VaR-Based Economic Capital Calculations

Introduction

A VaR-based capital calculation begins with a loss distribution and a confidence level. The confidence level identifies a quantile of that distribution: the loss threshold that is sufficient to cover losses in at least the specified proportion of modeled outcomes. Whether that threshold is itself called economic capital depends on the loss basis. If expected loss is funded separately, capital may be measured as the VaR less expected loss; if the calculation is intended to cover the whole loss, the VaR is the required amount.

The distinction matters in practice. A correct quantile can still produce an incorrect capital result if expected loss is subtracted when it should not be, or omitted when capital is explicitly defined above expected loss. The calculations below make both choices visible. This treatment is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: For a loss random variable $X$ with cumulative distribution function $F_X(x)=P(X\leq x)$, the Value-at-Risk at confidence level $\alpha$, where $0<\alpha<1$, is $$ \operatorname{VaR}_{\alpha}(X)=\inf\{x:F_X(x)\geq\alpha\}. $$ The $\alpha$-quantile is the smallest loss threshold whose cumulative probability is at least $\alpha$. If expected loss is funded separately, VaR-based economic capital is commonly defined as $\operatorname{VaR}_{\alpha}(X)-E[X]$. If capital is intended to cover the entire loss, the required amount is $\operatorname{VaR}_{\alpha}(X)$, without that subtraction.

What the Confidence Level Says

The confidence level $\alpha$ is a cumulative probability, not the probability of exceeding the VaR threshold. For example, $\alpha=0.99$ means the modeled loss is at or below the VaR threshold with probability at least 0.99. Equivalently, the probability of exceeding that threshold is at most 0.01. The word “at least” is important for distributions with discrete outcomes, where the cumulative probability can jump past the requested level.

For a continuous distribution with a strictly increasing cumulative distribution function, the quantile typically satisfies $F_X(\operatorname{VaR}_{\alpha}(X))=\alpha$. For a discrete distribution, equality need not hold. The definition uses the first loss value at which the cumulative probability reaches or exceeds $\alpha$; it does not interpolate between possible loss amounts.

A VaR calculation also requires a clearly stated loss variable and measurement basis. For example, changing the horizon or whether losses are gross or net changes the distribution being measured. The calculation here takes $X$ as given: it is the loss variable on the specified basis and horizon. A finite $E[X]$ is needed when calculating capital as VaR less expected loss.

From a Loss Quantile to a Capital Amount

Suppose expected loss is provided for separately, for example through a reserve or an explicit expected-loss allowance. Capital measured above that allowance is the extra amount needed to bring total loss resources up to the VaR threshold. This gives a short justification for the subtraction:

\begin{aligned} \text{total loss resources} &= E[X]+K,\\ P(X\leq E[X]+K)&\geq\alpha,\\ E[X]+K&\geq\operatorname{VaR}_{\alpha}(X),\\ K&\geq\operatorname{VaR}_{\alpha}(X)-E[X]. \end{aligned}

Thus, the minimum capital above expected loss is $\operatorname{VaR}_{\alpha}(X)-E[X]$, under this setup. If no separate expected-loss amount is included and the question asks for resources sufficient to cover the loss at confidence level $\alpha$, the required total resources are $\operatorname{VaR}_{\alpha}(X)$. These are different measures, not competing calculations. Follow the stated capital definition and report which amount is being calculated.

The quantile itself is found as follows. For a discrete model, accumulate probabilities in increasing order of loss. For a continuous model, solve $F_X(x)=\alpha$ when the cumulative distribution function is continuous and strictly increasing. Then calculate $E[X]$, if required, and apply the expected-loss adjustment only if the definition calls for it.

Specify the loss basis.
Identify the loss variable, measurement horizon, units, and confidence level from the problem.

Find the quantile.
Use the cumulative distribution and the definition of $\operatorname{VaR}_{\alpha}(X)$. For discrete outcomes, select the first loss whose cumulative probability is at least $\alpha$.

Apply the requested capital definition.
Report VaR alone for total loss resources, or subtract $E[X]$ when capital is defined above separately funded expected loss.

