From Product Cash Flows to Solvency Measures
A product projection may contain premiums, charges, expenses, investment income, benefits, and reserves. A solvency calculation asks what these items imply for the insurer’s balance sheet and capital position. The essential discipline is to trace each item once, with a consistent boundary between assets and liabilities. A premium credited to a policy account, for example, can increase both assets and the account-value liability; it does not automatically create an equal amount of profit.
This tutorial uses a simplified balance-sheet framework to connect product cash flows to own funds and a stated capital requirement. It also distinguishes those measures from an asset share, which is an accumulated product-level measure rather than the insurer’s solvency balance sheet. The calculations are hypothetical and use only the assumptions stated in each example.
Build a Consistent Cash-Flow Map
Before calculating a solvency result, label each item by where it appears. External premiums and expenses generally change the assets in the projection. A benefit paid from those assets reduces them. Investment income increases them. The liability changes separately as benefits are incurred, premiums or investment returns are reflected in policy obligations, and reserves are released or strengthened.
A deduction from a universal life policy account needs particular care. If the deduction reduces the account-value liability but does not bring new money into the insurer’s total assets, it is not an external cash receipt \(G\). Its effect is already captured in the lower closing liability. Counting it as cash received as well as recognizing the liability reduction would count the same item twice. Conversely, if the projection boundary treats an amount as an actual external receipt, it belongs in \(G\), and the related liability treatment must still be consistent.
| Item | Typical place in the bridge | Consistency check |
|---|---|---|
| Premium received from outside the insurer | External inflow \(G\) | Reflect any resulting policy obligation in \(R_1\) |
| Expense paid from projected assets | External outflow \(E\) | Do not subtract the same expense again from net cash flow |
| Claim paid from projected assets | External outflow \(B\) | Use the same claim timing and reserve basis |
| Investment income | Asset increase \(I\) | Apply the stated rate to the stated investment base and period |
| Charge deducted from a policy account | Usually reflected in the liability movement | Do not also count it as external cash unless it is actually received across the chosen boundary |
| Reserve strengthening | Increase in \(R_1-R_0\) | An increase in liability reduces own funds, all else equal |
Why the Own-Funds Bridge Works
The reconciliation follows directly from the definition of own funds; it is not a separate profit convention. The asset change is the external cash received, less external expenses and benefits, plus investment income. The liability change must then be deducted because own funds are assets minus liabilities.
With the stated convention, \(\Delta A=G-E-B+I\). Internal policy-account transfers are not added as external receipts.
The liability change is \(\Delta R=R_1-R_0\), using the same product scope and period as the asset projection.
Since \(\Delta(A-R)=\Delta A-\Delta R\), substitution gives \(\Delta(A-R)=G-E-B+I-\Delta R\).
The sign of the reserve movement is important. If the closing reserve exceeds the opening reserve, then \(\Delta R>0\), and own funds are lower than they would be with no reserve increase. A reserve release has the opposite effect. This is a balance-sheet reconciliation: it does not mean that every reserve increase is a cash payment.
Worked Example: Expected Asset Share for a Hypothetical Cohort
A hypothetical one-year policy cohort receives a premium of $600 per original policy at the start of the year and pays a start-of-year expense of $50 per original policy. The net amount is invested for one year at an effective annual return of 3%. At year-end, there is a 1% probability per original policy of paying a $400 claim; otherwise no claim is paid. There are no lapses, other cash flows, or opening assets. Find the expected year-end asset share per original policy. The method is to accumulate the invested amount and subtract the expected claim outflow. The common mistake to avoid is subtracting the claim probability as if it were a dollar amount.
Step 1: Amount invested at the start of the year. The premium less the expense is \(600-50=\$550\).
Step 2: Accumulate the invested amount. At 3% for one year, it becomes \(550(1.03)=\$566.50\).
Step 3: Find the expected claim. The probability of a $400 claim is 0.01, so the expected claim per original policy is \(0.01(400)=\$4.00\).
Step 4: Subtract the expected claim. The expected year-end asset share is \(566.50-4.00=\$562.50\) per original policy.
