Aggregating Risk Charges and Diversification

Why Risk Charges Are Aggregated

An insurer may face several distinct sources of risk, each with its own prescribed capital charge. Adding the charges gives a conservative undiversified total, but it does not reflect how the risks may move together. A prescribed correlation-based aggregation rule adjusts the total for the assumed dependence between risks.

The calculation is mechanical, but its interpretation matters. A correlation matrix is part of the stated aggregation basis; it is not enough to know the standalone charges alone. The resulting aggregate is meaningful only under the assumptions and scope of that rule. This method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $C_i$ be the nonnegative standalone charge for risk $i$, for $i=1,\ldots,n$. Let $R=(\rho_{ij})$ be the stated correlation matrix, where $\rho_{ii}=1$, $\rho_{ij}=\rho_{ji}$, and $\rho_{ij}$ is the prescribed correlation between risks $i$ and $j$. The correlation-aggregated charge is $$ C_{\mathrm{agg}}=\sqrt{\sum_{i=1}^{n}\sum_{j=1}^{n}\rho_{ij}C_iC_j} =\sqrt{\mathbf{C}^{\mathsf T}R\mathbf{C}}. $$ The undiversified total is $\sum_i C_i$, and the diversification reduction, when the aggregate is no greater than that total, is $\sum_i C_i-C_{\mathrm{agg}}$.

What the Formula Represents

The diagonal terms in the double sum are the squares of the standalone charges, since $\rho_{ii}=1$. Each distinct pair contributes twice in the double sum: once as $\rho_{ij}C_iC_j$ and once as $\rho_{ji}C_jC_i$. A convenient way to calculate is therefore

C_{\mathrm{agg}}^2 =\sum_i C_i^2+2\sum_{i<j}\rho_{ij}C_iC_j.

The factor of 2 is essential. It does not mean the pairwise correlation itself is doubled; it accounts for the two symmetric entries of the matrix. For a single risk, the formula gives $C_{\mathrm{agg}}=C_1$. For independent risks, all off-diagonal correlations are zero and the aggregate is the square root of the sum of squared charges, not their arithmetic sum.

A standard justification comes from variance aggregation. Suppose $Z_i$ are centered random changes in loss, with standard deviations $\sigma_i$, and $\operatorname{Corr}(Z_i,Z_j)=\rho_{ij}$. Then

\operatorname{Var}\left(\sum_i Z_i\right) =\sum_i\operatorname{Var}(Z_i)+2\sum_{i<j}\operatorname{Cov}(Z_i,Z_j) =\sum_i\sigma_i^2+2\sum_{i<j}\rho_{ij}\sigma_i\sigma_j.

If the standalone charges are proportional to those standard deviations using the same proportionality factor, taking the square root gives the correlation aggregation formula. A regulatory rule may prescribe this formula directly, without asserting that actual losses are normal or that the charge is an exact tail quantile. Correlations and standalone charges alone do not generally determine the full distribution of aggregate losses.

For a valid correlation matrix, $R$ is symmetric, has ones on the diagonal, and is positive semidefinite. Consequently $\mathbf{C}^{\mathsf T}R\mathbf{C}\geq 0$ for every charge vector. Pairwise correlations within $[-1,1]$ are necessary but are not, by themselves, enough to guarantee that a whole matrix is valid. In exam calculations, use the stated matrix and check that its row and column ordering matches the charge ordering.

Worked Calculations

Worked Example: Three Risks With Positive Correlations

A hypothetical insurer has standalone charges of $10 million for market risk, $6 million for longevity risk, and $4 million for expense risk. The prescribed correlations are 0.25 between market and longevity, 0.10 between market and expense, and 0.20 between longevity and expense. Calculate the aggregate and its reduction from the undiversified total. Report amounts in millions to two decimal places.

Method. Use the pairwise form of the quadratic aggregation formula. The charges are $C_1=10$, $C_2=6$, and $C_3=4$, with all amounts measured in millions.

First calculate the diagonal contribution:

$$ 10^2+6^2+4^2=100+36+16=152. $$

Then calculate each distinct pair contribution, including the factor of 2:

2(0.25)(10)(6)=30,\qquad 2(0.10)(10)(4)=8,\qquad 2(0.20)(6)(4)=9.6.

Thus the quadratic total is $152+30+8+9.6=199.6$, and the aggregate is $\sqrt{199.6}=14.1276$ million before rounding. To two decimal places, the aggregate is $14.13 million. The undiversified total is $10+6+4=20$ million, so the diversification reduction is $20-14.1276=5.8724$, or $5.87 million to two decimal places.

Check. The aggregate lies between the largest standalone charge, $10 million, and the undiversified total, $20 million. This is consistent with these positive correlations and this charge vector. The common mistake avoided here is to add the three charges and report $20 million as the correlation-aggregated result.

Worked Example: A Large Correlation in One Risk Pair

A hypothetical product portfolio has standalone charges of $8 million, $6 million, and $4 million. The prescribed correlations for pairs (1,2), (1,3), and (2,3) are 0.50, 0.10, and 0.20. Find the aggregate and the reduction from the undiversified total, in millions to two decimal places.

Method. Calculate the squared charges, then add the three pair terms. The diagonal contribution is $8^2+6^2+4^2=64+36+16=116$.

The pair contributions are

2(0.50)(8)(6)=48,\qquad 2(0.10)(8)(4)=6.4,\qquad 2(0.20)(6)(4)=9.6.

