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IFoA CS2 practice questions

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Question 1Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a Markov jump process used to model the health states of an income protection policyholder, the transition intensities are assumed constant over each policy year. Which quantity is the generator matrix Q used to derive directly?

  1. The vector of expected claim amounts payable on death in each state
  2. The stationary distribution of the process only, not the transient behaviour
  3. The matrix of one-year transition probabilities P(1) via the matrix exponential exp(Q) Correct answer.
  4. The cumulative hazard function for each individual transition, integrated over all future time

Explanation

For a time-homogeneous Markov jump process, the transition probability matrix over an interval of length t satisfies P(t) = exp(Qt), so the one-year matrix is obtained by exponentiating the generator. The off-diagonal entries of Q are the transition intensities and the diagonal entries are minus the total intensity of leaving each state. This is the standard Kolmogorov forward equation solution for constant intensities.

Question 2Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A survival model is fitted to a portfolio of term assurance policies. The force of mortality is assumed to follow mu(x) = A + B*c^x. Which name is given to this parametric mortality law?

  1. Gompertz law
  2. Makeham law Correct answer.
  3. Weibull law
  4. de Moivre law

Explanation

The Makeham law adds a constant term A to the exponentially increasing Gompertz term, so mu(x) = A + Bc^x. The constant A captures an age-independent background hazard. The pure Gompertz law has no constant additive term.

Question 3Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a binomial model for the number of deaths in a group of independent lives, the number of deaths D is Binomial(n, q). Which statement about the variance of the observed mortality ratio D/E, where E is the expected number of deaths n*q, is correct?

  1. The variance of D/E equals q(1-q)/n
  2. The variance of D/E equals q(1-q)/(n*q^2) Correct answer.
  3. The variance of D/E is independent of n
  4. The variance of D/E equals n*q*(1-q)

Explanation

Since Var(D) = n*q*(1-q) and E = n*q, the variance of D/E is Var(D)/E^2 = n*q*(1-q)/(n^2*q^2) = (1-q)/(n*q). This shows the ratio becomes more precise as the expected number of deaths increases.

Question 4Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A life insurer models the time to death of a policyholder using a random variable T with survival function S(t). Which expression gives the force of mortality at time t?

  1. S(t) divided by the density f(t)
  2. f(t) divided by S(t) Correct answer.
  3. The negative derivative of S(t) with respect to t
  4. The integral of S(u) from 0 to t

Explanation

The force of mortality is the instantaneous rate of failure given survival to time t, equal to f(t)/S(t). Equivalently it is minus the derivative of log S(t). This is the fundamental hazard-rate definition used throughout survival modelling.

Question 5Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A Cox proportional hazards model is fitted with a single covariate z. The fitted hazard is h(t|z) = h0(t)*exp(beta*z). Which interpretation of exp(beta) is correct when z increases by one unit?

  1. The hazard at every duration t is multiplied by exp(beta), holding the baseline hazard fixed Correct answer.
  2. The baseline hazard h0(t) is shifted in time by beta
  3. The expected survival time increases by beta units
  4. The cumulative hazard at time t is increased by an additive amount beta

Explanation

In a proportional hazards model the covariate acts multiplicatively on the baseline hazard, and this multiplier is constant over time. A one-unit increase in z therefore scales the hazard by exp(beta) at all durations. This proportionality assumption is the defining feature of the Cox model.

Question 6Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a survival analysis, right-censoring occurs when a policyholder leaves the observation window before the event of interest is observed. Which assumption is standard when fitting a Kaplan-Meier estimator?

  1. Censoring times are dependent on the survival times
  2. All censored lives are assumed to have died at the censoring time
  3. Censoring can only occur at the end of the study period
  4. Censoring is non-informative, so censored lives have the same future survival prospects as those still under observation Correct answer.

Explanation

The Kaplan-Meier estimator requires non-informative censoring, meaning the censoring mechanism carries no information about the underlying survival time. Under this assumption, lives at risk at each event time form a representative sample. If censoring were informative, the estimator would be biased.

Question 7Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A graduated mortality table is being tested for smoothness. Which diagnostic is most directly designed to detect excessive irregularity in the graduated rates?

  1. The signs test
  2. The chi-square test on individual ages
  3. The cumulative deviations test
  4. A test based on third differences of the graduated rates Correct answer.

