SOA Fundamentals of Actuarial Mathematics (FAM) Practice Test

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• Which statement best distinguishes discrete from continuous varying life insurance valuation?
• In the Black-Scholes framework, d2 is defined in relation to d1. Which expression correctly defines d2?
• If nEx = exp(-(mu+delta)n), what is nEx when n is increased by 1?
• How are censored observations treated in Kaplan-Meier or Nelson-Aalen estimators of survival?
• Structured settlements use two approaches to determine annuity payments. Which statement correctly identifies which approach is more suitable?
• If S(t) = exp(-H(t)), what is log S(t) in terms of H(t)?
• Which expression represents the base of the original expression before applying the exponent 20?
• In a surplus share treaty reinsurance that covers Y% in excess of X, who pays what?
• Which ADLs are listed as examples for LTC eligibility?
• Reserves are added to which quantity when determining ultimate losses, and why?
• If X has a transformed distribution Y = g(X) and you need F_Y(y) without seeing the source material, which method is generally appropriate?
• Which property makes the exponential distribution unique among continuous distributions?
• In the Delta of a replicating portfolio, which formula correctly defines Delta?
• How is the expense policy value defined in terms of EPV?
• The sum of n independent Geometric random variables is a geometric distribution with alpha equal to which value?
• In life tables, what does the notation _n p_x represent?
• When faced with an f(x), F(x), or S(x) that looks difficult to integrate, what is a recommended approach?
• If the sample mean equals the sample variance, which distribution is most consistent with the data?
• Incurred losses for a calendar year are defined as which of the following?
• In the Credibility Premium Pc formula, what does M denote?
• If the survival function S(t) tends to zero as t grows without bound, which statement is true under almost sure death assumption?
• How many ADLs must a policyholder be unable to perform to receive LTC benefits? and what are they?
• The continuous Ax integral value is commonly expressed by which integral?
• In the variance formula, the first bit is squared. Which option expresses this correctly?
• In discrete life annuities, Ax equals which expression?
• The Bornhuetter-Ferguson reserving method uses R = Premium earned × Loss ratio ultimate losses × (1 - 1/Chain-ladder age to ultimate factor). Which option expresses this?
• Under the same framework, what is a_x?
• When the death benefit equals the policy value, which technique helps compute the premium P?
• Which expression represents the variance of the loss variable given by (b + P/δ)^2 (2Abar_x - Abar_x^2)?
• When there is a deductible in MLE, L(x) should be computed as which expression?
• The delta of a put option is which of the following expressions?
• Replicating portfolio (B) equals which expression?
• Discrete varying insurance: for increasing whole life in discrete time, the present value is given by a sum. If the policy is term with term length n, what is the upper limit of the summation?
• Which of the following is covered by inland marine insurance?
• In MLE notes, when you estimate a parameter and then recalculate under a new condition, the original parameter is probably equal to:
• In the present value expression for a life policy with constant μ and δ, delta represents which concept?
• In life-table notation, p_x denotes what?
• In the bottom-up approach for determining structured settlement payments, what is the initial step?
• Let x = 1 and blah = 2. What is the value of (1 - x/(x+blah))^20?
• Under i = 0.05, the conversion of an m-period annuity due to an annual annuity due under the equivalence principle is given by which expression for a non-term annuity?
• Which statement best describes semicontinuous insurance?
• The variance of the loss variable L, where L = benefits + expenses - premiums, can be expressed as which of the following forms (in terms of B, F, V, P, d, Ax, and V)?
• For a whole life policy with constant force of mortality μ and constant force of interest δ, the present value is given by which integral?
• In SULT, the identity t+hEx = tEx * hEx+t holds. Which option expresses this correctly?
• Loss Elimination Ratio (LER) is defined as which ratio?
• Parallelogram method: which sequence describes the pricing steps?
• Which statement expresses Translation Invariance as defined among the four coherence rules?
• In the per-loss super with deductible d, policy limit u, and coinsurance alpha, m = (u/alpha) + d. The maximum covered loss m is given by which expression?
• Which statement correctly describes the domain of the expression (blah/(x+blah))^20?
• The Bernoulli shortcut for variance applies when the distribution has which characteristics?
• Under a constant force of mortality with force mu and force of interest delta, what is A_x?
• Let IA denote the present value of an increasing annuity and DA denote that of a decreasing annuity, both for n payments. Which equation expresses IA + DA as a multiple of a level annuity with n payments?
