Why a Reference Projection Comes First
A stress result is meaningful only in relation to a clearly defined baseline. Before changing an assumption, an actuary needs a reference projection: a consistent set of assumptions, cash-flow timings, valuation methods, and output measures that describes the business before the stress is applied. If, for example, the baseline and stressed calculations use different expense timing or reserve definitions, their difference cannot be attributed solely to the stress.
A reference projection is not a promise of future experience. It is a structured calculation based on a stated set of assumptions. It should project the same policies and cash flows on a coherent basis, and it should make clear which outputs are being compared. This topic is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Specify the Basis Before Calculating
The label “best estimate” does not, by itself, specify every modeling choice. In practice, the precise basis depends on the purpose and valuation framework. A useful reference specification records the assumptions and conventions that determine the projected results. It should say which policies are included, how decrements and expenses are projected, how investment income is earned, and how future cash flows are valued.
Timing conventions matter as much as assumption levels. If premiums arrive at the start of a year and benefits are paid at the end, the premium earns a year of investment income but the benefit does not. A projection that silently changes this timing is not a like-for-like comparison.
| Projection item | Specify before calculating |
|---|---|
| Business and boundary | Policies included, valuation date, projection horizon, and treatment of policies entering or leaving the block |
| Demographic assumptions | Mortality, morbidity, lapse, recovery, and any other relevant transition assumptions |
| Financial assumptions | Investment return, discount rate, inflation, and any guarantees or participation assumptions included |
| Cash-flow timing | When premiums, expenses, benefits, taxes, and other cash flows occur within each projection period |
| Valuation outputs | Reserve definition and basis, asset-share method, and whether amounts are per policy or for the whole block |
| Comparison measure | For example, projected profit, change in surplus, a ratio at a specified date, or discounted profit over a stated horizon |
A reference projection should not mix quantities that use different scopes or bases without identifying the difference. For example, an asset amount for an entire block cannot be compared directly with a reserve per surviving policy. Similarly, a reserve calculated on one set of future cash flows is not directly comparable with projected assets that omit some of those flows.
Assets, Reserves, Surplus, and Profit
At a valuation time \(t\), let \(A_t\) be the assets allocated to the projected block and let \(V_t\) be the corresponding reserve liability. Define surplus as assets less reserves. If there are no shareholder contributions or distributions during the period, the period’s profit emergence can be measured as the change in surplus. This is an accounting reconciliation on the stated projection basis; it is not a claim that every valuation framework defines reported profit identically.
If a shareholder contribution or distribution occurs, it must be identified separately. For example, if a distribution \(D_{t+1}\) is deducted from closing assets, profit before distribution is \(S_{t+1}-S_t+D_{t+1}\). Omitting the adjustment would treat a transfer of funds as if it were an operating loss.
The one-period reserve relation follows directly from that definition. If expected net outgo during the next year is paid at year-end and the expected reserve for continuing policies at year-end is \(V_{t+1}\), then the opening reserve is the present value of those two components. If the effective annual interest rate is \(i\), the one-year discount factor is \(v=1/(1+i)\).
This relation makes the timing explicit: both the year’s net outgo and the continuing reserve are measured at time \(t+1\), then discounted to time \(t\). If cash flows occur at different times, or if the valuation includes future premiums, their timing and signs must be reflected in the prospective calculation rather than forced into this simplified one-period form.
Worked Example: Reconciling a One-Year Reference Projection
A hypothetical block has 100 in-force single-premium protection policies at time 0. Assets are $15 per policy, or $1,500 for the block. The annual effective investment return is 4%. During year 1, the probability of a claim is 5% per policy; each claim pays $100 at year-end. Year-1 expenses are $3 per policy at year-end. Assume expected values, no lapses, and no additional premiums. At time 1, each surviving policy has a one-year remaining term with claim probability 4%, a $100 claim benefit, and $4 of year-end expense. Calculate the opening reserve, closing assets and reserve, and year-1 profit. Use the same expected cash flows and interest rate throughout.
