Auditing Integrated Product and Solvency Calculations

Why Integrated Calculations Need Independent Checks

A product projection can contain individually plausible numbers and still be internally inconsistent. A fund account may not match its unit ledger; an asset share may not reconcile to the projected cash flows; or reported own funds may not equal assets less liabilities. Auditing means checking the links between these quantities, rather than accepting each output in isolation.

The checks in this tutorial apply to hypothetical projections. They are accounting and valuation checks, not substitutes for validating the product design, assumptions, or applicable solvency rules. A reconciliation can show that calculations agree with one another while still using an inappropriate assumption. The auditor must therefore check both the arithmetic and the stated basis.

Definition: An audit identity is a relationship that must hold when specified quantities use the same timing, population, units, and valuation basis. A failure to reconcile identifies an inconsistency to investigate; it does not, by itself, identify which input or calculation is wrong.

Establish the Basis Before Comparing Outputs

Before comparing values, record the projection date, the policies included, the cash-flow timing, the currency and scale, and whether a figure is per policy or aggregate. These details matter. For example, an asset share per continuing policy cannot be compared directly with a reserve for the whole cohort. Likewise, a fund value measured before a charge is not directly comparable with one measured after the charge.

For a linked account, the unit ledger and monetary account provide two calculation routes. If $N_t$ units are held at time $t$, and the unit price is $U_t$, the account value is $N_tU_t$. When units are purchased or cancelled during a period, the account roll-forward must reflect the corresponding cash allocations and charges, with investment return applied according to the stated timing.

Audit identity: For a unit-linked account valued after all transactions at time $t$, the reported account value must equal the closing number of units multiplied by the closing unit price. The unit ledger must also reconcile opening units, unit purchases, and unit cancellations.

Worked Example: Reconciling Units, Charges, and Account Value

A hypothetical policy begins the year with 120 units priced at $10.00 each. A premium allocation of $570 buys units at $10.00 each. During the year, the unit price rises by 5%, and a charge then cancels 7 units at the closing price. Find the closing unit count and account value, and check the result by a monetary roll-forward. Assume no other transactions.

Method. First reconcile units; then value the closing units. For the independent monetary check, apply the 5% return to the account before the end-of-year unit cancellation. This timing is part of the stated hypothetical projection.

The premium purchases $570/10=57$ units. The closing unit count is therefore

120+57-7=170\text{ units}.

The closing unit price is $10.00(1.05)=\$10.50$. The unit-based account value is $170(10.50)=\$1,785.00$.

For the monetary roll-forward, the opening value is $120(10)=\$1,200.00$. Immediately after the premium allocation, it is $1,200+570=\$1,770.00$. Investment return is $1,770(0.05)=\$88.50$, giving $\$1,858.50$ before the charge. The cancelled units are worth $7(10.50)=\$73.50$. Thus the closing account value is $1,858.50-73.50=\$1,785.00$, agreeing with the unit calculation.

Common mistake avoided. Deducting the $73.50 charge before applying the investment return would use a different transaction timing. Under the stated timing, both calculations agree at $1,785.00.

Asset Shares, Reserves, and Profit Must Be Kept Distinct

An asset share tracks accumulated experience-related cash flows allocated to a policy or group, under a stated accumulation basis. A reserve is a liability valuation at a specified date and on a specified assumption basis. They answer different questions and need not be equal. If they are compared, align their date and population first, then interpret their difference as a surplus or shortfall under the chosen bases.

For a simple one-period audit, let $C$ be the cohort’s assets after the period’s cash flows, and let $V$ be the total reserve for policies remaining in force. The end-period surplus is $C-V$. If the beginning reserve is zero and the profit convention values the period’s net cash flow at period end, the period profit is that same difference. With a nonzero opening reserve, the reserve movement must also be included in the profit calculation.

Audit identity: Under a stated profit convention, profit must reconcile to the period’s net cash flows and the change in reserves. In the simple one-period case with zero opening reserve, end-period profit equals end-period assets less end-period reserves. Asset share and reserve are not interchangeable; their comparison is meaningful only when dates, populations, and units are aligned.

Worked Example: Checking Cohort Assets, Asset Share, and Profit

A hypothetical cohort has 100 policies at the start of a year. Each pays a $1,000 premium at the start of the year, and expenses total $10,000 at that time. During the year, 10 policies lapse and receive $200 each at year-end; 2 other policies make a $5,000 claim each at year-end. The remaining 88 policies stay in force. Assume the net opening cash flow earns 5% for the year, and year-end benefits earn no interest during the year. The reserve at year-end is $900 per continuing policy, and the opening reserve is zero. Find the cohort assets, asset share per continuing policy, and period profit.

Method. Roll the start-of-year net cash flow forward, deduct year-end outgo, and compare the resulting assets with the reserve for the continuing policies.

Start-of-year premiums are $100(1,000)=\$100,000$. After the $10,000 expense, the amount invested is $\$90,000$. Investment income is $90,000(0.05)=\$4,500$, so the fund before year-end payments is $\$94,500$.

Lapse payments total $10(200)=\$2,000$, and claims total $2(5,000)=\$10,000$. Total year-end outgo is $\$12,000$, leaving cohort assets of $94,500-12,000=\$82,500$.

The asset share per continuing policy is $82,500/88=\$937.50$. Total reserves are $88(900)=\$79,200$. The end-period surplus is $82,500-79,200=\$3,300$, or $37.50 per continuing policy.

Under the stated zero-opening-reserve convention, the period profit is also $\$3,300$. As a second check, the accumulated net cash flow before year-end payments is $94,500; after payments, it is $82,500. Subtracting the $79,200 closing reserve gives $3,300. The calculation uses the 88 continuing policies for both the per-policy asset share and reserve comparison.

