CMFAS Reinsurance Premium Quiz 58

First-order stochastic dominance (FOSD) implies that for any threshold value ‘y’, the probability of the first risk being greater than or equal to ‘y’ is less than or equal to the probability of the second risk being greater than or equal to ‘y’. This means the first risk has a lower or equal probability of exceeding any given claim amount. The question describes a scenario where an insurer is evaluating two potential portfolios of insurance policies. Portfolio A is preferred over Portfolio B if it consistently offers a lower probability of experiencing claims above any specified level. This directly aligns with the definition of FOSD, where a distribution F is preferred to F’ if F(y) >= F'(y) for all y, which is equivalent to P(S >= y)

Borch’s Theorem, a fundamental concept in the economics of uncertainty and optimal risk sharing, establishes a condition for Pareto efficiency in the context of multiple agents trading risks. The theorem states that an allocation of risks (yi(ω)) among N agents is Pareto efficient if and only if there exists a set of positive constants (λi) such that the ratio of the marginal utilities of any two agents is equal to the inverse ratio of these constants. Mathematically, this is expressed as \(u’_i(y_i) / u’_j(y_j) = \lambda_j / \lambda_i\) for all \(i, j\). This condition implies that the marginal rate of substitution between any two agents is constant across all states of the world, which is a hallmark of efficient risk sharing. The other options represent incorrect interpretations of this theorem or related concepts. Option B describes a situation where marginal utilities are equal, which is a specific case but not the general condition for Pareto efficiency. Option C incorrectly suggests that the ratio of marginal utilities should be equal to the ratio of the constants, reversing the correct relationship. Option D introduces the concept of aggregate wealth, which is relevant in market equilibrium but not the direct condition for Pareto efficiency as stated by Borch’s Theorem.

The Surplus Treaty, as described, operates on a risk-by-risk basis. The cession rate for each individual risk is determined by comparing the insured value (Ri) against the cedant’s retention limit (Ci) and the underwriting limit (Ki). The formula for the cession rate (1-ai) is min((Ri-Ci)+, (Ki-Ci)+) / Ri. This means that the reinsurer’s participation is not a fixed percentage of all business but is dynamically calculated for each policy based on its value relative to the agreed-upon limits. Therefore, the reinsurer’s share of premiums and claims will vary depending on the specific risk being underwritten, reflecting the principle of similarity between ceded premiums and ceded claims for each individual risk within the treaty’s parameters.

The question tests the understanding of the collective model in insurance, specifically how aggregate claims are represented. The collective model posits that the total claim amount is a function of both the number of claims (frequency) and the amount of each individual claim (severity). Therefore, the aggregate claim amount (S) is modeled as the sum of individual claim amounts, where the number of terms in the sum is determined by a frequency variable (N). This is mathematically represented as S = \sum_{i=1}^{N} X_i, where X_i are individual claim amounts and N is the frequency variable. Option (b) describes the individual model, where the total claim is simply the sum of claims from a fixed number of risks. Option (c) incorrectly suggests that the aggregate claim is the product of frequency and severity, which is not the standard collective model formulation. Option (d) is a misrepresentation of the collective model’s components.

The question tests the understanding of the recursive relationship for calculating the stop-loss transform, specifically how the value at retention ‘d’ relates to the value at retention ‘d-1’. The provided formula $\Pi(d) = \Pi(d-1) – (1 – F_S(d-1))$ shows that to find the stop-loss transform at retention ‘d’, one subtracts the probability that the total claim amount is less than ‘d’ (i.e., $1 – F_S(d-1)$) from the stop-loss transform at retention ‘d-1’. This means that as retention increases, the stop-loss transform decreases by the probability of claims falling below the new, higher retention level.

