Why Change One Assumption at a Time?
A profit projection can depend on mortality, lapse, expenses, and investment returns at once. If several assumptions change together, the resulting profit difference does not show which input caused it. A one-at-a-time assumption shock avoids that ambiguity: establish a baseline, change one specified input, hold the other inputs and conventions fixed, and compare the outputs.
The comparison should distinguish two related but different outputs. A projected profit test measures the effect of the shock on the year's cash flows and reserve movements under a stated reserve convention. A prospective reserve calculation measures how the liability changes when its valuation assumptions change. A reserve can be remeasured without changing the projected investment return, and a projected return can change without changing the valuation basis. State which output is being compared; do not silently combine them.
A One-Year Model for Profit and Reserve
Use a hypothetical annual model with mutually exclusive outcomes during the year. At the start of the year, an in-force policy has opening reserve \(V_0\), receives premium \(P\), and incurs an immediate expense \(e\). At year-end, death occurs with probability \(q\), lapse occurs with probability \(l\), and the policy continues with probability \(p=1-q-l\). Death benefit \(B\) and lapse benefit \(S\) are paid at year-end; a continuing policy carries closing reserve \(V_1\). The fund available at the start earns an effective annual investment return \(j\). Assume \(q\geq 0\), \(l\geq 0\), and \(q+l\leq 1\).
Here \(\Pi\) is expected year-end profit per policy in force at the start of the year. Death, lapse, and continuation are weighted by their probabilities because each outcome applies to a different portion of the starting in-force group. This is an annual, one-step formula: it does not by itself model the timing of events within the year, policy-specific expenses, or multiple lapse benefits.
For a reserve illustration, suppose the valuation is made just before the start-year premium and expense. The premium is received immediately, the expense is paid immediately, and the year-end obligations are \(B\) on death, \(S\) on lapse, and a continuation reserve \(W\) if the policy remains in force. With valuation effective annual interest \(i\), the prospective reserve is
This reserve formula assumes the same mutually exclusive outcome probabilities and year-end timing stated above. It includes no other future cash flows. In the examples, the projection convention holds the booked opening reserve and closing reserve fixed when measuring the direct profit effect of an experience shock. Reserve remeasurement is calculated separately with the prospective formula. This separation prevents an assumption's direct cash-flow effect from being mixed with a change in the amount booked as reserve.
Why the Formulas Work
The profit formula follows by accounting for the fund and then subtracting expected year-end outgo. The starting fund after premium and expense is \(V_0+P-e\); after investment return it is \((V_0+P-e)(1+j)\). Expected year-end outgo is the sum of the three mutually exclusive amounts weighted by their probabilities. Subtracting that outgo gives the formula:
For the reserve, discount the probability-weighted year-end obligations to the valuation date, then subtract the net amount received immediately. The immediate net receipt is \(P-e\), so it reduces the liability:
These are different calculations: the profit equation uses the investment return \(j\) and the reserve amounts used in the projection, while the prospective reserve uses the valuation rate \(i\) and future expected obligations. They may use the same numerical rate in a particular model, but that is an assumption, not an identity.
Worked One-at-a-Time Tests
The following hypothetical policy supplies a common baseline for the four tests. All monetary amounts are dollars per policy. The policy has \(P=10\), \(e=2\), \(B=100\), \(S=10\), \(W=20\), and \(V_1=20\). Baseline probabilities are \(q=0.02\) and \(l=0.03\), so \(p=1-0.02-0.03=0.95\). The valuation rate is \(i=0.05\), and the baseline investment return is \(j=0.05\).
The baseline prospective reserve is
The year-end fund before outgo is \((12.285714\ldots+10-2)(1.05)=21.30\). Expected outgo is \(0.02(100)+0.03(10)+0.95(20)=2+0.30+19=21.30\), so baseline expected profit is \(0.00\). The unrounded baseline reserve is retained in the calculations below.
