Why Isolate One Assumption at a Time?
A projected reserve, profit measure, or asset share depends on assumptions and on the timing of the cash flows to which those assumptions apply. A sensitivity test asks a controlled question: what changes in the result when one specified input changes and all other inputs are held fixed? This isolates a directional effect. It does not, by itself, say how likely the change is or how several changes would interact.
The reference projection is the result under a stated set of baseline assumptions. A one-factor test changes exactly one assumption by a specified amount, recalculates the projection, and compares the result with the reference. For an actuarial measure \(M\), with baseline input \(\theta_0\) and perturbed input \(\theta_1\), the absolute change is \(M(\theta_1)-M(\theta_0)\). Keeping the subtraction order explicit prevents the common error of reporting a change with the wrong sign.
A percentage change, when the baseline measure is nonzero, is \(100\Delta M/M(\theta_0)\%\). It is not the same as the per-unit sensitivity: one describes a relative change in the result, while the other scales the result change by the size of the input change. Report both the input perturbation and the output comparison; a sensitivity without its units and direction is difficult to interpret.
How a Perturbation Flows Through a Projection
The method is the same whether the output is a prospective reserve, a projected profit, or an asset share. First record the baseline assumptions and output. Next change only the stated input. Then recalculate every affected cash flow and accumulation or discounting step. Finally, subtract the baseline output from the perturbed output and explain the sign in context.
State the baseline assumption, the measure being tested, the projection horizon, and all relevant cash-flow timing and valuation conventions.
Change one input by the stated amount. Keep the product, all other assumptions, and the measurement basis fixed.
Propagate the change through each affected cash flow and through any subsequent discounting or accumulation.
Calculate perturbed minus baseline, state the units, and explain what the sign means for the particular measure.
For a present-value reserve, a simple deterministic model with payments \(C_t\) at the end of year \(t\) and constant effective annual rate \(i\) has
Here \(C_t\) is a net liability cash flow: positive for an outflow and negative for an inflow. This formula assumes the cash flows and discount rate are specified deterministically, with end-of-year timing. It does not represent a full multi-state valuation unless the expected cash flows appropriately include the state-contingent probabilities and payments.
The reason the reserve change can be traced through each cash flow is the linearity of present value. If a perturbation changes \(C_t\) by \(\Delta C_t\), while \(i\) and the timing stay fixed, then
The equality follows by subtracting the baseline sum term by term; the unchanged cash flows cancel. This is a useful check when the one-factor change affects only a known set of cash flows. If the changed assumption also alters probabilities, timing, or other projected quantities, those effects must also be recalculated rather than treated as fixed \(\Delta C_t\).
Worked Example: A Discount Assumption and a Single Reserve Payment
A hypothetical contract has one guaranteed payment of $121 at the end of year 2 and no other future cash flows. The reference effective annual interest rate is 10%. Test a one-factor change to 9%, holding the payment amount and timing fixed. Calculate both reserves, the absolute change, and the percentage change. Round money to the nearest cent and the percentage to two decimals.
Method. Discount the same payment under each rate, then calculate perturbed minus baseline. The only changed input is the annual rate.
At the reference rate, \(v=1/1.10\), so
Under the perturbed rate of 9%,
Therefore, the reserve change is \(101.8433-100=1.8433\), or $1.84 to the nearest cent. Relative to the reference reserve, the percentage change is \(100(1.8433/100)\%\), or 1.84% to two decimals. The reserve increases because the same positive payment is discounted at a lower rate. This is a comparison of two specified rates, not a claim that the one-percentage-point change is equally important at every starting rate or for every cash-flow pattern.
Check. The perturbed discount factor for year 2 is \(1/1.1881\), which is greater than \(1/1.21\). Thus the perturbed present value must exceed $100, consistent with the calculated positive change.
Profit Changes and Present Value
For a profit projection, keep the profit convention and timing consistent between the reference and perturbed cases. In the next example, each \(P_t\) is a net profit amount available at the end of year \(t\), and the profit present value discounts those amounts at a fixed effective annual rate. The example assumes the specified expense change reduces profit dollar for dollar in the year the expense is incurred; it does not change any other cash flow.
Worked Example: An Expense Perturbation to a Profit Vector
A hypothetical two-year projection has reference profits of $12 at the end of year 1 and $18 at the end of year 2. The effective annual discount rate is 5%. A revised expense assumption increases expenses by $2 in each year, with all other projected amounts unchanged. Find the reference and perturbed profit present values and their difference. Round money to the nearest cent.
Method. Adjust the profit in each affected year first, then discount each profit at its own year-end. The profit vector in the reference case is \((12,18)\); in the perturbed case it is \((12-2,18-2)=(10,16)\).
The reference present value is
The perturbed present value is
Thus the change is \(24.036282-27.755102=-3.718820\), or −$3.72 to the nearest cent. As a check using the change in profits directly, the profit changes are \((-2,-2)\), so their present value is
The small difference in the last decimal is due to rounding intermediate values; using unrounded terms gives the same result. Higher expenses reduce profit in both years, so the present-value change is negative. Applying the $2 change only once, or discounting both annual changes as if they occurred at year 1, would use the wrong cash-flow timing.