Check. The expected claim is only one percent of $400, not the full $400. The calculated asset share is below the accumulated $566.50 by exactly $4.00, as required.
Universal Life Account Deductions and the Liability Movement
For an account-based contract, one useful way to keep the bridge consistent is to separate the account-value component of the liability from any additional provision. The account value can change through premiums credited to it, investment crediting, and deductions. Those movements may affect the liability even when they are not external cash flows in the insurer’s consolidated projection.
The following example assumes the assets are pooled. A charge deducted from the policy account reduces the account-value liability; it does not leave the insurer’s pooled assets. The example also assumes the stated investment return applies to the stated asset and account-value bases for the full year. These are simplifying assumptions for the reconciliation, not universal product rules.
Worked Example: Reconciling a Hypothetical Universal Life Projection
At the start of a year, a hypothetical insurer has projected assets of $1,000, an account-value liability of $900, and an additional guarantee provision of $50. It receives a $200 premium at the start of the year and pays a $30 expense immediately. Assets earn 4% on the resulting $1,170 for the year. The policy account receives the $200 premium, earns 4% on its $1,100 pre-crediting balance, and has a $40 charge deducted after investment crediting. The closing guarantee provision is $25. Find the closing assets, closing liability, and change in own funds. There are no claims or other cash flows. The method is to project assets and liability components separately, then reconcile them. The common mistake is to count the $40 account deduction as a cash receipt in addition to recognizing its reduction of the account liability.
Step 1: Opening liability and own funds. The opening liability is \(R_0=900+50=\$950\). Opening own funds are \(1{,}000-950=\$50\).
Step 2: Project assets. After the premium and expense, the asset amount before investment income is \(1{,}000+200-30=\$1{,}170\). Investment income is \(1{,}170(0.04)=\$46.80\). Thus closing assets are \(A_1=1{,}170+46.80=\$1{,}216.80\).
Step 3: Project the account-value liability. The account balance before crediting is \(900+200=\$1{,}100\). Its investment credit is \(1{,}100(0.04)=\$44.00\), giving \(1{,}144.00\). After the $40 charge, the closing account value is \(1{,}144-40=\$1{,}104\).
Step 4: Add the closing guarantee provision. The closing total liability is \(R_1=1{,}104+25=\$1{,}129\).
Step 5: Calculate closing and changed own funds. Closing own funds are \(1{,}216.80-1{,}129=\$87.80\). The increase is \(87.80-50=\$37.80\).
Reconciliation check. The liability increase is \(1{,}129-950=\$179\). The bridge gives \(200-30+46.80-179=\$37.80\), matching the change in own funds. The $40 charge is already reflected in the account-value liability; adding it again as an external receipt would make the bridge inconsistent.
From Own Funds to a Solvency Result
A solvency result commonly compares own funds with a separately determined capital requirement. For a simplified illustration, define the solvency surplus as own funds less the stated requirement, and define the coverage ratio as own funds divided by that requirement. These calculations are meaningful only when own funds and the requirement are measured on compatible bases and dates. Actual regulatory frameworks may define eligible own funds, capital requirements, adjustments, and reporting rules differently.
A stress scenario changes one or more projected asset or liability values. Its effect on own funds can be traced through the same bridge: lower assets reduce own funds, and higher liabilities also reduce own funds. The capital requirement may itself change under stress or under a prescribed scenario; it should therefore be stated separately rather than inferred from the own-funds movement.
Worked Example: Translating a Stress into a Solvency Result
A hypothetical insurer’s base projection has closing assets of $5,360,000, closing liabilities of $4,750,000, and a stated capital requirement of $300,000. Under a specified hypothetical stress, additional claims reduce closing assets by $80,000, lower investment income reduces them by a further $25,000, and reserve strengthening increases closing liabilities by $60,000. The stressed capital requirement is $380,000. Calculate base and stressed own funds, solvency surplus, and coverage ratio. The method is to update assets and liabilities separately before comparing own funds with the requirement. The common mistake is to subtract the reserve strengthening from assets instead of increasing the liability.
Step 1: Base own funds. They are \(5{,}360{,}000-4{,}750{,}000=\$610{,}000\).
