Integrated Risk Measure and Capital Decision Case

From Aggregate Loss to a Capital Decision

A portfolio capital decision often needs more than an estimate of average loss. A measure of the upper tail can describe how severe losses may be at a chosen confidence level, while a capital measure above expected loss separates ordinary expected claims from protection against adverse outcomes. The decision also depends on how component losses occur together: dependence can change aggregate tail risk even when each component’s loss distribution is unchanged.

This case uses a one-year horizon and hypothetical losses measured in thousands of dollars. There is no discounting, interest, or mortality contingency in the model. The goal is to calculate aggregate value-at-risk (VaR), tail value-at-risk (TVaR), capital above expected loss, and a component allocation, then assess how an independence assumption changes the result.

Definition: For an aggregate loss $S$ with distribution function $F_S$, the VaR at confidence level $p$ is the lower quantile $ \operatorname{VaR}_p(S)=\inf\{s:F_S(s)\ge p\} $. The TVaR at level $p$ is the average of the upper quantiles: $ \operatorname{TVaR}_p(S)=\frac{1}{1-p}\int_p^1\operatorname{VaR}_q(S)\,dq $. For a discrete loss, this definition includes only the fraction of probability at the VaR threshold needed to form an upper tail of probability $1-p$. It is not generally the same as $E[S\mid S\ge \operatorname{VaR}_p(S)]$.

For a discrete $S$, let $v=\operatorname{VaR}_p(S)$. A useful equivalent expression is $ \operatorname{TVaR}_p(S)=v+\frac{E[(S-v)^+]}{1-p} $, where $(x)^+=\max(x,0)$. This form handles an atom at $v$ correctly: losses strictly above $v$ contribute their excess, and the remaining tail probability is filled by part of the mass at $v$.

Definition: In this case, capital above expected loss at level $p$ is $ K_p=\operatorname{TVaR}_p(S)-E[S] $. It is the tail-risk measure less the expected aggregate loss. It is not the total amount of capital held, and it is not automatically a regulatory or solvency requirement.

Worked Portfolio: VaR, TVaR, and Total Capital

Consider a hypothetical insurer with two one-year loss components, $X_1$ and $X_2$, in thousands of dollars. The components represent two hypothetical coverages. Their joint distribution is specified directly, so the dependence structure is part of the model rather than an assumption to infer.

Worked Example: A 95% Tail-Risk Capital Decision

Use the following joint outcomes to calculate $E[S]$, 95% VaR, 95% TVaR, and capital above expected loss, where $S=X_1+X_2$. A hypothetical capital limit is $150,000. Decide whether the modelled capital above expected loss is within that limit.

Probability	$X_1$, thousands	$X_2$, thousands	$S$, thousands
0.80	0	0	0
0.10	60	0	60
0.06	0	100	100
0.04	80	120	200

Method. First obtain the aggregate distribution and mean. Then identify the 95th percentile by cumulative probability and calculate TVaR using the excess-over-VaR formula. This avoids treating every loss equal to VaR as part of the tail.

Expected loss. The aggregate expectation is $E[S]=0(0.80)+60(0.10)+100(0.06)+200(0.04)=0+6+6+8=20$ thousand dollars. As a check, the component means are $E[X_1]=60(0.10)+80(0.04)=9.2$ and $E[X_2]=100(0.06)+120(0.04)=10.8$, and their sum is $20$.

VaR. The cumulative probabilities are $F_S(0)=0.80$, $F_S(60)=0.90$, $F_S(100)=0.96$, and $F_S(200)=1.00$. Since $0.90<0.95\le0.96$, $\operatorname{VaR}_{0.95}(S)=100$ thousand dollars.

TVaR. At $v=100$, only the 200 outcome has a positive excess, equal to $200-100=100$, with probability $0.04$. Thus $E[(S-v)^+]=100(0.04)=4$, and $\operatorname{TVaR}_{0.95}(S)=100+\frac{4}{0.05}=100+80=180$ thousand dollars.

Capital decision. Capital above expected loss is $K_{0.95}=180-20=160$ thousand dollars. The hypothetical $150,000 limit is exceeded by $10,000. The result is expressed in thousands throughout: the TVaR is $180,000, expected loss is $20,000, and the difference is $160,000.

Check. The upper 5% consists of all 4% probability at loss 200 and 1% probability at loss 100. Its mean is $[0.04(200)+0.01(100)]/0.05=9/0.05=180$ thousand dollars, confirming the TVaR calculation. The method avoids the mistake of including the full 6% mass at 100 in a conditional average.

