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Question 1Modern Actuarial Statistics II

Under the collective risk model, the aggregate claim amount S is the sum of a random number N of individual claim amounts X_i. Which statement best describes the standard independence assumption used to derive the compound distribution of S?

  1. The claim counts N are independent of the individual claim sizes X_i, but the X_i are allowed to be dependent within a single accident year.
  2. The claim counts N depend on the individual claim sizes because large claims tend to arrive in clusters.
  3. The claim counts N are independent of the individual claim sizes X_i, and the X_i are mutually independent and identically distributed. Correct answer.
  4. The individual claim sizes X_i are independent of each other, but N is deterministic across all portfolios.

Explanation

The collective risk model treats N as a counting random variable independent of the severity variables, which are i.i.d. copies of a common severity distribution. This independence is what allows the compound distribution to be built from the frequency and severity components separately.

Question 2Modern Actuarial Statistics II

An insurer models annual claim counts with a Poisson distribution having mean 4. Using the Panjer recursion for the compound Poisson aggregate loss with severity mass function f(x), which starting value P(S = 0) is correct?

  1. P(S = 0) = 4 exp(-4)
  2. P(S = 0) = exp(-4) / (1 - f(0))
  3. P(S = 0) = exp(-4) Correct answer.
  4. P(S = 0) = 1 - exp(-4)

Explanation

For a compound Poisson model the probability of zero aggregate claims comes only from having zero claims, which is exp(-lambda) = exp(-4). Severity values do not enter because no claim occurs.

Question 3Modern Actuarial Statistics II

In credibility theory, the Bühlmann model sets the credibility factor Z for a risk group as a function of the number of exposure periods n. Which expression gives Z in the Bühlmann model?

  1. Z = n / (n + k), where k = EPV / VHM Correct answer.
  2. Z = n / (n + k), where k = VHM / EPV
  3. Z = 1 - exp(-n k), where k = EPV / VHM
  4. Z = k / (n + k), where k = EPV / VHM

Explanation

Bühlmann credibility gives Z = n/(n+k) with k equal to the ratio of expected process variance to the variance of hypothetical means. A larger between-risk variance raises Z, while a larger process variance lowers it.

Question 4Modern Actuarial Statistics II

A Bayesian credibility analysis uses a Poisson frequency model with a gamma prior on the mean. Which statement about the posterior distribution is correct?

  1. The posterior is gamma only when the prior mean equals the sample mean.
  2. The posterior is gamma with updated shape and rate parameters obtained by adding total observed counts and total exposure to the prior parameters. Correct answer.
  3. The posterior is Poisson because the Poisson likelihood is conjugate to itself.
  4. The posterior is normal by the central limit theorem once exposure exceeds 30.

Explanation

The gamma prior is conjugate to the Poisson likelihood: observed counts add to the shape parameter and exposure adds to the rate parameter. The posterior mean is then a credibility-weighted average of prior and observed experience.

Question 5Modern Actuarial Statistics II

A generalized linear model for claim frequency uses a Poisson error with a log link. Which statement about the linear predictor is correct?

  1. The variance of the frequency is modeled as a linear function of the covariates.
  2. The expected frequency is modeled directly as a linear combination of the covariates.
  3. The logit of the expected frequency is modeled as a linear combination of the covariates.
  4. The log of the expected frequency is modeled as a linear combination of the covariates. Correct answer.

Explanation

With a log link, the model sets log(mu) equal to the linear predictor, so covariate effects multiply the mean on the original scale. This ensures fitted frequencies remain positive without constraints on the coefficients.

Question 6Modern Actuarial Statistics II

A Tweedie generalized linear model is fitted to pure premium data with power parameter p strictly between 1 and 2. What does this model accomplish?

  1. It assumes pure premiums follow a normal distribution with constant variance across all covariate values.
  2. It guarantees that the fitted pure premium equals the product of separately fitted frequency and severity models.
  3. It combines a point mass at zero with a continuous positive claim amount distribution, allowing zero and positive pure premiums in one model. Correct answer.
  4. It removes the need for a link function because the Tweedie family is scale invariant.

