Calculating VaR from Aggregate Loss Models

Introduction

A capital decision often asks how large aggregate losses could be at a specified confidence level. The answer depends on the distribution of total losses over a clearly defined period, not just on average claim size or the chance of a single large claim. Value at Risk (VaR) summarizes one point in that aggregate-loss distribution. Comparing it with expected losses gives a measure of capital above the expected amount.

The calculation may start with a frequency-severity model or with simulated aggregate outcomes. In either case, the key is to use the correct quantile convention, particularly when aggregate losses are discrete. This tutorial is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: For an aggregate loss $X$ with cumulative distribution function $F_X(x)=P(X\le x)$, the VaR at confidence level $\alpha$, where $0<\alpha<1$, is $$ \operatorname{VaR}_{\alpha}(X)=\inf\{x:F_X(x)\ge \alpha\}. $$ The capital amount above expected losses, under the convention used here, is $\operatorname{VaR}_{\alpha}(X)-E[X]$. The loss period and the basis of $X$ must be the same for both quantities.

From a Compound Model to Aggregate VaR

A standard aggregate-loss model has a claim count $N$ and claim amounts $Y_1,Y_2,\ldots$. The aggregate loss is $X=\sum_{j=1}^{N}Y_j$, with $X=0$ when $N=0$. The usual compound model assumes that the claim amounts are identically distributed and independent of one another, and that $N$ is independent of the claim amounts. The frequency distribution specifies $N$; the severity distribution specifies each $Y_j$.

To calculate an exact VaR, determine the aggregate distribution, then find the smallest loss value at which its cumulative probability reaches or exceeds $\alpha$. For a discrete distribution, the cumulative probability can jump over $\alpha$. In that case, VaR is the loss at the jump, not an interpolated value. This also means that $P(X>\operatorname{VaR}_{\alpha}(X))$ need not equal $1-\alpha$.

Result: If $N$ is independent of iid claim amounts $Y_j$, with finite means, then $$ E[X]=E[N]E[Y_1]. $$ This gives the expected-loss amount to compare with VaR. It does not by itself determine VaR: a quantile depends on the shape and tail of the full aggregate distribution, not only on its mean.

Worked Example: Poisson Frequency With a Fixed Claim Amount

A hypothetical one-year portfolio has $N\sim\operatorname{Poisson}(2)$ claims. Each claim is exactly $1,000, independently of the count. Calculate the 90% VaR and the capital amount above expected losses. Assume all amounts are for the same one-year period, with no discounting or other adjustments.

Method. Since each claim is $1,000, aggregate loss is $X=1{,}000N$. Thus the aggregate quantile is found by locating the 90% quantile of the Poisson count and then converting claims to dollars.

For a Poisson random variable with parameter 2, $P(N=k)=e^{-2}2^k/k!$. The cumulative probability through three claims is $$ P(N\le 3)=e^{-2}\left(1+2+\frac{2^2}{2!}+\frac{2^3}{3!}\right) =e^{-2}\left(1+2+2+\frac{4}{3}\right) =0.8571. $$ This is below 0.9000. Through four claims, $$ P(N\le 4)=P(N\le 3)+e^{-2}\frac{2^4}{4!} =0.8571+0.0902=0.9473. $$ This is at least 0.9000, so the smallest qualifying count is four. Therefore, $\operatorname{VaR}_{0.90}(X)=4(1{,}000)=$$4,000.

The expected claim count is 2, so $E[X]=2($1{,}000)=$2{,}000$. Capital above expected losses is $$ $4{,}000-$2{,}000=$2{,}000. $$ Check and common mistake. The cumulative probabilities satisfy $0.8571<0.9000\le0.9473$, confirming the quantile. Using three claims because its cumulative probability is “close” would give only 85.71% coverage and understate VaR.

Why the Quantile Rule Works

The definition is an inverse-CDF rule. It selects the first loss level whose cumulative probability includes at least an $\alpha$ fraction of outcomes. For a discrete aggregate distribution with ordered possible losses $x_1<x_2<\cdots$, let $F_X(x_j)=\sum_{k\le j}P(X=x_k)$. The VaR is the first $x_j$ for which this sum is at least $\alpha$. For the preceding value, $F_X(x_{j-1})<\alpha$. Consequently, the selected loss is the smallest threshold meeting the confidence requirement. If the cumulative probability jumps from below $\alpha$ to above it, no loss between the two possible outcomes has that cumulative probability; interpolation would not follow the inverse-CDF definition.

The expected aggregate loss follows from conditioning on the claim count. Given $N=n$, linearity of expectation gives $E[X\mid N=n]=nE[Y_1]$. Taking expectations over $N$ gives $E[X]=E[N]E[Y_1]$, where independence ensures the same claim severity distribution applies regardless of the count. This mean identity is useful for the capital subtraction, but the quantile still requires the distribution or simulated outcomes.

Worked Example: Building the Aggregate Distribution

A hypothetical one-period cover has $N\sim\operatorname{Binomial}(2,0.5)$. Given a claim, its amount is $100 or $300 with probability 0.5 each. Claim amounts are independent of the count and of one another. Find the 90% VaR and capital above expected losses. Assume a common one-period basis and no discounting.

Method. Enumerate count outcomes and conditional severity outcomes, combine probabilities for equal aggregate amounts, and apply the inverse-CDF rule.

The count probabilities are $P(N=0)=0.25$, $P(N=1)=0.50$, and $P(N=2)=0.25$. With no claims, aggregate loss is zero with probability 0.25. With one claim, aggregate loss is $100 or $300, each with unconditional probability $0.50(0.50)=0.25$.

