Sensitivity to Profit and Contingency Assumptions

Why Provision Sensitivity Matters

A profit provision or contingency provision is often expressed as a percentage of premium. Changing either assumption changes the share of premium available for losses and fixed expenses, so the resulting premium requirement does not generally move by the same percentage as the provision. The effect can be material when several provisions change together or when the available premium margin is narrow.

The purpose here is to isolate this effect: hold projected losses, fixed expenses, and variable expenses constant, then compare the indicated premium under alternative profit and contingency assumptions. This is useful for understanding the sensitivity of an indication, not for deciding which assumption is appropriate. A selected provision still requires support from the applicable assumptions and judgment.

Definition: Let $L$ be projected losses, $F$ be fixed expenses, $v$ be the variable expense ratio, $p$ be the target profit provision as a fraction of premium, and $c$ be the contingency provision as a fraction of premium. Define $A=L+F$ and $D=1-v-p-c$. Under the model below, the required premium is $P=A/D$, provided $D>0$.

The model assumes that $L$ and $F$ are dollar amounts for the same book and period, while $v$, $p$, and $c$ are proportions of the required premium. It also assumes the provisions are additive, no other premium-based load is omitted, and the projected dollar amounts do not change between scenarios. A contingency provision is a margin allowance in this calculation; it is not itself a probability of loss.

If $D\leq 0$, the model has no finite positive premium solution for $A>0$: the variable expense, profit, and contingency shares together leave no positive premium share for $A$. If a provision is instead specified as a fixed dollar amount, it belongs in the dollar requirement rather than in $p$ or $c$. The percentage formula does not apply unchanged to that case.

How a Provision Change Affects Premium

Starting from $P=A/D$, a higher profit or contingency provision reduces $D$ and increases the required premium, assuming $A$ and the other inputs remain fixed. For two scenarios with positive denominators, write their denominators as $D_0$ and $D_1$, and their required premiums as $P_0$ and $P_1$.

P_0=\frac{A}{D_0},\qquad P_1=\frac{A}{D_1}

Since the same $A$ applies in both scenarios, dividing the two requirements gives a convenient relative-change formula:

\frac{P_1}{P_0}=\frac{D_0}{D_1},\qquad \frac{P_1-P_0}{P_0}=\frac{D_0}{D_1}-1

The ratio formula is also the indicated percentage rate change when the same exposure base and rating basis apply and premium changes proportionally with the rate level. If comparing a required premium with an actual current premium, the indicated rate change is $P_{\text{required}}/P_{\text{current}}-1$; the current premium need not equal a previous scenario’s requirement.

Derivation: The premium dollars are allocated to variable expenses $vP$, profit $pP$, contingency $cP$, losses $L$, and fixed expenses $F$. Thus $P=vP+pP+cP+L+F$. Moving the premium-based terms to the left gives $P(1-v-p-c)=L+F=A$. If $D=1-v-p-c>0$, division by $D$ yields $P=A/D$. This explains why increasing $p$ or $c$ reduces the denominator rather than simply adding the same percentage to the original premium.

For a small change in a single provision, the local sensitivity follows by differentiating $P=A/(1-v-p-c)$, holding the other inputs fixed:

\frac{\partial P}{\partial p}=\frac{A}{D^2},\qquad \frac{\partial P}{\partial c}=\frac{A}{D^2}

Both derivatives are positive when $A>0$ and $D>0$. They describe the change in dollars of required premium per unit change in the provision ratio; a one-percentage-point change means a change of $0.01$, not $1$. The derivative is a local approximation. For an actual finite change, use the two-denominator formula rather than treating the derivative as exact.

Worked Example: Raising the Contingency Provision

A hypothetical commercial equipment book has projected losses of $480,000 and fixed expenses of $60,000. Its variable expense ratio is 22%, and the target profit provision is 5% of premium. Compare the required premium with a 2% contingency provision and with a 4% contingency provision. Calculate the dollar and percentage increase. Assume all other inputs and the exposure base remain unchanged.

Method. Use the premium requirement $P=(L+F)/(1-v-p-c)$. The common dollar amount is $A=480{,}000+60{,}000=\$540{,}000$.

With $c=0.02$, the denominator is $D_0=1-0.22-0.05-0.02=0.71$. Therefore,

P_0=\frac{\$540{,}000}{0.71}=\$760{,}563.380281\ldots

With $c=0.04$, the denominator is $D_1=1-0.22-0.05-0.04=0.69$. Therefore,

P_1=\frac{\$540{,}000}{0.69}=\$782{,}608.695652\ldots

The increase, calculated from the unrounded requirements, is $\$782{,}608.695652\ldots-\$760{,}563.380281\ldots=\$22{,}045.315371\ldots$. The percentage increase is $0.71/0.69-1=0.0289855\ldots$. Rounded only at the final step, the premium increase is $22,045.32, or 2.8986%.

Check. The revised denominator is smaller, so the revised premium must be larger. The ratio check gives $P_1/P_0=0.71/0.69=1.0289855\ldots$, consistent with the calculated increase. The common mistake avoided is treating a two-percentage-point increase in the premium provision as a two-percent increase in premium.

Changing Profit and Contingency Together

When both assumptions change, their combined effect on the denominator is the relevant quantity, provided the other inputs remain fixed. A change in profit from $p_0$ to $p_1$ and contingency from $c_0$ to $c_1$ changes the denominator by $-[(p_1-p_0)+(c_1-c_0)]$. The provisions are additive in this model, but the dollar premium change is not obtained by adding two independently calculated percentage increases: each change operates through the denominator.

Worked Example: Revising Both Target Provisions

A hypothetical specialty liability portfolio has projected losses of $360,000 and fixed expenses of $40,000. The variable expense ratio is 24%. Compare a target profit of 3% and contingency of 2% with a target profit of 5% and contingency of 4%. Find both required premiums and the percentage increase.