Explanation

The Tweedie family with 1 < p < 2 has a mass at zero and a continuous positive part, which suits pure premium data containing many zero-claim exposures. It models aggregate loss cost directly rather than as a product of separately fitted components.

Question 7Modern Actuarial Statistics II

In a frequency-severity model, the expected loss cost per exposure is computed as the product of expected claim frequency and expected claim severity. Which assumption is required for this product to equal the expected aggregate loss per exposure?

  1. The number of exposures is large.
  2. Claim frequency and claim severity are independent. Correct answer.
  3. Claim severity is normally distributed.
  4. Claim frequency follows a Poisson distribution.

Explanation

The expected value of a product of random variables equals the product of expectations when the variables are independent. Without independence, a covariance term would appear and the simple product would be biased.

Question 8Modern Actuarial Statistics II

An actuary applies a chain-ladder development method to a cumulative claims triangle. Which assumption is fundamental to the validity of the chain-ladder estimates?

  1. Claims development patterns are stable across accident years, so the same age-to-age factors apply to all cohorts. Correct answer.
  2. Claim amounts are normally distributed within each accident year.
  3. The most recent accident year is fully mature.
  4. There are no large claims in any development period.

Explanation

Chain-ladder relies on the assumption that development factors estimated from past cohorts apply to current cohorts. If development patterns shift across accident years, the projected ultimate losses will be biased.

Question 9Modern Actuarial Statistics II

A Bornhuetter-Ferguson estimate is used for an immature accident year. Compared with a pure chain-ladder estimate for the same year, what is the main behavioral difference?

  1. The Bornhuetter-Ferguson estimate ignores the reported-to-date losses entirely.
  2. The Bornhuetter-Ferguson estimate always exceeds the chain-ladder estimate.
  3. The Bornhuetter-Ferguson estimate requires a fully developed triangle before it can be applied.
  4. The Bornhuetter-Ferguson estimate is anchored to an a priori expected loss and is less sensitive to early erratic reported losses. Correct answer.

Explanation

Bornhuetter-Ferguson blends reported losses with an independent expected-loss benchmark, weighting the benchmark heavily when little has developed. This stabilizes estimates for immature years where chain-ladder can be volatile.

Question 10Modern Actuarial Statistics II

In reserving, the expected loss ratio method estimates ultimate losses for an accident year by multiplying an expected loss ratio by earned premium. Which situation makes this method most appropriate?

  1. A line where reported losses have fully developed to ultimate.
  2. A line where premium is unavailable but claim counts are known.
  3. A mature line with a long history of stable development.
  4. A new line of business with little or no credible loss experience. Correct answer.

Explanation

The expected loss ratio method relies on a benchmark loss ratio rather than the observed losses, so it is most useful when experience is thin or nonexistent. For mature lines, development-based methods are typically preferred because they use actual experience.

Question 11Modern Actuarial Statistics II

Mack's distribution-free chain-ladder model provides a key output beyond the point estimate of reserves. What does the Mack model estimate?

  1. The full posterior predictive distribution of reserves under a Bayesian prior.
  2. The probability that the insurer becomes insolvent in the next year.
  3. The correlation between accident years in the development triangle.
  4. The mean squared error of prediction of the chain-ladder reserve estimate. Correct answer.

Explanation

Mack derived an estimator for the conditional mean squared error of the chain-ladder reserve, decomposing it into process and parameter uncertainty. It does not assume a full distribution, so it stops at the second moment.

Question 12Modern Actuarial Statistics II

An actuary fits an over-dispersed Poisson model to a claims triangle and compares it with the chain-ladder method. What is a well-known result about the fitted values?