• VaR at level p% corresponds to which percentile of the underlying loss distribution?
• Bernoulli distribution is a special case of the Binomial distribution when which parameter equals 1?
• In binomial MLE, which constraint must hold for the number of trials m given observed data?
• Replicating portfolio (Delta) equals which expression?
• The density function f_x(t) for a life aged x in continuous time is related to tpx and the force of mortality μ_{x+t}. Which expression correctly represents f_x(t)?
• Under full credibility, how are nc and ne related?
• In discrete annuities, which probability appears in place of k|qx in the formulation for varying insurance?
• The ultimate rate change derived from pricing is given by ELR/PLR - 1. If PLR is given, which expression yields the required rate change?
• In the top-down approach for structured settlements, what is determined first?
• In continuous life annuities, Ax equals which expression?
• Incurred losses for Accident year or plan year (Pricing and reserving) consist of which components?
• What is the Black-Scholes price of a European call option with underlying price S_t, strike K, risk-free rate r, volatility sigma, and time to maturity T-t?
• Using the Uniform Distribution of Deaths (UDD) method, the continuous equivalent premium rate contAx is computed as which formula?
• Uniform(0,1) is a special case of which distribution, when the first two shape parameters are equal to 1?
• If the number of independent policies doubles from n to 2n, by what factor does the variance of the total benefit increase?
• The survival function S(t) is interpreted as the probability of which event?
• In a policy with four coverages, which item is covered under third party bodily injury and property when at fault in a tort jurisdiction?
• If policy value at time t is part of the death benefit at t and equivalence is not explicitly stated, which formula is commonly used to relate benefits and policy values?
• For annual policies issued uniformly throughout a year, where is the effective midpoint relative to the policy start?
• According to SULT, combining increasing continuous term with decreasing continuous term yields which type of term?
• Using the annuity method, the reserve for year n is given by which expression?
• Ratemaking: Loss Ratio Method. Indicated Average Rate Change equals?
• Variance of the aggregate when the A_j are independent and each is scaled by 1000, i.e., 1000*A_j, is which expression?
• In the Bornhuetter-Ferguson method, the chain-ladder age to ultimate factor represents which concept?
• In AB0 notation, the ratio pn/pn-1 is modeled as
• In a binomial risk-neutral stock model, the probability of a downward move is given by which expression?
• The sum of independent NegativeBinomial variables with the same beta parameter is distributed as which?
• Discrete varying insurance: decreasing term uses the same summation as increasing term but with what modification?
• When the death benefit is equal to the policy value, what technique can help determine the premium P?
• Given nEx = exp(-(mu+delta)n), which expression equals the term insurance present value?
• If the observed mean exceeds the observed variance, the data are most consistent with which distribution?
• For a level premium with rating factors such as age and risk class, how is the premium adjusted?
• According to SULT, what is the result of combining increasing term with decreasing term?
• Special distribution shortcuts if there's a deductible: the post-deductible distribution for a Pareto claim is which of the following?
• In the notation tqx, the expression t q_x equals which of the following in terms of F_x(t)?
• The variance of a whole life annuity is given by which expression?
• If X ~ Normal(mu, sigma^2) and Y = cX, what is the distribution of Y?
• Sum of independent Binomial variables with the same success probability q is distributed as which?
• The second moment relationship between continuous and discrete life insurance includes a factor (i^2 + 2i) / (2 delta). Which statement is correct?
• P(A|B) is equal to which expression?
• Which expression gives the full credibility for claims nc?
• The variance of the total present value of n independent whole life policies with benefit b is given by which expression?
• Variable & Fixed Expense Ratio is defined as which of the following?
• For per-loss super with deductible d, policy limit u, coinsurance alpha, and inflation rate r, the per-loss expected payment equals which expression? The formula uses m = (u/alpha) + d.
• If X ~ Lognormal(mu, sigma^2) and Y = cX, what is the distribution of Y?
• Which expression is a valid simplification of (1 - x/(x+blah))^20?
• In the term Ax + nEx · Ax+n, what does the symbol n represent?
• In the Replicating portfolio B, which expression is correct for B in the binomial model?
• Which expression correctly defines the cumulative hazard function H(x) in terms of the hazard rate h(t)?
• In discrete summations for varying insurance, what does the term 'kurtate' signify about the indexing of years?
