Why Stress Test a Price Plan?
A price plan can meet its targets under a central set of assumptions and still be vulnerable to adverse experience. Claims costs may rise more than expected, fewer customers may renew, or both may occur together. Stress testing evaluates what a candidate plan would produce under specified adverse conditions; it does not establish how likely those conditions are.
The useful question is not simply whether a plan has a positive margin in a stress. It is whether the plan continues to meet the controls that matter: for example, a minimum contribution, a minimum retained premium volume, or a maximum permitted price movement for an individual customer. A plan that fails a required control is not acceptable under that scenario, even if another metric remains strong.
A stress test should begin with a defined candidate plan and a defined measurement period. For each risk segment, state the number of eligible renewals, candidate price, stressed expected claim cost, expense, and scenario renewal rate. If a customer-level price-movement guardrail applies, compare the candidate price with that customer’s expiring price. Do not substitute portfolio averages for individual checks when the rule applies customer by customer.
Calculating Stressed Portfolio Results
For a simple one-period calculation, let segment \(j\) have \(N_j\) eligible renewal policies. Let \(p_j\) be the candidate premium per renewed policy, \(c_j\) the scenario expected claim cost per renewed policy, \(e_j\) the variable expense per renewed policy, and \(r_j\) the scenario renewal rate. The calculation assumes that \(r_j\) applies to the eligible policies and that \(c_j\) represents the expected claims cost of the policies that renew.
Here, contribution means premium less expected claims and the stated variable expense. It is not necessarily accounting profit: any omitted fixed expenses, capital costs, or other cash flows are outside this definition. If a plan is assessed over multiple periods, the timing and persistence of premiums, claims, and renewals must be specified separately rather than silently included in a one-period calculation.
The equations follow from the renewal indicator. For an eligible policy, let \(I\) equal 1 if it renews and 0 otherwise. Then \(E[I]=r_j\). If the premium is \(p_j\) for a renewal, expected premium for that policy is \(E[Ip_j]=r_jp_j\). If the expected claim cost conditional on renewal is \(c_j\), and expense is \(e_j\), expected contribution is \(r_j(p_j-c_j-e_j)\). Summing this quantity over \(N_j\) eligible policies gives the segment formula. This argument also shows why claims costs must describe the renewing group: if higher-risk or lower-risk customers renew at different rates, the claims assumption for all eligible policies may not be appropriate for those who remain.
State which assumptions change, the amount of each change, the affected period, and whether the changes apply to all segments or only some.
Use scenario renewal rates and scenario claims costs to calculate retained policies, written premium, and contribution for each segment.
Sum the segment results, then compare each total and each applicable customer price movement with its stated threshold.
Worked Example: A Combined Claims and Retention Stress
A hypothetical renewal portfolio has 600 lower-risk policies with an expiring premium of $1,000 and 400 higher-risk policies with an expiring premium of $1,500. A candidate plan charges $1,100 and $1,590, respectively. Under a combined adverse scenario, lower-risk expected claims rise from $700 to $770 per renewing policy and the renewal rate is 82%; higher-risk expected claims rise from $1,100 to $1,210 and the renewal rate is 70%. Variable expenses are $100 and $120 per renewing policy. The required controls are at least $950,000 of written premium, at least $180,000 of contribution, and no customer price increase above 12%. Evaluate the plan.
Method and assumptions. Apply the one-period segment formulas. The scenario rates are assumed to describe renewal under this candidate plan, and the scenario claims costs are assumed to be conditional on renewal. The customer-level price test uses each segment’s stated expiring premium.
Lower-risk segment. Expected retained policies are \(600(0.82)=492\). Written premium is \(492($1,100)=$541,200\). Contribution per retained policy is \($1,100-$770-$100=$230\), so segment contribution is \(492($230)=$113,160\).
Higher-risk segment. Expected retained policies are \(400(0.70)=280\). Written premium is \(280($1,590)=$445,200\). Contribution per retained policy is \($1,590-$1,210-$120=$260\), so segment contribution is \(280($260)=$72,800\).
Portfolio totals and check. Total written premium is \($541,200+$445,200=$986,400\). Total contribution is \($113,160+$72,800=$185,960\). Both exceed their respective minimums. The price increases are \(($1,100/$1,000)-1=10\%\) and \(($1,590/$1,500)-1=6\%\), each no greater than 12%. The plan meets all three stated controls under this scenario.
Common mistake avoided. Applying the claims increases but retaining the original renewal rates would calculate a different scenario. The calculation must use both adverse assumptions together because the problem specifies a combined stress.