  1. The over-dispersed Poisson model reproduces the chain-ladder reserve estimates when the model is saturated in accident and development effects. Correct answer.
  2. The over-dispersed Poisson model systematically produces reserves that are half the chain-ladder values.
  3. The over-dispersed Poisson model cannot be applied to cumulative claims triangles.
  4. The over-dispersed Poisson model requires the dispersion parameter to equal exactly one.

Explanation

The over-dispersed Poisson GLM with accident-year and development-year factors gives the same fitted values as chain-ladder. The dispersion parameter captures variance beyond the Poisson mean without changing the point estimates.

Question 13Modern Actuarial Statistics II

A risk measure rho is said to be coherent if it satisfies a set of axioms. Which axiom states that combining two risks never produces a requirement greater than the sum of their separate requirements?

  1. Monotonicity
  2. Positive homogeneity
  3. Translation invariance
  4. Subadditivity Correct answer.

Explanation

Subadditivity requires rho(X + Y) <= rho(X) + rho(Y), capturing the diversification benefit of pooling risks. It is the axiom most closely tied to the coherence debate around Value at Risk.

Question 14Modern Actuarial Statistics II

Tail Value at Risk at level alpha is often defined as the average of the worst outcomes beyond the Value at Risk threshold. Which property does TVaR possess that Value at Risk generally lacks?

  1. Subadditivity, making it a coherent risk measure. Correct answer.
  2. Monotonicity, which Value at Risk does not satisfy.
  3. Translation invariance, which Value at Risk does not satisfy.
  4. Positive homogeneity, which Value at Risk does not satisfy.

Explanation

TVaR is subadditive and therefore coherent, while VaR can violate subadditivity for non-elliptical loss distributions. VaR does satisfy monotonicity, translation invariance, and positive homogeneity.

Question 15Modern Actuarial Statistics II

A portfolio has two independent risks with identical loss distributions. Which statement correctly compares the Value at Risk of the combined portfolio with the sum of the individual Values at Risk?

  1. The combined VaR is generally less than the sum of the individual VaRs because of diversification. Correct answer.
  2. The combined VaR always exceeds the sum of the individual VaRs.
  3. The combined VaR always equals the sum of the individual VaRs regardless of dependence.
  4. The combined VaR equals the individual VaR because the risks are identical.

Explanation

For independent or imperfectly dependent risks, the quantile of the sum is typically below the sum of quantiles, reflecting diversification. Equality holds only under special dependence structures such as perfect positive dependence.

Question 16Modern Actuarial Statistics II

In extreme value theory, the generalized Pareto distribution is used to model exceedances over a high threshold. Which result justifies this choice?

  1. The central limit theorem shows that exceedances become normal as the sample grows.
  2. The Fisher-Tippett theorem states that block maxima follow a generalized Pareto distribution.
  3. The law of large numbers shows that exceedances converge to their mean.
  4. The Pickands-Balkema-de Haan theorem shows that threshold exceedances converge to a generalized Pareto distribution as the threshold rises. Correct answer.

Explanation

The Pickands-Balkema-de Haan theorem provides the asymptotic justification for fitting a generalized Pareto distribution to excesses over a high threshold. The Fisher-Tippett theorem concerns block maxima and the generalized extreme value distribution.

Question 17Modern Actuarial Statistics II

An actuary simulates aggregate losses using a compound Poisson model and wants to reduce Monte Carlo error for a tail quantile estimate. Which technique directly reduces variance by conditioning on the number of claims?

  1. Increasing the random seed value used in the simulation.
  2. Rao-Blackwellization by analytically computing the conditional expectation given the claim count. Correct answer.
  3. Rounding simulated losses to the nearest thousand.
  4. Using the same uniform random numbers for frequency and severity.

Explanation

Conditioning on the claim count and replacing the simulated severity sum with its analytic conditional expectation reduces variance, an application of Rao-Blackwellization. The estimator remains unbiased while requiring fewer simulations for a given precision.

Question 18Modern Actuarial Statistics II

A copula is used to model dependence between two lines of business. Which statement about the copula function is correct?

