Diagnosing Unstable Indications

Why Diagnose an Unstable Indication?

A rate indication is a calculation based on data and assumptions, not a self-validating answer. If a small, plausible change in an input causes a large movement in the indicated rate change, that movement deserves investigation before the indication is used. The source may be genuine uncertainty in a projection, a fragile formula, a mismatch in the underlying data, or a calculation error.

Sensitivity analysis asks what happens to an indication when an input changes while the other inputs are held fixed. Diagnosis goes further: it asks whether the change is plausible, whether the changed input is supported by evidence, and whether the calculation still compares consistent populations and periods. A large movement is a signal to investigate, not proof that the indication is wrong.

Definition: An indication is unstable with respect to an input when a small, plausible change in that input produces a material change in the indicated rate level or rate change. “Small” and “material” should be judged in the context of the decision; there is no universal cutoff. A controlled perturbation changes one input at a time so that its effect can be distinguished from the effects of other inputs.

A Model for Diagnosing Sensitivity

For a simple overall rate-level calculation, let $P$ be current on-level premium, $L$ be projected losses and loss adjustment expenses, $F$ be fixed expenses, $v$ be variable expenses as a proportion of premium, and $q$ be the target profit and contingency provision as a proportion of premium. Suppose $P>0$, $L+F\geq 0$, and the expense and profit provisions apply to the indicated premium on the same basis. If $1-v-q>0$, the premium required to cover losses and fixed expenses while providing for variable expenses and the target provision is $(L+F)/(1-v-q)$. The indicated rate-level factor relative to current premium is therefore:

R=\frac{L+F}{P(1-v-q)}, \qquad \text{indicated rate change}=R-1.

This formula assumes the inputs are measured on a consistent basis and that the provisions are proportional to premium as stated. It does not apply unchanged if, for example, an expense is a fixed dollar amount per policy rather than the fixed total $F$, or if the target provision is defined on a different base. If $1-v-q\leq 0$, the equation does not produce a finite positive required premium under these assumptions. Even when it is positive, a very small denominator can make the result highly sensitive.

A useful first check is a controlled one-way perturbation: calculate the baseline indication, change one assumption by a specified amount, and recalculate. The difference between the two indications is a finite change; it should be reported in percentage points when comparing rate changes. For a smooth function of an input, a derivative describes local sensitivity near the baseline, but it does not replace recalculation for a material finite change.

Why the Denominator Can Amplify Small Changes

The sensitivity of $R$ to the profit and contingency provision $q$ follows directly by differentiating the formula while holding $P,L,F,$ and $v$ fixed. The positive derivative means that increasing the target provision increases the indicated factor. Its magnitude also shows how strongly the indication responds at the current assumptions.

\frac{\partial R}{\partial q} =\frac{L+F}{P(1-v-q)^2}.

For a small change $\Delta q$, the local approximation is $\Delta R\approx \frac{\partial R}{\partial q}\Delta q$. This is a first-order approximation, not an exact result for a large change. The squared denominator explains the amplification: as $1-v-q$ gets smaller while remaining positive, the derivative can become large. The same kind of diagnostic can be made for other inputs, but the derivative must correspond to the model actually being used.

Worked Example: Testing a Small Change in the Target Provision

A hypothetical insurer has $P=\$10{,}000{,}000$ of current on-level premium, projected losses and loss adjustment expenses $L=\$7{,}200{,}000$, fixed expenses $F=\$800{,}000$, variable expenses $v=0.12$, and target profit and contingencies $q=0.04$. Calculate the indicated rate change, then test an increase in $q$ to 0.05. Use the percentage-point difference between the resulting rate changes.

Method and assumptions. Use the proportional-expense model above. All dollar amounts are on the same basis, and $v$ and $q$ are proportions of indicated premium. The baseline denominator is positive.

1. Combine projected losses and fixed expenses.

L+F=\$7{,}200{,}000+\$800{,}000=\$8{,}000{,}000.

2. Calculate the baseline indication. The denominator proportion is $1-0.12-0.04=0.84$. Required premium is $\$8{,}000{,}000/0.84=\$9{,}523{,}809.52$. Thus $R=\$9{,}523{,}809.52/\$10{,}000{,}000=0.95238095$, and the indicated rate change is $0.95238095-1=-0.04761905$, or $-4.7619\%$.

3. Recalculate with $q=0.05$. The denominator proportion is now $1-0.12-0.05=0.83$. Required premium is $\$8{,}000{,}000/0.83=\$9{,}638{,}554.22$. The factor is $0.96385542$, so the indicated rate change is $-3.6145\%$.

4. Compare. The rate change moves by $-3.6145\%-(-4.7619\%)=1.1474$ percentage points. The indication changes in the expected direction: a larger target provision requires more premium. The derivative at the baseline is $0.8/(0.84)^2=1.1338$. A one-percentage-point increase in $q$ therefore predicts a local increase of about $1.1338$ percentage points in $R$, close to the exact $1.1474$-point change.