Test Controls Separately, Then Together
Controls answer different questions. A minimum written-premium requirement concerns the total premium produced by the policies expected to renew. A contribution requirement concerns the margin after the specified claims and expenses. A customer price guardrail concerns each affected customer’s price movement. Passing one does not imply passing another.
For a customer with expiring premium \(q\) and candidate premium \(p\), the percentage price movement is \(p/q-1\). If the allowed upward movement is \(g\), the check is \(p/q-1\leq g\). Where a rule also limits decreases, test the lower bound separately. A portfolio-level average increase cannot prove that every customer passes an individual guardrail.
Portfolio totals should be calculated from the same population and scenario. In particular, when written premium is defined as premium from expected renewals, it is \(\sum_j N_jr_jp_j\), not \(\sum_jN_jp_j\). The latter treats every eligible policy as renewing and may materially overstate premium in a low-retention stress.
Worked Example: Compare Two Plans Against Three Controls
A hypothetical portfolio has 500 policies in each of two segments. Expiring premiums are $800 and $1,600. Under a specified adverse scenario, expected claims are $520 and $1,120 per renewing policy, and variable expenses are $80 and $100. The portfolio requires at least $860,000 of written premium and $250,000 of contribution. No customer may receive an increase above 12%.
Candidate Plan X charges $880 and $1,680, with scenario renewal rates of 75% and 65%. Candidate Plan Y charges $920 and $1,600, with scenario renewal rates of 70% and 70%. Evaluate both plans.
Method and assumptions. Calculate each segment independently using its plan-specific renewal rate, then sum the totals. The stated claims costs and expenses are assumed applicable to renewing policies under either plan.
Plan X, first segment. Retained policies are \(500(0.75)=375\). Premium is \(375($880)=$330,000\). Contribution per renewal is \($880-$520-$80=$280\), giving contribution of \(375($280)=$105,000\).
Plan X, second segment. Retained policies are \(500(0.65)=325\). Premium is \(325($1,680)=$546,000\). Contribution per renewal is \($1,680-$1,120-$100=$460\), giving contribution of \(325($460)=$149,500\). Plan X totals are $876,000 premium and $254,500 contribution. Its price movements are \(($880/$800)-1=10\%\) and \(($1,680/$1,600)-1=5\%\). It meets the premium, contribution, and customer controls.
Plan Y, first segment. Retained policies are \(500(0.70)=350\). Premium is \(350($920)=$322,000\). Contribution per renewal is \($920-$520-$80=$320\), giving $112,000 contribution.
Plan Y, second segment. Retained policies are \(500(0.70)=350\). Premium is \(350($1,600)=$560,000\). Contribution per renewal is \($1,600-$1,120-$100=$380\), giving $133,000 contribution. Plan Y totals are $882,000 premium and $245,000 contribution. The first-segment increase is \(($920/$800)-1=15\%\), above the 12% limit; the second-segment increase is 0%.
Conclusion and check. Plan Y passes the written-premium minimum but fails both the contribution minimum and the customer increase limit. Plan X passes all three controls. The check is consistent: Plan X’s premium exceeds $860,000 by $16,000 and its contribution exceeds $250,000 by $4,500.
Common mistake avoided. Choosing Plan Y because it has the larger written premium ignores the other binding controls. A plan must pass every required control, not merely the measure that looks strongest.
Stress Assumptions and Their Limits
A stress is informative only if its assumptions are coherent and its interpretation is clear. If claims inflation is specified as a percentage, apply it to the stated claims-cost base, not automatically to premium or to expenses. If renewal rates differ by segment or candidate plan, use the applicable rate for each calculation. When assumptions represent claims among renewing customers, explain how risk mix or selection is reflected in those costs.
Stress tests are conditional calculations, not guarantees. A plan that passes one adverse scenario may fail a more severe one, and a chosen stress does not establish that outcomes outside it are impossible. When several adverse factors could occur together, testing them separately alone can conceal the combined effect. Conversely, combining extreme assumptions without a rationale may produce a scenario that is not useful for decision-making. State what the scenario represents and report the result as conditional on it.
A second useful check is whether the scenario changes the population that generates the result. If retention falls more among lower-priced risks, the remaining portfolio’s expected claims may not equal the original average. In that case, a calculation using the old average claims cost can be internally consistent only if the assumption is explicitly maintained; it should not be presented as a realistic selection-adjusted result. Re-estimate the renewing-group cost or identify the limitation.