• In Woolhouse approximation, for a two-term approximation, which statement is correct?
• Are waiting periods common for LTC contracts? how many days if so?
• Geometric distribution is a special case of the Negative Binomial distribution when the first parameter alpha equals 1. What value does alpha take?
• Which expression correctly represents the On-Level Factor?
• Pure Premium is defined as the expected loss per exposure. Which option reflects this?
• In Black-Scholes, what does volatility represent?
• What does a standard fire policy cover?
• The force of mortality at age x+t, μ_{x+t}, can be expressed in several equivalent forms. Which expression is a correct representation of μ_{x+t}?
• Which coverage pays medical payments (in tort jurisdictions) or personal injury protection (no-fault jurisdictions)?
• In the per-loss formula, the term E[X ∧ m/(1+r)] caps X at what amount?
• Permissible Loss Ratio is defined as which expression?
• Which statement about the base of the simplified expression is true?
• In survival analysis, the survival function S(x) is related to the cumulative hazard function H(x) by which expression?
• Which expression represents the expected payment per payment in terms of the expected payment per loss and S(d)?
• Which formula correctly expresses Var[S] for an aggregate model with N and X?
• For the log-transformed confidence interval for H(t), Us depends on which quantity(s)?
• If the coinsurance factor applied to a loss exceeds 1, what is the resulting payment?
• With a franchise deductible D and deductible amount d, the expected value of Y under a super policy is E[X] - E[min(X, D)] + d*S(d).
• The exponential distribution is a special case of the gamma distribution with which shape parameter equal to 1?
• In survival analysis, S(t) equals which expression in terms of H(t)?
• Which action is recommended when solving a problem with dates on a timeline, according to problem-solving tips?
• The mean excess loss function e(d) is defined as which of the following?
• Regarding Value at Risk and Tail Value at Risk, which statement is true?
• Which statement about the Interim Reserve Exact method is true when s = 1?
• In Interim Reserve Exact method, the equation (Vt + P) (1 + i)^s = sqx+t v^(1-s) + spx+t t+sV is the general relationship between which quantities when time t+s is considered? Which statement is true about s?
• Pure premium method of ratemaking uses which expression?
• Under inland marine insurance, which item is typically considered an instrumentality of transportation and communication?
• Which statement correctly contrasts discrete and continuous varying life insurance valuations?
• In life contingencies notation, what concept does S_x denote conceptually?
• If X is Pareto distributed and Y = ln(1 + X/theta), what distribution does Y have?
• The expected payment amount beyond the deductible, conditional on a payment occurring, is which of the following?
• After applying a deductible d to a Uniform(a,b) claim with d > a, the remaining claim amount is distributed as which?
• Which of the following expresses the recursive relationship for ex using p_x and e_{x+1}?
• Which of the following is an equivalent expression to (1 - x/(x+blah))^20?
• Which of the following is equivalent to the base of the simplified expression?
• In right censoring at u, which survival function value is used in likelihood?
• The distribution of a sum of independent normal variables is which of the following?
• What is the simplified form of the expression (1 - x/(x+blah))^20, assuming x+blah ≠ 0?
• In UD D, the m-ly n-year term annuity formula is: m-ly n year term annuity = alpha(m) * annual version - beta(m)*( 1 - nEx). Which option matches this expression?
• The conditional density f_{X|j<X<k}(x) for j < X < k is given by which expression?
• In Kaplan-Meier estimation, the survival probability up to time T is given by which product?
• For the second moment of Ax in the continuous case, which formula is correct?
• Under 'Auto coverage: Other than collision,' which item is not covered?
• What does a fully discrete 15-pay whole life insurance policy of 200,000 on a 50-year-old mean?
• Which practice is recommended when deriving the value of insurance or annuities in terms of other annuities?
• Under the SULT framework, which expression correctly represents tEx?
• In zero-truncated distributions, the normalization factor is 1/(1-p0). Which statement best describes its effect?
• The present value, at interest rate i, of an n-payment annuity immediate of 1 per period is commonly expressed as which of the following in terms of v and d?
• In probability with grouped data, which expression is a correct form for P(a < X <= b) using survival or distribution functions?
• Loss ratio method of ratemaking uses which expression for price calculation?
• If the observed mean is less than the observed variance, the data are most consistent with which distribution?
• Which coverage component in standard automobile insurance is designed to cover injuries to other people in an accident (third-party bodily injury)?
