Why Expense Sensitivity Matters
A rate indication depends not only on projected losses but also on how expenses are represented. A fixed expense per exposure adds a dollar amount to the cost to be funded by premium. A variable expense provision takes a percentage of premium, so a change in that assumption affects the amount of premium available to fund losses and fixed expenses. These two mechanisms respond differently to assumption changes.
Sensitivity analysis quantifies those differences. It asks how much the indicated premium or rate level changes when an expense assumption changes while the other inputs are held constant. The result helps an actuary identify which assumptions are influential, check that the indication is behaving sensibly, and direct attention toward assumptions whose uncertainty could materially affect the selection. The calculations below use hypothetical insurance products and data.
The model assumes that \(L\) and \(A\) are amounts on the same per-exposure basis and that variable expenses and \(q\) are both proportions of the indicated premium. If a provision is instead specified as a fixed dollar amount, it belongs in the numerator, not the denominator. The formula is not appropriate without adjustment if an expense is calculated on a different base, such as a percentage of losses or of a premium measure other than the indicated premium.
If the current premium on the same exposure basis is \(P_0\), the indicated rate level change is \(P/P_0-1\). Comparing indicated premiums directly is useful for a per-exposure interpretation; comparing rate level changes requires the current premium base as well. A change in indicated premium and a change in the rate level factor are related, but they are not interchangeable quantities.
How the Two Expense Types Affect the Indication
A fixed expense change affects the numerator dollar for dollar. A variable expense ratio change affects the denominator. The distinction explains why it is essential to identify both the units and the premium base of each assumption before calculating an indication.
These derivatives describe the local effect of a small change, with the other inputs held constant. Since \(e\) is a proportion, a one-percentage-point increase means \(\Delta e=0.01\), not a change of one whole unit. The derivative with respect to \(e\) is therefore multiplied by 0.01 to approximate the dollar effect of a one-percentage-point increase. For larger changes, recalculate the indication using the new denominator rather than relying on a linear approximation.
Classify each expense as a fixed amount per exposure or as a percentage of premium. Confirm the applicable exposure and premium bases.
Calculate \(P_0=(L+A)/(1-e-q)\) using the original assumptions and record the result before making any changes.
Recalculate the indication with the alternative expense assumption. Holding other inputs constant isolates the effect being measured.
Report the dollar difference and, where relevant, the percentage difference from the baseline. Verify the direction: higher fixed expenses or a higher variable expense ratio should increase the indicated premium under this model.
Worked Sensitivity Calculations
Worked Example: A Fixed Expense Increase and a Higher Variable Ratio
A hypothetical specialty policy has projected losses and fixed expenses totaling $640 per exposure. The baseline variable expense ratio is 20%, and the provision \(q\) is 5%. An alternative expense review raises the numerator amount to $648 and the variable expense ratio to 21.5%. The provision \(q\) is unchanged. Calculate the baseline and alternative indicated premiums, their dollar difference, and the percentage increase from baseline.
Method. Apply the prospective premium model to each set of assumptions. The combined change is evaluated directly; the calculation does not attribute the total difference separately to the fixed and variable components.
Baseline. The denominator is \(1-0.20-0.05=0.75\), so $$ P_0=\frac{640}{0.75}=853.333333\ldots $$ The baseline indicated premium is $853.33 per exposure to the nearest cent.
Alternative. The denominator is \(1-0.215-0.05=0.735\), so $$ P_1=\frac{648}{0.735}=881.632653\ldots $$ The alternative indicated premium is $881.63 per exposure to the nearest cent.
Dollar impact. Before rounding, the difference is $$ \frac{648}{0.735}-\frac{640}{0.75} =881.632653\ldots-853.333333\ldots =28.299319\ldots $$ The impact is $28.2993 per exposure to four decimal places, or $28.30 to the nearest cent, above baseline.
Percentage impact. Relative to the unrounded baseline, $$ \frac{28.299319\ldots}{853.333333\ldots}=0.033163\ldots, $$ so the indicated premium increases by 3.3163% to four decimal places.