Worked Example: Increase the Mortality Assumption
Method. Increase only \(q\), from \(0.02\) to \(0.03\). Keep \(l=0.03\), the benefits, expenses, investment return, and booked reserves fixed for the direct profit comparison. Then calculate the reserve on the shocked probability basis separately.
The shocked continuation probability is \(p=1-0.03-0.03=0.94\). With the baseline opening reserve, the year-end fund remains \(21.30\). Expected outgo becomes \(0.03(100)+0.03(10)+0.94(20)=3+0.30+18.80=22.10\). Thus shocked profit is \(21.30-22.10=-0.80\), a change of \(-0.80\) from baseline.
For reserve remeasurement, use the shocked probabilities in the prospective formula:
The reserve change is \(13.047619\ldots-12.285714\ldots=0.761905\ldots\). To the nearest cent, the shocked reserve is \(\$13.05\), an increase of \(\$0.76\). The direct profit change is \(-\$0.80\). Common mistake avoided: increasing mortality while leaving the continuation probability at \(0.95\) would make the probabilities sum to more than one. Recalculate \(p\) whenever \(q\) or \(l\) changes.
Worked Example: Increase the Lapse Assumption
Method. Increase only \(l\), from \(0.03\) to \(0.05\). Keep \(q=0.02\), so the shocked continuation probability is \(p=1-0.02-0.05=0.93\). For the profit comparison, retain the baseline booked reserves.
Expected outgo is \(0.02(100)+0.05(10)+0.93(20)=2+0.50+18.60=21.10\). The fund before outgo is still \(21.30\), giving profit \(21.30-21.10=0.20\), or an increase of \(\$0.20\) from baseline.
The separate shocked reserve is
Its change from baseline is \(12.095238\ldots-12.285714\ldots=-0.190476\ldots\), so the reserve decreases by \(\$0.19\) to the nearest cent. In this hypothetical model, the lapse benefit is relatively small compared with the released continuation reserve, which explains the positive direct profit change. Common mistake avoided: a lapse shock does not necessarily reduce profit; compare the lapse payment with the reserve no longer held for continuing policies.
Worked Example: Increase the Start-Year Expense
Method. Increase only the immediate expense, from \(e=2\) to \(e=3\). Keep the baseline probabilities and both booked reserves fixed for the direct profit test. The year-end fund before outgo becomes \((12.285714\ldots+10-3)(1.05)=19.285714\ldots(1.05)=20.25\). Expected outgo remains \(21.30\), so profit is \(20.25-21.30=-1.05\). The direct profit change is \(-\$1.05\).
For the reserve remeasurement, the net immediate receipt is now \(P-e=10-3=7\), one dollar less than at baseline:
The remeasured reserve increases by exactly \(13.285714\ldots-12.285714\ldots=\$1.00\), to \(\$13.29\) to the nearest cent. Common mistake avoided: the immediate expense reduces the fund available to earn interest in the profit projection, so the profit reduction is \(1(1.05)=\$1.05\), not merely \(\$1.00\). In the reserve calculation it is paid immediately, so it increases the required reserve by \(\$1.00\).
Worked Example: Reduce the Investment Return
Method. Reduce only the projected investment return \(j\), from \(0.05\) to \(0.04\). Keep the valuation rate \(i=0.05\), probabilities, cash flows, and booked reserves unchanged. The starting fund is \(12.285714\ldots+10-2=20.285714\ldots\). At the shocked return it grows to \(20.285714\ldots(1.04)=21.097143\ldots\).
Expected outgo remains \(21.30\), so shocked profit is \(21.097143\ldots-21.30=-0.202857\ldots\), or \(-\$0.20\) to the nearest cent. The prospective reserve change is \(\$0.00\), because this test changes the projected earned return but leaves the valuation rate and reserve cash flows fixed. Common mistake avoided: do not also change the valuation rate just because the investment return changed. A valuation-rate shock is a different one-at-a-time test and requires a separate reserve calculation.
