Integrated Long-Term Solvency Stress Case

Introduction

An integrated solvency stress asks how several assumption changes affect a long-term product at the same time. Higher mortality or morbidity may increase claims; higher expenses reduce the assets available to meet obligations; a lower investment return reduces accumulation; and a change in lapse or persistence changes both the claims paid and the number of policies for which reserves remain. The result should be read as a linked calculation, not as a list of independent impacts.

This tutorial uses a deliberately simplified one-year model to make the reconciliation transparent. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The model is not a universal valuation formula: actual product cash flows may include surrender benefits, premiums after the valuation date, expenses contingent on policy status, or more detailed timing. Those must be included when they apply.

Model and definitions: At the start of the year, a cohort has opening assets $A_0$ per starting policy and receives premium $P$ while incurring expense $E$. Assume these amounts occur at the start of the year. They accumulate for one year at effective annual rate $i$. At year-end, mutually exclusive outcomes are death, morbidity, lapse, or persistence, with probabilities $q_d,q_m,q_l,p$, respectively, and $q_d+q_m+q_l+p=1$. Death and morbidity benefits $B_d,B_m$ are paid at year-end; in this model lapse has no payment. Let $V$ be the year-end reserve per persisting policy. Then end assets after claims, $C$, profit per starting policy, and asset share per persisting policy are

C=(A_0+P-E)(1+i)-q_dB_d-q_mB_m,\qquad \Pi=C-pV,\qquad AS=\frac{C}{p}\quad (p>0).

Here, $\Pi$ is the expected year-end surplus after setting aside the reserve for persisting policies. The asset share $AS$ is the assets remaining after the modeled claims, divided by the expected number of persisting policies, on a per-starting-policy basis. This pooling convention assumes the remaining assets are allocated equally across persisting policies. If the contract pays a lapse benefit, that payment must be subtracted in $C$; if costs or payments occur at different times, their timing must be reflected rather than automatically accumulating all items for a full year.

Why Profit and Asset Share Reconcile

The key distinction is the denominator. Profit above is per starting policy, whereas asset share is per persisting policy. Since $C=pAS$, the definitions imply

\Pi=pAS-pV=p(AS-V).

This identity is a useful check: the expected surplus per starting policy equals the persistence probability multiplied by the asset-share margin over the reserve per persisting policy. It does not say that asset share and reserve are the same quantity. Asset share describes assets allocated before deducting the reserve requirement; profit deducts the reserve burden for the expected persisting cohort.

The identity follows directly from the cohort counts. For one starting policy, expected persisting policies equal $p$. If each receives asset share $AS$, allocated assets total $pAS$. The reserve requirement for those policies totals $pV$. Their difference is $pAS-pV$, which is the same as $C-pV$ because $C=pAS$.

A Consistent Calculation Sequence

For an integrated case, keep the cash-flow calculation separate from the reserve calculation until the final reconciliation. This makes it easier to identify whether a change comes from claims and accumulation or from a changed valuation of future obligations.

Fix the measurement basis.
State the cohort, period, cash-flow timing, outcome probabilities, investment rate, and whether the result is per starting policy or per persisting policy.

Accumulate available assets.
Apply the stated return to opening assets plus start-of-year premiums less start-of-year expenses. Do not apply a full-year return to a cash flow that occurs at year-end.

Deduct modeled payments.
Subtract expected benefits using their outcome probabilities. Include any lapse payments or other cash flows specified in the case.

Apply the reserve to persisting policies.
Multiply the reserve per persisting policy by the persistence probability to obtain the reserve burden per starting policy.

Reconcile and check.
Calculate profit per starting policy and asset share per persisting policy. Verify that $\Pi=p(AS-V)$.

Under stress, repeat the sequence using the stressed assumptions together. Do not combine a stressed claim amount with a baseline persistence rate or reserve unless the question explicitly asks for a one-factor-at-a-time attribution. The combined result can be nonlinear: changing mortality or morbidity also changes the number of policies that remain in force.

Worked Example: Correcting the Asset-Share Denominator

A hypothetical one-year cohort has opening assets of $400 per starting policy, a start-of-year premium of $900, and start-of-year expenses of $80. The effective annual return is 3%. At year-end, death, morbidity, lapse, and persistence probabilities are 0.01, 0.04, 0.10, and 0.85. The death benefit is $12,000, the morbidity benefit is $1,500, lapse has no payment, and the reserve is $250 per persisting policy. Find profit per starting policy and asset share per persisting policy.

Method. First accumulate opening assets and the net start-of-year cash flow, then deduct expected claims. Use the resulting assets for the asset share, and separately deduct the expected reserve burden for profit.