Explanation

Smoothness tests examine the shape of the graduated curve, and a common approach uses third (or higher-order) differences of the graduated rates, which should be small for a smooth curve. The other tests listed assess goodness of fit or bias, not smoothness. Graduation must balance smoothness against fidelity to the crude data.

Question 8Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a multiple-state model for a critical illness policy, a policyholder can be Healthy (H), Critically Ill (C), or Dead (D). Which set of transition intensities is typically permitted in the standard model?

  1. H to C, C to D and D to H
  2. H to C, C to H, H to D and C to D Correct answer.
  3. H to C and H to D only
  4. H to D and C to D only

Explanation

In a critical illness model, a healthy life can become ill or die, and an ill life can recover to healthy or die. Transitions out of the dead state are not permitted because death is absorbing. This four-intensity structure is the standard starting point for CI multiple-state models.

Question 9Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A survival model uses a frailty random variable to capture unobserved heterogeneity between lives. Which effect does the presence of unobserved frailty typically have on the observed population hazard as age increases?

  1. The population hazard is unaffected because frailties average out
  2. The population hazard grows more slowly than the hazard for any individual, reflecting selection of robust lives Correct answer.
  3. The population hazard becomes constant at all ages
  4. The population hazard grows more quickly because frail lives dominate at older ages

Explanation

Frail individuals die earlier, so survivors at older ages are increasingly drawn from the robust subgroup. As a result, the observed population hazard rises more slowly than the hazard conditional on frailty. This selection effect is a key reason why aggregate mortality curves flatten at older ages.

Question 10Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a Markov chain used to model the states of a long-term care insurance, the transition matrix P is estimated from data. Which property must P satisfy to be a valid one-step transition matrix?

  1. P must be symmetric
  2. Each column sums to one and all entries are non-negative
  3. Each row sums to one and all entries are non-negative Correct answer.
  4. The diagonal entries must all equal zero

Explanation

A transition matrix has non-negative entries and each row sums to one, because the entries in a row give the probabilities of moving to each possible state from the current state. Column sums need not equal one. The diagonal entries give the probability of remaining in the same state and are generally positive.

Question 11Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A life insurer is modelling the number of claims from a portfolio where the number of policies is large and the claim probability per policy is small. Which distribution is the standard limiting approximation for the total number of claims?

  1. Uniform distribution
  2. Beta distribution
  3. Poisson distribution Correct answer.
  4. Normal distribution

Explanation

When the number of independent trials is large and the probability of success per trial is small, the binomial distribution converges to the Poisson distribution with mean equal to n*p. This Poisson limit is the standard model for claim counts in mortality and morbidity studies. It is used to derive the variance of the observed-to-expected ratio.

Question 12Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a Poisson model for the number of deaths D with mean E, the standardised deviation Z = (D - E)/sqrt(E) is used to test the null hypothesis that the true mortality equals the expected rate. Which distribution does Z approximately follow under the null hypothesis?

  1. A t-distribution with E degrees of freedom
  2. A chi-square distribution with one degree of freedom
  3. A standard normal distribution Correct answer.
  4. A Poisson distribution with mean E

Explanation

For a Poisson variable with mean E, the standardised deviation (D - E)/sqrt(E) is approximately standard normal when E is reasonably large. This underlies the individual standardised deviation test for comparing actual and expected deaths. The square of Z would follow a chi-square distribution with one degree of freedom.

Question 13Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A graduation is applied to crude mortality rates and the resulting graduated rates are compared with the crude rates using the chi-square test. Which statement best describes the null hypothesis of this test?

  1. The graduated rates are monotonically increasing
  2. The crude rates equal the graduated rates in expectation Correct answer.
  3. The graduated rates are smooth
  4. The crude rates are normally distributed

Explanation

The chi-square test for graduation assesses goodness of fit, testing whether the differences between actual and expected deaths are consistent with random variation. The null hypothesis is that the graduated rates are the true underlying rates, so the crude rates equal them in expectation. Smoothness and monotonicity are separate criteria assessed by other tests.

Question 14Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a defined benefit pension scheme, the actuarial liability is valued using a discount rate. Which effect does a decrease in the assumed discount rate have on the present value of the scheme's liabilities, all else equal?