This question tests the understanding of how risk tolerance influences the distribution of risk in a reinsurance market, specifically in the context of optimal risk sharing. The provided text states that relative income sensitivities to aggregate wealth are proportional to agents’ relative risk tolerances. For Constant Absolute Risk Aversion (CARA) utility functions, this translates to the proportion of aggregate wealth an agent retains being equal to their risk tolerance divided by the sum of all risk tolerances. Therefore, if an agent has a significantly higher risk tolerance, they will retain a proportionally larger share of the aggregate wealth, approaching full retention if their risk tolerance is infinitely higher than others (risk-neutral). This aligns with the concept of a risk-neutral individual bearing the entire risk.

An excess-of-loss treaty with parameters ‘a’ (guarantee) and ‘b’ (priority) means the reinsurer pays the portion of a claim that exceeds ‘b’, up to a maximum of ‘a’. Therefore, if a claim is ‘x’, the reinsurer pays min(max(x-b, 0), a). In this scenario, the claim is HK$150,000. The priority (b) is HK$50,000. The guarantee (a) is HK$100,000. The amount exceeding the priority is HK$150,000 – HK$50,000 = HK$100,000. Since this amount is equal to the guarantee (a), the reinsurer pays the full HK$100,000. The total claim cost is HK$150,000, and the reinsurer pays HK$100,000, leaving the cedant to retain HK$50,000 (which is the priority amount).

The Esscher principle calculates the premium by adjusting the probability distribution of the risk using an exponential tilting mechanism. Specifically, it recalculates the expected value of the claim amount (S) under a new probability measure G, which is derived from the original distribution F by multiplying by an exponential factor $e^{\alpha x}$ and normalizing. This process effectively overweights the more adverse states of nature, reflecting a higher degree of risk aversion for larger claims. The formula for the premium is given by $\Pi(S) = E(Se^{\alpha S}) / E(e^{\alpha S})$. This method is particularly useful for capturing the impact of extreme events and is consistent with the properties of translation invariance and additivity under certain conditions, making it a robust choice for premium calculation in actuarial science.

The question probes the understanding of how risk aversion influences the allocation of a risk among multiple insurers under the exponential pricing principle. The exponential principle, Π(S) = α ln(E[e^{α}S]), quantifies the premium as a function of the risk aversion coefficient (α) and the expected exponential of the risk. The optimal coinsurance strategy, as derived from the principle, dictates that each insurer takes a portion of the risk inversely proportional to their risk aversion coefficient. Specifically, if α_i is the risk aversion coefficient for insurer i, and α is the aggregate risk aversion (α = ∑ α_i), then the optimal share for insurer i is S*_i = (α_i / α)S. This means that insurers with higher risk aversion (α_i) will take a smaller share of the risk, and those with lower risk aversion will take a larger share, thereby minimizing the total premium. Therefore, an insurer with a higher risk aversion coefficient would be allocated a smaller portion of the total risk.

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 static model. Option C describes a scenario where claim frequency is fixed, which is not the core of the dynamic collective model. Option D describes a situation that is too simplistic and doesn’t capture the time-dependent nature of accumulated claims.

This question tests the understanding of how risk tolerance influences the distribution of risk in a reinsurance market, specifically in the context of optimal risk sharing. The provided text states that relative income sensitivities to aggregate wealth are proportional to agents’ relative risk tolerances. For Constant Absolute Risk Aversion (CARA) utility functions, this translates to the proportion of aggregate wealth an agent retains being equal to their risk tolerance divided by the sum of all risk tolerances. Therefore, if an agent has a significantly higher risk tolerance than others, they will retain a proportionally larger share of the risk, approaching the entire risk in the limiting case of being risk-neutral.

