VaR. Expected losses are provided separately, and the assessment assumes 20% realizable tax relief on unexpected-loss capital. No interest-rate or mortality assumption is used.
| Two-year aggregate loss | Probability |
|---|---|
| 0 | 0.9000 |
| 5,000 | 0.0600 |
| 20,000 | 0.0400 |
Step 1: Locate the 99% quantile. The cumulative probability at 0 is 0.9000; at 5,000 it is \(0.9000+0.0600=0.9600\); at 20,000 it is 1.0000. Since 0.9600 is below 0.99, the 99% VaR is 20,000 ($000s).
Step 2: Find expected two-year loss.
Step 3: Subtract separately provided expected loss. Pre-tax unexpected-loss capital is \(20{,}000-1{,}100=18{,}900\) ($000s).
Step 4: Apply the tax offset. After-tax capital is \((1-0.20)(18{,}900)=0.80(18{,}900)=15{,}120\) ($000s).
Answer and check. Pre-tax economic capital is $18,900,000 and after-tax economic capital is $15,120,000. These are two-year amounts. Do not multiply the supplied losses by two or apply square-root-of-time scaling: that would alter a distribution already specified for the two-year horizon. The check is that the tail quantile, expectation, and tax adjustment all use that same horizon and basis.
Worked Example: TVaR With Probability Mass at the Threshold
A hypothetical insurer uses the following one-year, pre-tax aggregate loss distribution in $000s. The required measure is 95% TVaR. Expected loss is provided separately, and a 30% realizable tax offset applies to unexpected-loss capital. Values are undiscounted; no mortality assumption is involved.
| One-year loss | Probability |
|---|---|
| 0 | 0.9000 |
| 2,000 | 0.0600 |
| 10,000 | 0.0300 |
| 30,000 | 0.0100 |
Step 1: Define the discrete-tail calculation. TVaR at level \(p\) is the average of the upper \(1-p\) fraction of the loss distribution. At a discrete threshold, only the required fraction of probability mass at the quantile is included; including the entire atom can make the tail probability exceed \(1-p\).
The cumulative probability through loss 0 is 0.9000 and through loss 2,000 is 0.9600, so the 95% VaR is 2,000. The upper 5% tail consists of all 3% probability at 10,000, all 1% at 30,000, and 1% of the 6% probability at 2,000. This is exactly 5% total probability.
Step 2: Calculate expected loss.
Step 3: Calculate and tax-adjust capital. Pre-tax unexpected-loss capital is \(12{,}400-720=11{,}680\) ($000s). After-tax capital is \((1-0.30)(11{,}680)=0.70(11{,}680)=8{,}176\) ($000s).
Answer and check. Pre-tax economic capital is $11,680,000 and after-tax economic capital is $8,176,000. The upper-tail numerator is 620, and \(620/0.05=12{,}400\), confirming the TVaR. A common error is to average all outcomes at or above 2,000: that includes 6% probability at 2,000, producing a tail larger than the required 5%.
Interpretation and Assessment Traps
An economic capital figure is not interpretable without its label. A useful reported result states, at minimum, the loss basis, horizon, risk measure and confidence level, whether expected loss is deducted, and whether the result is pre-tax or after-tax. For example, “after-tax one-year 95% VaR-based unexpected-loss capital, net of the stated recoveries” is more informative than “capital is 2,520.”
| Assessment choice | What to verify | Common error |
|---|---|---|
| Risk measure | Use the specified VaR or TVaR. | Substituting a tail average for a quantile, or vice versa. |
| Expected-loss treatment | Subtract expected loss only when the stated capital definition treats it separately. | Subtracting expected loss twice, or not subtracting it when instructed. |
| Horizon | Match both the risk measure and expected loss to the stated horizon. | Scaling a distribution that is already supplied for the required horizon. |
| Tax basis | Apply only the specified and supportable tax convention. | Applying a tax factor to total VaR before subtracting expected loss. |
| Discrete tail | Include exactly the required tail probability for TVaR. | Including the full probability mass at the quantile when only part is needed. |
These distinctions are central to a defensible assessment: the arithmetic is usually short, but the interpretation determines which arithmetic is appropriate. The resulting figure is an estimate under the supplied distribution and conventions, not a guarantee that realized losses will remain below the selected quantile.
Check Your Understanding
For each hypothetical assessment, identify the stated convention and calculate the requested capital. Amounts are in $000s; distributions apply to their stated horizons and are pre-tax unless specified.
- A one-year loss distribution has outcomes 0, 1,000, and 5,000 with probabilities 0.8000, 0.1500, and 0.0500. Calculate 95% VaR-based capital when expected losses are provided separately. (Answer: VaR = 1,000; expected loss = 400; capital = 600 $000s.)
- A supplied two-year distribution has losses 0, 300, and 1,800 with probabilities 0.9400, 0.0400, and 0.0200. Calculate 99% VaR-based capital if expected losses are separately provided; do not apply a tax adjustment. (Answer: VaR = 1,800; expected loss = 48; capital = 1,752 $000s.)
- A one-year loss distribution has outcomes 0, 100, and 700 with probabilities 0.8000, 0.1500, and 0.0500. Calculate 90% TVaR-based capital when expected loss is separately provided. (Answer: TVaR = 400; expected loss = 50; capital = 350 $000s.)
- A one-year distribution has outcomes 0, 200, and 1,000 with probabilities 0.9000, 0.0800, and 0.0200. Find 95% VaR-based after-tax capital if expected loss is separately provided and a 30% realizable tax offset applies to unexpected-loss capital. (Answer: VaR = 200; expected loss = 36; pre-tax capital = 164; after-tax capital = 114.8 $000s.)
- A supplied three-year distribution has outcomes 0, 2,000, and 10,000 with probabilities 0.9000, 0.0800, and 0.0200. Calculate 95% VaR-based capital under two conventions: expected loss is separately provided, or expected loss is not separately provided. (Answer: VaR = 2,000; expected loss = 360; capital is 1,640 $000s when expected loss is separately provided and 2,000 $000s when it is not.)