End-to-End Rate, Risk, and Ruin Decision Workshop

From Fitted Losses to a Decision

A pricing decision is not complete when an indicated premium has been calculated. The indicated rate must also support the chosen risk margin after expenses and reinsurance, while available capital must satisfy the applicable risk constraint. These calculations use different parts of the model, so an internally consistent decision keeps the assumptions and units aligned from start to finish.

This workshop uses a hypothetical annual portfolio. The fitted model is a compound Poisson model with independent claim counts and severities. No investment income is included ($i=0$); mortality is not applicable. The model and its parameters are stated explicitly below. The resulting calculations are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: For a portfolio with claim count $N$ and claim severities $X_1,X_2,\ldots$, the aggregate claim amount is $S=\sum_{j=1}^{N}X_j$. In a compound Poisson model, $N\sim\operatorname{Poisson}(\lambda)$, and the severities are independent and identically distributed, independent of $N$. A rate indication is the premium per exposure unit that satisfies a stated loss, expense, and risk-load balance. It is conditional on the model and pricing assumptions; it is not by itself a capital or ruin test.

Worked Example: Fit the Model and Indicate a Rate

A hypothetical insurer observes 240 claims over 12,000 exposure-years, with total claim payments of $1,440,000. Assume claim counts are Poisson and claim severities are exponential, independent of the counts. The insurer uses a 10% safety loading on expected claims, fixed expenses of $15 per exposure-year, and variable expenses equal to 20% of gross premium. Find the fitted expected loss per exposure-year and the gross indicated rate. Use annual units, $i=0$, and no mortality assumption.

Method. Estimate the frequency as claims divided by exposure-years. For an exponential severity model, the maximum likelihood estimate of the mean is the sample mean severity. Then solve the premium balance, rather than adding the variable expense percentage to the loss cost after the fact.

The fitted frequency is $ \hat{\lambda}=240/12{,}000=0.02 $ claims per exposure-year. The fitted exponential mean severity is $ \hat{\beta}=1{,}440{,}000/240=\$6{,}000 $. Thus $X\sim\operatorname{Exponential}(\text{mean } \$6{,}000)$, with $E[X]=\$6{,}000$, and expected losses per exposure-year are $0.02(\$6{,}000)=\$120$.

The target claim allowance including the 10% safety loading is $1.10(\$120)=\$132$. Let $P$ be the gross premium per exposure-year. Variable expenses are $0.20P$, and fixed expenses are $15, so the premium balance is $P-0.20P-15=132$. Therefore $0.80P=147$, and $P=\$183.75$.

Check. At this rate, variable expenses are $0.20(\$183.75)=\$36.75$. Gross premium less variable and fixed expenses is $\$183.75-\$36.75-\$15=\$132$, exactly the loaded claim allowance. The common mistake avoided here is treating the 20% variable expense as 20% of losses; it is specified as 20% of premium.

Why the Adjustment Coefficient Gives a Capital Test

For the classical infinite-horizon surplus model, surplus starts at $U$, premium income arrives continuously at rate $c$, and claims arrive according to a stationary Poisson process with rate $\lambda$. Claim amounts are independent and identically distributed, with moment-generating function $M_X(r)=E[e^{rX}]$ finite at the value used. The safety loading is positive: $c>\lambda E[X]$. These assumptions exclude, among other things, changing claim rates, dependence between claims, and an unmodelled investment return.

The positive adjustment coefficient $R$ solves Lundberg's equation $\lambda(M_X(R)-1)=cR$. Under the stated model, the infinite-horizon probability of ruin from initial capital $U$, denoted $\psi(U)$, satisfies the Lundberg upper bound $\psi(U)\le e^{-RU}$. The bound is conservative: passing it establishes that the model-based upper bound meets a target, not that the true ruin probability equals the bound.

Derivation: If $X$ is exponential with mean $\beta$, then $M_X(r)=1/(1-\beta r)$ for $0\le r<1/\beta$. Write the premium rate as $c=(1+\theta)\lambda\beta$, where $\theta>0$ is the safety loading on expected claims. Substituting into Lundberg's equation and taking the positive root gives $ \lambda\{\frac{1}{1-\beta R}-1\}=(1+\theta)\lambda\beta R $. For $R>0$, cancel $\lambda\beta R$ to obtain $1/(1-\beta R)=1+\theta$. Hence $R=\theta/((1+\theta)\beta)$. The zero root also solves the original equation but is not the positive adjustment coefficient. This closed form does not apply to an arbitrary severity distribution; in that case the positive root must be found from the appropriate moment-generating function, if it exists.