The Beekman convolution formula provides a method to calculate the probability of ruin without relying on the Lundberg coefficient, which is particularly useful when the distribution of individual claim sizes has a ‘fat tail’ where the Lundberg coefficient might not exist. This formula focuses on the maximum aggregate loss (L) experienced by the insurer. Ruin occurs when the cumulative claims exceed the initial surplus (u). Therefore, the probability of ruin, denoted by \(\psi(u)\), is equivalent to the probability that the maximum aggregate loss is greater than the initial surplus, i.e., \(\psi(u) = P(L > u)\). This is also expressed as \(\psi(u) = 1 – F_L(u)\), where \(F_L(u)\) is the cumulative distribution function of the maximum aggregate loss. The formula’s significance lies in its ability to handle situations where traditional ruin theory assumptions, like the existence of the Lundberg exponent, are violated.

This question tests the understanding of the relationship between the stop-loss order of claim size distributions and the probability of ruin in a Cramer-Lundberg model. Proposition 31 states that if one claim size distribution (X) is stochastically larger than another (Y) in the stop-loss sense (X \ge_2 Y), then the probability of ruin for the model with claim sizes X will be less than or equal to the probability of ruin for the model with claim sizes Y, for any initial capital u. This is because a larger claim size distribution, in the stop-loss sense, implies a lower probability of ruin, all other factors being equal. Therefore, if the claim sizes in Model A are stochastically larger than those in Model B under the stop-loss order, the ruin probability for Model A will be lower.

The question tests the understanding of how changes in the parameters of a lognormal distribution affect the stop-loss premium. The provided text states that the stop-loss premium, represented by E[(X-K)+], increases with the mean (m) and variance (σ^2) of the lognormal distribution. Specifically, the derivative of the stop-loss premium with respect to m is shown to be non-negative, indicating an increase. While the text doesn’t explicitly show the derivative with respect to σ, it states that it also demonstrates an increase with variance. Therefore, an increase in the mean or variance of the claim size, when modeled by a lognormal distribution, will lead to a higher stop-loss premium.

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 Beekman convolution formula provides a method to calculate the probability of ruin without relying on the Lundberg coefficient. This is particularly advantageous when the distribution of individual claim sizes has a ‘fat tail,’ meaning the probability of very large claims does not diminish rapidly. In such cases, the Lundberg coefficient may not exist. The formula focuses on the maximum aggregate loss (L), which is defined as the maximum surplus deficit over time. Ruin occurs when this maximum aggregate loss exceeds the initial surplus (u). Therefore, the probability of ruin, \(\psi(u)\), is equivalent to \(1 – F_L(u)\), where \(F_L(u)\) is the cumulative distribution function of the maximum aggregate loss. This approach bypasses the need for the Lundberg coefficient by directly analyzing the distribution of the maximum deficit.

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 of the retained risk (E[u(Z)]) preserves the stop-loss order. This means that if one retained risk is preferred to another under the stop-loss order, it will also be preferred under the expected utility criterion. Option A correctly identifies this principle. Option B is incorrect because minimizing the variance of net claims (E[(Z – E[Z])^2]) only preserves the stop-loss order under specific conditions (when the reinsurer prices according to the variance or standard deviation principle and the criterion is applied to the set I_mu). Option C is incorrect as maximizing expected utility is a criterion that preserves the stop-loss order, not minimizing it. Option D is incorrect because while minimizing ceded premiums can preserve the stop-loss order under certain pricing principles (like variance or standard deviation), it’s not a universally applicable rule without those conditions, and the question asks for a criterion that *always* preserves it.

A quota-share treaty involves the reinsurer accepting a fixed percentage of the cedent’s risks, premiums, and claims. This means that both the proportion of premiums ceded and the proportion of claims ceded are identical. The cedent retains a fixed percentage, and the reinsurer assumes the remaining percentage. This proportional sharing of both income and outgo is the defining characteristic of proportional reinsurance, and specifically, the quota-share treaty.

