This question tests the understanding of the different types of reinsurance treaties and the obligations of the parties involved. Facultative reinsurance involves the reinsurer having the option to accept or reject each risk ceded by the insurer. Facultative-obligatory reinsurance binds the reinsurer to accept risks within a defined category, but the cedent retains the option to cede. Obligatory reinsurance, on the other hand, obliges both the cedent to cede all risks within a defined category and the reinsurer to accept them. Therefore, a treaty where the reinsurer is bound to accept all risks within a specified category, but the cedent has the freedom to choose which risks to cede, aligns with the definition of facultative-obligatory reinsurance.

This question tests the understanding of the different types of reinsurance treaties and the obligations of the parties involved. Facultative reinsurance involves the reinsurer having the option to accept or reject each risk ceded by the insurer. Facultative-obligatory reinsurance binds the reinsurer to accept risks within a defined category, but the cedent retains the option to cede. Obligatory reinsurance, on the other hand, obligates both the cedent to cede all risks within a specified category and the reinsurer to accept them. Therefore, a scenario where the reinsurer is bound to accept all risks within a defined class, but the insurer has the choice to cede them, describes facultative-obligatory reinsurance.

The question tests the understanding of the dynamic collective model in insurance risk theory, specifically how it represents accumulated claims over time. The dynamic model, as described, models the stochastic process (St)t≥0, where St represents the accumulated claims from time 0 to time t. This is achieved by defining St as a sum of individual claim sizes (Xi) occurring within that period, with the number of claims (Nt) being a counting process. This contrasts with the static model, which only considers the aggregate claims at a single fixed point in time. Option B describes a scenario where claim frequency is fixed, which is not characteristic of the dynamic collective model. Option C describes a situation where claim severity is constant, also not a defining feature. Option D refers to a model where claims are independent of time, which is contrary to the dynamic nature of the model being assessed.

A quota-share reinsurance treaty involves the cedant ceding a fixed percentage of both premiums and claims to the reinsurer. This means the ratio of ceded premiums to gross premiums is identical to the ratio of ceded claims to gross claims. The reinsurer also typically provides a commission to the cedant to cover administrative expenses associated with managing the ceded portion of the portfolio. If this commission rate is set equal to the cedant’s expense rate, the treaty becomes ‘integrally proportional,’ meaning the net result for the cedant, relative to the gross result, mirrors the proportion of business retained. Therefore, if the cedant retains 60% of the business (a retention rate of 0.6), and the treaty is integrally proportional, they will also retain 60% of the net result after reinsurance.

This question tests the understanding of how a cedant using a mean-variance criterion for proportional reinsurance would adjust its retention based on the characteristics of a risk. The formula derived from the first-order conditions of the optimization problem shows that the retention proportion ‘a_i’ is directly proportional to the safety loading ‘L_i’ (premium minus expected claim) and inversely proportional to the variance of the claim ‘Var(S_i)’. Therefore, a risk with a higher safety loading (meaning it’s more profitable or less risky relative to its premium) will be retained more (lower ceded proportion), and a risk with higher volatility (higher variance) will be ceded more (lower retention). Option A correctly reflects this inverse relationship between variance and retention, and the direct relationship between safety loading and retention.

This question tests the understanding of criteria that preserve the stop-loss order in reinsurance. Proposition 47 states that if a utility function ‘u’ is increasing and convex, then minimizing the expected utility E[u(Z)] of the retained risk ‘Z’ preserves the stop-loss order. This means that a treaty leading to a lower retained risk according to the stop-loss order will also result in a lower expected utility value for a risk-averse insurer (represented by a convex utility function). Option B is incorrect because minimizing the variance of net claims is a specific instance of this principle, not the general rule. Option C is incorrect as it relates to the reinsurer’s pricing principles and not directly to the cedent’s optimization criterion for preserving the stop-loss order. Option D is incorrect because while minimizing the probability of ruin is a valid criterion under specific conditions (Cramer-Lundberg model and expected value pricing), it’s not the overarching principle for preserving the stop-loss order in all cases.

The question tests the understanding of the variance of a compound Poisson process. The formula for the variance of the total claim amount (St) in a compound Poisson process is given by Var(St) = \lambda t \times E[X^2], where \lambda is the intensity of the Poisson process and E[X^2] is the second moment of the individual claim size distribution. The provided options represent variations of this formula or related concepts. Option A correctly states this formula. Option B incorrectly uses \lambda^2 instead of \lambda. Option C incorrectly uses E[X] instead of E[X^2] and omits the time component ‘t’. Option D incorrectly uses E[X^2] without the time component ‘t’ and the intensity \lambda.

The question probes the understanding of how a reinsurer’s pricing, specifically the safety loading, influences the priority level in an excess-of-loss reinsurance arrangement, according to the provided text. The text states that ‘the priority is proportional to the safety loading: the more expensive the reinsurance, the higher the retention.’ This implies a direct relationship where a higher safety loading (making the reinsurance more expensive for the cedent) leads to a higher priority (meaning the cedent retains more risk before the reinsurance applies). Option A correctly reflects this direct proportionality. Option B is incorrect because the text explicitly states priorities depend on safety loadings, not directly on claim distributions themselves, although claim distributions influence the safety loading calculation. Option C is incorrect as it suggests an inverse relationship, which contradicts the text’s statement about proportionality. Option D is incorrect because while the model might predict uniform priority if the reinsurer sets the same price for all risks (implying a uniform safety loading), the fundamental principle described is proportionality to the safety loading, not a guaranteed uniform priority in all cases.

This question tests the understanding of proportional reinsurance, specifically the concept of the reinsurer sharing in both premiums and claims in a fixed ratio. In proportional reinsurance, the reinsurer’s participation is directly proportional to the original policy’s premium and the claims incurred. This means if the reinsurer accepts 50% of the risk, they also receive 50% of the premium and pay 50% of the claims. Non-proportional reinsurance, on the other hand, involves the reinsurer paying claims only when they exceed a certain predetermined level or retention, and the premium is typically a fixed amount or a rate based on the excess risk.

The Zero Utility Principle calculates the premium as the certainty equivalent of the risk, based on the reinsurer’s utility function. This means the reinsurer is indifferent between paying the premium and facing the uncertain outcome of the risk. The equation E[u(Π_u(S) – S)] = u(0) represents this indifference point, where Π_u(S) is the premium, S is the random claim amount, and u is the reinsurer’s utility function. The premium is the smallest amount the reinsurer would accept to cover the portfolio, reflecting their risk aversion.