Recording and Interpreting Results
A useful test record shows the baseline and shocked input as well as the output differences. For the hypothetical policy, the reserve column reports prospective reserve remeasurement; the profit column reports the direct one-year profit change with the baseline booked opening and closing reserves held fixed.
| Input changed | Baseline | Shocked value | Change in remeasured opening reserve | Change in direct expected profit |
|---|---|---|---|---|
| Mortality probability \(q\) | 0.02 | 0.03 | +\$0.76 | −\$0.80 |
| Lapse probability \(l\) | 0.03 | 0.05 | −\$0.19 | +\$0.20 |
| Immediate expense \(e\) | \$2.00 | \$3.00 | +\$1.00 | −\$1.05 |
| Investment return \(j\) | 0.05 | 0.04 | \$0.00 | −\$0.20 |
The signs depend on the hypothetical benefits, surrender amount, reserve, and timing; they are not universal rules. The table is useful because it preserves the identity of each test. If a reserve is actually rebooked in the projection, state that convention and calculate the resulting profit on that basis rather than adding a reserve change to the fixed-reserve profit result without explanation.
A valuation interest shock is distinct from the earned-return example. For instance, keeping the baseline expected year-end liability numerator \(21.30\) and changing only the valuation rate to \(i=0.04\) gives \(V_0=21.30/1.04-8=12.480769\ldots\), an increase of about \(\$0.20\). This is a reserve remeasurement; it does not change the projected investment return unless that is separately specified.
Common Traps in Controlled Tests
- Changing more than one input. If mortality and expenses both change, their individual contributions cannot be read directly from that single comparison.
- Forgetting the probability balance. When death or lapse probability changes, update \(p=1-q-l\) and check that probabilities remain valid.
- Mixing profit and remeasurement. Identify whether the projection holds booked reserves fixed or recalculates them. Report the reserve change separately if the purpose is to isolate the direct profit effect.
- Confusing earned and valuation interest. The projected investment return accumulates the year's fund; the valuation rate discounts future obligations. They may differ.
- Ignoring timing. An expense paid at the start of the year reduces the amount earning investment return. A year-end benefit is not accumulated for a full year in this model.
A disciplined one-at-a-time test is therefore more than changing a number in a spreadsheet. It states the baseline, defines the single shock, fixes the remaining conventions, checks probability and timing consistency, and reports the reserve and profit effects with their measurement bases clearly identified. This controlled approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
Use the hypothetical baseline in this tutorial. Unless a question says otherwise, hold booked reserves fixed for direct profit comparisons and recalculate reserves separately using the stated prospective formula.
- Increase mortality from \(q=0.02\) to \(q=0.025\), holding \(l=0.03\). Find the direct profit change and the change in the remeasured opening reserve. (Answer: profit change \(-\$0.40\); reserve change \(+\$0.38\), to the nearest cent.)
- Increase lapse from \(l=0.03\) to \(l=0.04\), holding \(q=0.02\). Find the direct profit and the remeasured opening reserve. (Answer: profit \(+\$0.10\); reserve \(\$12.19\), a change of \(-\$0.10\) to the nearest cent.)
- Increase the immediate expense from \(\$2.00\) to \(\$2.50\). Find the direct profit change and reserve change. (Answer: profit change \(-\$0.53\); reserve change \(+\$0.50\), to the nearest cent.)
- Increase the investment return from \(j=0.05\) to \(j=0.06\), leaving the valuation basis fixed. Find the direct profit change and reserve change. (Answer: profit change \(+\$0.20\); reserve change \(\$0.00\), to the nearest cent.)
- Change only the valuation rate from \(i=0.05\) to \(i=0.04\), retaining the baseline cash flows and earned return. Find the remeasured opening reserve and its change. (Answer: reserve \(\$12.48\); increase \(\$0.20\), to the nearest cent.)