Why Test Pricing Assumptions?
A customer-level price recommendation is only as dependable as the assumptions behind it. A modest change in predicted renewal probability, expected loss cost, or expense can change which price has the highest expected contribution. It can also change the portfolio’s expected premium, retained policies, and contribution. Sensitivity analysis makes these dependencies visible before a price plan is implemented.
The analysis here is deterministic: change an input to a specified alternative value, then recalculate the recommendation and outcomes. It does not assign probabilities to alternative assumptions or simulate a distribution of possible portfolio results. Each price is evaluated using a supplied renewal forecast, and recommendations are selected from the feasible candidate prices.
The formula is an expected-value calculation, not a guarantee of realized contribution. It assumes the supplied renewal probability is appropriate for the offered price and that the loss and expense estimates match the same policy and exposure basis. If expenses are incurred whether or not the customer renews, their expected amount must instead be included as a cost outside the renewal-probability multiplier. If expenses vary with premium, use the applicable \(E(p)\) for each candidate.
Why the Contribution Formula Works
Let the renewal indicator be \(I\), equal to 1 if the policy renews and 0 otherwise. Conditional on renewal, the contribution is \(p-L(p)-E(p)\); if there is no renewal, the contribution under the stated assumptions is zero. Thus the random contribution is \(I[p-L(p)-E(p)]\). Since \(P(I=1)=q(p)\), taking its expectation gives:
This also shows why a high renewal probability alone does not identify the best price. The relevant quantity is renewal probability multiplied by contribution per renewed policy. Changing an input can alter that product and potentially change which price ranks first.
Worked Example: A Single Policy and a Recommendation Switch
Worked Example: A Single Policy and a Recommendation Switch
A hypothetical insurer is considering three renewal premiums for one policy: $1,000, $1,100, and $1,200. The corresponding predicted renewal probabilities are 0.80, 0.70, and 0.55. Assume the expected loss cost is $600 at every price, the expense is $100 per renewed policy at every price, and no expense is incurred if the policy does not renew. Select the price with the greatest expected contribution, then test a $50 increase in expected loss cost.
Method and baseline. The total conditional cost is \(L+E=\$600+\$100=\$700\). For $1,000, the contribution per renewal is $1,000 − $700 = $300, so expected contribution per expiring policy is \(0.80(\$300)=\$240\).
At $1,100, the conditional contribution is $1,100 − $700 = $400, giving \(0.70(\$400)=\$280\). At $1,200, it is $1,200 − $700 = $500, giving \(0.55(\$500)=\$275\). The highest result is $280, so the recommended premium is $1,100.
Loss-cost sensitivity. Increase expected loss cost from $600 to $650 while holding expenses and renewal probabilities fixed. Total conditional cost is now $750. The three expected contributions are \(0.80(\$1,000-\$750)=\$200\), \(0.70(\$1,100-\$750)=\$245\), and \(0.55(\$1,200-\$750)=\$247.50\). The recommendation switches to $1,200.
Check. In the baseline, $280 exceeds $275 and $240. Under the higher loss cost, $247.50 exceeds $245 and $200. The calculations show how a $50 change in one assumption can alter the selected price even though the renewal forecasts have not changed.
Common mistake avoided: selecting a price using premium times renewal probability alone ignores the costs of renewed policies. For example, those products are $800, $770, and $660, which would incorrectly select $1,000. The correct comparison subtracts expected loss and expense before multiplying by renewal probability.
Measuring Sensitivity at the Customer and Portfolio Levels
For a portfolio with \(n\) expiring policies, let \(p_j\) be the selected price for policy \(j\), with corresponding renewal probability \(q_j\), loss cost \(L_j\), and expense \(E_j\). Under the same conditional-cost assumptions, expected total contribution is the sum of expected contributions:
Expected renewed policy count is \(\sum_j q_j\), and expected renewal premium is \(\sum_j q_jp_j\). These measures answer different questions: contribution accounts for costs, while expected premium does not. Recompute the recommended price for each policy under a changed assumption before aggregating if the objective is to measure the result of a re-optimized plan.