A surplus treaty is designed to reinsure risks on a policy-by-policy basis, where the cession rate is determined by the relationship between the risk’s insured value, the insurer’s retention limit, and the underwriting limit. The formula for the cession rate (1-a_i) is min((R_i – C_i)+, (K_i – C_i)+) / R_i, where R_i is the risk value, C_i is the retention limit, and K_i is the underwriting limit. This means that for a risk to be reinsured under a surplus treaty, its value must not exceed the underwriting limit (K_i). If the risk value (R_i) is less than or equal to the retention limit (C_i), no reinsurance is ceded as the insurer retains the entire risk. If R_i is greater than C_i but less than or equal to K_i, a portion of the risk is ceded based on the formula. Risks exceeding the underwriting limit (K_i) are excluded from the treaty altogether. Therefore, a risk valued at 1.2 million euros, with an underwriting limit of 1 million euros, would not be covered by this specific surplus treaty.

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 in the context of Lundberg’s approach, models the total claim amount as a stochastic process \(S_t\) indexed by time \(t\). This process is defined as the sum of individual claims occurring within a specific time frame, where the number of claims is governed by a counting process \(N_t\). Therefore, \(S_t\) is the sum of \(N_t\) independent and identically distributed claim sizes \(X_i\) occurring up to time \(t\). Option B incorrectly suggests a fixed number of claims, which is characteristic of a static model. Option C misrepresents the relationship between the counting process and claim sizes, implying a direct dependency that isn’t the core of the dynamic model. Option D introduces a concept of ‘average claim cost’ without specifying its role in the dynamic accumulation process, making it less accurate than the direct representation of accumulated claims.

First-order stochastic dominance (FOSD) implies that for any threshold value ‘y’, the probability of the first risk being greater than or equal to ‘y’ is less than or equal to the probability of the second risk being greater than or equal to ‘y’. This means the first risk has a lower or equal probability of exceeding any given claim amount. The question describes a scenario where an insurer is evaluating two potential portfolios of insurance policies. Portfolio A is preferred to Portfolio B if it dominates it in the first-order stochastic sense. This means that for any level of claim severity, Portfolio A has a lower or equal probability of experiencing a claim of that magnitude or greater compared to Portfolio B. Option A correctly states this relationship by asserting that the probability of a claim exceeding any given amount is less than or equal for Portfolio A compared to Portfolio B. Option B incorrectly reverses this relationship. Option C introduces the concept of second-order stochastic dominance, which is a different criterion focusing on risk aversion. Option D incorrectly suggests that FOSD is determined by the expected claims alone, ignoring the entire distribution of potential claims.

The Variance Principle for premium calculation adds a margin to the pure premium (expected value) that is directly proportional to the variance of the claim amounts. The formula is \Pi(S) = E(S) + \beta Var(S), where \beta is a positive constant. This means that as the variability (variance) of claims increases, the premium also increases, reflecting a greater need for a safety margin to cover potential deviations from the average. The other options are incorrect because the Expected Value Principle only considers the mean, the Standard Deviation Principle uses the square root of variance, and the Exponential Principle uses a logarithmic transformation of the expected value of an exponential function of the claims.

The question tests the understanding of how changes in the parameters of a lognormal distribution affect the stop-loss premium. The provided text states that the stop-loss premium, represented by E[(X-K)+], increases with the mean (m) and variance (σ^2) of the lognormal distribution. Specifically, the derivative of the stop-loss premium with respect to m is shown to be non-negative, indicating an increase. While the text doesn’t explicitly show the derivative with respect to σ, it states that it also demonstrates an increase with variance. Therefore, an increase in the mean or variance of the claim size distribution, when modeled by a lognormal distribution, will lead to a higher stop-loss premium.

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, also known as treaty reinsurance, binds both parties: the cedent must cede all risks within the agreed scope, and the reinsurer must accept them. Therefore, a scenario where the reinsurer is obligated to accept all risks within a specified category, and the cedent is bound to cede them, describes obligatory reinsurance.

This question tests the understanding of the relationship between the total expected claims (ES) and the expected number of claims (EN) and the expected severity of each claim (EX) within the collective risk model. Proposition 13 states that ES = EN * EX. This means the total expected cost is the product of how many claims are expected and the average cost per claim. Therefore, if the expected number of claims doubles while the expected severity remains constant, the total expected claims will also double.