  1. It captures the dependence structure separately from the marginal distributions of the two lines. Correct answer.
  2. It determines the marginal distributions of each line of business.
  3. It can only represent perfect positive dependence.
  4. It requires both lines to have normal marginal distributions.

Explanation

By Sklar's theorem, a joint distribution can be decomposed into its marginals and a copula that encodes the dependence. This separation lets the actuary choose marginals and dependence structure independently.

Question 19Modern Actuarial Statistics II

In a bivariate Archimedean copula, the generator function determines the dependence structure. What happens to the copula as the generator parameter approaches the independence limit?

  1. The copula approaches the product copula, giving independent marginals. Correct answer.
  2. The copula approaches the Fréchet lower bound of perfect negative dependence.
  3. The copula approaches the Fréchet upper bound of perfect positive dependence.
  4. The copula becomes undefined and no joint distribution exists.

Explanation

In the independence limit the Archimedean copula reduces to the product copula, so the variables behave independently. At the opposite extreme the same family can approach the Fréchet upper bound.

Question 20Modern Actuarial Statistics II

A survival model for time-to-claim-reporting uses a hazard function. Which quantity does the hazard function represent?

  1. The instantaneous rate of reporting at time t given that the claim has not yet been reported by time t. Correct answer.
  2. The cumulative probability that a claim is never reported.
  3. The expected total number of claims reported by time t.
  4. The probability that a claim is reported exactly at time t.

Explanation

The hazard is a conditional instantaneous rate: it describes the reporting intensity at time t among claims still unreported. Integrating the hazard gives the cumulative hazard and hence the survival function.

Question 21Modern Actuarial Statistics II

In a proportional hazards model for claim reporting, a covariate has an estimated coefficient of 0.5 with a log link on the hazard. How is the hazard ratio for a one-unit increase in that covariate interpreted?

  1. The survival probability is multiplied by exp(0.5).
  2. The hazard is multiplied by exp(0.5), approximately 1.65. Correct answer.
  3. The hazard is increased by 0.5 in absolute terms.
  4. The hazard is multiplied by 0.5.

Explanation

In a proportional hazards model the hazard ratio for a one-unit covariate change is exp(beta), here exp(0.5) which is about 1.65. The effect is multiplicative on the hazard, not additive.

Question 22Modern Actuarial Statistics II

An actuary uses a Markov chain to model transitions between claim states such as open, closed with payment, and closed without payment. Which matrix summarizes the one-step transition probabilities between these states?

  1. The covariance matrix of the state indicators.
  2. The generator matrix, whose diagonal entries are positive.
  3. The stationary distribution vector, whose entries sum to one.
  4. The transition probability matrix, whose rows sum to one. Correct answer.

Explanation

A discrete-time Markov chain is characterized by a transition matrix whose (i,j) entry gives the probability of moving from state i to state j. Each row is a probability distribution and therefore sums to one.

Question 23Modern Actuarial Statistics II

A health insurance model classifies individuals by an unobservable frailty that raises their claim frequency. Why is a mixed Poisson model with a gamma frailty often preferred over a simple Poisson model?

  1. It guarantees that claim counts are independent across individuals and over time.
  2. It forces the variance of claim counts to equal the mean.
  3. It eliminates the need to estimate any frequency parameters.
  4. It induces over-dispersion and positive correlation in claim counts across time for the same individual. Correct answer.

Explanation

Unobserved heterogeneity captured by a gamma frailty produces a negative binomial-like marginal with variance exceeding the mean. It also creates dependence across periods for the same individual, which the simple Poisson model cannot represent.

Question 24Modern Actuarial Statistics II

In a bonus-malus system for motor insurance, transition rules move policyholders between discount levels based on claim history. Which modeling framework is most natural for analyzing long-run premium levels under such a system?

  1. A linear regression of premium on policyholder age.
  2. A Markov chain with states defined by discount levels and transition probabilities driven by claim frequency. Correct answer.
  3. A compound Poisson process with no state dependence.
  4. A generalized Pareto model for large losses.

