Aggregating Policy Results into Portfolio Solvency

From Model Points to a Portfolio Result

A product projection often gives results for a small number of representative policies, or model points. A portfolio result must reflect how many actual policies each model point represents. This matters because two product groups with different asset shares, guarantees, or stress losses do not contribute equally unless their exposure weights are equal.

The central operation is weighting: multiply each model-point result by the number of policies it represents, add across model points, and then divide by total exposure only when a per-policy average is required. Portfolio totals and portfolio averages answer different questions. Solvency analysis also requires care: adding results under a specified stress is not the same operation as aggregating risk capital under a dependence assumption.

Definition: Suppose model point $j$ represents $n_j$ policies, and has per-policy quantity $x_j$. Its portfolio contribution is $n_jx_j$. The portfolio total is $\sum_j n_jx_j$, and the exposure-weighted per-policy average is $\frac{\sum_j n_jx_j}{\sum_j n_j}$, provided $\sum_j n_j>0$. If the model point represents different policy counts for different projection years, use the applicable year’s weights.

The quantity $x_j$ could be an asset share, liability, stressed loss, or own funds amount. For example, when $x_j$ is a per-policy surplus, the portfolio surplus is the sum of weighted surpluses. It is not generally the unweighted average of the model-point surpluses. If model points differ in policy size, benefit amount, or exposure duration, policy count may not be the appropriate weight; use the specified exposure measure instead.

Why Weighting Works

The weighting rule follows from additivity. If each of $n_j$ policies represented by a model point has the same projected quantity $x_j$, then their combined quantity is $x_j+x_j+\cdots+x_j=n_jx_j$. Adding these contributions across model points gives the portfolio total. Dividing by the total number of represented policies gives the average per policy.

\text{Portfolio average} = \frac{\sum_j n_jx_j}{\sum_j n_j}

This identity assumes that the model-point result is applicable to each policy represented by that point and that the quantity is additive across policies. It does not mean that every portfolio measure is a simple weighted average. Ratios, such as a solvency coverage ratio, must be calculated from their aggregate numerator and denominator; averaging the model-point ratios generally gives the wrong result.

Worked Example: Aggregating Asset Shares and Surplus

A hypothetical insurer has two model points at a valuation date. Model point A represents 1,200 policies with an asset share of $18,450 and liability of $17,900 per policy. Model point B represents 800 policies with an asset share of $22,100 and liability of $22,400 per policy. Calculate the total asset share, weighted average asset share, total liability, and portfolio surplus. The values are assumed constant within each model point.

Method. Multiply each per-policy amount by its policy count, then sum the totals. Calculate per-policy averages only after obtaining the portfolio totals.

Asset share. Model point A contributes $1{,}200\times \$18{,}450=\$22{,}140{,}000$. Model point B contributes $800\times \$22{,}100=\$17{,}680{,}000$. Therefore, total asset share is $\$22{,}140{,}000+\$17{,}680{,}000=\$39{,}820{,}000$. There are $1{,}200+800=2{,}000$ policies, so average asset share is $\$39{,}820{,}000/2{,}000=\$19{,}910$ per policy.

Liability. Model point A contributes $1{,}200\times \$17{,}900=\$21{,}480{,}000$. Model point B contributes $800\times \$22{,}400=\$17{,}920{,}000$. Total liability is $\$39{,}400{,}000$, or $\$39{,}400{,}000/2{,}000=\$19{,}700$ per policy on average.

Surplus and check. Total surplus is $\$39{,}820{,}000-\$39{,}400{,}000=\$420{,}000$. The per-policy surplus is $\$420{,}000/2{,}000=\$210$. Checking by model point gives $1{,}200(\$18{,}450-\$17{,}900)+800(\$22{,}100-\$22{,}400)=\$660{,}000-\$240{,}000=\$420{,}000$, agreeing with the aggregate calculation.

Common mistake avoided. The unweighted average asset share, $(\$18{,}450+\$22{,}100)/2=\$20{,}275$, incorrectly gives the two model points equal weight even though they represent different numbers of policies.

Stressed Own Funds and Risk Capital

A specified deterministic stress can be aggregated by weighting each model point’s stressed loss or change in own funds and then summing. This gives the portfolio result under that particular stress. A risk-capital calculation may instead combine standalone capital amounts using a stated dependence rule. Those are distinct calculations: a stress loss should not be substituted for standalone capital, and standalone capital should not be summed unless the prescribed method calls for it.