Check the result.
Confirm the cumulative probability at the chosen quantile is at least $\alpha$, the preceding cumulative probability is below $\alpha$ for a discrete distribution, and units are consistent.

Worked Calculations

Worked Example: Discrete Annual Losses and Two Capital Bases

A hypothetical insurer models one-year losses, in thousands of dollars, as follows: loss 0 with probability 0.50, loss 100 with probability 0.30, loss 300 with probability 0.15, and loss 800 with probability 0.05. Calculate the 95% VaR, total resources at that threshold, and economic capital above expected loss.

Method. Order the outcomes from smallest to largest and accumulate their probabilities:

\begin{array}{c|c|c} \text{Loss }x & P(X=x) & F_X(x)\\ 0 & 0.50 & 0.50\\ 100 & 0.30 & 0.80\\ 300 & 0.15 & 0.95\\ 800 & 0.05 & 1.00 \end{array}

The first loss with cumulative probability at least 0.95 is 300. Therefore, $\operatorname{VaR}_{0.95}(X)=300$ thousand dollars. The cumulative probability immediately below that loss is 0.80, which is below 0.95; at 300 it reaches 0.95. This verifies the discrete quantile boundary.

The expected loss is

$$ E[X]=0(0.50)+100(0.30)+300(0.15)+800(0.05) =0+30+45+40=115 $$

All amounts in this equation are thousands of dollars. Total loss resources at the VaR threshold are 300 thousand dollars. If expected loss is funded separately, capital above expected loss is $300-115=185$ thousand dollars. Check: $115+185=300$, so expected loss plus the calculated capital equals the VaR threshold. A common mistake is to report 300 thousand dollars as unexpected-loss capital even though the question asks for capital above expected loss.

Worked Example: VaR from a Normal Loss Model

A hypothetical portfolio’s one-year loss $X$, measured in thousands of dollars, is modeled as normal with mean 2,000 and standard deviation 500. Assume $X\sim N(2{,}000,500^2)$, so the second normal parameter is the variance. Calculate the 99.5% VaR and the capital above expected loss. Use the standard normal quantile $z_{0.995}=2.5758$.

Method. Under the stated normal model, standardize the loss using $Z=(X-\mu)/\sigma$, where $Z$ is standard normal. At the 99.5% quantile, $Z=2.5758$, so

\begin{aligned} \operatorname{VaR}_{0.995}(X) &=\mu+\sigma z_{0.995}\\ &=2{,}000+500(2.5758)\\ &=2{,}000+1{,}287.9\\ &=3{,}287.9 \end{aligned}

Thus, the 99.5% VaR is 3,287.9 thousand dollars. The expected loss is the normal mean, $E[X]=2{,}000$ thousand dollars. Capital above expected loss is $3{,}287.9-2{,}000=1{,}287.9$ thousand dollars. Check: $2{,}000+1{,}287.9=3{,}287.9$ thousand dollars, the VaR threshold. The calculation rounds only the final amounts to one decimal place in thousands of dollars. This formula relies on the specified normal model; it is not a general quantile formula for arbitrary loss distributions.

Worked Example: Aggregated Scenario Losses with a Discrete Boundary

A hypothetical firm assesses two loss sources together using five joint scenarios. The table gives each scenario’s total loss, in thousands of dollars, and its probability. Calculate the 95% VaR and capital above expected loss.

Scenario	Total loss (thousands of dollars)	Probability
A	0	0.70
B	100	0.10
C	200	0.10
D	300	0.06
E	600	0.04

Step 1: Verify the probability model. The probabilities sum to $0.70+0.10+0.10+0.06+0.04=1.00$. The losses are already in increasing order.

Step 2: Find the quantile. The cumulative probabilities at losses 0, 100, 200, 300, and 600 are 0.70, 0.80, 0.90, 0.96, and 1.00, respectively. Since 0.90 is below 0.95 and 0.96 is at least 0.95, $\operatorname{VaR}_{0.95}(X)=300$ thousand dollars. The 600-thousand-dollar outcome is not the 95% VaR: it is beyond the first loss value that reaches the required cumulative probability.

