Free SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) Credibility Practice Questions

Credibility theory on SOA Exam ASTAM covers Buhlmann and Buhlmann-Straub models, empirical Bayes estimation, and the mathematical connection between credibility weighting and linear regression.

145 Questions
44 Easy
63 Medium
38 Hard
2026 Syllabus

Sample Questions

Question 1 Easy
In the context of Bayesian credibility, the prior distribution π(θ)\pi(\theta) represents:
Solution
B is correct.

In the Bayesian credibility (greatest accuracy) framework, the prior π(θ)\pi(\theta) represents the distribution of the risk parameter Θ\Theta across the heterogeneous population of risks. It captures how different risks are before any data on a specific risk is observed. When we select a risk at random, Θ\Theta is drawn from this prior distribution.
Question 2 Medium
The semiparametric empirical Bayes approach to credibility differs from the nonparametric approach in which key way?
Solution
C is correct.

In the semiparametric empirical Bayes approach, the conditional distribution of losses given the risk parameter, f(x∣θ)f(x \mid \theta), is assumed to belong to a parametric family (e.g., Poisson, normal). This allows the structural parameters v=E[σ2(Θ)]v = E[\sigma^2(\Theta)] and a=Var[μ(Θ)]a = \text{Var}[\mu(\Theta)] to be estimated using the known functional form of σ2(θ)\sigma^2(\theta) and μ(θ)\mu(\theta). The prior π(θ)\pi(\theta) remains completely unspecified (nonparametric component).
Question 3 Hard
The Greatest Accuracy Credibility (Buhlmann) estimate minimizes the mean squared error E[(ZXˉ+(1−Z)μ−μ(Θ))2]E[(Z\bar{X}+(1-Z)\mu-\mu(\Theta))^2] over all linear estimates of the form c0+c1Xˉc_0 + c_1\bar{X}. Derive the optimal ZZ and explain why it equals n/(n+k)n/(n+k).
Solution
E is correct.

For a fixed risk Θ\Theta, the estimator μ^=ZXˉ+(1−Z)μ\hat{\mu} = Z\bar{X}+(1-Z)\mu has MSE
E[(ZXˉ+(1−Z)μ−μ(Θ))2].E\left[(Z\bar{X}+(1-Z)\mu-\mu(\Theta))^2\right].
Conditioning on Θ\Theta, E[Xˉ∣Θ]=μ(Θ)E[\bar{X}|\Theta] = \mu(\Theta) and Var(Xˉ∣Θ)=σ2(Θ)/n\text{Var}(\bar{X}|\Theta) = \sigma^2(\Theta)/n. The overall MSE decomposes as
MSE(Z)=Z2⋅vn+(1−Z)2⋅a\text{MSE}(Z) = Z^2 \cdot \frac{v}{n} + (1-Z)^2 \cdot a
where v=E[σ2(Θ)]v = E[\sigma^2(\Theta)] and a=Var[μ(Θ)]a = \text{Var}[\mu(\Theta)]. Differentiating:
ddZMSE=2Zvn−2(1−Z)a=0  ⟹  Zvn=(1−Z)a\frac{d}{dZ}\text{MSE} = 2Z\frac{v}{n} - 2(1-Z)a = 0 \implies Z\frac{v}{n} = (1-Z)a
Z(vn+a)=a  ⟹  Z∗=aa+v/n=nana+v=nn+v/a=nn+k.Z\left(\frac{v}{n}+a\right) = a \implies Z^* = \frac{a}{a+v/n} = \frac{na}{na+v} = \frac{n}{n+v/a} = \frac{n}{n+k}.
Create a Free Account to Access All 145 Questions →

About FreeFellow

FreeFellow is an AI-native exam prep platform for actuarial (SOA & CAS), CFA, CFP, CPA, CAIA, GARP FRM, IRS Enrolled Agent, IMA CMA, and FINRA / NASAA securities licensing candidates — built around modern AI as a core capability rather than as a bolt-on. Every lesson ships with AI-narrated audio. Every constructed-response item has a copy-to-AI prompt builder so candidates can paste their answer into their own ChatGPT or Claude for self-graded feedback. Fellow members get instant AI grading on essays against the official rubric (currently CFA Level III, expanding to other essay-bearing sections).

The 70% you need to pass — question bank, written solutions, lessons, formula sheet, mixed practice, readiness tracking — is free forever, with no trial period and no credit card. Become a Fellow ($59/quarter or $149/year per track) to unlock mock exams, flashcards with spaced repetition, performance analytics, AI essay grading, and a personalized study plan.