Scenario Analysis and Assumption Interactions

Why Joint Scenarios Matter

A one-way sensitivity changes one assumption while holding the others fixed. That isolates the model’s response to that assumption at the selected base point. A scenario analysis instead changes a set of related assumptions together to describe a coherent alternative view of the claims or their settlement. The joint result is not generally the sum of the separate one-way changes: reserve models often combine assumptions multiplicatively, and portfolio assumptions can also depend on one another.

For reserve reporting, this distinction matters when explaining how a combined change in claim severity, unpaid proportion, portfolio mix, or payment timing affects an estimate. A one-way sensitivity table remains useful, but its entries do not automatically predict the outcome of a joint scenario. The comparison is meaningful only when the base and alternative scenarios use the same valuation basis and the assumptions within each scenario fit together.

Definition: A reserve scenario is a specified set of assumptions used together to calculate a reserve estimate. A one-way change is the difference between the base estimate and an estimate in which one assumption changes while all other assumptions remain at their base values. For a specified joint scenario, the interaction residual is the joint reserve change minus the sum of the one-way changes for the assumptions being compared. It measures the part not captured by adding those one-way changes; it is not, by itself, a separate reserve component.

The examples below use simple expected-value reserve models so that the effect of combining assumptions can be seen directly. Their inputs are hypothetical and treated as deterministic scenario assumptions; no claim-level probability distribution or mortality model is assumed. Actual reserve analyses may use more detailed models and additional assumptions.

Multiplicative Assumptions and the Interaction Term

Suppose a hypothetical portfolio has $N$ claims, an average ultimate claim severity $S$, and an unpaid proportion $q$. In this simplified model, the reserve is $R=NSq$. The model treats $NS$ as the estimated ultimate loss and $q$ as the share of that loss still unpaid. It is an illustrative calculation, not a universal reserve formula.

$$ R_0=NS_0q_0 $$

If severity changes by the absolute amount $\Delta S$ and the unpaid proportion changes by $\Delta q$, while $N$ stays fixed, the joint estimate is $N(S_0+\Delta S)(q_0+\Delta q)$. Expanding the product shows exactly why the joint change need not equal the sum of the one-way changes.

\begin{aligned} R_{\text{joint}} &=N(S_0+\Delta S)(q_0+\Delta q)\\ &=NS_0q_0+Nq_0\Delta S+NS_0\Delta q+N\Delta S\Delta q. \end{aligned}

The one-way severity change is $Nq_0\Delta S$, and the one-way unpaid-proportion change is $NS_0\Delta q$. Their sum omits the cross-product $N\Delta S\Delta q$. Therefore, for this model, the interaction residual is $N\Delta S\Delta q$. Equivalently, if $x=\Delta S/S_0$ and $y=\Delta q/q_0$ are the relative changes, the residual is $R_0xy$. The relative-change form requires each change to be divided by its own base assumption.

Key result: For $R=NSq$ with $N$ fixed, adding the two one-way changes omits $N\Delta S\Delta q$. A positive residual means the joint change is greater than the sum of the one-way changes; a negative residual means it is smaller. Its sign depends on the directions of the two assumption changes.

Worked Example: Severity and Unpaid Proportion Move in Opposite Directions

A hypothetical book has 60 claims, average severity of $10,000, and an unpaid proportion of 0.40. A joint scenario increases severity by 4% and reduces the unpaid proportion to 0.35. Calculate the base reserve, both one-way changes, the joint reserve, and the interaction residual. All values are deterministic expected-value inputs; no claim distribution or mortality assumption is used.

Method. Apply the multiplicative reserve model $R=NSq$. First calculate the base estimate:

R_0=60(10{,}000)(0.40)=\$240{,}000.00.

Severity change alone. Severity becomes $10{,}000(1.04)=\$10{,}400$, while $N=60$ and $q=0.40$ stay fixed. The revised estimate is $60(10{,}400)(0.40)=\$249{,}600.00$, so the one-way change is $+\$9{,}600.00$.

Unpaid-proportion change alone. With severity still at $10,000, the estimate is $60(10{,}000)(0.35)=\$210{,}000.00$. The one-way change is $-\$30{,}000.00$. The absolute change in $q$ is $-0.05$; its relative change is $-0.05/0.40=-0.125$, or $-12.5\%$.

Joint scenario. Apply both alternative assumptions: $60(10{,}400)(0.35)=60(\$3{,}640)=\$218{,}400.00$. The joint change from base is $\$218{,}400-\$240{,}000=-\$21{,}600.00$, while the sum of the one-way changes is $\$9{,}600-\$30{,}000=-\$20{,}400.00$.

Interaction residual and check. The residual is $-\$21{,}600-(-\$20{,}400)=-\$1{,}200.00$. Using relative changes gives $R_0(0.04)(-0.125)=\$240{,}000(-0.005)=-\$1{,}200.00$. Using absolute changes gives $N\Delta S\Delta q=60(\$400)(-0.05)=-\$1{,}200.00$, the same result. Thus the joint change equals the sum of the one-way changes plus the negative residual.

