Why Reinsurance Selection Is a Multi-Objective Decision
A catastrophe reinsurance decision is not simply a search for the lowest premium. A lower-priced contract may leave the insurer exposed to a large modeled loss, while a more protective contract may cost enough to make its expected total cost unattractive. The selection should reflect the portfolio’s modeled loss distribution, the insurer’s risk appetite, and the capital objective the protection is intended to support.
The comparison in this tutorial uses annual modeled catastrophe losses and hypothetical reinsurance terms. It is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. Each example states its loss scenarios and probabilities; those scenarios are model inputs, not observations about a real insurer or market.
For a discrete model with outcomes \(y_j\) and probabilities \(p_j\), where \(\sum_j p_j=1\), the expected retained loss is \(\sum_j p_j y_j\). Adding the certain premium gives expected total cost. Retained volatility can be summarized with the standard deviation, and a selected quantile can represent a capital target. These are different measures: none alone describes every feature of the risk.
A Useful Relationship Between Premium, Cost, and Capital
A certain premium shifts every outcome of annual total cost by the same amount. It therefore affects expected cost and the VaR of total cost, but it does not change the dispersion of retained catastrophe losses. The distinction matters when a decision framework separately imposes a budget limit and a capital limit.
The first equality follows from linearity of expectation. Since \(P\) is constant, subtracting the mean gives \(C-E[C]=Y-E[Y]\), so the variance is unchanged. For VaR, \(F_C(c)=P(Y+P\le c)=F_Y(c-P)\). Thus the smallest \(c\) with \(F_C(c)\ge p\) is the smallest \(y\) with \(F_Y(y)\ge p\), plus \(P\). If capital is defined using \(\operatorname{VaR}_p(Y)-E[Y]\), adding the fixed premium to both VaR and mean leaves that difference unchanged. Actual capital frameworks may use other definitions or include additional risks.
Specify which outputs matter: expected total cost, a retained-loss quantile, volatility, capital, or a combination.
Use the modeled event outcomes and probabilities consistently, and identify the retained loss under each option.
Combine expected retained loss with premium for expected total cost; calculate VaR or volatility from the retained-loss distribution.
Reject alternatives that fail a required budget or capital limit, then compare the remaining options against the stated objective.
Comparing Expected Cost and a Tail-Risk Measure
Worked Example: Weighing a Lower Premium Against Retained Loss
A hypothetical insurer models annual catastrophe losses with three outcomes, in millions of dollars: \(X=12\) with probability \(0.80\), \(X=40\) with probability \(0.15\), and \(X=70\) with probability \(0.05\). Option A leaves retained losses of \(12,30,50\) million in those scenarios and has a certain premium of \(3\) million. Option B leaves \(12,27,42\) million and has a certain premium of \(4.5\) million. Compare expected total cost and 95% retained-loss VaR.
Method. Weight each retained loss by its scenario probability to obtain expected retained loss, add the premium, and find the first retained loss at which cumulative probability reaches 0.95.
Option A. Its expected retained loss is
Expected total cost is \(16.6+3=19.6\) million. The retained-loss cumulative probabilities are \(0.80\) at 12 million and \(0.95\) at 30 million, so \(\operatorname{VaR}_{0.95}=30\) million.
Option B. Its expected retained loss is
Expected total cost is \(15.75+4.5=20.25\) million. Its cumulative probability is \(0.80\) at 12 million and \(0.95\) at 27 million, so \(\operatorname{VaR}_{0.95}=27\) million.
Decision interpretation. Option A has the lower expected total cost by \(20.25-19.6=0.65\) million, while Option B reduces 95% VaR by \(30-27=3\) million. Neither option is best without knowing how the insurer values that extra tail protection. The common mistake this comparison avoids is selecting only by premium: Option A’s lower premium does not, by itself, establish that it better meets the insurer’s risk objective.
Check. The probabilities sum to \(0.80+0.15+0.05=1.00\). Option B’s retained losses do not exceed the corresponding gross losses, and each expected cost equals expected retained loss plus its stated premium.
Retained Volatility and Capital Constraints
Two options can have the same expected total cost but different retained-loss distributions. A capital constraint can distinguish between them even when their expected costs are tied. Conversely, a lower retained VaR does not automatically make an option preferable if it costs more than the available budget.
Worked Example: Equal Expected Cost, Different Capital Need
A hypothetical portfolio has modeled retained-loss scenarios, in millions, with probabilities \(0.70, 0.20,\) and \(0.10\). Alternative M leaves losses of \(8,28,48\) million and costs a certain premium of \(2.5\) million. Alternative N leaves \(8,38,43\) million and costs \(1\) million. Compare expected total cost and 90% VaR. The insurer requires 90% retained-loss VaR to be no more than 30 million.
Method. Calculate expected losses and cumulative probabilities for each option. Apply the capital limit to retained-loss VaR, not to expected cost.
Alternative M. Expected retained loss is
Expected total cost is \(16.0+2.5=18.5\) million. The cumulative probability is \(0.70\) at 8 million and \(0.90\) at 28 million, so \(\operatorname{VaR}_{0.90}=28\) million. M meets the 30-million limit.
Alternative N. Expected retained loss is
Expected total cost is \(17.5+1=18.5\) million. Its cumulative probability is \(0.70\) at 8 million and \(0.90\) at 38 million, so \(\operatorname{VaR}_{0.90}=38\) million. N fails the limit.