Asset Shares and Accumulated Effects
An asset share is a projected accumulation of amounts assigned to a policy or group under a stated convention. Its movement depends on the timing of premiums, expenses, benefits, and investment return. The following simple recurrence assumes all listed cash flows occur at the beginning of each year and the remaining balance earns the effective annual rate \(i\) for that year. It also assumes the policy remains in force throughout the projection; no decrements are modeled.
Worked Example: One Expense Change Carried Through an Asset Share
A hypothetical policy has an opening asset share of $100. At the beginning of each of two years, it receives a $20 premium, incurs a $5 expense, and pays a $10 benefit. The effective annual investment return is 4%. In the perturbed projection, only the year 1 expense increases by $2; the year 2 expense remains $5. Calculate the asset share at the end of each year in both projections and the final difference. Round money to the nearest cent.
Method. Apply the stated recurrence year by year. A year 1 change affects the end-of-year 1 balance and then earns one more year of return before the end-of-year 2 measure.
In the reference projection, the amount remaining after the year 1 beginning-of-year cash flows is \(100+20-5-10=105\). Therefore, \(A_1^{\mathrm{ref}}=105(1.04)=109.20\). At the beginning of year 2, the net addition is \(20-5-10=5\), so
In the perturbed projection, the year 1 expense is $7. The year 1 amount after cash flows is \(100+20-7-10=103\), giving \(A_1^{\mathrm{pert}}=103(1.04)=107.12\). Year 2 is unchanged apart from starting from the lower asset share:
The end-of-year 1 change is \(107.12-109.20=-2.08\). The end-of-year 2 change is \(116.6048-118.768=-2.1632\), or −$2.16 to the nearest cent. The extra $2 expense reduces the end-of-year 1 balance by \(2(1.04)=2.08\); that difference then accumulates for another year, giving \(2.08(1.04)=2.1632\). The later effect is larger in magnitude because of the positive return. The asset share is not itself a reserve: it is the accumulation produced by this particular cash-flow and investment convention.
Check. The final difference also equals \(-2(1.04)^2=-2.1632\), confirming that the single extra expense is carried through both years of accumulation.
Directional Effects, Limits, and Exam Traps
A one-factor result supports a directional statement only under its stated model. For example, with a fixed positive payment and otherwise unchanged deterministic cash flows, lowering the discount rate increases its present value. With a fixed positive expense increase that reduces profits dollar for dollar, projected profit decreases in the affected period. With a positive return and the asset-share timing used above, an earlier expense change also affects later accumulated balances. These directions need not hold for every output: a measure may combine inflows and outflows, and a changed assumption may affect amounts, probabilities, or timing in opposing ways.
For a small input change, the finite-difference sensitivity per unit, \(\Delta M/\Delta\theta\), can approximate the local rate of change. The approximation becomes less informative for a large perturbation or a strongly nonlinear projection. A central difference, \([M(\theta_0+h)-M(\theta_0-h)]/(2h)\), can approximate a local derivative when both perturbed projections are available and the function is sufficiently smooth. Neither calculation makes a one-factor test a joint stress: simultaneous changes can have interaction effects that are absent from separate tests.
The most frequent errors are procedural rather than computational:
- Changing more than one input. If the aim is to isolate one factor, every other assumption and convention must remain fixed.
- Reversing the subtraction. State whether the reported change is perturbed minus baseline or baseline minus perturbed. Here it is always perturbed minus baseline.
- Ignoring timing. A cash flow at year 2 is discounted for two years, and an asset-share difference can accumulate over later years.
- Mixing output measures. A reserve increase, a profit decrease, and an asset-share decrease are different statements; do not assume they are numerically equal.
- Overstating a direction. Explain the conditions behind the result. A direction established for a single positive payment need not describe a net liability containing both inflows and outflows.
Check Your Understanding
For each problem, use perturbed minus baseline and keep all unmentioned assumptions fixed.
- A hypothetical contract pays $84 at the end of year 2, with no other cash flows. The effective annual discount rate is 5%. The payment increases by 3%, while its timing and discount rate remain fixed. Find the change in the time-0 present value. Round to the nearest cent. (Answer: +$2.29)
- A two-year profit projection has profits of $7 and $11 at the ends of years 1 and 2. The effective annual discount rate is 10%. A changed expense assumption reduces year 1 profit by $1 and year 2 profit by $3. Find the change in profit present value, rounded to the nearest cent. (Answer: −$3.39)
- An asset share opens at $50. At the beginning of year 1, premium is $12, expense is $4, and benefit is $8. The effective annual return is 5%. In the perturbed projection, only the year 1 expense increases by $2. Find the change in end-of-year asset share. (Answer: −$2.10)
- A reserve has positive net outflows of $30 at the end of year 1 and $50 at the end of year 2, discounted at an effective annual rate of 4%. A one-factor change increases those outflows by $2 and $1, respectively. Find the reserve change, rounded to the nearest cent. (Answer: +$2.85)
- Under a fixed deterministic projection, a positive expense increase reduces the profit in each affected year, all else unchanged. What is the direction of change in the present value of those profits if the same positive discount rate and timing are retained? (Answer: It decreases; the present value of the negative profit changes is negative.)