Common mistake avoided: Calling $-0.05$ the relative change in $q$ and putting it into the relative-change formula would give the wrong residual. It is the absolute change; the relative change is $-0.125$.

Interactions From Portfolio Mix

Interactions also arise when the portfolio is split into groups. A mix change shifts the number of claims assigned to each group, so a change in a group’s severity or unpaid proportion applies to a different number of claims than it did in the base scenario. If group shares are used, they must sum to 1 in both scenarios. A scenario that changes a group’s share without specifying how the other shares adjust is incomplete.

Worked Example: Group Mix and Group Assumptions Change Together

A hypothetical portfolio has 200 claims. Group A has a 60% share, severity of $3,000, and unpaid proportion of 0.30. Group B has a 40% share, severity of $7,000, and unpaid proportion of 0.50. In the joint scenario, B’s share rises to 50% (so A’s share falls to 50%), B’s severity rises by 10%, and B’s unpaid proportion falls to 0.48. Calculate the base, each one-way change, the joint estimate, and the interaction residual. No claim distribution or mortality assumption is used.

Method and base estimate. Calculate each group’s claim count from its share. A has $200(0.60)=120$ claims and B has $200(0.40)=80$. Their reserve contributions are $120(3{,}000)(0.30)=\$108{,}000.00$ and $80(7{,}000)(0.50)=\$280{,}000.00$. Thus $R_0=\$388{,}000.00$.

Mix change alone. At the alternative 50%/50% mix with base group assumptions, each group has 100 claims. The estimate is $100(3{,}000)(0.30)+100(7{,}000)(0.50)=\$90{,}000+\$350{,}000=\$440{,}000.00$. The one-way change is $+\$52{,}000.00$.

B severity change alone. Keep the base mix and $q_B=0.50$, but set B severity to $7{,}000(1.10)=\$7{,}700$. The estimate is $\$108{,}000+80(7{,}700)(0.50)=\$108{,}000+\$308{,}000=\$416{,}000.00$, a change of $+\$28{,}000.00$.

B unpaid-proportion change alone. Keep the base mix and B severity at $7,000, but use $q_B=0.48$. The estimate is $\$108{,}000+80(7{,}000)(0.48)=\$108{,}000+\$268{,}800=\$376{,}800.00$, a change of $-\$11{,}200.00$. The three one-way changes sum to $\$52{,}000+\$28{,}000-\$11{,}200=+\$68{,}800.00$.

Joint scenario. Each group now has 100 claims. B’s revised severity is $7,700 and its revised unpaid proportion is 0.48. The joint estimate is $100(3{,}000)(0.30)+100(7{,}700)(0.48)=\$90{,}000+\$369{,}600=\$459{,}600.00$. Its change from base is $+\$71{,}600.00$.

Interaction residual and check. Subtract the sum of the one-way changes from the joint change: $\$71{,}600-\$68{,}800=+\$2{,}800.00$. The joint estimate also reconciles as $\$388{,}000+\$68{,}800+\$2{,}800=\$459{,}600$. The residual captures the combined effect of changing the mix and B’s assumptions; it is not an additional group reserve.

Common mistake avoided: Applying the alternative B severity or unpaid proportion to the base count of 80 while also using the new 50% share mixes scenario bases. Each estimate must use a complete, internally consistent set of shares, counts, and group assumptions.

Interactions When Payment Timing Changes

When a reserve estimate is discounted, a change in payment timing interacts with changes in the amount expected to be paid. For a single deterministic payment at time $t$, the present value is the payment divided by $(1+i)^t$, where $i$ is the effective annual interest rate. This illustration assumes one payment date, a constant effective annual rate, and no mortality or other contingencies. It does not prescribe when discounting is appropriate for a particular reporting basis.

Worked Example: Severity Increase and a One-Year Payment Delay

A hypothetical estimate has 50 claims, severity of $10,000, and an unpaid proportion of 0.40. The entire expected unpaid amount is assumed paid at the end of year 1. Use an effective annual interest rate of $i=5\%$, so $v=1/1.05$. A joint scenario raises severity by 10% and moves payment to the end of year 2; $N$, $q$, and $i$ otherwise remain fixed. Calculate the base and scenario present values and compare the joint change with the two one-way changes. The assumptions are deterministic, with no mortality model or claim severity distribution.

Method and base estimate. The undiscounted expected unpaid amount is $50(10{,}000)(0.40)=\$200{,}000$. Paying at the end of year 1 gives $R_0=200{,}000/1.05=\$190{,}476.190476$, or $190,476.19 to the cent.

Severity increase alone. Severity becomes $11,000, making the expected unpaid amount $50(11{,}000)(0.40)=\$220{,}000$. At year 1 its present value is $220{,}000/1.05=\$209{,}523.809524$. The one-way change is $+\$19{,}047.619048$.

Timing delay alone. Keep the expected unpaid amount at $200,000 but move payment to year 2. The present value is $200{,}000/(1.05)^2=200{,}000/1.1025=\$181{,}405.895692$. The one-way change is $-\$9{,}070.294784$.

