Comparing Sensitivity, Scenario, and Stress-Test Results

Why These Results Are Not Interchangeable

A solvency report may show a one-factor sensitivity, a combined scenario, and a prescribed stress side by side. Each can be useful, but they answer different questions. A sensitivity asks how a measure changes near the current assumptions when one input moves. A scenario asks what happens under a specified combination of changes. A prescribed stress applies a defined test, which may specify the shock size, direction, timing, and calculation basis.

The distinction matters because a reported impact depends not only on which assumptions change, but also on how far they move, whether they move together, how the model responds to combinations, and which output is being reported. These results are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus when the candidate can calculate and interpret them without treating unlike quantities as directly comparable.

Definition: A local one-factor sensitivity is the rate of change of a selected output with respect to one assumption, evaluated at a stated baseline while other assumptions are held fixed. A scenario impact is the change in that output when a specified set of assumptions changes together. A prescribed stress result is the change calculated under a defined stress specification and reporting basis. A stress is a test condition; by itself it is not a forecast or a probability.

To compare results, first identify the output and sign convention. In the examples below, $R$ is a risk measure in thousands of dollars, and a positive change in $R$ means a worse result. A liability measure might instead increase when conditions worsen, while a surplus measure might decrease. The direction must therefore be stated rather than inferred from the word “impact.”

Local Sensitivity and Finite One-Factor Impacts

Let $R(x,y)$ be an output determined by two assumptions. The local sensitivity to $x$ at the baseline $(x_0,y_0)$ is the partial derivative $\partial R/\partial x$ there. It measures the approximate change in $R$ per unit change in $x$, with $y$ held at its baseline value. A finite one-factor impact is different: it is the actual difference in $R$ after a specified, nonzero change in $x$.

\text{Local sensitivity to }x = \left.\frac{\partial R}{\partial x}\right|_{(x_0,y_0)}, \qquad \text{finite impact} = R(x_0+\Delta x,y_0)-R(x_0,y_0).

For a small change, the derivative can approximate the finite impact by multiplying it by the change. That approximation is not generally exact for a large change or a nonlinear model. A practical report should therefore identify whether it presents a derivative, a finite impact, or a normalized quantity such as impact per percentage point.

Worked Example: Separating a Local Sensitivity From a Finite Impact

A hypothetical insurer measures a capital-related risk amount $R$, in thousands of dollars, using the illustrative model $R(x,y)=100+6x+4y+2xy+x^2$. The baseline is $x=y=0$. Here $x$ is an expense-level increase in percentage points and $y$ is an adverse investment-assumption shock in percentage points. Calculate the local sensitivities, the finite impact of increasing $x$ by one percentage point alone, and the impact of increasing both factors by one unit.

Method and baseline. Evaluate the model at the stated baseline: $R(0,0)=100+0+0+0+0=100$ thousand dollars.

Local sensitivities. Differentiate with respect to each input: $\partial R/\partial x=6+2y+2x$, and $\partial R/\partial y=4+2x$. At the baseline, the sensitivities are $6$ thousand dollars per percentage point for $x$, and $4$ thousand dollars per percentage point for $y$.

Finite one-factor impact. With $x=1$ and $y=0$, $R(1,0)=100+6(1)+4(0)+2(1)(0)+1^2=107$. The finite impact is $107-100=7$ thousand dollars. It is not $6$: the extra $1$ thousand dollars is the $x^2$ term.

Combined scenario. With $x=y=1$, $R(1,1)=100+6+4+2+1=113$. The combined impact is $113-100=13$ thousand dollars. Adding the two finite one-factor impacts gives $7+4=11$, which misses the interaction term $2xy=2$. Thus the combined result exceeds that sum by $2$ thousand dollars.

Check. The model’s contributions above baseline are $6$ from $6x$, $4$ from $4y$, $2$ from $2xy$, and $1$ from $x^2$, totaling $13$. The common mistake avoided is treating local sensitivities or separate one-factor results as exact additive components of a combined result.