Decision interpretation. The options have equal expected total cost, but only M meets the stated capital constraint. As a further volatility comparison, M has \(E[Y^2]=0.70(64)+0.20(784)+0.10(2304)=432\), so \(\operatorname{Var}(Y)=432-16^2=176\), in squared millions of dollars. N has \(E[Y^2]=0.70(64)+0.20(1444)+0.10(1849)=518.5\), so \(\operatorname{Var}(Y)=518.5-17.5^2=212.25\). M also has lower retained standard deviation: \(\sqrt{176}\approx13.27\) million versus \(\sqrt{212.25}\approx14.57\) million.
The common mistake avoided is assuming the cheaper premium makes N cheaper overall: both expected total costs are 18.5 million. The capital limit, not the premium ranking, determines eligibility here. The variance calculations also confirm that M has lower retained volatility, but the stated limit is specifically on VaR.
Check. The probabilities sum to 1.00. At 90%, the cumulative probability reaches exactly 0.90 at the second outcome for both alternatives, so the VaRs are 28 and 38 million, respectively.
Using Constraints to Make a Defensible Selection
A selection process should make the trade-off explicit. Consider a hypothetical portfolio with gross losses of 10, 40, and 100 million at probabilities \(0.80, 0.15,\) and \(0.05\). Option P leaves retained losses of \(10,30,70\) million and costs a certain premium of 2 million. Option Q leaves \(10,25,55\) million and costs 5 million. Suppose the insurer’s expected-total-cost budget is 19.6 million and its 95% retained-loss VaR limit is 27 million.
Worked Example: Selecting the Option That Meets Both Limits
Method. Calculate expected total cost and 95% retained-loss VaR for each alternative, then test each against both limits. A candidate must meet both requirements to be eligible.
Option P. Expected retained loss is
Expected total cost is \(16.0+2=18.0\) million, which is within the 19.6-million budget. Cumulative probability reaches \(0.80\) at 10 million and \(0.95\) at 30 million, so 95% VaR is 30 million. P fails the 27-million capital limit.
Option Q. Expected retained loss is
Expected total cost is \(14.5+5=19.5\) million, which is also within budget. Its cumulative probability reaches \(0.95\) at 25 million, so 95% VaR is 25 million. Q meets the capital limit and is eligible.
Decision interpretation. Choose Q if both stated limits are binding requirements: P is cheaper in expected total cost, but fails the capital test. The common VaR trap is to report the largest modeled loss as 95% VaR. For P, that would give 70 million, but the cumulative probability is already 0.95 at 30 million. Under the stated discrete quantile definition, P’s 95% VaR is 30 million, not 70 million.
Check. Both expected costs are at or below 19.6 million; only Q has VaR at or below 27 million. The selected option is therefore Q.
What the Comparison Does—and Does Not—Establish
These calculations compare protection against the specified modeled losses and assumptions. A real selection also depends on whether the modeled scenarios adequately represent the portfolio and on contract terms, exclusions, limits, counterparty considerations, and the insurer’s broader risk appetite. A modeled improvement is not a guarantee that future losses will follow the model.
It is useful to keep three decision quantities distinct:
| Quantity | What it summarizes | How it is used here |
|---|---|---|
| Expected total cost | Mean retained loss plus certain premium | Cost comparison or budget test |
| Retained-loss VaR | A specified quantile of modeled retained loss | Tail-loss or capital constraint |
| Retained volatility | Dispersion of retained-loss outcomes | Additional comparison of uncertainty |
In particular, the premium is not a reduction in the modeled retained catastrophe loss. It is an expense added to retained loss to form total cost. Likewise, a VaR target is not the same as an expected-loss target: it describes a point in the modeled loss distribution, not an average.
Check Your Understanding
For each hypothetical problem, use the stated discrete loss model and the quantile definition given above. Monetary amounts are in millions of dollars.
- A contract leaves retained losses of 5, 20, and 60 with probabilities \(0.60,0.30,\) and \(0.10\), and has a certain premium of 1.5. Find expected total cost and 90% retained-loss VaR. (Answer: Expected retained loss \(=15.0\); expected total cost \(=16.5\); 90% VaR \(=20\).)
- Two options face the same modeled scenarios, with probabilities \(0.75,0.20,\) and \(0.05\). Option R leaves losses of 4, 24, and 64 and has premium 2; Option S leaves 4, 18, and 50 and has premium 4. The expected-cost budget is 13.2 and the 95% VaR limit is 20. Which option is eligible? (Answer: R has expected total cost 13.0 and 95% VaR 24, so it fails the VaR limit; S has expected total cost 13.1 and 95% VaR 18, so S is eligible.)
- A hypothetical option leaves losses of 10, 30, and 70 with probabilities \(0.50,0.30,\) and \(0.20\), and costs a certain premium of 3. Calculate expected total cost, retained-loss standard deviation, 80% VaR, and \(\operatorname{VaR}_{0.80}-E[Y]\). (Answer: Expected retained loss \(=28\), expected total cost \(=31\), variance \(=516\), standard deviation \(\approx22.72\), 80% VaR \(=30\), and the stated capital measure \(=2\).)
- Options A and B have retained losses of \(6,26,66\) and \(6,21,46\), respectively, at probabilities \(0.70,0.25,0.05\). Their certain premiums are 3 and 5. With an expected-cost budget of 17 and a 95% VaR limit of 24, identify the eligible option. (Answer: A has expected total cost \(17.0\) and 95% VaR \(26\), so it fails the VaR limit; B has expected total cost \(16.75\) and 95% VaR \(21\), so B is eligible.)