Method and timing. Project the expected year-1 outgo and survivors, then calculate the time-1 reserve for those survivors. Discount the year-1 outgo and time-1 reserve to obtain the opening reserve. Premiums are absent, and all year-1 claims and expenses occur at year-end.
Expected year-1 claims are \(100(0.05)=$5\) claims, so expected claim payments are \(5($100)=$500\). Year-1 expenses are \(100($3)=$300\). Total expected year-end outgo is therefore \(500+300=$800\), and expected survivors are \(100(1-0.05)=95\).
For each time-1 survivor, expected remaining-year outgo is \(0.04($100)+$4=$8\). The reserve per survivor at time 1 is \(8/1.04=$7.6923077\), so the aggregate time-1 reserve is \(95($7.6923077)=$730.7692\).
The opening reserve is the discounted value of the expected $800 year-end outgo plus the $730.7692 continuing reserve:
Opening surplus is \(S_0=$1{,}500-$1{,}471.8935=$28.1065\). Closing assets are \(A_1=$1{,}500(1.04)-$800=$760\). Closing surplus is \(S_1=$760-$730.7692=$29.2308\). Thus year-1 profit is \(S_1-S_0=$1.1243\), or $1.12 to the nearest cent.
Check and common mistake. The opening asset amount is $15 per policy times 100 policies, or $1,500 total; it is not $15,000 or $150,000. The asset roll-forward also checks: investment income is \(0.04($1,500)=$60\), and \(1{,}500+60-800=$760\). Applying the closing reserve then gives the stated profit.
Asset Shares and Their Relation to Reserves
An asset share tracks accumulated amounts allocated to a policy or cohort under specified cash-flow and investment conventions. For a cohort, it is often clearest to project total accumulated funds first and then divide by the relevant number of policies, such as the expected number surviving. An asset share is not itself a reserve: it summarizes accumulated assets under the chosen allocation method, whereas a reserve values future expected net obligations.
Worked Example: Comparing a Cohort Asset Share with a Reserve
A hypothetical cohort of 10 policies pays a single premium of $100 per policy at time 0. An acquisition expense of $20 per policy is also paid at time 0. The effective annual investment return is 5%. During the year, each policy has a 10% claim probability; a claim of $200 is paid at year-end. There are no other cash flows. Assume expected values. At time 1, each surviving policy has one year remaining, with a 10% claim probability, a $200 claim benefit, and a $10 year-end maintenance expense. Find the time-1 asset share per survivor and reserve per survivor.
Method. Project total cohort funds through the year, deduct expected year-end claims, and divide by expected survivors. Separately, value the remaining year’s expected outgo per survivor.
At time 0, premiums total \(10($100)=$1{,}000\), and acquisition expenses total \(10($20)=$200\). Funds after these immediate cash flows are \(1{,}000-200=$800\). Investment income is \(0.05($800)=$40\), so funds before claims at year-end are $840.
Expected year-1 claims are \(10(0.10)=1\), with expected payments of \(1($200)=$200\). Closing cohort funds are therefore \(840-200=$640\). Expected survivors are \(10(1-0.10)=9\), giving an asset share of \(640/9=$71.1111\), or $71.11 per survivor.
For each survivor, expected remaining-year claim payments are \(0.10($200)=$20\), and the maintenance expense is $10. The time-1 reserve per survivor is \((20+10)/1.05=$28.5714\), or $28.57. The asset share exceeds the reserve by \(71.1111-28.5714=$42.5397\), or $42.54 per survivor.
Check and common mistake. The asset share is based on $640 of closing cohort funds divided by 9 survivors, not divided by the original 10 policies. The reserve is the discounted expected future outgo, not the accumulated asset share. Their difference is a comparison of projected assets with liabilities, not a redefinition of either measure.
Choosing and Comparing Reference Outputs
A projection can produce many outputs; the comparison measure should be chosen before interpreting results. A period profit is the change in surplus over that period. A discounted profit measure can aggregate profit emergence over a stated horizon. A solvency ratio or capital amount is a different type of output and requires its own defined numerator, denominator, and valuation date. The same measure, timing, and business scope should be used for the reference and any later scenario.