Check and common mistake. Both changes increase the premium, so a positive difference is expected. Do not subtract rounded premiums to obtain a more precise impact: the difference of the unrounded indications is $28.299319\ldots$, not the result of subtracting $881.63 and $853.33 and then claiming extra precision.
Worked Example: Comparing a Fixed Expense Change With a Ratio Change
For a hypothetical commercial coverage, projected losses are $520 per exposure, fixed expenses are $80 per exposure, the variable expense ratio is 20%, and \(q=5\%\). Compare the indicated premium increase from (a) raising fixed expenses by $15 and (b) raising the variable expense ratio by one percentage point. Change one assumption at a time.
Baseline. The numerator is \(520+80=600\), and the denominator is \(1-0.20-0.05=0.75\). Thus $$ P_0=\frac{600}{0.75}=800. $$
(a) Fixed expense increase. The new numerator is \(520+95=615\), with the denominator unchanged: $$ P_A=\frac{615}{0.75}=820. $$ The increase is \(820-800=$20.00\) per exposure.
(b) Variable ratio increase. One percentage point means the ratio becomes \(0.21\). The new denominator is \(1-0.21-0.05=0.74\), so $$ P_e=\frac{600}{0.74}=810.810810\ldots $$ The increase is \(810.810810\ldots-800=$10.810810\ldots\), or $10.81 per exposure to the nearest cent.
The $15 fixed expense increase has the larger impact in these particular scenarios: $20.00 versus $10.81 per exposure. This does not mean fixed expenses are always more influential. The comparison depends on the size of the assumed changes and the baseline inputs.
Check and common mistake. For the ratio change, subtracting one percentage point from the denominator directly would be incorrect. The ratio enters the denominator through \(1-e-q\); increasing \(e\) by 0.01 decreases that denominator by 0.01. Also, a ratio change is not a fixed dollar charge: it must be applied through the denominator.
Worked Example: Comparing Several Expense Assumptions
A hypothetical equipment policy has projected losses of $554 per exposure and fixed expenses of $46 per exposure. Its variable expense components are 12% for acquisition, 3% for premium taxes, and 4% for other premium-sensitive costs. The provision \(q\) is 6%. Compare the impact of increasing acquisition expense from 12% to 14% with the impact of increasing fixed expenses by $10. Hold all other assumptions constant.
Baseline. Total variable expenses are \(0.12+0.03+0.04=0.19\). The numerator is \(554+46=600\), and the denominator is \(1-0.19-0.06=0.75\). Therefore $$ P_0=\frac{600}{0.75}=800. $$
Acquisition ratio change. The new total variable ratio is \(0.14+0.03+0.04=0.21\), giving a denominator of \(1-0.21-0.06=0.73\). The new indication is $$ P_{\mathrm{acq}}=\frac{600}{0.73}=821.917808\ldots. $$ The increase is $21.917808\ldots, or $21.92 per exposure to the nearest cent. Relative to baseline, the increase is \(21.917808\ldots/800=0.027397\ldots\), or 2.7397% to four decimal places.
Fixed expense change. Fixed expenses become \(46+10=56\), so the numerator is \(554+56=610\). The denominator remains 0.75: $$ P_{\mathrm{fixed}}=\frac{610}{0.75}=813.333333\ldots. $$ The increase is $13.333333\ldots, or $13.33 per exposure to the nearest cent. Relative to baseline, this is \(13.333333\ldots/800=0.016666\ldots\), or 1.6667% to four decimal places.
The acquisition ratio change has the larger effect in these specified scenarios. The result follows from the stipulated changes, not from comparing 2 percentage points with $10 as if they were units of the same kind.
Check and common mistake. The acquisition ratio changes by 2 percentage points, from 12% to 14%; it does not change by 2% of the old ratio. Applying a relative 2% increase would give 12.24%, a different assumption and a different indication.