Worked Example: Test Capital Under the Fitted Gross-Risk Model

Use the fitted frequency $\lambda=0.02$, exponential mean $\beta=\$6{,}000$, and 10% safety loading from the preceding hypothetical case. For 1,000 exposure-years, test whether $150,000 of initial capital meets an infinite-horizon ruin-bound target of 5%. Assume no reinsurance, annual units, $i=0$, and no mortality assumption.

The aggregate claim arrival rate is $1{,}000(0.02)=20$ claims per year. Expected aggregate claims are $20(\$6{,}000)=\$120{,}000$ per year, and the premium available for claims is $c=1.10(\$120{,}000)=\$132{,}000$ per year. The safety loading is positive, as required.

Using the exponential adjustment coefficient, $R=0.10/(1.10(\$6{,}000))=1/66{,}000$ per dollar. The capital-based upper bound is $e^{-RU}=e^{-150{,}000/66{,}000}=e^{-2.2727}=0.1030$, to four decimal places. Since $0.1030>0.05$, the $150,000 capital does not satisfy the 5% bound.

The required capital from the bound is $U\ge-\ln(0.05)/R$. Since $-\ln(0.05)=2.995732$, this gives $U\ge2.995732(\$66{,}000)=\$197{,}718.31$, rounded to cents. At this capital, the bound equals 0.0500. The common mistake avoided is comparing capital with expected annual losses alone: the ruin calculation depends on the safety-loaded surplus model and the full severity tail, not just the mean.

Putting Reinsurance, Rate, and Capital on One Basis

A quota-share arrangement retains a fixed proportion $q$ of each claim and cedes the rest. If gross severity is exponential with mean $\beta$, retained severity $qX$ is also exponential, with mean $q\beta$. Expected retained loss per exposure-year is $q\lambda\beta$; expected ceded loss is $(1-q)\lambda\beta$. In the case below, the reinsurer charges 115% of expected ceded losses. This is a hypothetical contractual pricing assumption, not a market estimate.

For each alternative, the gross premium must fund the retained claim allowance, fixed expenses, and the reinsurance premium, while variable expenses remain a percentage of gross premium. With a target safety loading $\theta$, the balance per exposure-year is $ (1-v)P-e_f-\pi_R=(1+\theta)q\lambda\beta $, where $v$ is the variable expense proportion, $e_f$ is fixed expense, and $\pi_R$ is the reinsurance premium. This balance is appropriate only if the risk load is defined on retained expected claims and the expense assumptions apply as stated.

Worked Example: Compare Quota Shares and Select an Action

A hypothetical insurer has 1,000 exposure-years, fitted gross claim frequency $\lambda=0.02$, and exponential severity with mean $\beta=\$6{,}000$. Use a 10% safety loading on retained expected claims, $15 fixed expense per exposure-year, 20% variable expense on gross premium, and a reinsurance premium equal to 115% of expected ceded losses. Compare no reinsurance, 20% quota share ($q=0.8$), and 40% quota share ($q=0.6$). The insurer has $150,000 capital and a 5% infinite-horizon ruin-bound target. Assume $i=0$, no mortality, and the stationary independent compound Poisson model.

Method. For each choice, calculate the ceded premium and solve the premium balance. Then use the retained exponential mean $q\beta$ in the adjustment coefficient. The aggregate claim frequency remains 20 per year; the quota share changes claim amounts, not claim counts.

With no reinsurance, expected losses are $120 per exposure-year and the indicated rate is $(1.10(120)+15)/0.80=\$183.75$. The reinsurance premium is zero, retained mean severity is $6,000, and $R=1/66{,}000$. The ruin upper bound at $150,000 is $e^{-150{,}000/66{,}000}=0.1030$, which fails the 5% target.

For $q=0.8$, expected ceded losses are $0.20(\$120)=\$24$ per exposure-year, so the reinsurance premium is $1.15(\$24)=\$27.60$. Retained expected losses are $\$96$, and the loaded retained claim allowance is $1.10(\$96)=\$105.60$. Thus $P=(105.60+15+27.60)/0.80=\$185.25$. The retained severity mean is $0.8(\$6{,}000)=\$4{,}800$, so $R=0.10/(1.10(\$4{,}800))=1/52{,}800$. The bound is $e^{-150{,}000/52{,}800}=0.0584$, which also fails the 5% target.

