SOCIETY OF ACTUARIES - EXAM STAM SHORT-TERM ACTUARIAL MATHEMATICS - EXAM STAM SAMPLE QUESTIONS

Q: What are the scope changes between Questions 1-307 and Questions 308-326?

Questions 1-307 are taken from a previous Exam C set, with irrelevant items deleted and topic weights noted as not exam-representative. Questions 308-326 are based on newly added material, with specific correction notes in the April 2018 and December 2018 updates.

Q: How is credibility premium or Bühlmann credibility addressed in these sample questions?

The document includes problems that ask for Bühlmann’s k for aggregate losses and for Bühlmann credibility estimates or credibility premiums using specified claim frequency and severity distributions and priors, such as Poisson/exponential setups and nonparametric empirical Bayes.

Q: What kinds of probability models and estimation methods appear in the questions?

The questions use distributions including Poisson, Pareto, exponential, binomial, and gamma priors, and they ask for maximum likelihood estimates, Bayesian estimates, and posterior probabilities, alongside chi-square goodness-of-fit testing for categorical claim types.

Sep 27, 2026

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Society of Actuaries STAM short-term actuarial mathematics sample questions covering topics such as Poisson, Pareto, exponential and binomial modeling, credibility theory (including Bühlmann k and credibility premium), Bayesian estimation, and maximum likelihood–based asymptotic variance. The questions include multiple-choice problems with specified parameterizations, prior distributions, and hypotheses such as chi-square goodness-of-fit. Updates are listed for April 2018 and December 2018, with deletions, corrections, and added questions noted.

LanguageEnglish

CategoryExam

FormatPDF

Pages167

Reading Time7 h 1 min

AccessFree to download

Questions 1-307

Updates (April 2018)

Updates (December 2018)

Questions 308-326

Subsequent additions (e.g., 327-328)

SOCIETY OF ACTUARIES EXAM STAM SHORT-TERM ACTUARIAL MATHEMATICS EXAM STAM SAMPLE QUESTIONS Questions 1-307 have been taken from the previous set of Exam C sample questions. Questions no longer relevant to the syllabus have been deleted. Questions 308-326 are based on material newly added. April 2018 update: Question 303 has been deleted. Corrections were made to several of the new questions, 308-326. December 2018 update: Corrections were made to questions 322, 323, and 325. Questions 327 and 328 were added. Some of the questions in this study note are taken from past examinations. The weight of topics in these sample questions is not representative of the weight of topics on the exam. The syllabus indicates the exam weights by topic. Copyright 2018 by the Society of Actuaries PRINTED IN U.S.A. 1. DELETED 2. You are given: (i) The number of claims has a Poisson distribution. (ii) Claim sizes have a Pareto distribution with parameters θ = 0.5 and α = 6 (iii) The number of claims and claim sizes are independent. (iv) The observed pure premium should be within 2% of the expected pure premium 90% of the time. Calculate the expected number of claims needed for full credibility. (A) Less than 7,000 (B) At least 7,000, but less than 10,000 (C) At least 10,000, but less than 13,000 (D) At least 13,000, but less than 16,000 (E) At least 16,000 3. DELETED 4. You are given: (i) Losses follow a single-parameter Pareto distribution with density function: f (x) = ~~ ~~xαα+1~~ ~~ , x > 1, 0 < α <∞ (ii) A random sample of size five produced three losses with values 3, 6 and 14, and two losses exceeding 25. Calculate the maximum likelihood estimate of α . (A) 0.25 (B) 0.30 (C) 0.34 (D) 0.38 (E) 0.42 5. You are given: (i) The annual number of claims for a policyholder has a binomial distribution with probability function: 􀂧 2􀂷 2 p (x | q) = 􀂨 􀂸 qx (1− q) −x , x = 0,1, 2 􀂩x 􀂹 (ii) The prior distribution is: π (q) = 4q3 , 0 < q < 1 This policyholder had one claim in each of Years 1 and 2. Calculate the Bayesian estimate of the number of claims in Year 3. (A) Less than 1.1 (B) At least 1.1, but less than 1.3 (C) At least 1.3, but less than 1.5 (D) At least 1.5, but less than 1.7 (E) At least 1.7 6. DELETED 7. DELETED 8. You are given: (i) Claim counts follow a Poisson distribution with mean θ . (ii) Claim sizes follow an exponential distribution with mean 10θ . (iii) Claim counts and claim sizes are independent, given θ . (iv) The prior distribution has probability density function: π (θ) = ~~ ~~5θ6~~ ~~ , θ > 1 Calculate Bühlmann’s k for aggregate losses. (A) Less than 1 (B) At least 1, but less than 2 (C) At least 2, but less than 3 (D) At least 3, but less than 4 (E) At least 4 9. DELETED 10. DELETED 11. You are given: (i) Losses on a company’s insurance policies follow a Pareto distribution with probability density function: f (x | θ) = ~~ ~~(x~~ ~~+θθ)2~~ ~~ , 0 < x <∞ (ii) For half of the company’s policies θ = 1 , while for the other half θ = 3 . For a randomly selected policy, losses in Year 1 were 5. Calculate the posterior probability that losses for this policy in Year 2 will exceed 8. (A) 0.11 (B) 0.15 (C) 0.19 (D) 0.21 (E) 0.27 12. You are given total claims for two policyholders: | | Year | | | | | --- | --- | --- | --- | --- | | Policyholder | 1 | 2 | 3 | 4 | | X | 730 | 800 | 650 | 700 | | Y | 655 | 650 | 625 | 750 | Using the nonparametric empirical Bayes method, calculate the Bühlmann credibility premium for Policyholder Y. (A) 655 (B) 670 (C) 687 (D) 703 (E) 719 13. A particular line of business has three types of claim. The historical probability and the number of claims for each type in the current year are: | Type | Historical
Probability | Number of Claims in Current Year | | --- | --- | --- | | X | 0.2744 | 112 | | Y | 0.3512 | 180 | | Z | 0.3744 | 138 |

FAQs

What are the scope changes between Questions 1-307 and Questions 308-326?

Questions 1-307 are taken from a previous Exam C set, with irrelevant items deleted and topic weights noted as not exam-representative. Questions 308-326 are based on newly added material, with specific correction notes in the April 2018 and December 2018 updates.

How is credibility premium or Bühlmann credibility addressed in these sample questions?

The document includes problems that ask for Bühlmann’s k for aggregate losses and for Bühlmann credibility estimates or credibility premiums using specified claim frequency and severity distributions and priors, such as Poisson/exponential setups and nonparametric empirical Bayes.

What kinds of probability models and estimation methods appear in the questions?

The questions use distributions including Poisson, Pareto, exponential, binomial, and gamma priors, and they ask for maximum likelihood estimates, Bayesian estimates, and posterior probabilities, alongside chi-square goodness-of-fit testing for categorical claim types.

Sep 27, 2026

Upload by Cipher