This question tests the understanding of how risk tolerance influences the distribution of risk in a reinsurance market, specifically in the context of optimal risk sharing. The provided text states that relative income sensitivities to aggregate wealth are proportional to agents’ relative risk tolerances. For Constant Absolute Risk Aversion (CARA) utility functions, this translates to the proportion of aggregate wealth an agent retains being equal to their risk tolerance divided by the sum of all risk tolerances. Therefore, if an agent has a significantly higher risk tolerance (or is risk-neutral, implying infinite risk tolerance), they will retain the entire risk, with others retaining nothing beyond a constant amount.

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 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.

The Esscher principle calculates the premium by adjusting the probability distribution of the risk using an exponential tilting method. Specifically, it recalculates the expected value of the loss under a new probability measure, G, which is derived from the original distribution, F, by multiplying the probability density function by $e^{\alpha x}$ and normalizing. This means that states of nature with higher losses (larger x) are given more weight in the expectation calculation, reflecting a form of risk aversion. The formula $\Pi(S) = E(Se^{\alpha S}) / E(e^{\alpha S})$ directly implements this concept, where the expectation is taken with respect to the original distribution F, but the tilting factor $e^{\alpha S}$ is incorporated into both the numerator and the denominator. The other options represent different premium calculation principles: the Mean Value Principle is a special case of the Swiss Principle with $\alpha=0$ and focuses on the expected value; the Maximal Loss Principle sets the premium to the maximum possible loss, representing extreme risk aversion; and the Swiss Principle is a more general utility-based approach that can encompass other principles but is not defined by the specific Esscher transform formula.

The question tests the understanding of the recursive relationship for the stop-loss transform, specifically how it changes when the retention level increases. The formula $\Pi(d) = \Pi(d-1) – (1 – F_S(d-1))$ shows that the stop-loss transform at retention level $d$ is derived from the stop-loss transform at retention level $d-1$ by subtracting the probability that the total claim amount is less than $d-1$. Therefore, to find $\Pi(3)$ from $\Pi(2)$, we need to subtract the probability that the total claim amount is less than or equal to 2, which is $F_S(2)$. Looking at the provided table, $F_S(2) = 0.782$. Thus, $\Pi(3) = \Pi(2) – (1 – F_S(2)) = 0.420 – (1 – 0.782) = 0.420 – 0.218 = 0.202$. However, the question asks for the relationship between $\Pi(d)$ and $\Pi(d-1)$ in general terms. The recursive formula $\Pi(d) = \Pi(d-1) – (1 – F_S(d-1))$ is the correct representation of this relationship. The other options incorrectly manipulate the formula or use incorrect components.

This question tests the understanding of how reinsurance impacts an insurer’s financial position, specifically focusing on the trade-off between reduced risk and the cost of reinsurance. The scenario describes a situation where an insurer purchases stop-loss reinsurance. The core concept is that while reinsurance reduces the insurer’s exposure to large claims (thus lowering the potential for catastrophic losses), it comes at a cost, typically in the form of a reinsurance premium. This premium, along with the retained portion of claims, reduces the insurer’s expected gain compared to a situation without reinsurance. The question asks about the primary consequence of this reinsurance arrangement on the insurer’s financial outcome. Option A correctly identifies that the insurer’s potential for large gains is reduced due to the cost of reinsurance and the transfer of risk. Option B is incorrect because while the insurer’s risk is reduced, the primary impact on the *gain* is a reduction, not an increase, due to the premium paid. Option C is incorrect as the question is about the insurer’s gain, not the reinsurer’s. Option D is incorrect because the insurer’s retained claims are still subject to the stop-loss limit, and the primary financial impact on the insurer’s gain is the cost of the reinsurance itself.