  1. The present value of the liabilities increases Correct answer.
  2. The present value of the liabilities decreases
  3. The effect depends only on the mortality assumption
  4. The present value of the liabilities is unchanged

Explanation

A lower discount rate reduces the factor applied to each future payment, so the present value of future benefit payments increases. This is why falling interest rates are a major risk for defined benefit schemes. Mortality assumptions affect the timing and amount of payments but do not remove the discounting effect.

Question 15Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

An insurer models the surplus of a with-profits fund using a stochastic asset model. Which feature of the asset model is most important for capturing the clustering of extreme returns in stress scenarios?

  1. A deterministic interest rate term structure
  2. A fixed correlation of one between all asset classes
  3. A model allowing volatility to vary over time, such as a stochastic volatility or GARCH-type structure Correct answer.
  4. A constant expected return with independent identically distributed normal returns

Explanation

Volatility clustering, where large returns tend to be followed by large returns, is captured by models in which the conditional variance changes over time. Stochastic volatility and GARCH-type structures are standard choices for this. Constant-variance models cannot reproduce this clustering.

Question 16Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In an AR(1) time series model X_t = c + phi*X_{t-1} + e_t, where e_t is white noise, which condition ensures the process is stationary?

  1. The absolute value of phi is less than one Correct answer.
  2. The variance of e_t is greater than one
  3. phi equals one
  4. c equals zero

Explanation

An AR(1) process is stationary when the autoregressive coefficient has absolute value less than one, so shocks decay over time. If phi equals one the process is a random walk and is non-stationary. The constant c and the noise variance do not determine stationarity.

Question 17Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A run-off triangle of cumulative paid claims is being analysed using the chain-ladder method. Which assumption underlies the basic chain-ladder development factors?

  1. Development factors are the same for all accident years Correct answer.
  2. The ultimate loss ratio is known in advance
  3. Claims are fully paid after one development period
  4. Development factors depend on the accident year

Explanation

The chain-ladder method assumes that the expected development pattern is common to all accident years, so a single set of age-to-age factors can be applied. This is what allows the projection of incomplete years to ultimate. If development factors varied systematically by accident year, the method would be inappropriate.

Question 18Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a collective risk model, the aggregate claim amount S is the sum of a random number N of individual claim amounts X_i. Which expression gives the variance of S when N and the X_i are independent?

  1. E[N]*Var(X)
  2. E[N]^2*Var(X)
  3. Var(N)*E[X]^2 + E[N]*Var(X) Correct answer.
  4. Var(N)*Var(X)

Explanation

By the conditional variance formula, Var(S) = Var(E[S|N]) + E[Var(S|N)]. This gives Var(N)*E[X]^2 + E[N]*Var(X). The first term captures variability in claim frequency and the second captures variability in claim severity.

Question 19Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A life table gives q_x, the probability that a life aged exactly x dies before age x+1. Which expression gives p_x, the probability of surviving one year from age x?

  1. q_x / (1 + q_x)
  2. 1 + q_x
  3. q_x
  4. 1 - q_x Correct answer.

Explanation

Since a life aged x either dies within the year or survives it, the two probabilities sum to one, so p_x = 1 - q_x. This is the basic relationship between the annual death probability and the annual survival probability. It holds under the standard life table conventions.

Question 20Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a mortality investigation, exposure to risk is often calculated using the central exposed to risk. Which statement best describes the central exposed to risk for an individual observed from time a to time b?

  1. The expected number of deaths between a and b
  2. The total time spent under observation between a and b Correct answer.
  3. The number of deaths occurring between a and b
  4. The number of complete years lived between a and b

Explanation

The central exposed to risk is the total time each life spends under observation, summed across lives. It is paired with the force of mortality in the relationship between deaths and exposure. The initial exposed to risk, by contrast, counts lives at the start of the period.

Question 21Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A survival model is fitted with a Weibull baseline hazard. Which feature distinguishes the Weibull hazard from the exponential hazard?

  1. The Weibull hazard is bounded above, whereas the exponential hazard is unbounded
  2. The Weibull hazard applies only to discrete time models
  3. The Weibull hazard is always constant, whereas the exponential hazard varies with time
  4. The Weibull hazard can increase or decrease with time, whereas the exponential hazard is constant Correct answer.