Why the Sensitivity Formulas Hold
The derivative results follow directly from differentiating the premium formula while holding the other inputs fixed. Write \(D=1-e-q\), so \(P=(L+A)/D\). Because \(D\) does not depend on \(A\), increasing \(A\) by a small amount changes the numerator at a rate of one dollar of numerator per dollar of expense. Hence $$ \frac{\partial P}{\partial A}=\frac{1}{D}. $$
For \(e\), the numerator is constant and the denominator \(D\) decreases as \(e\) increases. Differentiating \((L+A)(1-e-q)^{-1}\) gives $$ \frac{\partial P}{\partial e} =(L+A)(1-e-q)^{-2} =\frac{L+A}{D^2}. $$ Since \(P=(L+A)/D\), this is also \(P/D\). The positive derivative confirms that increasing the variable expense ratio raises the indicated premium, as long as the model remains defined and \(e+q<1\).
These are local sensitivity measures, not substitutes for recalculation when a change is large. They also have different units: \(\partial P/\partial A\) measures dollars of premium per dollar of fixed expense, while \(\partial P/\partial e\) measures dollars of premium per unit change in the expense proportion. To compare assumptions fairly, specify plausible changes for each and calculate their effects on the same output, such as dollars per exposure or percentage change in indicated premium.
Interpreting Results and Avoiding Common Traps
Sensitivity to a fixed expense is amplified by \(1/(1-e-q)\). If only 75% of premium is available to fund the numerator, for example, an additional dollar in fixed expense requires more than a dollar of premium. A variable ratio change can also have a substantial effect because it reduces the premium share available for the numerator. Its impact depends on both the starting indicated premium and the denominator.
A useful relative comparison, for small changes, follows by dividing the derivatives by \(P\): $$ \frac{dP}{P}\mathrel{\approx}\frac{dA}{L+A}+\frac{de}{1-e-q}. $$ The expression is a first-order approximation; for material changes, use the full formula. It makes clear that the effect of a fixed expense change depends on its size relative to the total numerator, while the effect of a ratio change depends on the remaining premium share in the denominator.
- Mixing expense bases. An expense stated per policy is not automatically a percentage of premium. Convert or place it in the model according to its actual basis.
- Changing multiple assumptions unintentionally. To isolate an assumption’s effect, change only that input. If several assumptions change, report the combined scenario or explicitly separate the effects.
- Using an inconsistent premium base. A percentage provision must be applied to the premium base specified by the model. If the assumed base differs from indicated premium, the model may require a different equation.
- Comparing raw changes with unlike units. A dollar change and a percentage-point change cannot be ranked by their numerical sizes alone. Translate each to the same indicated-premium impact.
- Rounding too early. Retain unrounded values through subtraction and division, then round the final dollar impact or percentage as requested.
Expense sensitivity is useful because it separates the impact of an assumption from other movements in projected losses or the rate indication. It does not, by itself, establish which assumption is most uncertain or which alternative is most appropriate. The actuary must also judge whether the alternative assumptions are plausible and whether the expense allocation and premium bases match the intended indication. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each problem, hold all assumptions constant except where a change is stated. Round final premiums and dollar impacts to the nearest cent.
- A hypothetical policy has \(L=$420\), \(A=$30\), \(e=18\%\), and \(q=4\%\). Find the indicated premium per exposure. (Answer: $576.92)
- A coverage has \(L=$680\), \(A=$70\), \(e=20\%\), and \(q=5\%\). Calculate the baseline indicated premium and the new premium if fixed expenses increase by $12. (Answer: Baseline $1,000.00; new premium $1,016.00; increase $16.00)
- A product has \(L=$390\), \(A=$60\), \(e=22\%\), and \(q=8\%\). Find the baseline premium and the premium impact if \(e\) increases by 1.5 percentage points. (Answer: Baseline $642.86; new premium $656.93; increase $14.08)
- A hypothetical line has \(L=$510\), \(A=$90\), \(e=17\%\), and \(q=3\%\). Compare an increase of $12 in fixed expenses with a 2-percentage-point increase in \(e\), changing one assumption at a time. Which change has the larger impact, and by how much? (Answer: Fixed-expense impact $15.00; variable-ratio impact $19.23; the variable-ratio change is larger by $4.23)
- A policy has \(L=$736\), \(A=$64\), \(e=20\%\), and \(q=5\%\). Fixed expenses are revised upward by 5%, with all other assumptions unchanged. Find the baseline premium, the revised premium, and the dollar increase. (Answer: Baseline $1,066.67; revised $1,070.93; increase $4.27)