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

Every Exam ASTAM formula you need on the test, grouped by topic and 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 Free forever
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All Exam ASTAM Formulas

Severity Models 3 items
Limited expected value — Pareto
E[X∧u]=θα−1[1−(θu+θ)α−1],α>1E[X\wedge u]=\dfrac{\theta}{\alpha-1}\left[1-\left(\dfrac{\theta}{u+\theta}\right)^{\alpha-1}\right],\quad\alpha>1
Limited expected value — general formula
E[X∧u]=∫0u[1−F(x)] dxE[X\wedge u]=\int_0^u [1-F(x)]\,dx
(for X≥0X\ge0)
Mean excess loss function
e(d)=E[X−d∣X>d]=E[X]−E[X∧d]S(d)e(d)=E[X-d\mid X>d]=\dfrac{E[X]-E[X\wedge d]}{S(d)}
Aggregate Models 4 items
Compound distribution — mean and variance
E[S]=E[N]⋅E[X]E[S]=E[N]\cdot E[X]
Var(S)=E[N] Var(X)+Var(N) (E[X])2\text{Var}(S)=E[N]\,\text{Var}(X)+\text{Var}(N)\,(E[X])^2
Panjer recursion
fS(x)=11−afX(0)∑y=1x(a+byx)fX(y) fS(x−y)f_S(x)=\dfrac{1}{1-a f_X(0)}\sum_{y=1}^{x}\left(a+\dfrac{by}{x}\right)f_X(y)\,f_S(x-y)
(a,b,0)(a,b,0) class: Poisson (0,λ)(0,\lambda), Binomial (−p/(1−p), (n+1)p/(1−p))(-p/(1-p),\,(n+1)p/(1-p))
Compound Poisson variance
Var(S)=λ E[X2]\text{Var}(S)=\lambda\,E[X^2]
(uses E[N]=Var(N)=λE[N]=\text{Var}(N)=\lambda)
Stop-loss expected value
E[(S−d)+]=E[S]−E[S∧d]E[(S-d)_+]=E[S]-E[S\wedge d]
Coverage Modifications 4 items
Loss elimination ratio
LER(d)=E[X∧d]E[X]LER(d)=\dfrac{E[X\wedge d]}{E[X]}
Fraction of expected loss eliminated by deductible dd
Payment per loss with deductible \(d\) and limit \(u\)
YL=min⁡(max⁡(X−d,0), u)Y^L=\min(\max(X-d,0),\,u)
E[YL]=E[X∧(d+u)]−E[X∧d]E[Y^L]=E[X\wedge(d+u)]-E[X\wedge d]
Payment per payment (excess loss variable)
YP=X−d∣X>dY^P=X-d\mid X>d
E[YP]=e(d)=E[X]−E[X∧d]1−F(d)E[Y^P]=e(d)=\dfrac{E[X]-E[X\wedge d]}{1-F(d)}
Payment per loss with ordinary deductible
YL=(X−d)+=max⁡(X−d,0)Y^L=(X-d)_+=\max(X-d,0)
E[YL]=E[X]−E[X∧d]E[Y^L]=E[X]-E[X\wedge d]
Construction and Selection of Parametric Models 3 items
Kolmogorov-Smirnov test statistic
D=sup⁡x∣Fn(x)−F(x;θ^)∣D=\sup_x|F_n(x)-F(x;\hat{\theta})|
FnF_n=empirical CDF; reject H0H_0 if DD exceeds critical value
MLE — likelihood and log-likelihood
L(θ)=∏i=1nf(xi;θ)L(\theta)=\prod_{i=1}^n f(x_i;\theta)
ℓ(θ)=∑i=1nln⁡f(xi;θ)\ell(\theta)=\sum_{i=1}^n \ln f(x_i;\theta)
Solve dℓdθ=0\dfrac{d\ell}{d\theta}=0
Chi-square goodness-of-fit statistic
χ2=∑j=1k(Oj−Ej)2Ej\chi^2=\sum_{j=1}^k\dfrac{(O_j-E_j)^2}{E_j}
df = cells −- 1 −- estimated parameters
Credibility 3 items
Buhlmann-Straub credibility
Z=mm+k,k=va,m=∑imiZ=\dfrac{m}{m+k},\quad k=\dfrac{v}{a},\quad m=\sum_i m_i
Weighted by exposures mim_i
Buhlmann credibility estimate
μ^=ZXˉ+(1−Z)μ0\hat{\mu}=Z\bar{X}+(1-Z)\mu_0
Z=nn+k,k=vaZ=\dfrac{n}{n+k},\quad k=\dfrac{v}{a}
v=E[σ2(θ)],  a=Var(μ(θ))v=E[\sigma^2(\theta)],\;a=\text{Var}(\mu(\theta))
Empirical Bayes — \(v\) and \(a\) estimates
v^=1r(n−1)∑i=1r∑j=1n(Xij−Xˉi)2\hat{v}=\dfrac{1}{r(n-1)}\sum_{i=1}^r\sum_{j=1}^n(X_{ij}-\bar{X}_i)^2
a^=v^(na^∗−v)…\hat{a}=\dfrac{\hat{v}(n\hat{a}^*-v)}{\ldots} (use unbiased moment estimators from Buhlmann-Straub setup)
Reserving and Pricing for Short-Term Insurance Coverages 4 items
Bornhuetter-Ferguson ultimate loss
U^i=Ci,current+(1−qi)⋅ELR⋅Pi\hat{U}_i=C_{i,\text{current}}+(1-q_i)\cdot ELR\cdot P_i
qiq_i=% reported, PiP_i=premium, ELRELR=expected loss ratio
IBNR reserve
IBNR=U^−CcurrentIBNR = \hat{U} - C_{\text{current}}
(estimated ultimate minus cumulative paid/reported losses)
Loss ratio
LR=Losses IncurredEarned PremiumLR = \dfrac{\text{Losses Incurred}}{\text{Earned Premium}}
Chain-ladder (development) method
C^i,k+1=f^k⋅Ci,k\hat{C}_{i,k+1}=\hat{f}_k\cdot C_{i,k}
f^k=∑iCi,k+1∑iCi,k\hat{f}_k=\dfrac{\sum_i C_{i,k+1}}{\sum_i C_{i,k}} (volume-weighted average development factor)
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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.
What else is free at FreeFellow for Exam ASTAM candidates?
The full original question bank is free with an account, subject to usage limits. Worked solutions, written lessons, mixed practice, and your readiness score stay free. The formula sheet is free too. Fellow is $39 per month or $79 per quarter, per exam family (USD). Fellow Plus is $49 per month, $99 per quarter, or $199 per year, per exam family (USD). Every annual plan is Fellow Plus. Fellow adds timed mock exams, spaced-repetition flashcards, performance analytics, and a personalized study plan. AI grading: 5 attempts a day on Fellow; Fellow Plus removes that allowance, subject to grading rate and usage limits.

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