The following example demonstrates a two-year profit comparison. The reserves are stipulated as outputs of the same best-estimate continuation valuation at each date; the calculation does not assume that the two-year projection horizon is the end of the contracts.
Worked Example: Setting a Two-Year Profit Comparison Measure
A hypothetical block has opening assets of $500 and a time-0 reserve of $450. The annual effective investment return is 4%. In year 1, a premium of $100 is received at the start of the year and $80 of expected outgo is paid at year-end. The time-1 reserve is $480. In year 2, a premium of $50 is received at the start of the year and $90 of expected outgo is paid at year-end. The time-2 reserve is $500. Calculate each year’s profit and the present value at time 0 of those two profits, discounting at 4%.
Method. Apply the same asset timing convention in both years: add the start-of-year premium before earning interest, deduct year-end outgo after interest, and define profit as the change in surplus. Discount each period’s profit to time 0 for the stated comparison measure.
At time 0, surplus is \(S_0=$500-$450=$50\). Year-1 closing assets are \((500+100)(1.04)-80=$544\). Therefore \(S_1=$544-$480=$64\), and year-1 profit is \(64-50=$14\).
Year-2 closing assets are \((544+50)(1.04)-90=$527.76\). Therefore \(S_2=$527.76-$500=$27.76\), and year-2 profit is \(27.76-64=-$36.24\).
The present value of the two profits is:
The selected two-year present-value measure is therefore -$20.04 to the nearest cent. This is a discounted measure of profit emergence over the stated horizon and on the stated basis. Because the time-2 reserve is $500, the horizon does not represent full run-off. Adding terminal surplus to this measure would require an explicitly defined comparison formula; it should not be added automatically.
Check and common mistake. The year-1 premium earns a full year of interest, so \((500+100)(1.04)\) is used. Applying interest only to the opening $500 would give a different result and would contradict the stated timing. The reserve is also deducted at each valuation date when calculating surplus, not treated as a cash payment.
Common Projection Traps
- Changing the basis between runs. A difference in results is not attributable to one changed assumption if the projection horizon, expense definition, or reserve basis also changes.
- Mixing per-policy and aggregate amounts. State the number of policies and whether each asset, reserve, or cash flow is per policy or for the block.
- Applying investment income at the wrong time. A cash flow received at the start of the year and one paid at year-end do not earn or lose the same amount of interest.
- Confusing asset share and reserve. Asset share accumulates allocated funds; reserve values future net obligations. Their difference can be informative, but the measures are not interchangeable.
- Ignoring boundary or terminal values. A truncated projection can leave material future obligations. Identify the terminal reserve and avoid adding it twice to a profit-based comparison.
- Calling an expected projection a realized result. Expected claims and survival counts are model quantities. Actual policy outcomes may differ.
Check Your Understanding
Use the stated timing and basis in each problem. Amounts are hypothetical and assume expected values where probabilities are given.
- A block has opening assets of $2,000. A premium of $300 is received at the start of the year, the annual effective return is 3%, and benefits and expenses of $600 are paid at year-end. Find closing assets. (Answer: $1,769)
- A one-year reserve covers a 2% claim probability, a $500 claim benefit, and a $5 year-end expense per policy. The annual effective interest rate is 5%. Find the reserve per policy. (Answer: $14.29)
- Twenty policies each pay a single premium of $50 at time 0. An expense of $5 per policy is paid immediately. The annual effective return is 4%, the year-end claim probability is 5%, and each claim pays $100. Find the time-1 asset share per expected survivor. (Answer: $44.00)
- Opening assets are $1,000 and the opening reserve is $900. At the end of the year, assets are $1,050 and the reserve is $980. With no shareholder contributions or distributions, find the year’s profit using change in surplus. (Answer: -$30)
- A reference projection produces profits of $25 at time 1 and -$10 at time 2. The comparison discount rate is 5%. Find the time-0 present value of the two profits, rounding to the nearest cent. (Answer: $14.74)