1. Check the outcome probabilities. They sum to $0.01+0.04+0.10+0.85=1.00$, so the stated outcomes form a complete mutually exclusive set.

2. Accumulate assets before claims. The amount invested at the start of the year is $400+900-80=1{,}220$. After the 3% return, assets are $1{,}220(1.03)=\$1{,}256.60$ per starting policy.

3. Deduct expected claims. Expected death claims are $0.01(12{,}000)=\$120$; expected morbidity claims are $0.04(1{,}500)=\$60$. Total expected claims are $120+60=\$180$. Thus $C=1{,}256.60-180=\$1{,}076.60$.

4. Calculate profit. The reserve burden per starting policy is $0.85(250)=\$212.50$. Therefore $\Pi=1{,}076.60-212.50=\$864.10$ per starting policy.

5. Calculate asset share. Divide the assets after claims by the persistence probability: $AS=1{,}076.60/0.85=\$1{,}266.5882\ldots$, or $1,266.59 per persisting policy to the nearest cent.

Check. The identity gives $0.85(1{,}266.5882\ldots-250)=\$864.10$, agreeing with the profit calculation. The common mistake is to use the starting-cohort amount $C$ as though it were already per persisting policy. Dividing by 0.85 is required for asset share; the reserve deduction belongs in the profit calculation, not in the numerator of this asset-share definition.

Worked Integrated Stress: Several Assumptions Change Together

The next example changes investment, expense, mortality, morbidity, lapse, persistence, and the reserve in one stressed scenario. The assumed outcome probabilities remain mutually exclusive and exhaustive in each scenario. The scenario values are hypothetical and are intended to illustrate a reconciliation, not to represent an actual product or portfolio.

Worked Example: Combined Mortality, Morbidity, Lapse, Expense, and Investment Stress

A hypothetical cohort has opening assets of $500 and a start-of-year premium of $1,000 per starting policy. In the baseline, expenses are $100 and the effective annual return is 4%. Year-end probabilities of death, morbidity, lapse, and persistence are 0.02, 0.03, 0.10, and 0.85. The death benefit is $10,000, the morbidity benefit is $2,000, and the reserve is $300 per persisting policy.

In the combined stress, expenses rise to $120, the return falls to 2%, and the probabilities become 0.03, 0.05, 0.12, and 0.80. Benefits are unchanged, while the reserve rises to $360 per persisting policy. Calculate baseline and stressed profit and asset share, then reconcile the profit change.

Baseline calculation. The probabilities sum to $0.02+0.03+0.10+0.85=1.00$. Start-of-year invested assets are $500+1{,}000-100=\$1{,}400$. Accumulation gives $1{,}400(1.04)=\$1{,}456$. Expected claims are $0.02(10{,}000)+0.03(2{,}000)=200+60=\$260$, so $C=\$1{,}456-\$260=\$1{,}196$.

The baseline reserve burden per starting policy is $0.85(300)=\$255$. Baseline profit is $1{,}196-255=\$941$ per starting policy. Baseline asset share is $1{,}196/0.85=\$1{,}407.0588\ldots$, or $1,407.06 per persisting policy to the nearest cent.

Stressed calculation. The stressed probabilities sum to $0.03+0.05+0.12+0.80=1.00$. Start-of-year invested assets are $500+1{,}000-120=\$1{,}380$. At 2%, these accumulate to $1{,}380(1.02)=\$1{,}407.60$. Expected claims are $0.03(10{,}000)+0.05(2{,}000)=300+100=\$400$, giving stressed $C=1{,}407.60-400=\$1{,}007.60$.

The stressed reserve burden per starting policy is $0.80(360)=\$288$. Stressed profit is $1{,}007.60-288=\$719.60$ per starting policy. Stressed asset share is $1{,}007.60/0.80=\$1{,}259.50$ per persisting policy.

Reconciliation. End assets after claims fall by $1{,}196-1{,}007.60=\$188.40$ per starting policy. The reserve burden rises by $288-255=\$33$. Therefore profit falls by $188.40+33=\$221.40$, from $941 to $719.60. The asset-share change is not the profit change: its denominator changes from 0.85 to 0.80, and it is calculated before deducting the reserve.

Check. Baseline $0.85(1{,}407.0588\ldots-300)=\$941$; stressed $0.80(1{,}259.50-360)=\$719.60$. Both reconcile. This combined comparison does not assign the $221.40 change to individual causes; doing that requires a stated attribution order or method, because the risks interact through claims, persistence, and reserves.

Reserve Revaluation Is Not a Cash-Flow Change

A reserve change can reduce measured profit even when the assets produced by the period’s cash flows do not change. Conversely, a change in asset share can arise from a change in expected claims or investment accumulation without a reserve revaluation. Keeping these effects distinct prevents a reserve movement from being counted as though it were an additional benefit payment.

