Attributing Stress Effects Across Reserves, Profits, and Asset Shares

Why Reconcile Stress Effects?

A stress may affect several measures at once. An expense shock can reduce accumulated assets, while a change in valuation assumptions can increase the reserve held for the same policy. Looking only at the reserve change, the profit change, or the ending asset share can therefore give an incomplete account of the solvency effect. The measures need to be reconciled on the same valuation dates, for the same business, and under a clearly stated profit convention.

The central accounting idea is that surplus is assets less reserves. A stress changes period profit according to its effect on both ending and opening surplus. In particular, the difference between stressed and base ending surplus is not automatically the difference between stressed and base profit: the opening surplus may also differ.

Definition: At a valuation time, let $A$ be the assets allocated to a specified block of business and $V$ its reserve on a stated basis. Define the block's surplus as $S=A-V$. In a period with no shareholder distributions, capital injections, or other transfers excluded from the block, define period profit as $P=S_1-S_0$, where subscripts 0 and 1 denote the opening and ending valuation times. If such transfers occur, their treatment must be stated and consistently allowed for.

A reserve is a valuation of obligations, not an asset account. An asset share is an accumulation of a policy's allocated cash flows under a specified investment and timing convention. They can be compared to assess a surplus position, but they are not interchangeable: the reserve is measured on a valuation basis, while the asset share follows its stated accumulation basis.

The Stress Attribution Identity

Use superscripts $b$ and $s$ for the base and stressed scenarios. The scenarios must refer to the same business and valuation dates. Define each scenario's period profit as the change in its own surplus. Subtracting base profit from stressed profit gives the attribution identity:

$$ P^s-P^b =(A_1^s-A_1^b)-(V_1^s-V_1^b) -(A_0^s-A_0^b)+(V_0^s-V_0^b). $$

The first two terms describe the change in ending surplus. The last two terms correct for any change in opening surplus. If opening assets are identical in both scenarios, the identity simplifies to:

$$ P^s-P^b =(A_1^s-A_1^b)-(V_1^s-V_1^b)+(V_0^s-V_0^b). $$

Derivation: In either scenario, $P=A_1-V_1-(A_0-V_0)$. Subtract the base expression from the stressed expression, grouping each stressed-minus-base difference by valuation time. This yields the identity above. The opening reserve term has a positive sign because a higher opening reserve reduces opening surplus and therefore increases the change in surplus, all else equal. The result relies on comparing profits defined with the same scope and accounting convention; it does not treat an ending-surplus difference alone as a profit difference.

When a stress changes the opening reserve, make that movement visible. For example, if opening assets are unchanged and the stressed opening reserve is higher, stressed opening surplus is lower. The scenario's period profit is calculated from its own opening surplus; the opening revaluation must not be silently omitted from the comparison.

Worked Stress Reconciliations

Worked Example: Expense Stress and Reserve Revaluation

A hypothetical closed policy block has opening assets of $100 and an opening reserve of $92 on the base basis. Under stress, the opening reserve is $94; opening assets are unchanged. A premium of $12 is received at the start of the period and invested immediately. The annual effective investment return is 4%. A claim and expense payment is made at year-end: $8 in the base scenario and $10 under stress. Ending reserves are $96 base and $99 stressed. Calculate each scenario's profit and the stressed-minus-base profit change.

Method. First accumulate the opening assets and premium to year-end, then deduct the year-end payment. Calculate opening and ending surplus separately in each scenario. This avoids treating the ending-surplus difference as the profit difference.

Base ending assets: the amount invested is $100+12=112$. Its year-end value is $112(1.04)=116.48$. After the $8 payment, $A_1^b=116.48-8=\$108.48$.

Stressed ending assets: the same $112 is invested, and the return remains 4%. After the $10 payment, $A_1^s=112(1.04)-10=116.48-10=\$106.48$.

Base profit: opening surplus is $100-92=\$8$. Ending surplus is $108.48-96=\$12.48$. Thus $P^b=12.48-8=\$4.48$.

Stressed profit: opening surplus is $100-94=\$6$. Ending surplus is $106.48-99=\$7.48$. Thus $P^s=7.48-6=\$1.48$. The stressed-minus-base profit change is $1.48-4.48=-\$3.00$.