• Which formula represents the better term insurance and annuity expression given as Ax + nEx * Ax+n?
• How is the current cumulative rate level calculated from rate changes r_i?
• Under auto coverage 'Other than collision,' which perils are typically covered?
• In UD D, the m-ly whole life annuity is represented as which expression?
• In endowment-type benefits, the final benefit equals which of the following?
• Which coverage provides protection against damage caused by Unidentified, Uninsured, underinsured motorists?
• Put-call parity equation: Which expression holds?
• In a one-period binomial model with up factor u, down factor d, and risk-free rate r over a period h, the replicating portfolio that reproduces payoffs V_u and V_d has initial value Delta*S0 + B, where Delta = (V_u - V_d)/(S_u - S_d) and B = e^{-r h}*(u V_d - d V_u)/(u - d). Which option correctly specifies Delta and B?
• Which statement about Incurred But Not Paid (IBNP) is consistent with the notes?
• Which expression represents f(x) in terms of other quantities?
• In a discrete-time framework, what does v represent in life insurance valuations?
• Which statement justifies the simplification to blah/(x+blah)?
• In UD D method, a m-ly n-year term annuity is given by which expression?
• Which statement about the base of (blah/(x+blah))^20 is true?
• For continuous varying increasing whole life, the present value is given by the integral ∫_0^∞ t v^t tpx μ_{x+t} dt. For a term policy with finite term n, what is the corresponding integral?
• What condition ensures the sum of gamma variables is still a gamma distribution?
• Rate Making Pure Premium Method. The Indicated Average Rate Change is based on what denominator?
• Which of the following correctly defines TVaRp(x) in terms of X and VaR_p?
• The distribution of the sum of independent Poisson variables is which of the following?
• In the life contingencies framework, what does μ denote?
• Evaluate the expression for x = 2 and blah = 3: (1 - x/(x+blah))^20.
• What is the purpose of the on-level factor in the Parallelogram method?
• In homeowners insurance with coinsurance, if the coinsurance requirement isn't met, the claim payment is given by which formula?
• Which statement correctly describes the sum of gamma variables when the scale parameters differ?
• In zero-modified Pn, E[N] Modified equals which expression?
• Suppose a deductible d is chosen so that p% of X are below d. Then TVaR_p(X) equals:
• If Y = cX and X ~ Normal(mu, sigma^2), which statement describes the distribution of Y?
• In the Parallelogram method, which step corresponds to calculating the premium portion in each square for rate levels?
• In life contingencies notation, tpx denotes the probability that a life aged x survives t years. Which statement best describes tpx?
• In Woolhouse approximations, ax:n equals which of the following relationships?
• In the binomial model, the symbols u and d represent which concept?
• In actuarial notation, what does u|t q_x represent?
• The hazard function h(x) in terms of the density f(x) and the survival function S(x) is given by which expression?
• In the Nelson-Aalen estimator, the cumulative hazard H(t) is estimated as the sum at each event time of which term?
• Renewable term insurance allows you to purchase another term at the end without proving your health status.
• As time t increases, the cumulative hazard H(t) is expected to be which?
• How is the average cumulative rate level defined?
• What is Var(S(t)) in terms of S(t) and Var(H(t))?
• Which statement about the exponent 20 is true?
• Unearned premium for CY is computed as which?
• Which expression gives nyearterm, the n-year term annuity under constant force of mortality?
• In a log-transformed confidence interval for S(t), the endpoints are given by which expression?
• Sum of gamma variables with a common scale parameter theta has distribution Gamma(sum alpha, theta) under which condition?
• If X is Uniform(a,b), what is Var(X)?
• If policies start on April 1, the midpoint is not July; which date represents the midpoint?
• For a distribution with probability p0 of zero, the zero-truncated probability for k > 0 is Pn_truncated(k) = Pn(k) / (1 - p0). Which describes this relationship for k > 0?
• VaRp is the tail value at risk at level p. Which description is correct?
• Under which condition is the original expression defined for real numbers?
• If x = 0 and blah ≠ 0, what is the value of the expression?
• With left truncation at d, the likelihood contribution for an observed value x > d is proportional to
• Gompertz law, compared to Makeham's law, has which characteristic regarding A terms in the mortality functions?
• How many characteristics are generally cited as making a risk insurable?
• Which of the following is the credibility exposure formula ne?