Sensitivity of Indications to Exposure and Loss Selections

Why Selection Sensitivity Matters

A pure premium indication depends on two selections: the projected losses in the numerator and the projected exposures in the denominator. Each selection can be reasonable on its own, yet changing either one can materially alter the result. Sensitivity analysis makes that dependence visible: it compares a defined baseline with one or more alternatives and identifies which selection drives the difference.

The purpose is not to choose whichever result is most favorable. It is to understand what changes, why it changes, and whether the alternative selections use consistent coverage, valuation, and projection assumptions. This is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For a specified experience set and projection basis, let $L$ be projected losses and $E$ be projected exposures on a consistent basis. The indicated pure premium is $\mathrm{PP}=L/E$, or projected losses per exposure unit. For a baseline selection, write $\mathrm{PP}_0=L_0/E_0$; for an alternative, write $\mathrm{PP}_1=L_1/E_1$.

To express a pure premium as a percentage indication, compare it with a reference pure premium $P_{\mathrm{ref}}$, representing the current loss-cost benchmark for the same coverage and exposure unit. In a pure loss-cost comparison, with other rate components absent or held unchanged, the indicated change is $c=\mathrm{PP}/P_{\mathrm{ref}}-1$. A positive result indicates an increase in loss cost relative to the benchmark; a negative result indicates a decrease. This comparison alone does not determine a total rate change when expenses, profit provisions, or other rate components also change.

All examples below use hypothetical, deterministic dollar amounts at a consistent valuation and coverage level. No interest, mortality, probability distribution, or parameter assumption is needed for these arithmetic comparisons.

Exact Sensitivity Relationships

The exact comparison is easiest to see by expressing each alternative as a factor relative to the baseline. Let $a=L_1/L_0$ be the loss factor and $b=E_1/E_0$ the exposure factor. Then:

\frac{\mathrm{PP}_1}{\mathrm{PP}_0} = \frac{L_1/E_1}{L_0/E_0} = \frac{L_1/L_0}{E_1/E_0} = \frac{a}{b}.

This identity explains the direction of the effects. Increasing selected losses raises pure premium if exposures are unchanged. Increasing selected exposures lowers pure premium if losses are unchanged. If both selections change, the effect is not generally the simple difference between their percentage changes: the exact factor is the loss factor divided by the exposure factor.

First-order sensitivity: For small changes in positive $L$ and $E$, the relative change in pure premium is approximately the relative change in losses minus the relative change in exposures: $$ \frac{\Delta \mathrm{PP}}{\mathrm{PP}} \approx \frac{\Delta L}{L}-\frac{\Delta E}{E}. $$ This is a local approximation, not an exact formula for large changes.

The approximation follows by differentiating $\mathrm{PP}=L/E$. The differential is $d\mathrm{PP}=dL/E-(L/E^2)dE$. Dividing both sides by $\mathrm{PP}=L/E$ gives $d\mathrm{PP}/\mathrm{PP}=dL/L-dE/E$. For finite changes, the exact factor $a/b$ should be used; the differential expression is useful for interpreting small changes and checking whether a result is plausible.

For an indicated percentage change against a fixed reference $P_{\mathrm{ref}}$, the difference in indications is measured in percentage points:

c_1-c_0 = \left(\frac{\mathrm{PP}_1}{P_{\mathrm{ref}}}-1\right) - \left(\frac{\mathrm{PP}_0}{P_{\mathrm{ref}}}-1\right) = \frac{\mathrm{PP}_1-\mathrm{PP}_0}{P_{\mathrm{ref}}}.

Thus a 2% baseline indication and a 1% alternative indication differ by negative 1 percentage point, not by negative 1%. The distinction matters when describing the impact of a selection change.

Worked Sensitivity Comparisons

Worked Example: Changing Loss and Exposure Selections Together

A hypothetical small-commercial portfolio has baseline projected losses of $2,500,000 and baseline projected exposures of 30,000 units. An alternative selection gives projected losses of $2,640,000 and projected exposures of 32,340 units. The current pure premium benchmark is $82 per exposure unit. Calculate both pure premiums, both indicated changes, and the shift in indication.