For illustration, suppose two risk components have standalone capital amounts $C_1$ and $C_2$, with an explicitly prescribed correlation parameter $\rho$. A common square-root aggregation form is:

C_{\mathrm{agg}}=\sqrt{C_1^2+C_2^2+2\rho C_1C_2}

This formula follows by considering the variance of a sum: $\operatorname{Var}(Z_1+Z_2)=\operatorname{Var}(Z_1)+\operatorname{Var}(Z_2)+2\operatorname{Cov}(Z_1,Z_2)$. If the capital amounts are treated as standard-deviation measures and their correlation is $\rho$, the covariance term is $\rho C_1C_2$. Taking the square root gives the displayed aggregation form. Its use requires the stated model assumptions and prescribed correlation; it is not a universal rule for every solvency framework.

Worked Example: Stressed Own Funds and Aggregated Capital

Consider a hypothetical portfolio of 1,000 traditional policies and 600 linked policies. Before stress, own funds are $1,200 per traditional policy and $900 per linked policy. A specified stress reduces own funds by $350 and $250 per policy, respectively. Separately, the standalone capital amounts are $500 per traditional policy and $400 per linked policy. For this illustration, aggregate the two portfolio-level standalone capital amounts using the square-root formula with $\rho=0.25$. Calculate post-stress own funds and the illustrative coverage ratio, defined here as post-stress own funds divided by aggregated capital.

Method. First aggregate own funds and stress losses using policy counts. Then aggregate standalone capital using the given dependence formula. The correlation assumption applies to the capital aggregation, not to the deterministic stress-loss calculation.

Own funds before stress. Traditional policies contribute $1{,}000\times \$1{,}200=\$1{,}200{,}000$. Linked policies contribute $600\times \$900=\$540{,}000$. Total own funds before stress are $\$1{,}740{,}000$.

Stress loss and post-stress own funds. The traditional-policy loss is $1{,}000\times \$350=\$350{,}000$. The linked-policy loss is $600\times \$250=\$150{,}000$. Total loss is $\$500{,}000$, so post-stress own funds are $\$1{,}740{,}000-\$500{,}000=\$1{,}240{,}000$.

Standalone capital totals. Traditional capital is $1{,}000\times \$500=\$500{,}000$. Linked capital is $600\times \$400=\$240{,}000$. Applying the stipulated formula:

C_{\mathrm{agg}} = \sqrt{500{,}000^2+240{,}000^2+2(0.25)(500{,}000)(240{,}000)} = \sqrt{367{,}600{,}000{,}000} \approx \$606{,}300

The illustrative coverage ratio is $\$1{,}240{,}000/\$606{,}300\approx 2.045$, or about 204.5%. Rounding is to the nearest dollar for capital and to three decimal places for the ratio. As a check, the aggregated capital lies below the simple sum of standalone capital, $\$740{,}000$, as expected here for a positive correlation below 1.

Common mistake avoided. The total stress loss of $500,000 is not the capital denominator. The question specifies separate standalone capital amounts and a correlation formula, so those must be aggregated independently.

Product Mix, Guarantees, and Persistency

Portfolio solvency depends not only on per-policy results but also on which policies remain in force. A guarantee can make an in-force policy’s liability relatively large compared with its asset share. If that product has weak per-policy surplus, a higher number of policies persisting can increase the portfolio deficit. Conversely, persistency changes in a product with positive per-policy surplus can reduce the amount of surplus contributed.

Persistency assumptions therefore affect both the scale and composition of projected exposure. For a projected valuation point, if $N_j$ policies are initially exposed and $p_j$ is the proportion expected to remain in force, the projected in-force count is $N_jp_j$. This simple calculation assumes the stated persistency proportion already reflects the relevant policy exits for the projection. If separate decrements are modeled, their treatment must follow the specified projection basis; do not apply an additional decrement to a proportion that already includes it.

Worked Example: Persistency Changes the Portfolio Deficit

A hypothetical portfolio initially contains 1,000 universal life policies and 600 participating policies. At the projection date, the universal life asset share is $10,500 per in-force policy and its liability, including a guarantee, is $11,200. The participating-policy asset share is $13,000 and its liability is $12,600. Under the first persistency assumption, 80% of universal life policies and 90% of participating policies remain in force. Under an alternative assumption, the corresponding proportions are 90% and 80%. Calculate asset shares, liabilities, and surplus under each assumption.

