Three Capital Quantities, Three Questions
A capital model can report several numbers for the same business units. A unit’s standalone capital answers, “How much capital would this unit require if considered by itself?” Diversified capital answers, “How much capital does the combined portfolio require?” Allocated capital answers, “How much of that portfolio total is assigned to each unit under a specified allocation rule?” These quantities are related, but they are not interchangeable.
The comparisons are meaningful only when the losses are measured on a consistent basis: the same time horizon, valuation basis, confidence level or other risk-measure settings, and treatment of expenses and recoveries. For example, comparing a one-year standalone measure with a multi-year portfolio measure does not isolate diversification.
A common choice of risk measure is Value-at-Risk at confidence level \(q\). For loss \(X\), define \(\operatorname{VaR}_q(X)\) as the smallest loss threshold at which the cumulative probability of loss is at least \(q\). In a discrete loss model, this definition matters: the VaR is a loss value, not an average of the losses in the worst outcomes.
Using VaR throughout the examples makes the comparisons concrete. Other capital measures can produce different numerical results, so a capital figure is incomplete unless its measure and settings are understood.
Standalone and Diversified Capital From a Joint Loss Model
Standalone capital depends on each unit’s own loss distribution. Diversified capital depends on the distribution of the sum, which in turn depends on how the units’ losses occur together. The individual distributions alone do not specify that joint behavior.
Worked Example: Independent Unit Losses at the 95% VaR Level
Consider two hypothetical insurance portfolios over one year. Unit A has a loss of $100 with probability 0.10 and otherwise has no loss. Unit B has a loss of $80 with probability 0.10 and otherwise has no loss. Assume the two losses are independent. Calculate the standalone 95% VaR for each unit, the portfolio’s diversified 95% VaR, and the diversification benefit.
Method. Find each unit’s loss quantile, then construct the joint distribution of the total loss. The independence assumption is used to multiply the probabilities of the two units’ outcomes.
For Unit A, the loss is $0 with probability 0.90 and $100 with probability 0.10. Thus \(F_{X_1}(0)=0.90\), which is below 0.95, while \(F_{X_1}(100)=1.00\). Therefore, \(K_1=\operatorname{VaR}_{0.95}(X_1)=\$100\).
For Unit B, the loss is $0 with probability 0.90 and $80 with probability 0.10. Thus \(F_{X_2}(0)=0.90\) and \(F_{X_2}(80)=1.00\). Therefore, \(K_2=\operatorname{VaR}_{0.95}(X_2)=\$80\). The sum of standalone capitals is \(100+80=\$180\).
Under independence, the four joint outcomes and their probabilities are:
- Neither unit has a loss: probability \(0.90(0.90)=0.81\), total loss $0.
- Only Unit B has a loss: probability \(0.90(0.10)=0.09\), total loss $80.
- Only Unit A has a loss: probability \(0.10(0.90)=0.09\), total loss $100.
- Both units have losses: probability \(0.10(0.10)=0.01\), total loss $180.
The probabilities sum to \(0.81+0.09+0.09+0.01=1.00\). In increasing order of total loss, the cumulative probabilities are 0.81 at $0, 0.90 at $80, 0.99 at $100, and 1.00 at $180. Since 0.90 is below 0.95 and 0.99 is at least 0.95, the portfolio’s diversified capital is \(K_P=\operatorname{VaR}_{0.95}(X_1+X_2)=\$100\).
The diversification benefit, defined here as the sum of standalone capitals minus diversified capital, is \(100+80-100=\$80\). The check is that the portfolio VaR of $100 is below the $180 sum of standalone VaRs by exactly $80.
Common mistake avoided: Adding the standalone VaRs and reporting $180 as the portfolio VaR skips the joint-distribution calculation. The sum is the standalone total, not the portfolio’s 95% VaR in this model.
Why the Joint Distribution Determines Portfolio Capital
The calculation above follows directly from the definition of VaR. For any candidate threshold \(x\), the cumulative probability for the portfolio is \(F_{X_P}(x)=P(X_1+X_2\leq x)\). The portfolio VaR is the first threshold at which this cumulative probability reaches the selected confidence level.
The marginal loss distributions describe each unit separately; an assumption or model for how the losses occur together is also needed.
For each possible joint outcome, add the unit losses and assign the probability of that outcome.
Accumulate probabilities in order of total loss and select the first loss amount whose cumulative probability is at least \(q\).
This is also why standalone capital figures are not sufficient to recover diversified capital. They summarize the units’ individual distributions at a chosen quantile, but they do not show whether high losses occur at the same time. The joint model supplies that information.
Allocated Capital Is an Assignment of the Portfolio Total
An allocation is a further step: it assigns the portfolio capital total to units under a specified basis, constraints, and reconciliation requirement. The allocated amount for a unit is not automatically equal to its standalone capital. Nor does a unit’s standalone figure, by itself, determine its allocated share.
The basic full-allocation check is arithmetic: the unit allocations must add to the diversified portfolio capital. That check confirms reconciliation, but it does not prove that the chosen allocation basis is appropriate. A complete interpretation therefore distinguishes the model’s capital outputs from the policy or method used to assign the total.
Worked Example: Interpreting Reconciled Allocated Shares
Use the independent-loss model from the first example, so Unit A has standalone 95% VaR of $100, Unit B has standalone 95% VaR of $80, and the portfolio’s diversified 95% VaR is $100. Suppose a hypothetical insurer’s approved allocation process assigns $64 to A and $36 to B. Check the allocation and compare each allocated amount with its standalone capital.