Explanation

The Weibull hazard has the form k*lambda*t^(k-1), which increases, decreases, or stays constant depending on the shape parameter k. The exponential hazard is the special case k=1 and is constant over time. This flexibility makes the Weibull model widely used in survival analysis.

Question 22Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a Markov model for a permanent health insurance, the transition intensity from the Sick state back to the Healthy state is set to zero. What does this imply about the model?

  1. Recovery from sickness is impossible in the model Correct answer.
  2. The model has no stationary distribution
  3. The Healthy state is transient
  4. The Sick state is absorbing

Explanation

Setting the recovery intensity to zero means that once a life enters the Sick state it cannot return to Healthy, so recovery is not modelled. The Sick state is not absorbing because transition to Dead may still occur. The Healthy state remains transient if death is reachable.

Question 23Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a generalised linear model for mortality rates, the log link is used with a Poisson error structure. Which statement about the fitted model is correct?

  1. The log of the expected mortality rate is a linear function of the covariates Correct answer.
  2. The variance of the response is constant across all ages
  3. The model requires the response to be normally distributed
  4. The linear predictor is equal to the mortality rate itself

Explanation

With a log link, the linear predictor eta = X*beta is equated to log of the mean, so the expected rate is exp(X*beta). The Poisson error structure makes the variance proportional to the mean. This is the standard GLM formulation for mortality rate modelling.

Question 24Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A stochastic mortality model such as the Lee-Carter model is used to project future mortality. Which feature of the model makes it suitable for generating confidence intervals for projected annuity liabilities?

  1. It incorporates a stochastic time index whose future values can be simulated Correct answer.
  2. It ignores the age pattern of mortality
  3. It produces a single deterministic projection of mortality
  4. It assumes mortality improvement rates are constant forever

Explanation

The Lee-Carter model decomposes log mortality into an age effect, a time index, and an age-time interaction, with the time index modelled as a stochastic process. Simulating future paths of the time index produces a distribution of projected mortality and hence of liabilities. This allows quantification of uncertainty in annuity costs.

Question 25Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A life insurer is pricing a term assurance policy and uses a survival model with a select period of two years. What does the select period represent in this context?

  1. The period during which policyholders may surrender without penalty
  2. The period after underwriting during which mortality is lower than for the general population of the same age Correct answer.
  3. The period during which the policy cannot be cancelled by the insurer
  4. The period over which premiums are guaranteed

Explanation

A select period reflects the effect of underwriting: recently accepted lives have lower mortality than lives of the same age who were accepted some years earlier. Mortality rates typically depend on both age and duration since selection, and after the select period the rates converge to the ultimate table. This matters for pricing and reserving.

Question 26Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a Markov jump process used to model transitions between healthy and sick states, the holding time in a state with constant total transition intensity lambda is exponentially distributed. Which property of the exponential distribution makes this model tractable for actuarial calculations?

  1. Its hazard rate increases linearly with duration, capturing ageing within the state.
  2. It guarantees that at most one transition can ever occur.
  3. It has the memoryless property, so future transitions do not depend on time already spent in the state. Correct answer.
  4. Its mean equals its variance, matching Poisson claim counts.

Explanation

With constant intensities the holding times are exponential, and the memoryless property means the transition probabilities depend only on the current state, not on elapsed duration. This is exactly the Markov assumption, which allows the Kolmogorov forward equations to be solved for transition probabilities.

Question 27Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

Under the binomial model for a mortality investigation, the number of deaths D among n independent lives each with death probability q is modelled as Binomial(n, q). Which expression gives the variance of the crude mortality estimate D/n?

  1. nq(1-q)
  2. q(1-q)/n Correct answer.
  3. q(1-q)
  4. q/n

Explanation

For a binomial variable the variance of D is nq(1-q), so dividing by n to obtain D/n scales the variance by 1/n squared, giving q(1-q)/n. The variance therefore shrinks as the number of lives exposed increases.

Question 28Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A life table has a radix of 100,000 at age 0 and l_40 = 95,000. Using the standard life-table approximation, which value gives the probability that a life aged exactly 40 survives to age 41?