Worked Example: Isolating a Reserve Revaluation

For a hypothetical cohort, end assets after claims are $1,210.20 per starting policy and the persistence probability is 0.80. The reserve is first valued at $1,400 per persisting policy and is then revalued at $1,500 per persisting policy. All cash-flow assumptions and end assets are unchanged. Calculate profit and asset share under each reserve basis.

Method. Calculate the reserve burden per starting policy under each valuation basis. Keep $C$ fixed because the example changes the reserve only.

At the first reserve basis, the reserve burden is $0.80(1{,}400)=\$1{,}120$, so profit is $1{,}210.20-1{,}120=\$90.20$ per starting policy. Asset share is $1{,}210.20/0.80=\$1{,}512.75$ per persisting policy.

At the revalued reserve basis, the reserve burden is $0.80(1{,}500)=\$1{,}200$, so profit is $1{,}210.20-1{,}200=\$10.20$ per starting policy. Asset share remains $1{,}512.75$ because neither end assets nor persistence changed. The $80 profit reduction equals $0.80(1{,}500-1{,}400)=\$80$, the increase in reserve burden per starting policy.

Check. The asset-share margin is $112.75 under the first reserve basis and $12.75 under the revalued basis. Multiplying each by 0.80 gives $90.20 and $10.20, respectively. The common mistake is to subtract the reserve from $C$ and then call the result asset share; that would mix reserve-adjusted profit with the pre-reserve asset-share measure.

Common Exam Traps

Mixing cohort bases. Profit is per starting policy in these formulas; asset share is per persisting policy. Use the persistence probability when changing between them.
Applying interest at the wrong time. In the stated model, opening assets, premium, and expense are invested for the year. A year-end benefit is deducted after accumulation, not accumulated for a year.
Forgetting probability consistency. If death, morbidity, lapse, and persistence are the complete set of outcomes, their probabilities must sum to one in each scenario.
Omitting a specified payment. A lapse benefit or other exit payment must be included in end assets if the contract pays it; it cannot be treated as zero merely because the simplified model above has no lapse payment.
Counting a reserve change twice. Deduct expected benefits when calculating end assets, then deduct the reserve burden to calculate profit. Do not also treat a reserve increase as an extra claim payment.
Claiming a unique risk attribution from a combined result. The difference between baseline and combined stress is a total scenario effect. Separate contributions depend on the chosen sequence or attribution method.

Key takeaway: Reconcile the scenario in two stages: first calculate assets after investment and modeled cash flows, then deduct the reserve burden for persisting policies. Keep the starting-policy and persisting-policy bases explicit, and use $\Pi=p(AS-V)$ as a final consistency check.

Check Your Understanding

For each problem, use the stated one-year timing convention and report money to the nearest cent where needed.

A hypothetical cohort has opening assets of $200, start-of-year premium $500, expenses $50, and effective annual return 4%. Death, morbidity, lapse, and persistence probabilities are 0.01, 0.02, 0.07, and 0.90. Benefits are $5,000 on death and $1,000 on morbidity; the reserve is $300 per persisting policy, and lapse has no payment. Find profit per starting policy and asset share per persisting policy. (Answer: profit $336.00 per starting policy; asset share $673.33 per persisting policy.)
Opening assets are $300, premium $700, expenses $80, and the effective annual return is 3%. The probabilities of death, morbidity, lapse, and persistence are 0.02, 0.03, 0.15, and 0.80. Benefits are $6,000 on death and $1,000 on morbidity; the reserve is $400 per persisting policy. Find profit and asset share. (Answer: profit $477.60 per starting policy; asset share $997.00 per persisting policy.)
End assets after claims are $1,050 per starting policy and persistence is 0.75. The reserve per persisting policy increases from $1,200 to $1,300, with all cash flows unchanged. Find the change in profit per starting policy and the asset share. (Answer: profit decreases by $75.00 per starting policy; asset share remains $1,400.00 per persisting policy.)
Opening assets are $400, premium $800, expenses $100, and the effective annual return is 2%. Death, morbidity, lapse, and persistence probabilities are 0.03, 0.04, 0.13, and 0.80. Benefits are $8,000 on death and $1,500 on morbidity; the reserve is $500 per persisting policy. Find profit and asset share. (Answer: profit $422.00 per starting policy; asset share $1,027.50 per persisting policy.)
A cohort has persistence probability 0.72, asset share $1,600 per persisting policy, and reserve $1,500 per persisting policy. Find profit per starting policy and end assets after claims per starting policy. (Answer: profit $72.00 per starting policy; end assets after claims $1,152.00 per starting policy.)

What You'll Learn