Method. Calculate each loss-to-exposure ratio, then compare each result with the same $82 benchmark. This avoids mixing a numerator from one selection with a denominator from the other.

Baseline pure premium.

\mathrm{PP}_0 = \frac{\$2{,}500{,}000}{30{,}000} = \$83.333333\ldots

Alternative pure premium.

\mathrm{PP}_1 = \frac{\$2{,}640{,}000}{32{,}340} = \$81.632653\ldots

Baseline indication.

c_0 = \frac{83.333333\ldots}{82}-1 = 0.01626016\ldots = 1.6260\%.

Alternative indication.

c_1 = \frac{81.632653\ldots}{82}-1 = -0.00447984\ldots = -0.4480\%.

The change in indication is $-0.4480\%-1.6260\%=-2.0740$ percentage points, using unrounded calculations before rounding the displayed results. The alternative loss factor is $2{,}640{,}000/2{,}500{,}000=1.056$; the exposure factor is $32{,}340/30{,}000=1.078$. Their ratio is $1.056/1.078=0.9795918\ldots$, which agrees with $81.632653\ldots/83.333333\ldots$. The alternative pure premium is therefore lower despite higher projected losses, because the selected exposures increased by a larger proportion.

Check and common mistake avoided. The alternative denominator is 32,340, so the ratio is about $81.63, not a value calculated with a rounded or mismatched exposure count. A second trap is to report the indication shift as $-2.0740\%$; it is $-2.0740$ percentage points. Pure premiums and indications are rounded only at the final step, to cents and four decimal places, respectively.

Worked Example: Isolating Exposure Selection Sensitivity

A hypothetical policy portfolio has projected losses of $1,200,000 under both selections. The baseline exposure estimate is 15,000 units, while an alternative audited estimate is 15,600 units. The current pure premium benchmark is $78 per unit. Find the pure premiums, indicated changes, and indication shift.

Method. Because the loss selection is fixed, this comparison isolates the denominator effect.

The baseline pure premium is $1{,}200{,}000/15{,}000=\$80.00$. Its indicated change is $80/78-1=0.0256410256\ldots=2.5641\%$.

The alternative pure premium is $1{,}200{,}000/15{,}600=\$76.9230769\ldots$. Its indicated change is $76.9230769\ldots/78-1=-0.0138067\ldots=-1.3807\%$.

The shift is $-1.3807\%-2.5641\%=-3.9448$ percentage points, calculated from unrounded values. As a check, the exposure factor is $15{,}600/15{,}000=1.04$, so the pure premium factor must be $1/1.04=0.9615385\ldots$. Indeed, $76.9230769\ldots/80=0.9615385\ldots$. The increased denominator reduces pure premium by about 3.8462% relative to baseline; the indication shift in percentage points is not that same percentage.

Common mistake avoided. More exposure does not mean more loss cost per exposure. With projected losses fixed, a larger exposure denominator lowers the pure premium. Rounding the alternative pure premium to $76.92 before calculating its indication would also introduce a small avoidable discrepancy.

Worked Example: Checking a Combined Change With an Approximation

A hypothetical equipment portfolio has baseline projected losses of $3,360,000 and exposures of 42,000 units. An alternative selection increases projected losses by 5% and decreases projected exposures by 2.5%. The current pure premium benchmark is $82. Calculate the alternative indication and compare the exact change in pure premium with the first-order approximation.

Baseline. The baseline pure premium is $3{,}360{,}000/42{,}000=\$80.00$. The baseline indicated change is $80/82-1=-0.02439024\ldots=-2.4390\%$.

Alternative selections. The losses are $3{,}360{,}000(1.05)=\$3{,}528{,}000$. The exposures are $42{,}000(0.975)=40{,}950$. Thus the alternative pure premium is $3{,}528{,}000/40{,}950=\$86.153846\ldots$. Its indication is $86.153846\ldots/82-1=0.05065666\ldots=5.0657\%$. The shift from baseline is $5.0657\%-(-2.4390\%)=7.5047$ percentage points, based on unrounded values.

