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∧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)

Frequently Asked Questions

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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.
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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.
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