Check by attribution: ending assets change by $106.48-108.48=-\$2$; ending reserves change by $99-96=\$3$; opening reserves change by $94-92=\$2$. Since opening assets are unchanged, the profit change is $-2-3+2=-\$3.00$, as calculated directly.

Common mistake avoided: the ending-surplus change is $7.48-12.48=-\$5$, not the profit change. Opening surplus also falls by $2, so the profit change is $-5-(-2)=-\$3$.

Using an Asset Share in the Bridge

For a policy cohort, an asset share can provide the accumulated asset measure, provided the cash flows, timing, and investment basis are defined. For a one-year illustration, suppose the opening asset share is $AS_0$, a premium and expense occur at the start of the year, a return is credited over the year, and a benefit is paid at year-end. With effective annual return $i$, the recurrence is:

AS_1=(AS_0+\text{start-of-year premium}-\text{start-of-year expense})(1+i) -\text{year-end benefit}.

The timing in this expression matters: the start-of-year cash flows earn a full year's return, while the year-end benefit does not. If the asset share is the asset measure used for the cohort's surplus comparison, calculate $AS-V$ at both dates and apply the same profit bridge. Any mismatch in scope—for example, comparing a cohort asset share with a reserve for a different set of policies—makes the resulting surplus attribution invalid.

Worked Example: Asset-Share Stress and the Profit Bridge

A hypothetical cohort has an opening asset share of $50. At the start of the year it receives a $20 premium and incurs an expense. The base expense is $4 and the stressed expense is $6. The annual effective return is 5% base and 4% stressed. A benefit of $12 is paid at year-end in both scenarios. Opening and ending reserves are $60 and $55 base, and $61 and $58 stressed. Find the asset shares, each period profit, and the profit change.

Method. Apply the asset-share recurrence with the given cash-flow timing. Then define surplus as asset share less reserve at each valuation time. This keeps investment accumulation and reserve valuation distinct.

Base asset share: the amount invested at the start is $50+20-4=\$66$. Accumulating it at 5% gives $66(1.05)=\$69.30$. Deducting the year-end benefit gives $AS_1^b=69.30-12=\$57.30$.

Stressed asset share: the amount invested is $50+20-6=\$64$. Accumulating it at 4% gives $64(1.04)=\$66.56$. After the benefit, $AS_1^s=66.56-12=\$54.56$.

Base profit: opening surplus is $50-60=-\$10$. Ending surplus is $57.30-55=\$2.30$. Therefore $P^b=2.30-(-10)=\$12.30$.

Stressed profit: opening surplus is $50-61=-\$11$. Ending surplus is $54.56-58=-\$3.44$. Therefore $P^s=-3.44-(-11)=\$7.56$. The profit change is $7.56-12.30=-\$4.74$.

Check by attribution: ending asset share changes by $54.56-57.30=-\$2.74$; the ending reserve increases by $58-55=\$3$; and the opening reserve increases by $61-60=\$1$. Thus the profit change is $-2.74-3+1=-\$4.74$.

Common mistake avoided: the asset-share reduction of $2.74 is not the profit reduction. The stressed ending reserve is also higher, while the opening reserve is higher too. All three movements are needed for the profit bridge.

Attributing a Block-Level Solvency Change

For a larger block, the same identity applies to totals, as long as asset and reserve amounts cover the same policies and use consistent valuation dates. The bridge identifies the net change in surplus and the period profit impact, but it does not, by itself, identify which underlying cash flow caused an asset movement. For this tutorial's purpose, the key is to reconcile the measures without mixing valuation effects and period changes.

Fix the comparison.
Specify the block, opening and ending dates, base and stressed assumptions, and the profit convention. Record whether external transfers are included or separately allowed for.

Calculate assets consistently.
Use the scenario's investment and cash-flow assumptions to determine opening and ending assets or asset shares. Apply each cash flow at its stated time.

Calculate reserves separately.
Use each scenario's reserve basis at both valuation dates. Do not substitute an asset share for a reserve.

Build the surplus and profit bridge.
Calculate $A-V$ at opening and ending times in each scenario, then compare the changes in surplus. Confirm the answer with the stress attribution identity.

