Capital as Risk Measure Above Expected Loss

What Capital Measures

A risk measure applied to an aggregate loss describes a selected level of risk for a stated period and portfolio. Capital, under the convention used here, is not that risk measure itself: it is the excess of the selected risk measure over expected loss. Keeping these quantities separate matters when comparing a risk measure with the expected claims that are already anticipated.

Throughout, let $X$ be a nonnegative aggregate loss over one specified period, and let $\rho(X)$ be the selected risk measure for that same loss. The expected loss is $E[X]$. Unless stated otherwise, amounts are in dollars and all calculations refer to the same portfolio and period.

Definition: Capital associated with the selected risk measure $\rho$ is $$ C_\rho(X)=\rho(X)-E[X]. $$ The risk measure $\rho(X)$ is the total amount indicated by that measure. Under this definition, it can be written as expected loss plus capital: $$ \rho(X)=E[X]+C_\rho(X). $$

This is an accounting decomposition, not a claim that a particular premium or regulatory framework must fund exactly these amounts. It separates the expected-loss component from the additional amount implied by the chosen risk measure. If the measure is VaR, the result is VaR-based capital; if it is TVaR, the result is TVaR-based capital.

This treatment of capital is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus. It is important to state the selected measure and confidence level, the loss being measured, and the expected loss used in the subtraction. A capital figure without those details is difficult to interpret or compare.

VaR- and TVaR-Based Capital

For confidence level $\alpha$, where $0<\alpha<1$, define the value-at-risk of $X$ as the lower $\alpha$-quantile:

\operatorname{VaR}_\alpha(X)=\inf\{x:F_X(x)\geq\alpha\}.

For this tutorial, TVaR is defined by the quantile-integral convention. This convention also handles discrete distributions, where the probability at the quantile may straddle the $\alpha$ boundary:

\operatorname{TVaR}_\alpha(X) = \frac{1}{1-\alpha}\int_\alpha^1 \operatorname{VaR}_u(X)\,du.

The corresponding capital amounts are obtained by subtracting the same expected loss in each case:

C_{\operatorname{VaR},\alpha}(X) = \operatorname{VaR}_\alpha(X)-E[X], \qquad C_{\operatorname{TVaR},\alpha}(X) = \operatorname{TVaR}_\alpha(X)-E[X].

The key point is that the risk measure is the total indicated amount, while capital is the difference between that amount and expected loss. These are not interchangeable labels. In particular, a high confidence level does not guarantee that VaR exceeds the mean for every possible distribution. Under the definition above, capital can therefore be negative; a separate rule that floors capital at zero would be an additional convention, not part of this formula.

Why the Subtraction Works

The formula follows directly from the intended decomposition. Suppose the total amount indicated by a selected measure is divided into an expected-loss component and a residual component called capital. If the total is $\rho(X)$ and the expected-loss component is $E[X]$, then

\rho(X)=E[X]+C_\rho(X).

Subtracting $E[X]$ from both sides gives $C_\rho(X)=\rho(X)-E[X]$. Thus the subtraction does not produce a new risk measure; it isolates the portion of the selected total that is above the expected loss. The decomposition is meaningful only when both terms refer to the same loss definition, portfolio, period, and currency.

For discrete losses, the TVaR calculation also has a direct interpretation. The quantile integral averages the losses in the upper $1-\alpha$ fraction of probability. If a probability mass at the quantile crosses the boundary, only the portion needed to fill that upper fraction is included. Dividing the resulting weighted loss by $1-\alpha$ gives the average loss in that tail. This is why the calculation may require a fractional part of an atom rather than simply averaging all outcomes at or above VaR.

Worked Example: Discrete Losses With a Quantile Boundary

A hypothetical equipment insurer models one-year aggregate claim loss $X$ as follows. Calculate VaR-based and TVaR-based capital at $\alpha=0.75$, and distinguish each capital amount from the corresponding total indicated amount.

Aggregate loss $X$	Probability
$0	0.50
$40	0.30
$160	0.20

Method and expected loss. First calculate the probability-weighted mean:

$$ E[X]=(0)(0.50)+(40)(0.30)+(160)(0.20)=0+12+32=44. $$

The cumulative probabilities are $F_X(0)=0.50$, $F_X(40)=0.80$, and $F_X(160)=1.00$. Since the first loss value with cumulative probability at least 0.75 is 40,

\operatorname{VaR}_{0.75}(X)=40, \qquad C_{\operatorname{VaR},0.75}(X)=40-44=-4.

The VaR-based total indicated amount is $40, and its capital under this definition is \(-$4. The negative sign is retained; it is not silently replaced by zero.

TVaR and total amount. The upper 25% of probability consists of all 20% at loss 160 and 5% of probability at loss 40. The quantile-integral average is therefore

\operatorname{TVaR}_{0.75}(X) = \frac{(0.20)(160)+(0.05)(40)}{0.25} = \frac{32+2}{0.25} = 136.

Thus $C_{\operatorname{TVaR},0.75}(X)=136-44=92$. The TVaR-based total indicated amount is $136, comprising $44 of expected loss and $92 of capital. As a check, $44+92=136$.

Common mistake avoided. Averaging only losses strictly above VaR would use the loss 160 alone and give 160. That is not the quantile-integral TVaR here: the upper 25% includes 5% of the probability at 40, so the correct result is 136.

Worked Example: VaR and TVaR for a Uniform Loss

For a hypothetical one-year loss $X$ uniformly distributed from $0 to $200, calculate the expected loss and the VaR- and TVaR-based capital amounts at $\alpha=0.95$. Assume the uniform model applies over that full interval.