For two claims, the possible totals are $200, $400, and $600. Conditional on two claims, their probabilities are 0.25, 0.50, and 0.25, respectively. Multiplying by $P(N=2)=0.25$ gives unconditional probabilities 0.0625, 0.1250, and 0.0625. The complete distribution is:

Aggregate loss	Probability	Cumulative probability
$0	0.2500	0.2500
$100	0.2500	0.5000
$200	0.0625	0.5625
$300	0.2500	0.8125
$400	0.1250	0.9375
$600	0.0625	1.0000

The cumulative probability first reaches 0.9000 at $400, so $\operatorname{VaR}_{0.90}(X)=$400$. The mean severity is $0.5($100)+0.5($300)=$200$, and $E[N]=2(0.5)=1$. Therefore $E[X]=1($200)=$200$, and capital above expected losses is $400 − $200 = $200.

Check and common mistake. The listed probabilities sum to 1.0000, and $0.8125<0.9000\le0.9375$, confirming the VaR. A common error is to treat the $300 outcome as the 90% quantile because it is the largest one-claim amount. That ignores the probability of two claims and the aggregate distribution.

VaR From Simulation Output

When aggregate outcomes are simulated, an empirical quantile can be calculated from the ordered simulation results. For this tutorial, use the empirical inverse-CDF convention: sort $m$ simulated aggregate losses from smallest to largest, and take the observation with rank $\lceil m\alpha\rceil$, counting from 1. This convention corresponds to the inverse of the empirical cumulative distribution function. Other software may use interpolation conventions, so a reported simulation quantile should state the convention used.

The simulated VaR is an estimate, not an exact model quantile. Its precision depends on the number of simulations and the amount of information in the upper tail. The capital calculation also requires an expected-loss amount on the same basis. If the expected loss is supplied from the model, do not replace it with the simulated sample mean unless the task specifically calls for that estimate.

Worked Example: Empirical VaR From Simulated Annual Losses

A hypothetical insurer simulates ten aggregate annual losses, recorded in thousands of dollars: 0.4, 0.8, 0.9, 1.0, 1.1, 1.2, 1.4, 1.7, 2.1, and 3.0. The model’s expected annual loss, calculated separately, is $1,200. Estimate the 90% VaR and capital above expected losses using the empirical inverse-CDF convention. No discounting applies.

Method. The ten results are already in increasing order. The selected rank is $\lceil m\alpha\rceil=\lceil10(0.90)\rceil=9$. The ninth result is 2.1 thousand dollars, so the empirical 90% VaR is $2,100.

Subtract the supplied expected annual loss on the same basis: $$ \text{Capital above expected losses}=$2{,}100-$1{,}200=$900. $$ The simulated VaR is the ninth observation, not the average of the ninth and tenth. The result is consistent with the stated convention: eight of the ten simulated outcomes are below $2,100, and nine are at or below it. The small simulation size makes this an illustrative estimate, not a highly stable tail estimate.

Common mistake avoided. Using rank 10 would select the largest simulated loss and estimate the 100% sample maximum, not the empirical 90% quantile under the specified inverse-CDF convention.

Interpreting the Capital Amount

VaR and expected loss answer different questions. Expected loss is the probability-weighted average across all outcomes. VaR is a threshold associated with a selected cumulative probability. Their difference is a capital amount above the expected loss under the stated convention; it is not itself a probability, and it is not a forecast of the loss that will occur.

For a high confidence level and a loss distribution with a substantial upper tail, VaR will generally exceed the mean. If a problem produces a quantile below the mean, the subtraction is still mathematically defined, but it should not be described as a positive capital buffer. A capital policy may impose a nonnegative floor, but that is an additional policy rule and should only be applied when specified.

Match the basis: VaR and expected loss must refer to the same period, covered losses, currency, and gross or net treatment.
Respect discreteness: choose the smallest loss with cumulative probability at least $\alpha$; do not interpolate unless a different convention is explicitly required.
Use the right simulation rank: state the empirical quantile convention and apply its indexing rule consistently.
Keep the two calculations separate: first calculate the quantile, then subtract expected loss.

Key takeaway: Calculate aggregate VaR by finding the smallest loss threshold whose cumulative probability reaches the specified confidence level. Then subtract expected aggregate loss on the same basis to obtain capital above expected losses.

Check Your Understanding

Use the inverse-CDF definition for model distributions and the empirical inverse-CDF convention for simulation results.

A hypothetical portfolio has $N\sim\operatorname{Poisson}(1)$, with every claim equal to $500. Find the 80% VaR and capital above expected losses. (Answer: VaR = $1,000; capital = $500.)
A one-period model has $N\sim\operatorname{Binomial}(2,0.4)$, with every claim equal to $1,000. Find the 90% VaR and capital above expected losses. (Answer: VaR = $2,000; capital = $1,200.)
Exactly two independent claims occur. Each is $100 with probability 0.8 and $500 with probability 0.2. Find the 95% VaR and capital above expected losses. (Answer: VaR = $600; capital = $240.)
Ten simulated aggregate losses, in thousands of dollars, are 0.5, 0.7, 0.8, 1.0, 1.1, 1.3, 1.6, 2.0, 2.4, and 3.2. The model’s expected loss is $1,460. Find the empirical 80% VaR and capital above expected losses. (Answer: VaR = $2,000; capital = $540.)
A hypothetical portfolio has $N\sim\operatorname{Poisson}(2)$, with every claim equal to $250. Find the 95% VaR and capital above expected losses. (Answer: VaR = $1,250; capital = $750.)

What You'll Learn