Worked Example: Aggregated Solvency Surplus Attribution

A hypothetical block is measured in thousands of dollars. Opening assets are 1,000 in both scenarios. The opening reserve is 940 base and 955 stressed. Ending assets are 1,060 base and 1,038 stressed; ending reserves are 1,020 base and 1,045 stressed. Calculate base and stressed period profit, the change in ending surplus, and the profit attribution.

Method. Work in thousands throughout. Calculate each opening and ending surplus first; this exposes whether the opening position also changes under stress.

Base: opening surplus is $1{,}000-940=60$. Ending surplus is $1{,}060-1{,}020=40$. Therefore base profit is $40-60=-20$, or $-\$20{,}000$.

Stress: opening surplus is $1{,}000-955=45$. Ending surplus is $1{,}038-1{,}045=-7$. Therefore stressed profit is $-7-45=-52$, or $-\$52{,}000$.

The stressed-minus-base change in ending surplus is $-7-40=-47$, or $-\$47{,}000$. The opening surplus change is $45-60=-15$, or $-\$15{,}000$. Hence the profit change is $-47-(-15)=-32$, or $-\$32{,}000$.

Check by attribution: ending assets change by $1{,}038-1{,}060=-22$; ending reserves change by $1{,}045-1{,}020=25$; opening reserves change by $955-940=15$. Opening assets do not change, so the profit change is $-22-25+15=-32$, or $-\$32{,}000$.

Common mistake avoided: reporting $-\$47{,}000$, the change in ending surplus, as the profit change ignores the $-\$15{,}000$ change in opening surplus. The profit change is $-\$32{,}000$.

Interpretation and Common Traps

A sound attribution distinguishes three questions: how ending assets changed, how ending reserves changed, and whether the opening position also changed. A concise report can show all three, along with the resulting profit change. This is especially important when a stress revalues opening reserves: a higher stressed opening reserve lowers opening surplus, so it enters the profit comparison with the opposite sign to the ending-reserve change.

Do not equate reserve movement with profit movement. Profit is based on the change in surplus, and surplus includes assets as well as reserves.
Do not equate ending-surplus change with profit change. They are equal only when opening surplus is the same in both scenarios.
Keep the reserve signs straight. A higher ending reserve reduces ending surplus; a higher opening reserve reduces opening surplus and, other things equal, increases the measured change in surplus.
Respect cash-flow timing in asset shares. A start-of-year premium or expense earns the specified year's return; a year-end benefit in the recurrence shown does not.
Keep scope and units consistent. Compare the same cohort and dates, and do not mix dollars with thousands of dollars.

This reconciliation is useful because it explains how a specified assumption shock appears in valuation, accumulation, and profit measures without treating those measures as synonyms. A complete bridge is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use a consistent base-to-stress comparison and check the answer against the surplus identity.

A block has opening assets of $80 in both scenarios. Opening reserves are $74 base and $76 stressed. Ending assets are $88 base and $85 stressed; ending reserves are $82 base and $86 stressed. Find each profit and the stressed-minus-base profit change. (Answer: base profit $0; stressed profit $-\$5$; change $-\$5$)
A cohort has opening asset share $30, receives a $10 premium and incurs a $2 expense at the start of the year, earns an effective annual return of 6%, and pays a $5 benefit at year-end. Opening reserve is $34 and ending reserve is $33. Find ending asset share and period profit. (Answer: ending asset share $35.28; profit $6.28)
Opening assets are identical in both scenarios. Ending assets increase by $4 under stress relative to base, the ending reserve decreases by $3, and the opening reserve decreases by $1. What is the stressed-minus-base profit change? (Answer: $6)
A cohort has opening asset share $40, a start-of-year premium of $8, a start-of-year expense of $1, a 5% effective annual return, and a year-end benefit of $6. Opening and ending reserves are $45 and $42. Find ending asset share and period profit. (Answer: ending asset share $43.35; profit $6.35)
In thousands of dollars, opening assets are 180 in both scenarios. Opening reserves are 175 base and 178 stressed. Ending assets are 200 base and 196 stressed; ending reserves are 190 base and 194 stressed. Find each profit and the stressed-minus-base profit change. (Answer: base profit $5, stressed profit $0, change $-\$5$, all in thousands of dollars)

What You'll Learn