Method. The common dollar requirement is $A=\$360{,}000+\$40{,}000=\$400{,}000$. Initially, $D_0=1-0.24-0.03-0.02=0.71$, so $P_0=\$400{,}000/0.71=\$563{,}380.281690\ldots$.

Under the revised assumptions, $D_1=1-0.24-0.05-0.04=0.67$. Thus $P_1=\$400{,}000/0.67=\$597{,}014.925373\ldots$. Subtracting the unrounded values gives an increase of $\$33{,}634.643683\ldots$. The relative increase is $0.71/0.67-1=0.05970149\ldots$.

Rounded only at the final step, the initial premium is $563,380.28, the revised premium is $597,014.93, and the increase is $33,634.64, or 5.9701%.

Check. Profit and contingency together rise by four percentage points, reducing the denominator from 0.71 to 0.67. The premium increase is greater than four percent because it is the ratio $0.71/0.67-1$, not simply the change in provision points. The common mistake avoided is adding provision-point changes directly to the premium percentage change.

Working Backward From a Premium Target

Sensitivity can also be evaluated in reverse. If a premium target is fixed, the model can identify the maximum contingency provision consistent with that premium, given the other assumptions. This is an algebraic constraint, not a recommendation that the resulting provision should be selected.

Solving $P=A/(1-v-p-c)$ for $c$ gives $c=1-v-p-A/P$. Because a higher $c$ increases the required premium, the premium target is met or exceeded in adequacy terms only when the chosen $c$ is no greater than this limit, assuming the other inputs are fixed.

Worked Example: Finding the Maximum Contingency at a Premium Target

A hypothetical property portfolio has projected losses of $450,000, fixed expenses of $50,000, a variable expense ratio of 23%, and a target profit provision of 4%. The premium target is $700,000. Find the largest contingency provision for which the required premium does not exceed the target, then check two candidate provisions: 1.5% and 1.6%.

Method. The common dollar requirement is $A=\$450{,}000+\$50{,}000=\$500{,}000$. At the target premium, $A/P=\$500{,}000/\$700{,}000=0.714285714\ldots$. Therefore the limiting contingency is

c_{\max}=1-0.23-0.04-\frac{500{,}000}{700{,}000} =0.0157142857\ldots

Thus $c_{\max}=1.5714\%$ when expressed as a percentage and rounded to four decimal places. At $c=1.5\%$, $D=1-0.23-0.04-0.015=0.715$, so the requirement is $\$500{,}000/0.715=\$699{,}300.699301\ldots$, which is below the target. At $c=1.6\%$, $D=1-0.23-0.04-0.016=0.714$, so the requirement is $\$500{,}000/0.714=\$700{,}280.112045\ldots$, which exceeds the target.

Rounded to cents, the two requirements are $699,300.70 and $700,280.11, respectively. The limiting provision is correct: at $c_{\max}$, $D=500{,}000/700{,}000$, so $P=700{,}000$. The common mistake avoided is subtracting a provision change from the target premium rather than solving for its effect through the denominator.

Common Traps and Interpretation

Confusing provision points with premium change. A two-point rise in $c$ changes $D$ by 0.02; the resulting relative premium change is $D_0/D_1-1$, not automatically 2%.
Using the wrong base. The model treats $p$ and $c$ as percentages of premium. If a problem defines either provision on another base, translate it to the stated basis before applying this formula.
Comparing inconsistent inputs. A scenario comparison isolates profit and contingency only if losses, fixed expenses, and variable expenses remain on the same basis. If any of these change, include the changes explicitly.
Rounding too early. Retain unrounded denominators and premium requirements through the calculation. Round dollar outputs to cents and percentage changes to the requested precision only at the end.
Misreading the result as a selected rate. A sensitivity calculation shows how an indication responds to assumptions. It does not establish that the alternative assumption is appropriate or that the indicated change should be implemented without further consideration.

Key takeaway: For fixed projected losses and fixed expenses, profit and contingency provisions expressed as percentages of premium affect the requirement through $D=1-v-p-c$. Compare scenarios using $P_1/P_0=D_0/D_1$, and distinguish the change in a provision from the resulting change in premium.

Check Your Understanding

For each problem, assume provision ratios apply to premium and that all inputs not explicitly changed remain constant.

A hypothetical fleet book has projected losses of $240,000, fixed expenses of $20,000, variable expenses of 20%, and profit of 4%. Find the required premium at a 3% contingency and at a 5% contingency, plus the dollar and percentage increase. (Answer: $356,164.38 and $366,197.18; increase $10,032.80, or 2.8169%)
A hypothetical insurer has projected losses of $125,000 and fixed expenses of $25,000. Variable expenses are 18% and profit is 6%. Calculate the premium at a 2% contingency and at a 4% contingency, and find the percentage increase. (Answer: $202,702.70 and $208,333.33; increase 2.7778%)
A hypothetical book has projected losses of $360,000, fixed expenses of $40,000, and variable expenses of 25%. Profit and contingency change together from 3% and 2% to 5% and 4%. Calculate both premium requirements and the percentage increase. (Answer: $571,428.57 and $606,060.61; increase $34,632.03, or 6.0606%)
A hypothetical program has projected losses of $300,000, fixed expenses of $40,000, variable expenses of 20%, and profit of 4%. If its premium target is $500,000, what is the largest contingency provision consistent with a requirement no greater than that target? (Answer: 8.0000%)
For the book in question 1, use the 3% contingency case as the starting scenario. If the contingency rises by one percentage point, state the exact premium ratio formula and determine the percentage increase to four decimal places. (Answer: $0.73/0.72-1$; 1.3889%)

What You'll Learn