Method. Apply each persistency proportion to its own initial model-point count, and weight that model point’s asset share and liability by the resulting in-force count. The guarantee is reflected in the stated universal life liability.

First assumption. The in-force counts are $1{,}000(0.80)=800$ universal life policies and $600(0.90)=540$ participating policies. Universal life asset shares total $800\times \$10{,}500=\$8{,}400{,}000$, and liabilities total $800\times \$11{,}200=\$8{,}960{,}000$. Participating asset shares total $540\times \$13{,}000=\$7{,}020{,}000$, and liabilities total $540\times \$12{,}600=\$6{,}804{,}000$. Thus, total asset shares are $\$15{,}420{,}000$, total liabilities are $\$15{,}764{,}000$, and surplus is $-\$344{,}000$.

Alternative assumption. The in-force counts are $1{,}000(0.90)=900$ universal life policies and $600(0.80)=480$ participating policies. Universal life asset shares total $900\times \$10{,}500=\$9{,}450{,}000$, and liabilities total $900\times \$11{,}200=\$10{,}080{,}000$. Participating asset shares total $480\times \$13{,}000=\$6{,}240{,}000$, and liabilities total $480\times \$12{,}600=\$6{,}048{,}000$. Total asset shares are $\$15{,}690{,}000$, total liabilities are $\$16{,}128{,}000$, and surplus is $-\$438{,}000$.

Check and interpretation. Under the first assumption, universal life contributes a deficit of $800(\$10{,}500-\$11{,}200)=-\$560{,}000$, while participating business contributes $540(\$13{,}000-\$12{,}600)=\$216{,}000$. Their sum is $-\$344{,}000$. Under the alternative, the corresponding amounts are $-\$630{,}000$ and $+\$192{,}000$, summing to $-\$438{,}000$. The alternative increases persistency in the guaranteed product with a per-policy deficit and decreases it in the product with a per-policy surplus, so the aggregate deficit grows.

Common mistake avoided. Applying one portfolio-wide persistency rate to all policies would erase the difference in product mix. Persistency must be applied to the appropriate model point before aggregating.

Checks Before Reporting a Portfolio Result

Before presenting an aggregate result, verify that the weights, valuation basis, and measure are consistent. In particular, distinguish policy counts from amounts at risk or premium volumes; use the exposure basis specified for the model points. Keep assets, liabilities, and surplus on the same valuation date and currency basis. If the question asks for an aggregate ratio, calculate it from aggregate amounts rather than averaging ratios.

Check the weights: confirm that model-point counts sum to the stated portfolio exposure.
Check the sign: define surplus consistently, for example as asset share less liability.
Check the stress: multiply each per-policy stressed result by the relevant exposure before adding.
Check the capital method: use the specified dependence rule and do not silently replace it with simple addition.
Check the interpretation: explain which model points drive the change, especially when guarantees and persistency interact.

A portfolio total can conceal offsetting contributions. Reporting the aggregate alone may therefore obscure a material deficit in one product that is being partly offset by another. A clear analysis shows the weighted contribution by model point as well as the total, then explains whether the result is driven by product mix, per-policy economics, or an assumption about which contracts persist. These aggregation methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, aggregate model-point amounts using the stated exposure weights. Answers are shown for checking.

A portfolio has 750 policies with asset share $9,600 each and 250 policies with asset share $12,800 each. Find total asset share and average asset share per policy. (Answer: $10,400,000 total; $10,400 per policy.)
There are 400 policies with own funds of $1,500 each and 300 policies with own funds of $700 each. A specified stress reduces these amounts by $200 and $100 per policy, respectively. Find post-stress own funds. (Answer: $700,000.)
A group has 500 policies with persistency 80%, asset share $14,000, and liability $14,500 per in-force policy. A second group has 400 policies with persistency 90%, asset share $10,000, and liability $9,600. Find the aggregate surplus at the projection date. (Answer: $-\$56,000$.)
Two standalone capital amounts are $300,000 and $400,000, with prescribed correlation 0.50. Using the square-root formula, find aggregate capital. If post-stress own funds are $900,000, find the coverage ratio to three decimal places. (Answer: approximately $608,276; coverage ratio 1.480.)
A model point with 600 policies contributes $250 surplus per policy, while another with 400 policies contributes a deficit of $500 per policy. Calculate portfolio surplus and average surplus per policy. State which model point drives the deficit. (Answer: $-\$50,000$ total; $-\$50$ per policy; the 400-policy model point drives the deficit.)

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