Why Combined Scenarios Can Differ

The difference between a combined scenario and the sum of separate impacts can arise from interaction and curvature. The interaction effect occurs when the effect of changing one assumption depends on the value of another. Curvature means the marginal effect of an assumption changes as the assumption moves away from baseline. Even if the model is smooth, a linear sensitivity measured at baseline need not represent the effect of a large change.

Justification: For small changes around $(x_0,y_0)$, a second-order approximation is $$ \Delta R \approx \frac{\partial R}{\partial x}\Delta x + \frac{\partial R}{\partial y}\Delta y + \frac{1}{2}\frac{\partial^2 R}{\partial x^2}(\Delta x)^2 + \frac{\partial^2 R}{\partial x\partial y}\Delta x\Delta y + \frac{1}{2}\frac{\partial^2 R}{\partial y^2}(\Delta y)^2. $$ The first two terms are the local, one-factor contributions. The remaining terms represent curvature and interaction. They vanish only under particular conditions, such as a locally linear response with no cross-effect. This explains why adding one-factor sensitivities is an approximation, not a general identity.

The approximation also clarifies why shock size matters. A derivative is a rate at one point, whereas a stress commonly moves assumptions by a finite amount. With material curvature, a larger stress may have a much larger or smaller average impact per unit change than the local sensitivity suggests.

Worked Example: How the Shock Size Changes the Average Impact

Suppose a hypothetical equity-linked product has a risk measure $R(q)=80+5q+2q^2$, in thousands of dollars. The variable $q$ is the size of an adverse equity shock, measured in percentage points, and the baseline is $q=0$. Calculate the local sensitivity, the finite impacts for shocks of one and five percentage points, and the average impact per percentage point for each finite shock.

Method. Differentiate and evaluate at baseline: $dR/dq=5+4q$, so the local sensitivity is $5$ thousand dollars per percentage point at $q=0$.

For a one-percentage-point shock, $R(1)=80+5(1)+2(1)^2=80+5+2=87$. The impact is $87-80=7$ thousand dollars, and the average impact per percentage point is $7/1=7$ thousand dollars.

For a five-percentage-point shock, $R(5)=80+5(5)+2(5)^2=80+25+50=155$. The impact is $155-80=75$ thousand dollars, and the average impact per percentage point is $75/5=15$ thousand dollars.

Check and interpretation. Both finite impacts exceed the baseline linear estimates of $5(1)=5$ and $5(5)=25$, respectively, because the quadratic term is positive. The average impact rises from $7$ to $15$ thousand dollars per percentage point as the shock grows. The common mistake avoided is multiplying the baseline derivative by a large shock and presenting that approximation as the exact stress result.

Prescribed Stresses and Comparison Discipline

A prescribed stress is defined by its specification, not by the word “stress” alone. The specification may set the size and direction of assumption changes, whether changes occur simultaneously, their timing, and the output to report. A prescribed stress may therefore differ from a custom scenario even when both are described as adverse. Unless the stress definition says otherwise, do not infer that it is a probability-weighted outcome or a likely future event.

Before comparing two reported impacts, check that they use the same:

Measure: for example, capital requirement, liability, or surplus.
Sign convention and units: an increase in a liability and a decrease in surplus can describe deterioration but have opposite signs.
Baseline and shock definition: a one-percentage-point change is not the same as a relative percentage change.
Combination and timing: one-factor changes, simultaneous changes, and shocks applied over different periods are distinct tests.
Calculation basis: changes in assumptions, methods, or reporting definitions can prevent a like-for-like comparison.

Worked Example: Comparing a Custom Scenario With a Prescribed Test

A hypothetical insurer reports a risk amount $R$ in thousands of dollars, with larger values indicating a worse result. Its illustrative model is $R(m,e)=200+4m+6e+2me$, where $m$ counts units of a 10% relative increase in a mortality assumption and $e$ counts percentage points of expense inflation. The baseline is $(0,0)$. A custom scenario uses $m=1,e=1$; a prescribed test for this hypothetical exercise uses $m=2,e=2$. Calculate both impacts and show why adding separate effects at the stress sizes is not exact.