For $q=0.6$, expected ceded losses are $0.40(\$120)=\$48$, and the reinsurance premium is $1.15(\$48)=\$55.20$ per exposure-year. Retained expected losses are $\$72$, and their loaded allowance is $1.10(\$72)=\$79.20$. Therefore $P=(79.20+15+55.20)/0.80=\$186.75$. Retained severity has exponential mean $0.6(\$6{,}000)=\$3{,}600$, giving $R=0.10/(1.10(\$3{,}600))=1/39{,}600$. The bound is $e^{-150{,}000/39{,}600}=0.0227$, which passes.

The required capital for the 40% quota share is $2.995732(\$39{,}600)=\$118{,}630.99$, below the available $150,000. As a rate-balance check, at $P=\$186.75$, the 20% variable expense is $37.35 per exposure-year; subtracting variable expense, $15 fixed expense, and $55.20 reinsurance premium leaves $\$79.20$, exactly 110% of the $72 retained expected loss. The supported action is therefore the 40% quota share with a gross indicated rate of $186.75 per exposure-year, subject to the stated model and assumptions. The common mistake avoided is using the gross mean severity in the reinsured ruin test: the adjustment coefficient must use the retained severity distribution.

Decision Checks and Common Traps

A sound recommendation reports the rate and capital conclusions separately, then states how they fit together. In the worked case, the larger quota share increases the indicated gross premium because the assumed ceded premium includes a load. At the same time, it reduces the retained severity mean enough to meet the stated capital constraint. A higher indicated premium is not necessarily a sign of worse risk protection; it can reflect the cost of the risk transfer.

Fix the model basis.
Keep the exposure period, fitted frequency, severity distribution, and units consistent. Distinguish a per-exposure quantity from a portfolio total.

Build the rate balance.
Apply variable expenses to premium when they are specified as a premium percentage; include fixed expenses and reinsurance cost in the correct units.

Recalculate the retained risk.
After reinsurance, use the retained claim distribution and the premium income available after expenses and ceded premium.

Test the constraint and state the decision.
Compare the ruin bound with its target, check the rate balance, and identify which assumptions or thresholds drive the recommendation.

Mixing portfolio and exposure units. The rate calculation is per exposure-year, while the ruin test uses the aggregate portfolio claim rate and total capital. Scale both consistently.
Treating the Lundberg bound as an exact probability. It is an upper bound under the classical model assumptions, not a direct estimate of actual ruin probability.
Ignoring the reinsurance premium in pricing. Reinsurance reduces retained losses but has a price; leaving that price out understates the indicated gross rate.
Changing only the mean in a non-exponential model. The closed-form adjustment coefficient derived here relies on exponential retained severity. A different retained distribution requires its own moment-generating function and root calculation.

These calculations do not remove the need to assess the quality of the fitted model, the stability of experience, or uncertainty in its parameters. They make the decision transparent conditional on the assumptions: the indicated rate funds the stated allowances, and the selected capital and reinsurance combination satisfies the specified model-based ruin bound.

Check Your Understanding

For each hypothetical case, use the stated assumptions and keep annual and portfolio units distinct.

A portfolio has 180 claims over 9,000 exposure-years and total payments of $900,000. Under a Poisson frequency and exponential severity model, find the fitted frequency, fitted mean severity, and expected loss per exposure-year. (Answer: $0.0200$ claims per exposure-year; $5,000 mean severity; $100 expected loss per exposure-year.)
Claim severity is exponential with mean $4,000 and the premium rate has a 25% safety loading over expected claims. Find the adjustment coefficient and the capital required for the Lundberg bound to be at most 1%. (Answer: $R=1/20{,}000$ per dollar; capital at least $92,103.40.)
Gross severity is exponential with mean $4,000, the insurer retains 75% under quota share, and the retained risk has a 20% safety loading. Find the adjustment coefficient. (Answer: $R=1/18{,}000$ per dollar.)
Expected retained losses are $140 per exposure-year. Fixed expenses are $12, variable expenses are 25% of gross premium, and the target loading is 8% of retained expected losses. Find the indicated gross rate. (Answer: $217.60 per exposure-year.)
Retained severity is exponential with mean $5,000, the safety loading is 15%, and available capital is $90,000. Find the Lundberg ruin bound to four decimal places and determine whether it meets an 8% target. (Answer: $R=1/38{,}333.33$ per dollar; bound $0.0956$, so it fails the 8% target.)

What You'll Learn

From Fitted Losses to a Decision

Worked Example: Fit the Model and Indicate a Rate

Worked Example: Fit the Model and Indicate a Rate

Why the Adjustment Coefficient Gives a Capital Test

Worked Example: Test Capital Under the Fitted Gross-Risk Model

Putting Reinsurance, Rate, and Capital on One Basis

Worked Example: Compare Quota Shares and Select an Action

Decision Checks and Common Traps

Check Your Understanding

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