The exact pure premium factor is $1.05/0.975=1.0769231\ldots$, an increase of 7.6923% relative to baseline. The first-order approximation is $5\%-(-2.5\%)=7.5\%$. The approximation is close but not exact. The indication shift is 7.5047 percentage points because it compares the pure premium difference with the $82 benchmark; it is not the 7.6923% relative increase from the $80 baseline.

Common mistake avoided. Subtracting a negative exposure change as though it lowered pure premium gives the wrong direction. Since exposures fall, the denominator effect increases pure premium. The exact factor confirms the direction and provides the final calculation.

A Practical Selection Review

Sensitivity results are most useful when the comparisons isolate one source of change where possible, then assess the combined alternative. A concise review can proceed as follows:

Fix the comparison basis.
Confirm that coverage, exposure units, valuation level, and projection period are consistent. Otherwise, a ratio difference may reflect a basis mismatch rather than selection sensitivity.

Record the baseline.
State the selected losses, exposures, resulting pure premium, and reference benchmark. Keep the baseline explicit so the direction and size of each comparison are unambiguous.

Change one selection at a time when diagnosing.
Hold exposures fixed to measure a loss-selection effect, or hold losses fixed to measure an exposure-selection effect. Then calculate the combined alternative if both selections are under review.

Calculate exact ratios and interpret the difference.
Use $L/E$ for each case, apply the same benchmark to each indication, and state whether the comparison is a relative percentage change or a percentage-point shift.

A result is not automatically more credible because its indication is closer to zero, nor is the higher or lower selection necessarily preferable. The analyst should explain the basis for the alternatives and whether the change is supported by the data and projection assumptions. Sensitivity analysis describes how the indication responds; it does not by itself decide which assumption is best.

Common Exam Traps

Changing numerator and denominator inconsistently. Pair each loss selection with its intended exposure selection. If an alternative is a complete scenario, do not combine its losses with the baseline exposures unless that is explicitly the comparison being tested.
Reversing the exposure effect. Since $\mathrm{PP}=L/E$, increasing $E$ with $L$ fixed lowers pure premium; decreasing $E$ raises it.
Using an approximation as an exact result. The difference of percentage changes is only a first-order approximation. For finite changes, divide the loss factor by the exposure factor.
Confusing percentage change with percentage points. A shift from one indication to another is their arithmetic difference in percentage points.
Comparing unlike bases. Different coverage, exposure units, or projection periods can make a numerical comparison misleading even when the division is correct.

Key takeaway: Pure premium sensitivity is governed exactly by the ratio of the loss-selection factor to the exposure-selection factor. Use a fixed, consistent benchmark to translate pure premiums into indications, retain full precision during calculations, and describe indication differences in percentage points.

Check Your Understanding

For each problem, calculate with unrounded intermediate values and round dollar pure premiums to cents and percentage results to four decimal places.

A hypothetical repair portfolio has baseline projected losses of $945,000 and exposures of 11,250 units. The benchmark pure premium is $80. Find the pure premium and indicated change. (Answer: $84.00; 5.0000%.)
A baseline selection gives losses of $1,500,000 and exposures of 20,000 units. An alternative gives losses of $1,560,000 and exposures of 19,500 units. With a benchmark of $78, calculate both indications and their shift. (Answer: baseline $-3.8462\%$, alternative $2.5641\%$; shift $6.4103$ percentage points.)
A baseline has losses of $2,400,000 and exposures of 30,000. An alternative increases losses by 8% and exposures by 4%. With a benchmark of $82, find the alternative pure premium, alternative indication, and exact relative pure premium change. (Answer: $83.08; 1.3133%; 3.8462% increase.)
A baseline has losses of $4,200,000 and exposures of 50,000. An alternative uses losses of $4,074,000 and exposures of 47,250. With a benchmark of $85, find both indications and the shift. (Answer: baseline $-1.1765\%$, alternative $1.4379\%$; shift $2.6144$ percentage points.)
Losses increase by 4% while exposures decrease by 1%. Calculate the exact relative change in pure premium and its first-order approximation. (Answer: exact increase $5.0505\%$; first-order approximation $5.0000\%$.)

What You'll Learn