A useful local check follows directly from the contribution formula. If a price remains selected and only its inputs change, an increase \(\Delta L\) in loss cost changes expected contribution by \(-q\Delta L\); an increase \(\Delta E\) in expense changes it by \(-q\Delta E\). A change in renewal probability by \(\Delta q\) changes contribution by \(\Delta q[p-L-E]\), provided the other inputs are fixed. These relationships apply while the selected price and the other assumptions are held fixed. They do not, by themselves, capture a switch to another candidate price.
Worked Example: Expense Sensitivity Across Two Customers
A hypothetical insurer has one expiring policy in each of two customer groups. Customer A has candidate premiums $900, $1,000, and $1,100, with renewal probabilities 0.85, 0.75, and 0.62. A’s expected loss cost is $500 and expense is $80 at every candidate price. Customer B has candidate premiums $1,200, $1,300, and $1,400, with renewal probabilities 0.80, 0.70, and 0.56. B’s expected loss cost is $800 and expense is $100. Choose each customer’s price, calculate portfolio contribution and expected renewal premium, then increase B’s expense by $50 and re-optimize.
Customer A. The conditional cost is $500 + $80 = $580. The expected contributions are:
- At $900: \(0.85(\$900-\$580)=0.85(\$320)=\$272\).
- At $1,000: \(0.75(\$1,000-\$580)=0.75(\$420)=\$315\).
- At $1,100: \(0.62(\$1,100-\$580)=0.62(\$520)=\$322.40\).
A’s recommendation is $1,100. Its expected renewal premium at that price is \(0.62(\$1,100)=\$682\).
Customer B. The conditional cost is $800 + $100 = $900. The expected contributions are \(0.80(\$1,200-\$900)=\$240\), \(0.70(\$1,300-\$900)=\$280\), and \(0.56(\$1,400-\$900)=\$280\). The last two tie, so apply the stated lower-price tie rule: B’s recommendation is $1,300. Its expected renewal premium is \(0.70(\$1,300)=\$910\).
Baseline portfolio. Expected total contribution is \(\$322.40+\$280=\$602.40\). Expected renewal premium is \(\$682+\$910=\$1,592\), and expected renewed policy count is \(0.62+0.70=1.32\).
Expense sensitivity for B. Increase B’s expense from $100 to $150, making its conditional cost $950. Its candidate contributions become \(0.80(\$1,200-\$950)=\$200\), \(0.70(\$1,300-\$950)=\$245\), and \(0.56(\$1,400-\$950)=\$252\). B’s recommendation changes to $1,400. A’s inputs and recommendation are unchanged, so portfolio contribution is \(\$322.40+\$252=\$574.40\). Expected renewal premium is \(\$682+0.56(\$1,400)=\$1,466\), and expected renewed count is \(0.62+0.56=1.18\).
Check. The increase in B’s expense reduces its contribution at the original $1,300 price by \(0.70(\$50)=\$35\), from $280 to $245. But after re-optimization, B selects $1,400 and earns $252 under the changed assumption. The portfolio result must therefore use the new recommendation, not just subtract $35 from the original portfolio total.
Method and common mistake avoided: this is customer-level re-optimization followed by portfolio aggregation. A frequent error is to change an input, adjust the contribution at the old selected prices, and call that the result of the revised plan. That calculation measures a fixed-price response, not the result after recommendations are recalculated.
Finding a Price-Switch Threshold
Sensitivity analysis is more informative when it identifies how much an input must change to alter a recommendation. For two candidate prices \(p_a\) and \(p_b\), with the same constant conditional cost \(K=L+E\), the higher price \(p_b\) is preferred when:
Rearranging this inequality gives a threshold for the higher-price renewal probability, when \(p_b-K>0\):
The threshold depends on the cost assumption as well as on the renewal forecast. When the conditional margin at the higher price is positive, a higher common cost reduces both margins but may change their relative attractiveness. Recalculate the threshold for the changed cost; do not assume a fixed renewal-probability change has the same effect under every cost level.
Worked Example: Renewal-Probability Threshold and Robustness
For one hypothetical policy, compare premiums of $1,000, $1,100, and $1,200. Renewal probabilities are 0.80, 0.72, and 0.61, respectively. Expected loss cost is $500 and expense is $100 at every price. Determine the recommendation, calculate the renewal-probability threshold for preferring $1,200 over $1,100, and assess a revised forecast of 0.59 at $1,200.