Check and mistake avoided. The baseline factor is below 1, so the negative rate change is consistent with required premium below current premium. Do not call the movement a 1.1474% change: it is 1.1474 percentage points between two rate changes. The derivative is an approximation, while the two recalculated indications give the exact finite comparison.

Use Perturbations as a Diagnostic, Not a Verdict

Sensitivity tables and scenarios are useful only if the perturbations are interpretable. Vary one input at a time to identify which assumptions drive movement; then test combinations if interactions are relevant. A perturbation should be tied to a plausible range, such as a range supported by historical experience, an alternative projection assumption, or a documented judgment. Arbitrary changes can demonstrate that a formula responds to its inputs, but they do not establish the uncertainty that should be assigned to the indication.

For each test, record the baseline, the changed input, the changed indication, and the difference. If comparing rate changes, use percentage points. If comparing factors, state the factor difference or relative percentage change explicitly; these are not interchangeable measures. Also check that the result has the expected direction. A movement in an unexpected direction may indicate an incorrectly signed input, an inconsistent denominator, or a mistaken implementation.

Confirm the calculation.
Reproduce the baseline from the stated formula and check units, signs, bases, and arithmetic. Make sure the reported output is a factor, a rate change, or a dollar amount as intended.

Define a plausible perturbation.
State which assumption or data item changes, by how much, and why that change is relevant. Hold other inputs fixed for a one-way test.

Recalculate and compare.
Report both indications and their difference on a clearly stated scale. Compare the finite change with the input range and with any local sensitivity approximation.

Trace the driver.
Inspect the formula and underlying inputs for amplification, mismatched data populations, period or unit inconsistencies, and implementation errors.

Classify what remains.
If the calculation and data reconcile, describe the remaining movement as assumption or projection uncertainty. If they do not, resolve the defect before treating the movement as evidence about the rate level.

Small Denominators and Structural Fragility

A sharp response can be mathematically correct and still be a warning about the structure of the calculation. In a model with a denominator $1-v-q$, a small positive denominator magnifies both the indication and its sensitivity to the assumptions that reduce that denominator. The first diagnostic is to display the denominator itself, not only the final indicated rate change. Then test plausible changes in $v$ and $q$, and verify that the provisions are defined on the stated premium base.

Worked Example: A Narrow Premium Provision

For a hypothetical specialized program, use amounts in thousands of dollars: $P=1{,}000$, $L+F=600$, variable expense provision $v=0.72$, and target profit and contingencies $q=0.25$. Calculate the indication, then test $q=0.26$ and $q=0.24$. State what the sensitivity says and what it does not say.

Method and assumptions. Apply the same proportional-premium model, with all amounts on the same basis. For each tested value, verify $1-v-q>0$ before calculating.

1. Baseline. The denominator proportion is $1-0.72-0.25=0.03$, which is positive. Required premium is $600/0.03=20{,}000$ thousand dollars. Dividing by current premium gives $R=20{,}000/1{,}000=20$, an indicated rate change of $20-1=19$, or $1{,}900\%$.

2. Increase $q$ to 0.26. The denominator proportion is $1-0.72-0.26=0.02$. Required premium is $600/0.02=30{,}000$ thousand dollars, so $R=30$ and the rate change is $2{,}900\%$.

3. Decrease $q$ to 0.24. The denominator proportion is $1-0.72-0.24=0.04$. Required premium is $600/0.04=15{,}000$ thousand dollars, so $R=15$ and the rate change is $1{,}400\%$.

4. Diagnose. A one-percentage-point increase in $q$ raises the rate change by 1,000 percentage points in this calculation; a one-point decrease lowers it by 500 points. The asymmetry is expected because the denominator is reciprocal, not linear. The extreme results reveal a structurally fragile specification under these inputs. They do not by themselves establish that the assumptions are wrong or identify a replacement rate. The definitions and bases of the provisions, and the suitability of the model for this hypothetical program, need review.

Check and mistake avoided. All three denominator proportions are positive, so each displayed result is finite under the model. Do not assume that a computed indication is decision-ready just because the arithmetic is correct; also do not silently cap or otherwise alter a result without a justified method.

Reconcile Data Before Calling a Change Uncertainty

A sensitivity test is only meaningful when the numerator and denominator describe compatible business. For example, required premium for one policy population cannot be divided by current premium for a different population and interpreted as a rate-level indication for either one. Period definitions, exposure counts, premium basis, and inclusion rules should be checked together. A large movement that disappears when records are matched is a data problem, not evidence of genuine projection uncertainty.