Step 2: Base solvency measures. The base surplus is \(610{,}000-300{,}000=\$310{,}000\). The base coverage ratio is \(610{,}000/300{,}000=2.0333\), or 203.3% when expressed as a percentage.
Step 3: Stressed assets and liabilities. Stressed assets are \(5{,}360{,}000-80{,}000-25{,}000=\$5{,}255{,}000\). Stressed liabilities are \(4{,}750{,}000+60{,}000=\$4{,}810{,}000\).
Step 4: Stressed own funds and solvency measures. Stressed own funds are \(5{,}255{,}000-4{,}810{,}000=\$445{,}000\). The stressed surplus is \(445{,}000-380{,}000=\$65{,}000\). The stressed coverage ratio is \(445{,}000/380{,}000=1.1711\), or 117.1%.
Check. Own funds fall by \(610{,}000-445{,}000=\$165{,}000\). This equals the total asset reduction of $105,000 plus the liability increase of $60,000. The stressed surplus remains positive because stressed own funds exceed the stated stressed requirement by $65,000.
Asset Share, Profit, and Solvency Are Not Interchangeable
An asset share accumulates allocated premiums and investment returns, less allocated expenses and benefits, for a defined policy group or policy. A reserve is a liability measured on a specified valuation basis. Own funds compare assets with liabilities on a balance-sheet basis, while a solvency surplus or ratio also uses a capital requirement. These quantities can be related, but they are not synonyms and need not be equal.
The projection basis matters. Experience assumptions used to project premiums, claims, lapses, expenses, and investment returns may differ from the assumptions used to value liabilities. Any comparison must identify which basis each amount uses. The earlier material on experience assumptions versus valuation assumptions and modified reserve bases is relevant when interpreting differences between projected profit and reserve movements.
A reliable calculation assigns each flow to one place, projects assets and liabilities using their stated timing, and then reconciles the result. This cash-flow discipline is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Common Mistakes and Exam Checks
- Subtracting outflows twice. If net cash flow already means receipts less expenses and benefits, do not subtract expenses and benefits again. For example, receipts of $100, expenses of $10, and benefits of $5 give net cash flow of $85. Subtracting the $10 and $5 again gives the incorrect $70. With no investment income or reserve change, the correct asset increase is $85.
- Counting an account deduction on both sides. If a policy-account charge reduces the liability and is not external cash received, do not also add it to \(G\).
- Using the wrong sign for a reserve change. An increase from \(R_0\) to \(R_1\) reduces own funds by \(R_1-R_0\); it is not an asset receipt.
- Mixing timing or investment bases. A rate applied to a full-year opening balance is not interchangeable with the same rate applied to a balance after a start-of-year expense or premium.
- Calling an asset share a solvency result. An asset share does not, by itself, include the insurer’s full balance sheet or capital requirement.
Check Your Understanding
Work each problem using the stated timing and projection boundary. Amounts are hypothetical.
- A cohort receives a $500 premium and pays a $20 expense at the start of a year. The net amount earns 5% for one year. At year-end, a $300 claim is paid with probability 0.02 per original policy. Find expected asset share per original policy. (Answer: $498.00)
- A universal life account has an opening value of $800. A $100 premium is credited at the start of the year; the account earns 3%, and a $25 charge is deducted after crediting. An additional closing guarantee provision is $18. Find the closing total liability. (Answer: $920.00)
- Opening assets are $2,000,000 and opening liabilities are $2,000,000. During the period, external receipts are $150,000, expenses are $20,000, benefits are $60,000, and investment income is $70,000. Closing liabilities are $2,075,000. Find the change in own funds. (Answer: an increase of $65,000)
- Closing assets are $1,240,000, closing liabilities are $1,100,000, and the stated capital requirement is $100,000. Find own funds, the solvency surplus, and the coverage ratio. (Answer: own funds $140,000; surplus $40,000; coverage ratio 140.0%)
- Base own funds are $220,000. In a stress, assets fall by $45,000 and liabilities increase by $35,000. The stressed capital requirement is $125,000. Find stressed own funds, surplus, and coverage ratio. (Answer: own funds $140,000; surplus $15,000; coverage ratio 112.0%)