  1. 1 - (l_40 - l_41)/l_40
  2. 95,000/100,000
  3. l_41/l_40 Correct answer.
  4. (l_40 - l_41)/l_40

Explanation

The one-year survival probability p_40 is the ratio of lives at age 41 to lives at age 40, that is l_41/l_40. The other expressions either give the wrong quantity or require l_41, which is not supplied in the stem.

Question 29Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In the collective risk model the aggregate claim S has mean E[N]E[X] and variance E[N]Var(X) + Var(N)(E[X])^2. Which term represents the extra variability contributed by the randomness of the claim count?

  1. Var(N)(E[X])^2 Correct answer.
  2. E[N]Var(X)
  3. E[N]E[X]
  4. Var(X)E[N]^2

Explanation

The variance of a compound sum splits into a term for claim-size variability, E[N]Var(X), and a term for count variability, Var(N)(E[X])^2. The second term vanishes only when the claim count is deterministic.

Question 30Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a generalised linear model for claim frequency the linear predictor is linked to the mean by a log link and the variance function is proportional to the mean. Which error distribution does this describe?

  1. Gamma
  2. Poisson Correct answer.
  3. Inverse Gaussian
  4. Normal

Explanation

The Poisson distribution has variance equal to its mean, so its variance function is proportional to mu, and the log link is the canonical link for Poisson counts. This makes it the standard choice for modelling claim frequencies.

Question 31Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A graduated mortality table is required to follow the pattern of the crude rates without reproducing random noise. Which criterion best captures this trade-off?

  1. Minimising the number of ages in the table
  2. Balancing goodness of fit against smoothness of the graduated rates Correct answer.
  3. Ensuring graduated rates exactly equal crude rates at every age
  4. Maximising the number of parameters in the graduation formula

Explanation

Graduation must satisfy competing desiderata: adherence to the data, smoothness, and suitability of the resulting table for its purpose. A good graduation balances fit against smoothness rather than pursuing either extreme.

Question 32Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A survival distribution has force of mortality mu(x) = 1/(100 - x) for 0 <= x < 100. Which expression gives the survival function S(x)?

  1. 1 - x/100
  2. (100 - x)/100 Correct answer.
  3. 100/(100 - x)
  4. exp(-x/100)

Explanation

Integrating the force of mortality from 0 to x gives the negative log survival, here -log(100 - x) + log 100, so S(x) = (100 - x)/100. This is the survival function of a uniform distribution on (0, 100).

Question 33Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a two-state model where a healthy life transitions to a sick state at rate sigma and sick lives recover at rate rho, with no mortality, what does the stationary distribution represent?

  1. The long-run proportion of time spent in each state Correct answer.
  2. The probability of never becoming sick
  3. The number of transitions per policyholder
  4. The expected time to first sickness

Explanation

For an irreducible finite-state Markov process the stationary distribution gives the long-run fraction of time the process spends in each state. It is found by solving the balance equations, here sigma times the healthy proportion equals rho times the sick proportion.

Question 34Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A portfolio has 500 independent policies, each producing a claim with probability 0.02 in a year. Using a Poisson approximation for the total number of claims, which mean should be used?

  1. 250
  2. 10 Correct answer.
  3. 500
  4. 0.02

Explanation

The Poisson approximation to the binomial uses the same mean, n times q, which is 500 multiplied by 0.02, giving 10. This approximation is appropriate when n is large and q is small.

Question 35Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a model for a critical illness policy with states Healthy, Critically Ill and Dead, the transition intensity from Critically Ill to Dead is increased. All else equal, which effect does this have on the expected present value of a lump-sum critical illness benefit paid on first diagnosis?

  1. It makes the expected present value equal to the premium.
  2. It increases the expected present value.
  3. It has no effect on the expected present value.
  4. It decreases the expected present value. Correct answer.

Explanation

A higher intensity out of the Critically Ill state shortens the time spent in that state before death, reducing the chance of surviving long enough to claim. Since the benefit is paid on diagnosis and the diagnosis has already occurred, the effect works through lives who would otherwise have been diagnosed later, so the value falls.

Question 36Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A survival model is fitted using the Cox proportional hazards form with covariate vector z. Which statement correctly describes the role of the baseline hazard h_0(t)?