Common mistake avoided. Dividing the $82,500 cohort assets by 100 would produce $825 per original policy, not the asset share per continuing policy. The denominator must match the question and the reserve population.

Profit Vectors and Solvency Totals Provide Further Checks

A profit vector records period profits under a stated projection and profit definition. It should be traceable to the same projected cash flows and reserve movements used elsewhere in the model. If a reported profit is inconsistent with those inputs, check whether it uses a different reserve basis, timing convention, or allocation before assuming the arithmetic is wrong.

Solvency outputs also have basic reconciliation requirements. At a valuation date, own funds are assets less liabilities on the applicable valuation basis. If a coverage ratio is defined as own funds divided by the capital requirement, the numerator and denominator must refer to the same entity and date. If capital modules are aggregated with a correlation formula, the reported aggregate should reproduce that formula before rounding.

Worked Example: Auditing Aggregated Own Funds and Capital

A hypothetical insurer reports amounts in thousands of dollars. Product A has assets of $1,200 thousand and liabilities of $1,050 thousand. Product B has assets of $800 thousand and liabilities of $740 thousand. Their standalone capital requirements are $100 thousand and $80 thousand, respectively, and the specified correlation parameter is 0.25. Aggregate the balance sheet, calculate own funds, aggregate capital using the stated two-module formula, and find the coverage ratio. Report money to the nearest thousand dollars and the ratio to three decimals.

Method. First sum assets and liabilities, then calculate own funds as their difference. Separately aggregate the two capital requirements using the specified formula:

SCR=\sqrt{SCR_A^2+SCR_B^2+2\rho SCR_A SCR_B}.

Total assets are $1,200+800=2,000$ thousand dollars. Total liabilities are $1,050+740=1,790$ thousand dollars. Therefore own funds are $2,000-1,790=210$ thousand dollars. The product-level check gives the same result: Product A has $1,200-1,050=150$ thousand dollars of own funds, and Product B has $800-740=60$ thousand dollars; $150+60=210$.

For capital, the expression under the square root is $100^2+80^2+2(0.25)(100)(80)=10,000+6,400+4,000=20,400$, in squared thousands of dollars. Thus aggregate capital is $\sqrt{20,400}=142.8286$ thousand dollars, or $143,000 to the nearest thousand.

Using the unrounded capital amount, the coverage ratio is $210/142.8286=1.4703$, or 1.470 to three decimals. The ratio exceeds 1 because own funds are greater than the calculated capital requirement; this arithmetic check does not establish that the assumptions or regulatory basis are appropriate.

Common mistake avoided. Adding the standalone capital requirements gives $180 thousand, but that does not reproduce the specified correlated aggregation. The formula, rather than a simple sum, governs this hypothetical calculation.

A Practical Audit Sequence

A disciplined sequence helps narrow down where a mismatch originates. Keep unrounded amounts during intermediate steps, and record any rounding only after the reconciliations are complete.

Fix the comparison basis.
Identify the valuation date, policy population, timing, currency, and whether each result is per policy or aggregate.

Reconcile the product ledger.
Check opening balance, premiums, allocations, unit purchases, investment returns, charges, and closing units or account value.

Rebuild accumulated assets.
Trace the relevant premium and expense flows, investment accumulation, and benefits or surrender payments to the stated asset-share date.

Reconcile liabilities and profit.
Confirm the reserve population and basis, then check that the profit vector reflects cash flows and reserve movements under its stated convention.

Reconcile solvency outputs.
Check that assets less liabilities equal own funds, that capital aggregation follows the specified method, and that the coverage ratio uses consistent amounts.

Common Audit Traps

Mixing populations. A cohort asset total and a per-policy reserve cannot be compared until they are put on the same basis.
Comparing different dates. A closing account value after a charge is not the same measure as a value before the charge.
Using inconsistent scales. A figure in thousands must not be added directly to a figure in dollars.
Treating asset share as reserve. One accumulates experience-related cash flows; the other values a liability on a stated basis.
Checking only the final ratio. A plausible coverage ratio can conceal errors in either own funds or capital. Reconcile both components independently.

These cross-checks are especially useful when product projections combine guarantees, investment performance, lapses, expenses, bonuses, and capital calculations. If two routes to the same quantity disagree, isolate the first point where the figures diverge. That is usually more informative than adjusting a final result to make the report balance. The methods here are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each hypothetical case, apply the stated basis and check the linked quantities.

A linked account has 80 units at $12.00, receives a $480 allocation at that price, and then has 5 units cancelled after a 10% rise in unit price. Find the closing unit count and account value. (Answer: 115 units; $1,518.00)
A cohort begins with 50 policies, each paying $1,000 at the start of the year. Start-of-year expenses are $4,000. The net amount earns 5% for one year; at year-end, claims total $3,000 and lapse payments total $1,000. If 40 policies remain and their reserve is $1,050 each, find total assets, asset share per remaining policy, and surplus. (Answer: $44,300; $1,107.50; $2,300)
A valuation reports assets of $750 thousand and liabilities of $690 thousand. Find own funds. If the applicable capital requirement is $40 thousand, calculate own funds divided by capital requirement to three decimals. (Answer: $60 thousand; 1.500)
Two capital modules are $90 thousand and $60 thousand, with correlation parameter 0.5. Using the specified square-root aggregation formula, calculate aggregate capital to the nearest thousand dollars. (Answer: $131 thousand)
A fund has 200 units at $8.00. A $400 premium allocation buys units at $8.00, the unit price then rises by 2%, and 3 units are cancelled at the closing price. Find the closing unit count and account value. (Answer: 247 units; $2,015.52)

What You'll Learn