Why Break Stress Results Down by Cohort and Duration?
A single portfolio-level stress result can conceal where the exposure sits. Two products may have the same aggregate adverse impact even though one is driven by many policies with modest effects and the other by a small group with large effects. A breakdown by issue cohort and policy duration makes those patterns visible and supports a more useful explanation of projected outcomes.
An issue cohort groups policies by when they were issued, such as calendar year or a defined underwriting generation. Policy duration is the time elapsed since issue, measured consistently in years or policy anniversaries. A cohort is followed through time; its policies move from shorter to longer durations as the projection advances. This is different from grouping policies by their current attained age or by the calendar year in which a projected cash flow occurs.
For example, if the metric is projected loss in dollars and the exposure is a count of policies, \(\Delta L_{c,d}\) must be dollars per policy over the same projection period and on the same valuation basis. A positive value then means the stress increases the loss. If the metric is solvency surplus instead, define the sign carefully: stressed surplus minus baseline surplus is negative when the stress reduces surplus. Analysts often report the adverse reduction as a positive amount, but that is a different sign convention and must be stated.
Constructing a Cohort-Duration Summary
A useful table reports more than the dollar impact. Include the exposure in each cell, the per-unit stress impact, and the weighted cell impact. Row and column totals then answer different questions: cohort totals show which issue generations contribute most to the portfolio result, while duration totals show where in policy life the impact is concentrated.
The share formula is meaningful as a contribution percentage when the total is nonzero and the cell impacts use a consistent sign convention. If some cells have favorable impacts and others adverse impacts, a signed share can be negative or exceed 100%. In that case, show signed dollar impacts as well; percentages alone can obscure offsetting effects. Exposure share, \(N_{c,d}/\sum_{c,d}N_{c,d}\), is not the same as impact share.
Worked Example: Weighting Cell Impacts Correctly
A hypothetical insurer assesses a one-year expense and morbidity stress for four cells. Each cell impact is stated as the increase in projected loss per policy over that same year. The baseline and stressed valuations use the same valuation date, benefit basis, and discounting method.
| Issue cohort | Duration | Policies \(N_{c,d}\) | Increase per policy |
|---|---|---|---|
| 2022 | 1 | 100 | $120 |
| 2022 | 2 | 80 | $200 |
| 2023 | 1 | 60 | $80 |
| 2023 | 2 | 40 | $300 |
Method. Multiply each cell's exposure by its per-policy increase, then add the cell impacts. All values are adverse loss increases, so they have the same positive sign.
For cohort 2022, the duration-1 impact is \(100\times \$120=\$12{,}000\), and the duration-2 impact is \(80\times \$200=\$16{,}000\). Its cohort total is \(\$12{,}000+\$16{,}000=\$28{,}000\).
For cohort 2023, the duration-1 impact is \(60\times \$80=\$4{,}800\), and the duration-2 impact is \(40\times \$300=\$12{,}000\). Its cohort total is \(\$4{,}800+\$12{,}000=\$16{,}800\).
The portfolio total is \(\$28{,}000+\$16{,}800=\$44{,}800\). By duration, duration 1 contributes \(\$12{,}000+\$4{,}800=\$16{,}800\), while duration 2 contributes \(\$16{,}000+\$12{,}000=\$28{,}000\). Thus duration 2 contributes \(28{,}000/44{,}800=0.625\), or 62.5% of the adverse impact. Cohort 2022 contributes \(28{,}000/44{,}800=62.5\%\).
Check. The row totals add to \(\$28{,}000+\$16{,}800=\$44{,}800\), and the duration totals add to \(\$16{,}800+\$28{,}000=\$44{,}800\). The two views reconcile to the same portfolio result. The common mistake avoided here is reporting the unweighted average of the four per-policy impacts as though it were the portfolio impact.