Why the Discrete TVaR Formula Works

The definition of TVaR averages quantiles over probability levels from $p$ to 1. Imagine moving through this upper tail in probability order. If the distribution has a discrete jump at $v=\operatorname{VaR}_p(S)$, only the fraction of that jump needed to complete tail probability $1-p$ is included. The contribution of all included losses can be written as $v$ for each unit of tail probability, plus the excess above $v$ for outcomes larger than $v$.

\int_p^1 \operatorname{VaR}_q(S)\,dq =(1-p)v+E[(S-v)^+]

To see the excess term, a loss $S>v$ contributes $S-v$ over its full probability mass; a loss at or below $v$ contributes no excess. Dividing both sides by $1-p$ gives $ \operatorname{TVaR}_p(S)=v+\frac{E[(S-v)^+]}{1-p} $. This derivation also explains why simply averaging observations satisfying $S\ge v$ can be wrong: that conditional average may include more than $1-p$ probability when there is mass at $v$.

Allocating Tail Risk to Components

For a discrete portfolio, a clear allocation of TVaR can be made by using the same fractional-tail rule for each component. Include all outcomes strictly above the aggregate VaR and the required fraction of outcomes at VaR, weighting each component by its loss in those selected outcomes. With this rule, the component allocations add to aggregate TVaR. Subtracting each component’s expected loss then allocates capital above expected loss.

Result: If a tail of probability $1-p$ is formed by selecting all outcomes with $S>v$ and the necessary fraction of probability at $S=v$, define $C_i$ as the selected-tail average of component loss $X_i$. Then $\sum_i C_i=\operatorname{TVaR}_p(S)$, because $S=\sum_i X_i$ in every selected outcome. The corresponding capital allocation is $C_i-E[X_i]$, and these allocations sum to $K_p$.

Worked Example: Allocating the Portfolio’s Tail Capital

Use the joint distribution in the preceding case to allocate 95% TVaR and capital above expected loss to the two components. Outcomes are measured in thousands of dollars.

Method. The 95% VaR is 100. The upper 5% tail contains the entire 4% outcome $(80,120)$, plus 1% of the 6% probability at $(0,100)$. The latter is one-sixth of that outcome’s probability mass.

Allocate TVaR. Component 1 contributes $0.04(80)+0.01(0)=3.2$ thousand-dollar probability-weighted units. Dividing by tail probability $0.05$ gives $C_1=3.2/0.05=64$ thousand dollars. Component 2 contributes $0.04(120)+0.01(100)=4.8+1=5.8$, giving $C_2=5.8/0.05=116$ thousand dollars.

Check the allocation. $C_1+C_2=64+116=180$, equal to aggregate TVaR. Expected losses are $E[X_1]=9.2$ and $E[X_2]=10.8$, so allocated capital above expected loss is $64-9.2=54.8$ for component 1 and $116-10.8=105.2$ for component 2. Their sum, $54.8+105.2=160$, matches total capital.

Common mistake avoided. Averaging component losses only in outcomes with $S\ge100$ would include the full 6% probability at aggregate loss 100 as well as the 4% at 200. That is a 10% conditional sample, not the required upper 5% tail, and it produces a different allocation.

Dependence Sensitivity with the Same Marginals

A portfolio decision should identify whether its aggregate distribution depends on an assumed dependence structure. The previous joint model gives $X_1$ marginal probabilities $P(X_1=0)=0.86$, $P(X_1=60)=0.10$, and $P(X_1=80)=0.04$. It gives $X_2$ marginal probabilities $P(X_2=0)=0.90$, $P(X_2=100)=0.06$, and $P(X_2=120)=0.04$. Independence preserves these component distributions and their expected losses, but changes which losses occur together.

Worked Example: Recalculating Capital Under Independence

Assume the same component marginals as above, but now assume $X_1$ and $X_2$ are independent. Calculate the aggregate distribution, 95% VaR, 95% TVaR, and capital above expected loss. Compare the capital with the dependent case.

Method. Under independence, each joint probability is the product of the corresponding marginal probabilities. For example, $P(X_1=0,X_2=100)=0.86(0.06)=0.0516$, and $P(X_1=60,X_2=120)=0.10(0.04)=0.004$. Add probabilities when different component pairs produce the same total.