Step 3: Calculate expected loss.

$$ E[X]=0(0.70)+100(0.10)+200(0.10)+300(0.06)+600(0.04) =0+10+20+18+24=72 $$

Expected loss is 72 thousand dollars. Therefore, capital above expected loss is $300-72=228$ thousand dollars; total loss resources at the VaR threshold are 300 thousand dollars. Check: $72+228=300$, and the cumulative probability at 300 is 0.96, which is at least 0.95. The common error avoided here is choosing the loss whose cumulative probability equals or first exceeds the tail probability $1-\alpha=0.05$; VaR is located using the cumulative probability $\alpha=0.95$.

Common Mistakes and Checks

A reliable answer states both the quantile and the capital basis. In particular, do not let the phrase “economic capital” obscure whether the question treats expected loss as already funded. Read the wording: “capital above expected loss” calls for subtraction, while “total resources to cover losses at the stated confidence level” calls for the VaR amount itself.

Using the wrong side of the probability. At confidence level $\alpha$, locate the $\alpha$-quantile of the loss distribution. The exceedance probability is $1-\alpha$, but that is not the cumulative probability to use when accumulating losses from smallest to largest.
Skipping a discrete outcome. Choose the first loss at which cumulative probability is at least $\alpha$. Do not interpolate between discrete loss values unless a model explicitly defines losses between them.
Subtracting expected loss twice. Calculate $E[X]$ once, and subtract it only when the requested capital is above separately funded expected loss.
Assuming a formula beyond its model. The normal quantile expression uses a normal loss model with the stated mean and standard deviation. Other distributions require their own quantile calculation.
Mixing units. If losses and expected loss are reported in thousands of dollars, keep both in thousands through the subtraction and label the final result accordingly.

Key takeaway: First find $\operatorname{VaR}_{\alpha}(X)$ from the loss distribution. Then report VaR as the total loss threshold, or calculate $\operatorname{VaR}_{\alpha}(X)-E[X]$ when the requested capital is specifically measured above expected loss.

Check Your Understanding

For each problem, identify the quantile before applying any expected-loss adjustment. All monetary amounts are hypothetical.

A one-year loss, in thousands of dollars, is 0 with probability 0.60, 50 with probability 0.25, 200 with probability 0.10, and 500 with probability 0.05. Find the 90% VaR and capital above expected loss. (Answer: VaR = 200 thousand dollars; expected loss = 57.5 thousand dollars; capital = 142.5 thousand dollars.)
A hypothetical loss is normal with mean 750 thousand dollars and standard deviation 120 thousand dollars. Using $z_{0.975}=1.9600$, find the 97.5% VaR and capital above expected loss. (Answer: VaR = 985.2 thousand dollars; capital = 235.2 thousand dollars.)
A loss distribution, in thousands of dollars, has outcomes 20, 80, 140, and 400 with probabilities 0.50, 0.30, 0.15, and 0.05, respectively. Find the 90% VaR, expected loss, and capital above expected loss. (Answer: VaR = 140 thousand dollars; expected loss = 75 thousand dollars; capital = 65 thousand dollars.)
A hypothetical risk committee asks for total loss resources at the 95% confidence level, not capital above expected loss. Losses in thousands of dollars are 0 with probability 0.70, 100 with probability 0.20, and 500 with probability 0.10. Find the required resources and, for comparison, calculate the capital above expected loss. (Answer: VaR and required total resources = 500 thousand dollars; expected loss = 70 thousand dollars; capital above expected loss = 430 thousand dollars.)
A portfolio’s one-year loss is modeled as normal with mean 1,200 thousand dollars and standard deviation 300 thousand dollars. Using $z_{0.99}=2.3263$, calculate the 99% VaR and state the capital above expected loss. (Answer: VaR = 1,897.89 thousand dollars; capital above expected loss = 697.89 thousand dollars.)

What You'll Learn