Joint scenario. With the higher severity and year-2 payment, the present value is $220{,}000/1.1025=\$199{,}546.485261$. The joint change is $199{,}546.485261-190{,}476.190476=+\$9{,}070.294785$. The one-way changes sum to $19{,}047.619048-9{,}070.294784=+\$9{,}977.324264$. Therefore the interaction residual is approximately $-\$907.03$, rounded to cents. Values are combined before rounding; independently rounded displayed changes may differ by a cent.

Check and interpretation. Relative to the base present value, the severity factor is $1.10$, while the timing factor changes from $v$ to $v^2$, a relative factor of $v=1/1.05$. The cross-term is $R_0(0.10)(v-1)=\$190{,}476.190476(0.10)(-1/21)=-\$907.029478$. This agrees with the residual from the present-value estimates. The delayed payment reduces the present value of the severity increase, which is why the residual is negative.

Common mistake avoided: Comparing an undiscounted joint amount with a discounted base reserve confounds the severity and timing changes. Apply the same valuation basis to every estimate in the comparison.

Building and Explaining a Coherent Scenario

A scenario is useful only if its assumptions describe a plausible, clearly specified alternative. A higher severity assumption might reasonably be paired with a different settlement pattern, but the relationship should be stated rather than presumed. Likewise, a mix scenario must specify how all group shares change, and a timing scenario must specify the payment basis and discount convention. The calculations do not establish whether a scenario is likely; they show its consequences conditional on the stated assumptions.

State the base.
List the base assumptions, reserve estimate, and valuation basis, including whether amounts are discounted.

Specify each alternative.
For each one-way result, change only its named assumption. For the joint scenario, list every changed assumption and any linked adjustment, such as group shares summing to 1.

Recalculate the joint estimate.
Substitute the joint assumptions together into the reserve model. Do not use the sum of one-way changes as a substitute for this calculation.

Reconcile and interpret.
Calculate the joint change, the sum of one-way changes, and their difference. Explain which assumptions combine to produce the residual.

For more than two changing assumptions, the residual can include several pairwise interactions and, in nonlinear models, higher-order effects. It is still defined by the comparison with the chosen one-way changes, but it should not automatically be assigned to one assumption. The result can also depend on the selected base: a one-way effect is measured at that base, and a different base may yield different sensitivities.

Common Traps in Scenario Comparisons

Confusing absolute and relative changes. A change from 0.40 to 0.35 is $-0.05$ in absolute terms and $-12.5\%$ relative to the base, not $-5\%$.
Adding one-way changes and calling the sum the joint result. That omits interaction terms when assumptions combine in the model.
Using inconsistent scenario inputs. For example, applying new group assumptions to old group counts while claiming to evaluate the new mix.
Changing the valuation basis between estimates. Discounting only the alternative estimate makes the comparison impossible to interpret as a clean assumption change.
Treating the residual as a causal allocation. It is the difference under a specified comparison, not proof that one assumption alone produced that amount.

When reporting results, show the base estimate, the one-way changes included in the comparison, the joint estimate, and the residual. That makes the reconciliation auditable and prevents a reader from mistaking the sum of isolated sensitivities for a forecast of the combined outcome. This scenario-analysis approach is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, calculate the requested values using the stated hypothetical assumptions. Round dollar answers to the nearest cent when needed.

A reserve model is $R=NSq$. At base, $N=80$, $S=\$5{,}000$, and $q=0.30$. A joint scenario increases $S$ by 8% and changes $q$ to 0.27. Find the joint reserve and interaction residual relative to the two one-way changes. (Answer: Joint reserve $116,640.00; one-way changes sum to $-\$2,400.00$; interaction residual $-\$960.00$.)
A portfolio has 100 claims split 60%/40% between A and B. A has $S_A=\$2{,}000$, $q_A=0.25$; B has $S_B=\$6{,}000$, $q_B=0.40$. In a joint scenario, B’s share becomes 50%, its severity rises 5%, and its $q$ becomes 0.38. Find the base reserve, joint reserve, sum of the three one-way changes, and residual. (Answer: Base $126,000.00; joint $144,700.00; one-way changes sum to $19,000.00; residual $-\$300.00$.)
A deterministic unpaid amount is $N S q$, with $N=100$, $S=\$12{,}000$, and $q=0.25$, paid at the end of year 1. At $i=5\%$, severity rises 10% and payment moves to year 2. What is the base present value, joint present value, and interaction residual versus the two one-way changes? (Answer: Base $285,714.29; joint $299,319.73; interaction residual approximately $-\$1,360.54$.)
For $R=NSq$, $N=120$, $S=\$5{,}000$, and $q=0.20$, a joint scenario raises $N$ to 132 and lowers $q$ to 0.18, with severity fixed. Find the joint change, the sum of the one-way changes, and the residual. (Answer: Joint change $-\$1,200.00$; one-way changes sum to $0.00; interaction residual $-\$1,200.00$.)

What You'll Learn