Why Exposure Weighting Is Necessary
The aggregation rule follows from addition across policies. Suppose every policy \(j\) in cell \((c,d)\) has the same stress impact \(\Delta L_{c,d}\). The cell's total impact is the sum of that amount for each of its \(N_{c,d}\) policies:
If impacts vary between policies, the exact cell result is \(\sum_j\Delta L_{c,d,j}\). The product \(N_{c,d}\Delta L_{c,d}\) remains valid when \(\Delta L_{c,d}\) is the exposure-weighted mean impact per policy. Thus the method does not require identical policies, but it does require a representative, correctly weighted per-unit amount.
The exposure measure must match the quantity being aggregated. A count of policies is appropriate for a per-policy one-year loss change. If the result is a loss rate per policy-year, use policy-years as exposure. If the model output is already a total cell amount, do not multiply it by exposure again. Define the valuation date, projection horizon, stress scenario, and exposure basis before comparing cells.
Worked Example: Separating Exposure Share from Impact Share
A hypothetical block has four cells under a specified one-year stress. The reported impacts are increases in projected loss per policy, and the counts are in-force policies at the valuation date.
| Cohort | Duration | Policies | Impact per policy |
|---|---|---|---|
| 2024 | 1 | 300 | $40 |
| 2024 | 2 | 100 | $180 |
| 2023 | 1 | 20 | $100 |
| 2023 | 2 | 80 | $275 |
The duration-1 cell impacts are \(300\times \$40=\$12{,}000\) and \(20\times \$100=\$2{,}000\), giving a duration-1 total of \(\$14{,}000\). The duration-2 impacts are \(100\times \$180=\$18{,}000\) and \(80\times \$275=\$22{,}000\), giving a duration-2 total of \(\$40{,}000\). Total exposure is \(300+100+20+80=500\) policies, and total impact is \(\$14{,}000+\$40{,}000=\$54{,}000\).
Cohort 2023 has \(20+80=100\) policies, or \(100/500=20\%\) of exposure. Its impact is \(\$2{,}000+\$22{,}000=\$24{,}000\), or \(\$24{,}000/\$54{,}000=44.44\%\) of the total. It therefore has a larger impact share than exposure share. Its average impact per policy is \(\$24{,}000/100=\$240\), compared with \(\$30{,}000/400=\$75\) for cohort 2024.
Duration 2 has \(100+80=180\) policies, or 36% of total exposure, but its \(\$40{,}000\) impact is \(40{,}000/54{,}000=74.07\%\) of the total. These results point to higher per-policy stress sensitivity at longer durations in this hypothetical block; they do not, by themselves, identify why that sensitivity occurs.
Check. The cohort impacts, \(\$30{,}000\) and \(\$24{,}000\), sum to \(\$54{,}000\); the duration impacts, \(\$14{,}000\) and \(\$40{,}000\), do as well. The common mistake avoided is describing the cohort with the largest impact share as the cohort with the largest exposure. Here cohort 2023 has only one-fifth of exposure but nearly half of the impact.
Following Cohorts Through a Projection
When results are projected over several years, an issue cohort remains the same cohort while its duration changes. A policy issued in 2024 is at duration 1 in the first projection year after issue and duration 2 in the next, subject to the model's duration convention. The number exposed may change through death, lapse, maturity, or other exits. Therefore, a projection table should identify both calendar projection year and policy duration when both are relevant.
Do not add amounts from different projection years unless they are amounts defined to be additive, such as annual loss flows. A point-in-time capital amount, reserve change, or solvency measure at one projection date is not automatically a cash flow that can be summed across dates. Keep the reported metric and time basis explicit.