The quadratic total is $116+48+6.4+9.6=180$. Therefore the aggregate is $\sqrt{180}=13.4164079$ million, which rounds to $13.42 million. The undiversified total is $8+6+4=18$ million. The reduction is $18-13.4164079=4.5835921$ million, which rounds to $4.58 million.

Check. Squaring the unrounded aggregate returns 180. Using 14.00 as the aggregate would fail this check: $14^2=196$, not 180. The common mistake avoided is estimating the square root or reduction without checking it against the quadratic total.

Worked Example: The Effect of a Negative Correlation

A hypothetical insurer has standalone charges of $5 million, $7 million, and $9 million for three risks. The prescribed correlations for pairs (1,2), (1,3), and (2,3) are 0, $-0.20$, and 0.10. Calculate the aggregate and the diversification reduction, in millions to two decimal places.

Method. Use the same quadratic formula; a negative correlation makes its pair term negative. The diagonal contribution is $5^2+7^2+9^2=25+49+81=155$.

The pair terms are

2(0)(5)(7)=0,\qquad 2(-0.20)(5)(9)=-18,\qquad 2(0.10)(7)(9)=12.6.

The quadratic total is $155+0-18+12.6=149.6$, which is nonnegative. The aggregate is $\sqrt{149.6}=12.231107$ million, or $12.23 million to two decimal places. The undiversified total is $5+7+9=21$ million, so the reduction is $21-12.231107=8.768893$ million, or $8.77 million.

Check. The negative pair contribution lowers the quadratic total by 18 relative to the calculation with that pair correlation set to zero. It does not subtract $18 million directly from the final aggregate; it changes the quantity under the square root. The common mistake avoided is to treat a negative correlation as a negative standalone charge.

A Reliable Calculation Sequence

Match the data.
Write the charges in a fixed order and label the corresponding rows and columns of the correlation matrix. Keep units consistent.

Form the quadratic total.
Square each standalone charge, then add $2\rho_{ij}C_iC_j$ once for every distinct pair $i<j$. Preserve negative signs.

Take the square root.
Check that the quadratic total is nonnegative, then take its square root. Do not round intermediate terms unnecessarily.

Compare and report.
Add the standalone charges separately, subtract the aggregate to find the reduction, and round the requested monetary results only at the end.

Common Exam Traps and Interpretation

A frequent error is to confuse an undiversified sum with a correlation-aggregated charge. For example, with charges 8, 6, and 4 and correlations 0.50, 0.10, and 0.20, the wrong calculation is $8+6+4=18$ million as the aggregate. That is the undiversified total. The correct quadratic total is $180$, so the prescribed aggregate is $\sqrt{180}=13.42$ million, and the reduction is $18-13.42=4.58$ million, to two decimal places.

Omitting the factor of 2. Each distinct pair contributes twice in $\mathbf{C}^{\mathsf T}R\mathbf{C}$. In the pairwise calculation, use $2\rho_{ij}C_iC_j$.
Using a correlation in the wrong pair. Identify which charge belongs to each matrix row and column before calculating. A correct formula with mismatched ordering gives an incorrect result.
Mixing units. If all charges are in millions, the quadratic total is in squared millions and its square root is in millions. Do not mix dollars with millions.
Rounding too early. Keep the quadratic total and square root at full calculator precision, then round the aggregate and the reduction at the end.
Assuming every matrix produces diversification. Under the usual nonnegative charges and valid correlations in $[-1,1]$, the aggregate is no greater than the sum, but it need not be much smaller. If a calculation contradicts that expectation, check the matrix, signs, and arithmetic.

The aggregate is a rule-based combination of the stated charges and correlations. It is not automatically a universal measure of economic risk, and a correlation matrix should not be treated as a substitute for the underlying risk analysis. For a prescribed calculation, use the stated basis precisely and distinguish the result from the undiversified sum.

Key takeaway: Aggregate prescribed charges by forming $\mathbf{C}^{\mathsf T}R\mathbf{C}$, including every pairwise cross term, and taking the square root. Compare that result with $\sum_i C_i$ only after computing the aggregate, and state the units and rounding clearly.

Check Your Understanding

For each problem, use the stated aggregation rule. Unless otherwise stated, report money to two decimal places and retain full precision until the final rounding.

Three hypothetical risk charges are $3 million, $4 million, and $12 million. The correlations for pairs (1,2), (1,3), and (2,3) are 0.20, 0.10, and 0.30. Find the aggregate and the reduction from the undiversified total. (Answer: aggregate $14.48 million; reduction $4.52 million.)
Charges are $8 million, $6 million, and $4 million, with pair correlations 0.50, 0.10, and 0.20 in that order. Find the aggregate and the reduction. (Answer: aggregate $13.42 million; reduction $4.58 million.)
Two risk charges are $10 million and $15 million, with prescribed correlation 0.40. Find the aggregate and its reduction from the undiversified total. (Answer: aggregate $21.10 million; reduction $3.90 million.)
Two charges are $9 million and $12 million, with prescribed correlation $-0.25$. Calculate the aggregate and the reduction. (Answer: aggregate $13.08 million; reduction $7.92 million.)
Three charges, measured in thousands of dollars, are 2.5, 3.0, and 1.5. Their pair correlations (1,2), (1,3), and (2,3) are 0.20, 0.40, and 0.10. Find the aggregate and the reduction, in thousands of dollars. (Answer: aggregate 4.94 thousand dollars; reduction 2.06 thousand dollars.)

What You'll Learn