Baseline and local sensitivities. The baseline is $R(0,0)=200$. The derivatives are $\partial R/\partial m=4+2e$ and $\partial R/\partial e=6+2m$. At baseline, they are $4$ thousand dollars per unit of $m$, and $6$ thousand dollars per percentage point of expense inflation.

Custom scenario. At $m=1,e=1$, $R(1,1)=200+4+6+2=212$. The impact is $212-200=12$ thousand dollars. The separate baseline linear contributions would sum to $4+6=10$, leaving $2$ thousand dollars attributable to the interaction term.

Prescribed test. At $m=2,e=2$, $R(2,2)=200+4(2)+6(2)+2(2)(2)=200+8+12+8=228$. The prescribed-test impact is $228-200=28$ thousand dollars. At those shock sizes, adding the separate linear terms gives $8+12=20$, which omits the $8$ thousand dollar interaction contribution.

Check and interpretation. The custom scenario impact is $12$, and the prescribed-test impact is $28$, both in thousands of dollars on the same model and baseline. Their difference is consistent with the larger prescribed shocks and the interaction term. This prescribed test is illustrative, not an official stress specification. The common mistake avoided is labeling every combined result “additive” or treating a prescribed test as though it were the local sensitivity.

Reporting and Exam Interpretation

A sound comparison states what each number measures before explaining why the numbers differ. For a local sensitivity, name the assumption, the baseline point, the direction of change, and the units per unit change. For a scenario or stress impact, state the full change from baseline and the assumptions changed together. If a report gives an “impact per unit,” establish whether that is a derivative, a finite difference divided by the shock size, or a normalized result calculated by another method.

When results appear inconsistent, check the calculation basis before concluding that the model behaves unexpectedly. For example, two values can differ because one is a liability change and the other is a surplus change, or because one reflects a single-factor change and the other includes simultaneous changes. The arithmetic comparison is meaningful only after the output, sign, units, baseline, and stress definition have been reconciled.

The central exam discipline is to avoid turning a local approximation into an exact forecast, adding one-factor impacts without checking interaction, or reading a prescribed stress as a probability statement. Report what was actually calculated, then explain the contribution of shock size, combination, and response shape.

Check Your Understanding

For each hypothetical model, calculate the requested result and check that the units and baseline are clear.

A risk measure in thousands of dollars is $R(x)=50+3x+2x^2$, where $x$ is an expense shock in percentage points and the baseline is zero. Find the local sensitivity at baseline, the impact of a two-point shock, and its average impact per point. (Answer: $3$ thousand dollars per point; impact $14$ thousand dollars; average $7$ thousand dollars per point.)
A risk measure is $R(a,b)=120+5a+4b+ab$, in thousands of dollars. At baseline $a=b=0$, $a$ is a mortality shock in units of a 10% relative increase and $b$ is expense inflation in percentage points. For $a=2,b=1$, find the total impact and the interaction contribution. (Answer: total impact $16$ thousand dollars; interaction contribution $2$ thousand dollars.)
A liability measure is $V(q)=300-8q+q^2$, in thousands of dollars, where $q$ is a downward interest-rate shock in percentage points and $q=0$ is baseline. Find the local sensitivity at baseline and the finite impact at $q=4$. (Answer: $-8$ thousand dollars per point; impact $-16$ thousand dollars.)
A hypothetical balance sheet has baseline assets of $\$500$ thousand and liabilities of $\$450$ thousand. Under a stated stress, assets are $\$470$ thousand and liabilities are $\$438$ thousand. Calculate baseline and stressed surplus, then the surplus impact. (Answer: baseline surplus $\$50$ thousand; stressed surplus $\$32$ thousand; impact $-\$18$ thousand.)
A risk measure is $R(u,v)=90+2u+3v+uv$, in thousands of dollars, with baseline $u=v=0$. A combined test uses $u=3,v=2$. Find the total impact, the sum of the separate linear contributions at those shock sizes, and the difference between them. (Answer: total impact $18$ thousand dollars; linear contributions sum to $12$ thousand dollars; difference $6$ thousand dollars.)

What You'll Learn