Worked Example: Detecting a Population Mismatch

A hypothetical portfolio has two groups, with amounts in thousands of dollars. Group A has current premium $800$ and required premium $840$. Group B has current premium $200$ and required premium $180$. A second extract reports Group B current premium of only $20$, but retains its full required premium of $180$. Calculate the full indication, the indication from the mismatched extract, and the indication for the consistently matched subset consisting of all of Group A and the portion of Group B represented by $20$ of premium. Assume Group B’s required premium is proportional to its premium at the stated group indication.

Method and assumptions. For these groups, calculate the overall factor as total required premium divided by total current premium. The subset calculation must include required premium only for the subset’s matching premium records.

1. Full portfolio. Total current premium is $800+200=1{,}000$. Total required premium is $840+180=1{,}020$. The factor is $1{,}020/1{,}000=1.02$, so the indicated rate change is $2\%$.

2. Mismatched extract. The reported current premium is $800+20=820$, but the required premium remains $840+180=1{,}020$. The calculated factor is $1{,}020/820=1.243902$, giving an indicated rate change of $24.3902\%$. This is not a valid like-for-like indication because the Group B numerator includes business omitted from its denominator.

3. Matched subset. Group B’s full factor is $180/200=0.9$. For the $20$ of Group B premium in the subset, corresponding required premium is $20(0.9)=18$. The matched subset therefore has current premium $800+20=820$ and required premium $840+18=858$. Its factor is $858/820=1.046341$, giving a rate change of $4.6341\%$.

4. Check and diagnosis. The mismatched result of $24.3902\%$ differs sharply from both the full portfolio’s $2\%$ and the consistently matched subset’s $4.6341\%$. The discrepancy is traced to different Group B populations in the numerator and denominator. It should be resolved by matching records or recalculating both amounts on a common population, not described as uncertainty in the rate indication.

Common Traps and Interpretation

Confusing sensitivity with uncertainty. Sensitivity describes how an output changes when an input is changed. Uncertainty concerns which input values are plausible and how much support they have. A large sensitivity alone does not quantify uncertainty.
Changing several inputs at once. A combined scenario can be useful, but it cannot identify the contribution of each input unless the effects are separately analyzed. Begin with controlled tests.
Comparing unlike units. A difference of 2 percentage points is not the same as a relative increase of 2 percent. Label the scale of every comparison.
Ignoring a fragile denominator. Review the denominator and the assumptions that form it, particularly when it is small. A valid-looking final number can obscure strong amplification.
Treating mismatched data as a sensitivity result. Reconcile the population, period, and premium basis before interpreting a changed indication.
Overlooking implementation errors. Recompute a sample independently, verify signs and units, and compare the implemented formula with the intended model. A repeatable spreadsheet output is not proof of correct implementation.

After validation, instability may remain because projected losses or other assumptions are genuinely uncertain. In that case, report the relevant range of inputs and the resulting range of indications, and identify which assumptions drive the movement. If the movement instead comes from a model weakness or data defect, disclose and address that issue before relying on the result. This diagnostic discipline is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, identify the calculation and the diagnostic conclusion. Amounts and assumptions are hypothetical.

A model has $P=\$5{,}000{,}000$, $L+F=\$4{,}000{,}000$, $v=0.10$, and $q=0.10$. Calculate the factor and rate change. If $q$ rises to 0.11, calculate the new rate change and the change in percentage points. (Answer: Baseline $R=1.0000$, rate change $0.00\%$; at $q=0.11$, $R=1.0127$, rate change $1.27\%$, an increase of about 1.27 percentage points.)
In the same model structure, $P=2{,}000$, $L+F=1{,}500$, $v=0.60$, and $q=0.35$, with amounts in thousands. Calculate the baseline factor and rate change. Is a one-point increase in $q$ feasible under the formula? (Answer: The denominator is $0.05$, so $R=15$ and the rate change is $1{,}400\%$. At $q=0.36$, the denominator is $0.04>0$, so the calculation remains finite; it gives $R=18.75$ and a rate change of $1{,}775\%$.)
A full portfolio has premium $500$ and required premium $525$. An extract has premium $400$ but retains required premium $525$. Calculate both rate changes and explain which check is needed. (Answer: Full factor $525/500=1.05$, rate change $5\%$; extract factor $525/400=1.3125$, rate change $31.25\%$. Reconcile whether numerator and denominator cover the same population, period, and premium basis.)
For $R=(L+F)/[P(1-v-q)]$, let $L+F=900$, $P=1{,}000$, $v=0.15$, and $q=0.05$. Calculate $\partial R/\partial q$ at these assumptions and estimate the factor change for a 0.01 increase in $q$. (Answer: The denominator proportion is $0.80$; $\partial R/\partial q=0.9/(0.80)^2=1.40625$. The estimated factor increase is $1.40625(0.01)=0.0140625$, or about 1.41 percentage points.)

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