Worked Example: Choosing a Plan That Clears a Severe Stress
A hypothetical book has 300 policies with expiring premiums of $1,000 and 200 policies with expiring premiums of $2,000. Expected claims are $650 and $1,300, with variable expenses of $100 and $150. The severe scenario raises claims by 8% in the first segment and 15% in the second. A plan must produce at least $450,000 of written premium and $115,000 of contribution; neither customer group may receive an increase above 10%.
Plan C charges $1,050 and $2,100 and has scenario renewal rates of 78% and 55%. Plan D charges $1,080 and $2,200 and has scenario renewal rates of 75% and 52%. Evaluate both.
Method and assumptions. First adjust each claims cost by its segment-specific stress factor, then calculate contribution and premium using the plan-specific renewal rates. The stated renewal rates already reflect the candidate plan under this scenario.
Stressed costs. First-segment claims are \($650(1.08)=$702\); second-segment claims are \($1,300(1.15)=$1,495\).
Plan C. Retained counts are \(300(0.78)=234\) and \(200(0.55)=110\). Segment contributions per renewal are \($1,050-$702-$100=$248\) and \($2,100-$1,495-$150=$455\). Total contribution is \(234($248)+110($455)=$58,032+$50,050=$108,082\). Written premium is \(234($1,050)+110($2,100)=$245,700+$231,000=$476,700\). The price increases are 5% in both segments. Thus, Plan C passes the premium and customer controls but falls short of the contribution minimum by \($115,000-$108,082=$6,918\).
Plan D. Retained counts are \(300(0.75)=225\) and \(200(0.52)=104\). Segment contributions per renewal are \($1,080-$702-$100=$278\) and \($2,200-$1,495-$150=$555\). Total contribution is \(225($278)+104($555)=$62,550+$57,720=$120,270\). Written premium is \(225($1,080)+104($2,200)=$243,000+$228,800=$471,800\). The price increases are 8% and 10%, so both are within the customer limit. Plan D passes all controls, with contribution $5,270 above its minimum and premium $21,800 above its minimum.
Conclusion and check. Under the stated severe scenario, Plan D passes all three controls; Plan C does not. Although Plan D has fewer expected renewals, its per-renewal contribution is sufficient to meet the portfolio threshold. This comparison is conditional on the supplied plan-specific renewal rates and claims costs.
Common mistake avoided. Applying the first segment’s 8% claims increase to both segments would understate the second segment’s stressed claims. Use the stress factor stated for each segment.
Common Exam Traps
- Using eligible policies instead of expected renewals. For renewal-based premium and contribution, multiply by the scenario renewal rate.
- Mixing scenario assumptions. Do not combine stressed claims with unstressed retention, or vice versa, unless the scenario explicitly calls for that combination.
- Using a single claims average despite selection. Make clear whether claims costs apply to all eligible risks or specifically to the customers expected to renew.
- Checking only portfolio totals. A portfolio can pass aggregate controls while individual customer price movements breach a guardrail.
- Calling a stress result an expected outcome. A conditional scenario result has no probability interpretation unless scenario probabilities are supplied and used.
Check Your Understanding
For each problem, use the stated one-period assumptions and test the requested controls.
- A hypothetical segment has 240 eligible renewals, a candidate premium of $900, stressed claims of $560 per renewal, variable expense of $70, and a stressed renewal rate of 75%. Calculate expected retained policies, written premium, and contribution. (Answer: 180 retained policies; $162,000 written premium; $48,600 contribution.)
- A two-segment plan has 300 and 200 eligible policies. Candidate premiums are $1,100 and $1,800; stressed claims are $700 and $1,250; expenses are $100 and $150; renewal rates are 80% and 65%. Expiring premiums are $1,000 and $1,600. If the customer increase cap is 12%, calculate total premium and contribution and determine whether the customer control passes. (Answer: $498,000 premium; $124,000 contribution; the customer control fails because the second-segment increase is 12.5%.)
- A hypothetical segment has 400 eligible policies, expiring premium $1,000, claims $620 increased by 10% under stress, and expense $90. Plan A charges $1,050 with a 80% renewal rate; Plan B charges $1,100 with a 74% renewal rate. The minimum contribution is $95,000, the minimum premium is $320,000, and the price increase cap is 12%. Which plan passes all controls? (Answer: Plan B; stressed claims are $682, contribution is $97,088, premium is $325,600, and its price increase is 10%.)
- A portfolio has 100 eligible policies in each of two segments. Candidate premiums are $900 and $1,500; claims are $600 and $1,000 before a 5% stress; expenses are $80 and $120; renewal rates are 90% and 60%. Determine stressed written premium and contribution, and test minimums of $165,000 premium and $35,000 contribution. (Answer: $171,000 written premium and $36,900 contribution; both minimums pass.)