  1. It must be constant over time for the model to be valid.
  2. It equals the hazard for a life with all covariates set to their means.
  3. It cancels from the partial likelihood and is estimated separately afterwards. Correct answer.
  4. It is estimated parametrically as part of the partial likelihood.

Explanation

The partial likelihood conditions on the risk sets and the baseline hazard cancels, so the regression coefficients can be estimated without specifying it. The baseline hazard is then recovered non-parametrically, for example by the Breslow estimator.

Question 37Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A mortality table is graduated by a parametric formula and the crude rates are compared with the graduated rates using the chi-square test statistic. What is the null hypothesis of this test?

  1. The graduated rates are smoother than the crude rates.
  2. The observed deaths follow the graduated rates with the given exposures. Correct answer.
  3. The graduated rates are exactly equal to the crude rates.
  4. The exposures are proportional to the graduated rates.

Explanation

The chi-square test of fit compares observed deaths with those expected under the graduated rates, treating the graduated rates as the true underlying rates. A large statistic indicates that the deviations are too big to be explained by binomial or Poisson sampling variation.

Question 38Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a defined contribution pension scheme, the member bears the investment risk. Which statement best describes the actuarial implication of this design compared with a defined benefit scheme?

  1. The member's retirement income depends on the accumulated fund and annuity rates at retirement. Correct answer.
  2. The retirement benefit is a fixed proportion of final salary regardless of fund performance.
  3. The sponsor holds a contingent liability for investment shortfalls.
  4. The scheme requires a statutory actuarial valuation of the sponsor's funding level.

Explanation

In a defined contribution arrangement the contributions are defined and the resulting pension depends on investment returns and the terms available at retirement, so the member carries the risk. The sponsor has no ongoing funding obligation beyond the agreed contributions.

Question 39Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A time series of annual mortality improvement rates is modelled as a stationary AR(2) process. Which feature of the autocorrelation function is characteristic of an AR(2) with complex roots?

  1. It decays exponentially without oscillation.
  2. It remains constant at all lags.
  3. It decays sinusoidally with a damping pattern. Correct answer.
  4. It cuts off sharply to zero after lag two.

Explanation

Complex roots of the AR(2) characteristic equation produce a damped sine-wave pattern in the autocorrelation function. Real roots instead give a monotone or mixed exponential decay.

Question 40Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a Lee-Carter type stochastic mortality model, the time-varying index is projected forward as a random walk with drift. Which consequence does the drift term have for long-horizon projections?

  1. It forces the index to revert to its long-run mean.
  2. It ensures mortality rates converge to a constant.
  3. It removes all uncertainty from the projection.
  4. It produces an approximately linear trend in the log mortality rate with horizon. Correct answer.

Explanation

A random walk with drift has a deterministic linear trend in the mean, so projected log mortality follows an approximately linear path with horizon, with uncertainty growing as the square root of the horizon. This matches the observed pattern of steady improvement over long periods.

Question 41Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a surplus model for a life insurance company, the surplus at the end of the year is the opening surplus plus premiums plus investment income minus claims and expenses. Which change increases the volatility of surplus?

  1. Increasing the proportion of assets held in equities Correct answer.
  2. Reducing the variability of claim amounts
  3. Shortening the duration of the liabilities
  4. Increasing the proportion of assets held in fixed-interest bonds

Explanation

Equities have higher return volatility than bonds, so shifting the asset mix towards equities increases the variability of investment income and hence of surplus. The other changes either reduce volatility or affect its level rather than its spread.

Question 42Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A survival model is fitted to grouped data with central exposed to risk E_x^c and observed deaths d_x. Under the assumption of a constant force of mortality over each age interval, which estimator of the force of mortality is used?

  1. d_x / (E_x^c)^2
  2. E_x^c / d_x
  3. 1 - d_x / E_x^c
  4. d_x / E_x^c Correct answer.

Explanation

With a constant force of mortality over the interval, the number of deaths is Poisson with mean equal to the force of mortality times the central exposed to risk, so the maximum likelihood estimator is the ratio of deaths to central exposure. This is the standard actuarial estimator.

Question 43Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A portfolio's claim counts are modelled with a negative binomial distribution rather than a Poisson. Which feature of the data most strongly motivates this choice?