Explanation

Bonus-malus systems are naturally modeled as Markov chains whose states are the discount classes and whose transitions depend on claim occurrence. The stationary distribution then yields long-run average premiums.

Question 25Modern Actuarial Statistics II

An insurer estimates the ultimate loss for an accident year using a stochastic reserving model and reports a 75th percentile of the reserve distribution. Which statement about this percentile is correct?

  1. It is the maximum possible reserve under the model.
  2. It is a value such that the reserve is estimated to be no greater than that amount with 75 percent probability. Correct answer.
  3. It is the reserve estimate with the highest likelihood.
  4. It is the average of the reserve distribution.

Explanation

A 75th percentile is the quantile below which 75 percent of the modeled reserve outcomes fall. It is a risk margin measure, distinct from the mean or the mode of the distribution.

Question 26Modern Actuarial Statistics II

A generalized linear model for claim counts uses a Poisson error structure with a log link. Which variance function does this specification imply?

  1. The variance equals the mean squared, so Var(Y) = mu^2.
  2. The variance is a constant independent of the mean.
  3. The variance equals phi times mu cubed, so Var(Y) = phi*mu^3.
  4. The variance equals the mean, so Var(Y) = mu. Correct answer.

Explanation

In the exponential dispersion family used by GLMs, the Poisson distribution has variance function V(mu) = mu, so the variance equals the mean. Over-dispersion is handled by adding a scale parameter phi, giving Var(Y) = phi*mu, not by changing the power.

Question 27Modern Actuarial Statistics II

An actuary fits a Tweedie GLM with power parameter p = 1.5 to pure premium data. Which combination of distributions does this choice represent?

  1. A Poisson claim count compounded with inverse Gaussian claim sizes.
  2. A degenerate distribution with a point mass at zero only.
  3. A Poisson claim count compounded with gamma claim sizes. Correct answer.
  4. A binomial claim count compounded with lognormal claim sizes.

Explanation

For Tweedie power parameters strictly between 1 and 2, the distribution is a compound Poisson-gamma: a Poisson number of claims with gamma-distributed severities. This produces a point mass at zero pure premium and a continuous positive part.

Question 28Modern Actuarial Statistics II

In a GLM for claim frequency, an actuary includes a categorical variable for vehicle class with four levels and uses treatment coding with a reference level. How many indicator columns does this variable contribute to the design matrix?

  1. Two
  2. Three Correct answer.
  3. One
  4. Four

Explanation

With treatment coding and an intercept in the model, a categorical variable with k levels contributes k minus 1 indicator columns to avoid perfect collinearity. For four levels this gives three indicators, with the fourth level absorbed into the reference.

Question 29Modern Actuarial Statistics II

An actuary compares a Poisson GLM and a negative binomial GLM fitted to the same claim count data. The Poisson model shows a deviance far exceeding its degrees of freedom. What does this indicate?

  1. The dispersion parameter is exactly one, confirming the Poisson fit.
  2. The data are over-dispersed relative to the Poisson assumption. Correct answer.
  3. The linear predictor is perfectly specified and no change is needed.
  4. The data are under-dispersed and the Poisson model is conservative.

Explanation

For a correctly specified Poisson model, the deviance is approximately chi-squared with degrees of freedom equal to the residual degrees of freedom, so the ratio should be near one. A deviance much larger than its degrees of freedom signals over-dispersion, which the negative binomial can accommodate through an extra parameter.

Question 30Modern Actuarial Statistics II

A GLM uses a log link and the fitted coefficient on a binary covariate is 0.4. Holding other covariates fixed, how does the fitted mean change when the covariate switches from 0 to 1?

  1. It is multiplied by exp(0.4), about 1.49. Correct answer.
  2. It increases by 0.4 in absolute terms.
  3. It is multiplied by 0.4.
  4. It increases by exp(0.4) in absolute terms.

Explanation

With a log link, the linear predictor enters as the log of the mean, so a one-unit change in a covariate multiplies the fitted mean by the exponential of its coefficient. Here exp(0.4) is approximately 1.49, a 49 percent increase.