Baseline calculation. The common conditional cost is \(K=\$500+\$100=\$600\). Expected contributions are \(0.80(\$400)=\$320\) at $1,000, \(0.72(\$500)=\$360\) at $1,100, and \(0.61(\$600)=\$366\) at $1,200. The baseline recommendation is $1,200.
Threshold calculation. For $1,200 to exceed $1,100, its renewal probability must satisfy \(q(\$1,200)(\$1,200-\$600)>0.72(\$1,100-\$600)\). Thus \(q(\$1,200)(\$600)>\$360\), or \(q(\$1,200)>0.60\). The forecast of 0.61 is above this threshold, consistent with the recommendation.
Revised forecast. If the $1,200 renewal probability falls to 0.59, its expected contribution is \(0.59(\$600)=\$354\). The $1,100 contribution remains $360 and the $1,000 contribution remains $320, so the new recommendation is $1,100.
Check. The revised forecast is below the 0.60 switch threshold, and the recomputed contributions confirm the switch. The recommendation is sensitive to this input: the baseline margin between the two leading options was only \(\$366-\$360=\$6\).
Common mistake avoided: comparing the renewal probabilities alone would favor $1,000, which has the highest probability, and miss the premium contribution from customers who renew at higher prices. The correct comparison uses each price’s probability and conditional contribution together.
How to Design a Useful Sensitivity Analysis
Choose alternative values that are plausible for the decision at hand, and state which inputs are held fixed. A simple set of tests might include a lower and higher renewal forecast, a revised loss-cost estimate, and a revised expense assumption. Change one input at a time to understand its individual effect; then, if useful, combine specified changes to see their joint effect. A combined scenario is still deterministic and should not be described as an expected outcome unless probabilities have been assigned and used appropriately.
For each feasible candidate price, retain its renewal probability, expected loss cost, expense, and calculated expected contribution.
Specify the changed assumption and its amount, unit, and scope. For example, identify whether an expense change applies per renewed policy or per expiring policy.
Recompute contribution for all feasible prices, apply the same constraints and tie rule, and record any change in the selected recommendation.
Calculate portfolio contribution, expected premium, and expected renewed count from the revised recommendations. Identify which input changes produced a material result.
A recommendation is more robust when it remains unchanged over a reasonable range of assumptions and its leading candidate has a meaningful contribution advantage. A narrow difference between the top candidates signals that small forecast revisions may change the selected price. That is a reason to report the sensitivity clearly, not a reason to imply that one precise recommendation is certain.
Check Your Understanding
For each problem, use the stated assumption changes and compare expected contribution at every listed candidate price.
- A policy has candidate prices of $1,000 and $1,200, with renewal probabilities 0.80 and 0.75. Expected loss cost is $500 and expense is $100 at either price, both conditional on renewal. Which price maximizes expected contribution? (Answer: $1,200; contributions are $320 and $450.)
- A policy has prices of $900 and $1,000, renewal probabilities 0.80 and 0.65, expected loss cost $550, and expense $50. Which price is recommended? (Answer: $1,000; contributions are $240 and $260.)
- At a selected price of $1,200, renewal probability is 0.70, expected loss cost is $700, and expense is $100. If loss cost increases by $50 and the recommendation is held fixed, what is the change in expected contribution? (Answer: a decrease of $35, from $280 to $245.)
- Two policies have already been assigned prices. Policy A has price $1,000, renewal probability 0.80, loss cost $500, and expense $100. Policy B has price $1,500, renewal probability 0.60, loss cost $800, and expense $100. What are expected total contribution, expected renewed count, and expected renewal premium? (Answer: $680, 1.40 policies, and $1,700.)
- Customer A has prices $900, $1,000, and $1,100 with renewal probabilities 0.90, 0.80, and 0.65. Customer B has prices $1,300, $1,400, and $1,500 with renewal probabilities 0.70, 0.58, and 0.50. For A, loss cost is $400 and expense is $80; for B, loss cost is $700 and expense is $100. Choose each customer’s price using the lower-price tie rule, then calculate expected total contribution. (Answer: A selects $1,000 and contributes $416; B selects $1,300 and contributes $350; total expected contribution is $766.)