Method and expected loss. For a uniform distribution on $[0,200]$, the mean is the midpoint:

E[X]=\frac{0+200}{2}=100.

On this interval, $F_X(x)=x/200$, so its quantile at probability $u$ is $200u$. In particular,

\operatorname{VaR}_{0.95}(X)=200(0.95)=190.

The VaR-based capital is $190-100=90$. The total indicated amount is $190, consisting of $100 expected loss and $90 capital.

For TVaR, average the quantiles from 0.95 to 1 and divide by the tail probability 0.05:

\operatorname{TVaR}_{0.95}(X) = \frac{1}{0.05}\int_{0.95}^{1}200u\,du = \frac{100(1^2-0.95^2)}{0.05} = \frac{9.75}{0.05} = 195.

Therefore $C_{\operatorname{TVaR},0.95}(X)=195-100=95$. The TVaR-based total is $195, made up of $100 expected loss and $95 capital. The check $100+95=195$ confirms the decomposition.

Common mistake avoided. Subtracting the expected loss from the quantile $190$ gives VaR-based capital, not TVaR-based capital. For TVaR, first calculate the average of the upper 5% of losses; that average is 195.

Worked Example: Comparing Two Risk Measures for a Discrete Portfolio

A hypothetical portfolio has the following annual aggregate-loss distribution. Calculate expected loss, VaR and TVaR at $\alpha=0.85$, and the capital and total indicated amount associated with each measure.

Aggregate loss $X$	Probability
$0	0.70
$100	0.20
$300	0.10

Method and expected loss. The mean is

$$ E[X]=(0)(0.70)+(100)(0.20)+(300)(0.10)=20+30=50. $$

The cumulative probabilities at losses 0, 100, and 300 are 0.70, 0.90, and 1.00. Since 0.90 is the first cumulative probability to reach 0.85, $\operatorname{VaR}_{0.85}(X)=100$. Hence

C_{\operatorname{VaR},0.85}(X)=100-50=50.

The VaR-based total is $100: $50 expected loss plus $50 capital.

For TVaR, the upper 15% consists of all 10% of probability at $300 and 5% of probability at $100. Therefore,

\operatorname{TVaR}_{0.85}(X) = \frac{(0.10)(300)+(0.05)(100)}{0.15} = \frac{30+5}{0.15} = 233.333\ldots.

Rounding money to the nearest cent only at the final step gives TVaR of $233.33 and TVaR-based capital of $233.333\ldots-50=$$183.33. The TVaR-based total is $233.33. Using unrounded amounts, $50+183.333\ldots=233.333\ldots$, as required.

Common mistake avoided. The VaR-based capital is $100-50=50$, not $100; $100 is the VaR-based total. For TVaR, the selected tail includes only 5% of the probability at $100, not all 20%.

Common Interpretive Traps

Capital calculations often involve a short subtraction, but errors in the meaning of the quantities can change the answer substantially. Keep the selected risk measure separate from the amount obtained after subtracting expected loss.

Calling the risk measure “capital.” If TVaR is $233.33 and expected loss is $50, the TVaR-based capital is $183.33. The total indicated amount remains $233.33.
Subtracting the wrong expected loss. Use $E[X]$ for the same aggregate loss being measured, not the expected loss of a different period, portfolio, or coverage definition.
Assuming capital must be nonnegative. In the first example, VaR is below the mean, giving capital of $-$4$. The stated definition does not impose a zero floor. If a setting requires a floor, state and apply that separate convention explicitly.
Ignoring a discrete quantile boundary. For TVaR, an atom at VaR may be only partly included in the upper tail. Use the quantile-integral convention or its equivalent fractional-tail calculation.
Comparing amounts with inconsistent units or horizons. A one-year risk measure cannot be compared directly with an expected loss stated for a different period without an appropriate model and conversion.

Key takeaway: State the loss, confidence level, and risk measure; calculate $E[X]$; then subtract it from that risk measure to obtain capital. The risk measure is the total indicated amount, while capital is only its excess over expected loss.

Check Your Understanding

For each problem, identify the total indicated amount as well as the capital when requested. All loss amounts are hypothetical.

A loss is $0 with probability 0.40, $50 with probability 0.40, and $200 with probability 0.20. Find $E[X]$, VaR-based capital, and the VaR-based total at $\alpha=0.80$. (Answer: $E[X]=\$60$; VaR $=\$50$; capital $=-\$10$; total $=\$50$.)
A loss is uniform on $[0,120]$. At $\alpha=0.90$, find VaR-based and TVaR-based capital. (Answer: $E[X]=\$60$; VaR $=\$108$, capital $=\$48$; TVaR $=\$114$, capital $=\$54$.)
A loss is $0 with probability 0.60, $75 with probability 0.25, and $225 with probability 0.15. At $\alpha=0.85$, find the VaR-based capital and total. (Answer: $E[X]=\$52.50$; VaR $=\$75$; capital $=\$22.50$; total $=\$75$.)
A loss is $0 with probability 0.50, $80 with probability 0.30, and $200 with probability 0.20. At $\alpha=0.75$, find TVaR-based capital using fractional inclusion at the quantile. (Answer: TVaR $=\$176$; $E[X]=\$64$; capital $=\$112$; total $=\$176$.)
Portfolio A has loss $0 with probability 0.50, $100 with probability 0.40, and $300 with probability 0.10. Portfolio B has loss $0 with probability 0.40, $100 with probability 0.50, and $250 with probability 0.10. At $\alpha=0.90$, calculate VaR-based capital for each and identify the lower capital amount. (Answer: VaR is $100 for both; A has $E[X]=\$70$ and capital $30, while B has $E[X]=\$75$ and capital $25; Portfolio B has lower VaR-based capital.)

What You'll Learn