Question 31Modern Actuarial Statistics II

An actuary uses a Bayesian model for claim frequency with a gamma prior on the Poisson mean and observes claim counts for several years. What is the conjugate posterior distribution for the Poisson mean?

  1. A beta distribution
  2. A lognormal distribution
  3. A normal distribution
  4. A gamma distribution Correct answer.

Explanation

The gamma distribution is conjugate to the Poisson likelihood when the parameter of interest is the Poisson mean. The posterior is gamma with updated shape and rate parameters combining prior and observed counts.

Question 32Modern Actuarial Statistics II

In credibility theory, an actuary combines an insured's own experience with a manual rate. As the number of exposure units for the insured increases, how does the credibility weight assigned to the insured's experience typically behave?

  1. It increases toward one. Correct answer.
  2. It becomes negative for very large exposures.
  3. It remains constant regardless of exposure.
  4. It decreases toward zero.

Explanation

Credibility weights increase with the volume of the insured's own data, approaching full credibility as exposure grows. Limited-fluctuation credibility sets the weight as a function of the ratio of observed exposure to the standard for full credibility.

Question 33Modern Actuarial Statistics II

An actuary applies Buhlmann credibility to estimate the expected claim frequency for a group. The between-group variance is large relative to the within-group variance. What does this imply about the credibility assigned to the group's own experience?

  1. The credibility is higher because groups differ substantially. Correct answer.
  2. The credibility is unaffected by the variance components.
  3. The credibility is lower because groups differ substantially.
  4. The credibility equals zero in all cases.

Explanation

The Buhlmann credibility factor is the exposure divided by exposure plus the ratio of within-group to between-group variance. A larger between-group variance reduces that ratio and raises the credibility weight, since the group's own data are more informative about its true mean.

Question 34Modern Actuarial Statistics II

A survival model is fitted to time-to-claim-reporting data, and the actuary wants to estimate the probability that a claim is still unreported at time t. Which function provides this directly?

  1. The probability density function f(t)
  2. The survival function S(t) Correct answer.
  3. The hazard function h(t)
  4. The cumulative hazard function H(t)

Explanation

The survival function S(t) is defined as the probability that the event time exceeds t, so it directly gives the probability of remaining unreported at time t. The hazard and density functions describe rates and local probabilities rather than the tail probability itself.

Question 35Modern Actuarial Statistics II

In a Cox proportional hazards model for claim reporting, a covariate has coefficient 0.5. How is the hazard ratio for a one-unit increase in that covariate interpreted?

  1. The survival probability increases by exp(0.5).
  2. The reporting hazard increases by 0.5 percentage points.
  3. The reporting hazard is multiplied by 0.5.
  4. The reporting hazard is multiplied by exp(0.5), about 1.65. Correct answer.

Explanation

In a proportional hazards model, the hazard is multiplied by exp(beta) for each one-unit increase in the covariate. With beta equal to 0.5, the hazard ratio is exp(0.5), approximately 1.65, indicating a higher reporting hazard.

Question 36Modern Actuarial Statistics II

An actuary models transitions among claim states using a continuous-time Markov chain with a generator matrix. What do the off-diagonal entries of the generator matrix represent?

  1. Discount factors applied to future payments
  2. Expected sojourn times in each state
  3. Instantaneous transition intensities between states Correct answer.
  4. Long-run stationary probabilities of each state

Explanation

The off-diagonal entries of a continuous-time Markov generator are the transition intensities, or hazard rates, governing movement between states. The diagonal entries are the negatives of the row sums, ensuring rows sum to zero.

Question 37Modern Actuarial Statistics II

A multi-state model tracks claim states, and the actuary wants the probability that a claim currently open will be settled within one year. Which quantity must be computed from the transition intensity matrix?