Worked Example: Tracking Duration Across Two Projection Years
A hypothetical stress produces an increase in annual loss for two issue cohorts. Counts are in force at the start of each year. The per-policy amounts are annual loss increases for that year, so annual impacts may be added to obtain a two-year cumulative loss impact. Assume the specified stress and valuation conventions remain consistent in both years.
| Projection year | Cohort | Duration | Policies | Annual increase per policy |
|---|---|---|---|---|
| 2025 | 2024 | 1 | 200 | $100 |
| 2026 | 2024 | 2 | 180 | $160 |
| 2025 | 2023 | 2 | 100 | $180 |
| 2026 | 2023 | 3 | 90 | $220 |
In 2025, cohort 2024 contributes \(200\times \$100=\$20{,}000\). Cohort 2023 contributes \(100\times \$180=\$18{,}000\). The 2025 total is \(\$20{,}000+\$18{,}000=\$38{,}000\).
In 2026, cohort 2024 has moved to duration 2 and contributes \(180\times \$160=\$28{,}800\). Cohort 2023 has moved to duration 3 and contributes \(90\times \$220=\$19{,}800\). The 2026 total is \(\$28{,}800+\$19{,}800=\$48{,}600\).
Because these are annual loss flows, the cumulative two-year impact is \(\$38{,}000+\$48{,}600=\$86{,}600\). The contribution of cohort 2024 over the two years is \(\$20{,}000+\$28{,}800=\$48{,}800\); cohort 2023 contributes \(\$18{,}000+\$19{,}800=\$37{,}800\). The sum is again \(\$86{,}600\).
Check. The table records the same cohorts at longer durations in 2026, with exposure counts reduced as specified. The calculation adds annual flows, not two point-in-time solvency values. Treating each cohort as if it remained at its original duration would misclassify the 2026 impacts.
Interpreting Concentration Without Overstating It
A cohort-duration table describes where the modeled stress result is concentrated. It does not prove that duration caused the difference. Cells may differ in benefit design, underwriting rules, premium structure, guarantee terms, policyholder behavior, or data quality. When possible, compare like with like or include explanatory dimensions that are relevant to the product. Avoid causal claims based only on a cross-sectional table.
Also distinguish absolute impact from sensitivity per exposure. A cell can dominate the total because it contains many policies, because each policy is highly sensitive, or both. Reporting exposure, mean impact per exposure, and total impact together makes the distinction clear. For signed results, retain the sign in the underlying calculations and explain any conversion to an “adverse impact” presentation.
Specify the baseline, stress scenario, valuation date, metric, projection horizon, and sign convention.
State how issue cohorts and durations are assigned, and whether exposure means policies, policy-years, or another suitable unit.
Compute cell impacts using compatible units, then check that row and column totals agree with the portfolio total.
Separate exposure share from impact share and identify possible product or assumption differences without treating association as proof of cause.
A common exam trap is to multiply a cell's total dollar impact by the policy count because the table labels that total as “impact.” First determine whether the given value is per policy or already an aggregate. A second trap is to compare a one-year flow in one cell with a multi-year present value in another. These quantities cannot be ranked or aggregated meaningfully until they have been placed on a consistent basis.
Check Your Understanding
For each problem, use the stated sign and exposure conventions and check that cell totals reconcile.
- A cell contains 250 policies and has a stress loss increase of $36 per policy for one year. What is its total impact? (Answer: $9,000)
- Two cohorts have 120 and 80 policies, with respective per-policy increases of $50 and $125. Find the total impact and the second cohort's impact share. (Answer: $16,000 total; 62.5%)
- At duration 1, 40 policies have an impact of $200 each; at duration 2, 60 policies have an impact of $150 each. Find the total impact and duration-2 impact share. (Answer: $17,000 total; 52.9412%)
- A cohort has 90 policies in one year with an annual loss increase of $80 per policy. In the next year it has 75 policies at the next duration with an annual increase of $120 per policy. If these are annual flows, what is the two-year cumulative impact? (Answer: $16,200)
- A cohort accounts for 15% of policy exposure and $27,000 of a $90,000 total stress impact. What is its impact share, and is that share greater or less than its exposure share? (Answer: 30%; greater by 15 percentage points)