  1. The counts are bounded above by the number of policies.
  2. The mean of the counts exceeds the variance.
  3. The variance of the counts exceeds the mean. Correct answer.
  4. The counts take only the values zero and one.

Explanation

The negative binomial arises as a Poisson mixture with a gamma-distributed mean and has variance greater than its mean, a property known as overdispersion. Excess variability relative to the Poisson is the usual reason for preferring it.

Question 44Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a multiple decrement model for a life insurance policy with decrements of death and withdrawal, the independent rate of decrement is required for pricing. Which statement correctly relates the independent rate to the dependent rate?

  1. The independent rate cannot be derived from the dependent rate.
  2. The two rates are always equal.
  3. The independent rate is always smaller than the dependent rate.
  4. The independent rate is always larger than the dependent rate. Correct answer.

Explanation

The dependent rate is reduced by competing decrements, so removing the competition increases the rate, making the independent rate larger. The relationship is recovered using the associated single decrement table.

Question 45Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A run-off triangle is analysed by the Bornhuetter-Ferguson method. Which feature distinguishes it from the chain-ladder method?

  1. It requires the triangle to be complete before use.
  2. It produces the same reserve as chain-ladder in all cases.
  3. It blends an expected loss ratio with the claims developed to date. Correct answer.
  4. It ignores the observed claims to date entirely.

Explanation

Bornhuetter-Ferguson combines an a priori expected ultimate based on exposure and an assumed loss ratio with the claims reported so far, weighting by the proportion still to emerge. This stabilises reserves for immature accident years where chain-ladder is volatile.

Question 46Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A life insurer models the number of policies in force as a Markov chain with states In Force, Lapsed and Claimed. Which matrix property is required for the transition matrix to be a valid stochastic matrix?

  1. The matrix is symmetric.
  2. The matrix has determinant equal to one.
  3. Each row sums to one and all entries are non-negative. Correct answer.
  4. Each column sums to one and all entries are non-negative.

Explanation

For a Markov chain, each row gives the conditional distribution of the next state given the current state, so entries must be non-negative and each row must sum to one. Column sums equal to one define a different property, double stochasticity.

Question 47Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a survival analysis of a pension scheme, a member's death is observed but the exact date of death is only known to lie within a calendar year. Which term describes this form of incomplete observation?

  1. Left-truncation
  2. Interval-censoring Correct answer.
  3. Immortal time bias
  4. Right-censoring

Explanation

Interval-censoring arises when the event is known to occur within an interval but the exact time is unknown. It differs from right-censoring, where the event occurs after the last observation.

Question 48Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A mortality model uses a Makeham law with force of mortality mu(x) = A + Bc^x. Which statement about this model is correct?

  1. It models only the senescent component of mortality.
  2. It is identical to the Gompertz law for all parameter values.
  3. It combines a constant component with an exponentially increasing senescent component. Correct answer.
  4. It implies a decreasing force of mortality with age.

Explanation

The Makeham law adds a constant term A to the Gompertz term Bc^x, allowing for a background level of mortality independent of age. Setting A to zero recovers the Gompertz law.

Question 49Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

In a GLM for claim severity, the deviance residuals are plotted against fitted values and show a pronounced funnel shape. Which diagnosis is most appropriate?

  1. The variance function is misspecified and a different error distribution may be needed. Correct answer.
  2. The regression coefficients are unbiased but the intercept is wrong.
  3. The link function is correct and the model needs no change.
  4. The model is overparameterised and terms should be removed.

Explanation

A funnel pattern in residual plots indicates that the variance depends on the mean in a way the assumed variance function does not capture. Changing the error distribution, for example from Poisson to gamma, is the usual remedy.

Question 50Actuarial Mathematics - core modelling and risk content (domain breakdown not enumerated in retrieved source)

A survival model is fitted with covariates and the proportional hazards assumption is doubted. Which diagnostic most directly assesses this assumption?

  1. A plot of the Kaplan-Meier estimator for the whole sample.
  2. A histogram of the observed survival times.
  3. A chi-square test of the overall model fit.
  4. A plot of Schoenfeld residuals against time. Correct answer.

Explanation

Schoenfeld residuals should show no systematic trend with time if proportional hazards holds, so plotting them against time reveals violations. A clear trend indicates that a coefficient varies with time.