Free SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) Formula Sheet (2026)

Every Exam ASTAM formula you need on the test, grouped by topic, rendered with full math notation. 21 formulas across 6 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

21 Formulas

6 Topics

2026 Syllabus

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All Exam ASTAM Formulas

Severity Models 3 items

Limited expected value — Pareto

E [X \land u] = \frac{θ}{α - 1} [1 - (\frac{θ}{u + θ})^{α - 1}], α > 1

Limited expected value — general formula

E [X \land u] = \int_{0}^{u} [1 - F (x)] d x

(for

X \geq 0

)

Mean excess loss function

e (d) = E [X - d ∣ X > d] = \frac{E [ X ] - E [ X \land d ]}{S ( d )}

Aggregate Models 4 items

Compound distribution — mean and variance

E [S] = E [N] \cdot E [X]

Var (S) = E [N] Var (X) + Var (N) (E [X])^{2}

Panjer recursion

f_{S} (x) = \frac{1}{1 - a f _{X} ( 0 )} \sum_{y = 1}^{x} (a + \frac{b y}{x}) f_{X} (y) f_{S} (x - y)

(a, b, 0)

class: Poisson

(0, λ)

, Binomial

(- p / (1 - p), (n + 1) p / (1 - p))

Compound Poisson variance

Var (S) = λ E [X^{2}]

(uses

E [N] = Var (N) = λ

)

Stop-loss expected value

E [(S - d)_{+}] = E [S] - E [S \land d]

Coverage Modifications 4 items

Loss elimination ratio

L E R (d) = \frac{E [ X \land d ]}{E [ X ]}

Fraction of expected loss eliminated by deductible

d

Payment per loss with deductible $d$ and limit $u$

Y^{L} = min (max (X - d, 0), u)

E [Y^{L}] = E [X \land (d + u)] - E [X \land d]

Payment per payment (excess loss variable)

Y^{P} = X - d ∣ X > d

E [Y^{P}] = e (d) = \frac{E [ X ] - E [ X \land d ]}{1 - F ( d )}

Payment per loss with ordinary deductible

Y^{L} = (X - d)_{+} = max (X - d, 0)

E [Y^{L}] = E [X] - E [X \land d]

Construction and Selection of Parametric Models 3 items

Kolmogorov-Smirnov test statistic

D = sup_{x} ∣ F_{n} (x) - F (x; θ^) ∣

F_{n}

=empirical CDF; reject

H_{0}

D

exceeds critical value

MLE — likelihood and log-likelihood

L(θ)=∏i=1n​f(xi​;θ)
ℓ(θ)=∑i=1n​lnf(xi​;θ)
Solve dθdℓ​=0

Chi-square goodness-of-fit statistic

χ^{2} = \sum_{j = 1}^{k} \frac{( O _{j} - E _{j} ) ^{2}}{E _{j}}

df = cells

-

-

estimated parameters

Credibility 3 items

Buhlmann-Straub credibility

Z = \frac{m}{m + k}, k = \frac{v}{a}, m = \sum_{i} m_{i}

Weighted by exposures

m_{i}

Buhlmann credibility estimate

μ^​=ZXˉ+(1−Z)μ0​
Z=n+kn​,k=av​
v=E[σ2(θ)],a=Var(μ(θ))

Empirical Bayes — $v$ and $a$ estimates

v^= \frac{1}{r ( n - 1 )} \sum_{i = 1}^{r} \sum_{j = 1}^{n} (X_{ij} - X ˉ_{i})^{2}

a^= \frac{v ^ ( n a ^ ^{*} - v )}{\dots}

(use unbiased moment estimators from Buhlmann-Straub setup)

Reserving and Pricing for Short-Term Insurance Coverages 4 items

Bornhuetter-Ferguson ultimate loss

U^_{i} = C_{i, current} + (1 - q_{i}) \cdot E L R \cdot P_{i}

q_{i}

=% reported,

P_{i}

=premium,

E L R

=expected loss ratio

IBNR reserve

I B N R = U^- C_{current}

(estimated ultimate minus cumulative paid/reported losses)

Loss ratio

L R = \frac{Losses Incurred}{Earned Premium}

Chain-ladder (development) method

C^_{i, k + 1} = f^_{k} \cdot C_{i, k}

f^_{k} = \frac{\sum _{i} C _{i, k + 1}}{\sum _{i} C _{i, k}}

(volume-weighted average development factor)

Frequently Asked Questions

Is the Exam ASTAM formula sheet free?

Yes. The full Exam ASTAM formula sheet is free, with no signup, no email, and no credit card required. 21 formulas across 6 topics, all rendered with the same KaTeX math notation used in the FreeFellow study app.

Can I download the Exam ASTAM formula sheet as a printable PDF?

Yes. A 1080x1350 portrait PDF (Instagram and LinkedIn carousel native size, also great for tablet study) is linked at the top of this page. The PDF is fully self-contained: math is pre-rendered, fonts are embedded, no internet connection needed once downloaded.

What's covered on the Exam ASTAM formula sheet?

Every formula is grouped by official syllabus topic, with the formula in math notation plus a one-line note on when to use it (or a watch-out from CAIA, CFA, or other prep-provider commentary). Coverage is calibrated to the 2026 syllabus and refreshed when the corpus changes.

What is FreeFellow's relationship with SOA?

No. FreeFellow is not affiliated with the SOA or any examination body. This is an independent study aid covering the published syllabus.

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The full question bank with detailed solutions, mixed practice, readiness tracking, lessons (where available), and the formula sheet are all free forever. Fellow ($79/quarter or $199/year per track) unlocks timed mock exams, spaced-repetition flashcards, performance analytics, AI essay grading, and a personalized study plan.

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Free SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) Formula Sheet (2026)

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