  1. The one-year transition probability matrix obtained by exponentiating the generator Correct answer.
  2. The stationary distribution of the chain
  3. The eigenvalues of the generator matrix alone
  4. The initial state vector

Explanation

Transition probabilities over a finite interval are obtained from the matrix exponential of the generator multiplied by the interval length. The resulting matrix gives the probability of moving from the open state to the settled state within one year.

Question 38Modern Actuarial Statistics II

A health insurance model includes an unobservable frailty that raises individual claim frequency. Why does a marginal count model that ignores frailty tend to understate the variance of aggregate claims?

  1. Unobserved heterogeneity induces positive correlation among counts within individuals that the marginal model omits. Correct answer.
  2. Ignoring frailty always produces a negative binomial distribution.
  3. Frailty reduces the mean claim frequency below the observed average.
  4. Frailty has no effect on the variance of aggregate counts.

Explanation

When frailty is unobserved, individual counts are correlated conditional on the covariates, and this extra variation inflates the variance of the aggregate. A marginal model that omits frailty assumes independent counts and therefore understates the true variance.

Question 39Modern Actuarial Statistics II

In a bonus-malus system, a policyholder at the maximum discount level makes one claim. Under a typical transition rule, what happens to the policyholder's level?

  1. The policyholder moves to an even more favorable level.
  2. The policyholder is removed from the system permanently.
  3. The policyholder moves to a less favorable level, losing part of the discount. Correct answer.
  4. The policyholder remains at the maximum discount indefinitely.

Explanation

Bonus-malus systems penalize claims by moving policyholders toward higher premium levels, so a claim at the maximum discount typically causes a drop to a less favorable level. The exact transition depends on the system's rules but the direction is a reduction in discount.

Question 40Modern Actuarial Statistics II

An insurer reports the 75th percentile of the reserve distribution from a stochastic reserving model. Which risk measure does this percentile correspond to?

  1. The standard deviation of reserves
  2. Tail Value at Risk at the 75 percent level
  3. Expected shortfall at the 75 percent level
  4. Value at Risk at the 75 percent level Correct answer.

Explanation

The 75th percentile of a loss distribution is by definition the Value at Risk at the 75 percent confidence level. Tail Value at Risk and expected shortfall instead average outcomes beyond a threshold.

Question 41Modern Actuarial Statistics II

An actuary wants a coherent risk measure that accounts for the severity of losses beyond a threshold. Which measure satisfies coherence and averages the worst outcomes beyond a given quantile?

  1. Value at Risk
  2. The mode of the loss distribution
  3. Tail Value at Risk Correct answer.
  4. Variance

Explanation

Tail Value at Risk averages losses beyond the Value at Risk threshold and satisfies the coherence axioms, including subadditivity. Value at Risk is not subadditive in general and can penalize diversification.

Question 42Modern Actuarial Statistics II

An actuary simulates aggregate losses and wants to reduce Monte Carlo error in the estimated tail probability. Which technique uses a correlated antithetic variable to reduce variance?

  1. Using a coarser time grid
  2. Increasing the number of simulations only
  3. Antithetic variates, pairing each uniform draw with its complement Correct answer.
  4. Ignoring the tail and estimating the mean instead

Explanation

Antithetic variates pair each simulated uniform with its complement, inducing negative correlation between paired outcomes and reducing the variance of the average. This lowers Monte Carlo error for a given number of simulations.

Question 43Modern Actuarial Statistics II

A copula model is used to combine marginal loss distributions for two lines of business. Which statement about the copula function is correct?

  1. It requires the marginals to be normally distributed.
  2. It determines the marginal distributions of each line.
  3. It fixes the correlation coefficient at one.
  4. It captures the dependence structure separately from the marginal distributions. Correct answer.

Explanation

Sklar's theorem shows that a joint distribution can be decomposed into its marginals and a copula that encodes the dependence structure. The copula is invariant to strictly increasing transformations of the marginals.

Question 44Modern Actuarial Statistics II

In a bivariate Archimedean copula, the generator function is parameterized by theta. As theta approaches the independence limit, what happens to the copula?