Aggregate distribution. The possible aggregate losses and probabilities are: 0 with probability $0.86(0.90)=0.774$; 60 with probability $0.10(0.90)=0.090$; 80 with probability $0.04(0.90)=0.036$; 100 with probability $0.86(0.06)=0.0516$; 120 with probability $0.86(0.04)=0.0344$; 160 with probability $0.10(0.06)=0.006$; 180 with probability $0.10(0.04)+0.04(0.06)=0.004+0.0024=0.0064$; and 200 with probability $0.04(0.04)=0.0016$. These probabilities sum to $0.774+0.090+0.036+0.0516+0.0344+0.006+0.0064+0.0016=1.0000$.

VaR. In ascending order, the cumulative probabilities are 0.774 at 0, 0.864 at 60, 0.900 at 80, and 0.9516 at 100. Therefore $\operatorname{VaR}_{0.95}(S)=100$ thousand dollars.

TVaR. The positive excesses above 100 are $20$ at probability 0.0344, $60$ at probability 0.006, $80$ at probability 0.0064, and $100$ at probability 0.0016. Their weighted sum is $20(0.0344)+60(0.006)+80(0.0064)+100(0.0016)=0.688+0.360+0.512+0.160=1.720$. Hence $\operatorname{TVaR}_{0.95}(S)=100+1.720/0.05=134.4$ thousand dollars.

Capital and comparison. The expected aggregate loss remains $9.2+10.8=20$ thousand dollars, so capital above expected loss is $134.4-20=114.4$ thousand dollars. This is $160-114.4=45.6$ thousand dollars lower than under the specified dependent joint model. The component marginals and expected losses are unchanged; the aggregate tail differs because the losses occur together less often under independence.

Check. The upper 5% under independence contains all outcomes above 100, with probability $0.0484$, and a further $0.0016$ probability at 100. Its mean is $[100(0.0016)+120(0.0344)+160(0.006)+180(0.0064)+200(0.0016)]/0.05=6.72/0.05=134.4$ thousand dollars. The method avoids assuming that unchanged marginal distributions imply unchanged portfolio risk.

Interpreting the Result and Avoiding Traps

In the hypothetical decision, the specified dependent model gives capital above expected loss of $160,000, above the $150,000 limit. The independence model gives $114,400, below that limit. This comparison does not establish which dependence model is appropriate; it shows that the capital conclusion is sensitive to that modelling choice. A decision-maker should therefore state the joint-loss assumption and assess whether it is defensible for the portfolio and horizon being modelled.

Do not confuse VaR and TVaR. VaR marks a percentile threshold; TVaR averages the losses in the upper tail.
Handle probability at VaR carefully. For discrete losses, include only the fraction at the threshold that completes a tail of probability $1-p$.
Keep units consistent. If losses are in thousands of dollars, means, tail measures, allocations, and capital must also be reported in thousands until converted.
Do not assume independence without checking its effect. The aggregate distribution depends on how component losses occur together, not only on their separate distributions.
Check allocations add up. Component TVaR allocations should sum to aggregate TVaR, and component capital allocations should sum to total capital above expected loss.

These calculations are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus: they combine aggregate tail-risk measures, capital above expected loss, component allocation, and sensitivity to dependence in one decision framework.

Check Your Understanding

For each hypothetical one-year loss model, use the lower-quantile VaR convention and the fractional-tail definition of TVaR. Monetary amounts are in thousands of dollars unless stated otherwise.

A portfolio loss is 0 with probability 0.70, 40 with probability 0.20, and 100 with probability 0.10. Calculate 80% VaR and 80% TVaR. (Answer: VaR = 40; TVaR = 70, in thousands.)
A loss is 0 with probability 0.50, 20 with probability 0.30, and 80 with probability 0.20. Calculate expected loss and 90% capital above expected loss. (Answer: $E[S]=22$, $\operatorname{TVaR}_{0.90}=80$, and capital = 58, in thousands.)
A two-component portfolio has outcomes $(X_1,X_2)=(0,0)$ with probability 0.80, $(40,0)$ with probability 0.10, and $(20,80)$ with probability 0.10. At 90%, calculate aggregate VaR, aggregate TVaR, and each component’s TVaR allocation. (Answer: aggregate VaR = 40; aggregate TVaR = 100; allocations are 20 to $X_1$ and 80 to $X_2$, in thousands.)
A portfolio has a 95% TVaR of $134,400 and expected loss of $20,000 under a specified dependence model. A decision limit for capital above expected loss is $120,000. Does the modelled amount meet the limit, and by how much? (Answer: capital is $114,400, so it is $5,600 below the limit.)

What You'll Learn