Method. First reconcile the assigned shares to portfolio capital. Then compare each assigned amount with the corresponding standalone result; do not treat the standalone figures as the allocation rule.
The total allocated capital is \(64+36=\$100\), equal to the diversified portfolio capital of $100. The allocation therefore fully reconciles.
Unit A is assigned $64, which is \(100-64=\$36\) below its standalone capital of $100. Unit B is assigned $36, which is \(80-36=\$44\) below its standalone capital of $80. The allocated shares sum to $100, whereas the standalone capitals sum to $180.
The calculation does not establish that $64 and $36 are the uniquely correct shares. They are the outputs of the stated hypothetical allocation process. Assessing whether that process is suitable requires evaluating its allocation basis and constraints, in addition to checking reconciliation.
Common mistake avoided: Replacing the stated allocation with the standalone amounts would assign \(100+80=\$180\), which does not reconcile to the portfolio capital of $100. Standalone capital is a unit-level risk measure, not an automatic allocation.
Dependence Can Change Diversified Capital Without Changing Standalone Capital
The next example keeps the individual loss distributions unchanged but changes their dependence. This isolates why a portfolio result cannot be inferred from standalone results alone.
Worked Example: The Same Marginals With a Common Loss Event
Use the same hypothetical units and one-year losses: A loses $100 with probability 0.10, and B loses $80 with probability 0.10. Now assume both losses occur together with probability 0.10, and neither occurs with probability 0.90. Calculate standalone and diversified 95% VaR, and show two possible full allocations of portfolio capital.
Method. Preserve the standalone distributions, construct the new joint distribution from the common-event assumption, and evaluate the 95% quantiles. Then check that each proposed allocation sums to the portfolio total.
The standalone distributions have not changed, so the standalone 95% VaRs remain \(K_1=\$100\) and \(K_2=\$80\). The portfolio loss is $0 with probability 0.90 and \(100+80=\$180\) with probability 0.10. Its cumulative probability is 0.90 at $0 and 1.00 at $180. Since 0.90 is below 0.95, the diversified 95% VaR is \(K_P=\$180\).
The diversification benefit is \(100+80-180=\$0\). Under the specified common-event model, the two high losses coincide, so there is no reduction in the 95% VaR relative to the sum of standalone VaRs.
For example, allocations of $100 to A and $80 to B reconcile because \(100+80=\$180\). Allocations of $90 to A and $90 to B also reconcile because \(90+90=\$180\). These are two different full allocations of the same portfolio capital. Without a specified allocation rule or additional constraints, the portfolio total does not select one of them.
Common mistake avoided: Keeping the independent-model portfolio VaR of $100 after changing the dependence assumption ignores that the joint outcomes have changed. The standalone VaRs remain the same, but the portfolio VaR becomes $180.
Interpreting Diversification and Avoiding Exam Traps
For a consistent risk measure, the diversification benefit is commonly expressed as the sum of standalone capitals less the diversified portfolio capital. This is a comparison, not an assumption that the benefit must be positive. Its sign and size depend on the capital measure and the modeled joint losses.
In the independent-loss example, the benefit was $80; in the common-event example, it was $0. Those results arose from different joint distributions, despite identical standalone loss distributions. Depending on the risk measure and model, a calculated benefit can also be negative. Therefore, do not describe diversification as a guaranteed reduction without checking the actual model result.
Keep these distinctions in view when reading a capital report:
- Standalone capital is calculated for a unit considered individually under the chosen measure.
- Diversified capital is calculated for the combined loss distribution, including the modeled dependence.
- Allocated capital assigns the portfolio total across units under a specified allocation basis. Full allocation requires the assigned amounts to sum to the portfolio total.
A frequent exam trap is to see several capital numbers and add or compare them without checking what each represents. Before calculating, identify the loss horizon, risk measure, confidence level, dependence assumption, and whether a reported unit amount is standalone or allocated. Then verify any full allocation by summing its shares.
Check Your Understanding
For each problem, state whether the requested figure is standalone, diversified, or allocated capital, and check any required reconciliation.
- Unit C has a loss of $60 with probability 0.10 and zero otherwise. Unit D has a loss of $40 with probability 0.10 and zero otherwise. The losses are independent. Find each standalone 95% VaR, the portfolio 95% VaR, and the diversification benefit. (Answer: Standalone VaRs are $60 and $40; portfolio VaR is $60; diversification benefit is $40.)
- Two units each have a loss of $75 with probability 0.08 and zero otherwise. Their losses occur together with probability 0.08 and are both zero otherwise. Find the standalone and portfolio 95% VaRs, then calculate the diversification benefit. (Answer: Standalone VaRs are $75 each; portfolio VaR is $150; diversification benefit is $0.)
- A portfolio has diversified capital of $120. Its allocation report assigns $68 to Unit M and $52 to Unit N. Check full allocation and state whether those figures can be called the units’ standalone capitals based on the information given. (Answer: They reconcile because $68 + $52 = $120; they cannot be identified as standalone capitals without separate unit-level calculations.)
- Two units have standalone capital of $90 and $70, while the portfolio’s diversified capital is $125. The allocation rule assigns $75 and $50. Calculate the diversification benefit and check the allocations. (Answer: Diversification benefit is $35; allocations reconcile because $75 + $50 = $125.)
- Two units have standalone capital of $100 and $80, and portfolio capital of $150. An analyst reports allocated shares of $85 and $70. Identify the reconciliation error and give the amount by which the total allocation differs from portfolio capital. (Answer: The shares total $155, so they exceed portfolio capital by $5 and do not fully reconcile.)