  1. It approaches the comonotonic copula.
  2. It approaches the countermonotonic copula.
  3. It approaches the independence copula. Correct answer.
  4. It becomes undefined.

Explanation

For common Archimedean families such as Clayton or Gumbel, a limiting value of the dependence parameter yields the independence copula. Increasing the parameter away from that limit increases dependence toward comonotonicity.

Question 45Modern Actuarial Statistics II

An actuary fits a frequency-severity model and wants the expected aggregate loss per exposure. Which expression gives this quantity?

  1. Expected severity divided by expected frequency
  2. The maximum of expected frequency and expected severity
  3. Expected frequency multiplied by expected severity Correct answer.
  4. Expected frequency plus expected severity

Explanation

Under the standard frequency-severity decomposition, the expected aggregate loss is the product of expected claim frequency and expected claim severity, assuming independence between counts and sizes. This product is the pure premium.

Question 46Modern Actuarial Statistics II

In a collective risk model, the aggregate loss is the sum of a random number of individual claim amounts. If claim counts and claim sizes are independent, which quantity is needed in addition to the mean and variance of the count and severity distributions to compute the variance of aggregate losses?

  1. The fourth moment of the count distribution
  2. The skewness of the severity distribution
  3. The copula between counts and sizes
  4. Only the means and variances of the count and severity distributions Correct answer.

Explanation

Under independence between counts and sizes, the variance of the aggregate loss is the variance of the count times the squared mean severity plus the mean count times the variance of severity. Only the first two moments of each component are required.

Question 47Modern Actuarial Statistics II

An actuary uses the expected loss ratio method to set reserves for an immature accident year. Which input is required in addition to the expected loss ratio and earned premium?

  1. The expected ultimate loss ratio applied to earned premium Correct answer.
  2. The reported incurred losses to date
  3. The tail factor beyond the latest diagonal
  4. The chain-ladder development factors

Explanation

The expected loss ratio method multiplies earned premium by an expected ultimate loss ratio to estimate ultimate losses, without relying on reported losses. Development factors and tail factors belong to triangle-based methods.

Question 48Modern Actuarial Statistics II

An actuary applies the Bornhuetter-Ferguson method to an immature accident year. Compared with the chain-ladder estimate for the same year, how does the Bornhuetter-Ferguson estimate typically behave?

  1. It ignores the expected loss ratio entirely.
  2. It is less sensitive to early reported losses and more stable. Correct answer.
  3. It is always larger than the chain-ladder estimate.
  4. It requires a fully developed triangle.

Explanation

Bornhuetter-Ferguson blends an expected loss estimate with reported losses weighted by the unreported proportion, so it dampens the influence of volatile early data. This makes it more stable than chain-ladder for immature years.

Question 49Modern Actuarial Statistics II

Mack's distribution-free chain-ladder model provides an estimate of the prediction error of reserves. What does this quantity represent?

  1. The discount rate applied to reserves
  2. The expected value of future claim payments
  3. The uncertainty of the ultimate reserve estimate including process and parameter risk Correct answer.
  4. The number of development periods in the triangle

Explanation

Mack's model estimates the mean squared error of prediction for the reserve, capturing both process variance and parameter estimation error. It does not require distributional assumptions beyond the chain-ladder structure.

Question 50Modern Actuarial Statistics II

An actuary fits an over-dispersed Poisson model to a claims triangle and compares it with the chain-ladder method. Which statement about the relationship is correct?

  1. The over-dispersed Poisson model always produces smaller reserves.
  2. The over-dispersed Poisson model reproduces the chain-ladder reserve estimates when the model fits. Correct answer.
  3. The over-dispersed Poisson model cannot be applied to triangles.
  4. The over-dispersed Poisson model ignores development patterns.

Explanation

The over-dispersed Poisson model, when it fits the triangle, yields reserve estimates identical to the chain-ladder method while providing a stochastic framework. This equivalence links deterministic and stochastic reserving.