Free SOA Exam FAM (Fundamentals of Actuarial Mathematics) Formula Sheet (2026)

Every Exam FAM formula you need on the test, grouped by topic, rendered with full math notation. 23 formulas across 9 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

23 Formulas
9 Topics
2026 Syllabus
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All Exam FAM Formulas

Severity, Frequency, and Aggregate Models 7 items
Compound distribution — mean
E[S]=E[N]⋅E[X]E[S]=E[N]\cdot E[X]
S=X1+⋯+XNS=X_1+\cdots+X_N, NN=frequency, XiX_i=severity (iid, indep of NN)
Compound distribution — variance
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
Compound Poisson — variance
Var(S)=λ E[X2]\text{Var}(S)=\lambda\,E[X^2]
(since E[N]=Var(N)=λE[N]=\text{Var}(N)=\lambda)
Stop-loss expected payment
E[(S−d)+]=E[S]−E[S∧d]E[(S-d)_+]=E[S]-E[S\wedge d]
dd=aggregate retention
Panjer recursion
fS(x)=11−a fX(0)∑y=1x(a+b yx)fX(y) fS(x−y)f_S(x)=\dfrac{1}{1-a\,f_X(0)}\sum_{y=1}^{x}\left(a+\dfrac{b\,y}{x}\right)f_X(y)\,f_S(x-y)
Applies to (a,b,0)(a,b,0) frequency class
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
Mean excess loss — Pareto
e(d)=d+θα−1e(d)=\dfrac{d+\theta}{\alpha-1}
Pareto: excess loss variable is also Pareto
Parametric Estimation 1 item
MLE for exponential (\(\theta\) parameterization)
θ^=xˉ=1n∑i=1nxi\hat{\theta}=\bar{x}=\dfrac{1}{n}\sum_{i=1}^n x_i
(MLE=sample mean for exponential mean parameter)
Introduction to Credibility 2 items
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))
Buhlmann-Straub credibility estimate
Z=mm+k,k=vaZ=\dfrac{m}{m+k},\quad k=\dfrac{v}{a}
where m=∑imim=\sum_i m_i (total exposure)
Pricing and Reserving for Short-Term Insurance Coverages 1 item
Bornhuetter-Ferguson ultimate loss
U=Actual Dev.+Expected UnreportedU = \text{Actual Dev.} + \text{Expected Unreported}
=C+(1−q) ELR⋅Premium= C + (1-q)\,ELR\cdot Premium
qq=reported fraction, ELRELR=expected loss ratio
Option Pricing Fundamentals 3 items
Black-Scholes call price
C=S0N(d1)−Ke−rTN(d2)C=S_0 N(d_1)-Ke^{-rT}N(d_2)
d1=ln⁡(S0/K)+(r+σ2/2)TσT,d2=d1−σTd_1=\dfrac{\ln(S_0/K)+(r+\sigma^2/2)T}{\sigma\sqrt{T}},\quad d_2=d_1-\sigma\sqrt{T}
Black-Scholes put price
P=Ke−rTN(−d2)−S0N(−d1)P=Ke^{-rT}N(-d_2)-S_0 N(-d_1)
Same d1,d2d_1,d_2 as call formula
Put-call parity
C−P=S0−Ke−rTC - P = S_0 - Ke^{-rT}
CC=call, PP=put, same KK and TT
Long-Term Insurance Coverages and Retirement Financial Security Programs 2 items
Curtate future lifetime — mean
ex=∑k=1∞kpxe_x = \sum_{k=1}^{\infty} {}_k p_x
kpx=P(Tx>k){}_k p_x=P(T_x>k)
Complete future lifetime — mean
e˙x=∫0∞tpx dt\dot{e}_x = \int_0^{\infty} {}_t p_x\,dt
Mortality Models 1 item
Survival function from force of mortality
tpx=exp⁡ ⁣(−∫0tμx+s ds){}_t p_x = \exp\!\left(-\int_0^t \mu_{x+s}\,ds\right)
Present Value Random Variables for Long-Term Insurance Coverages 3 items
Whole life insurance APV (discrete)
Ax=∑k=0∞vk+1 kpx qx+kA_x = \sum_{k=0}^{\infty} v^{k+1}\,{}_k p_x\,q_{x+k}
Variance of whole life insurance
Var(Z)=2 ⁣Ax−(Ax)2\text{Var}(Z)={}^2\!A_x-(A_x)^2
2 ⁣Ax{}^2\!A_x evaluated at v∗=v2v^* = v^2 (i.e., rate i∗=(1+i)2−1i^*=(1+i)^2-1)
Term insurance APV (discrete, \(n\)-year)
Ax:n‾∣1=∑k=0n−1vk+1 kpx qx+kA^1_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^{k+1}\,{}_k p_x\,q_{x+k}
Premium and Policy Value Calculation for Long-Term Insurance Coverages 3 items
Insurance-annuity relationship
Ax=1−d a¨xA_x = 1 - d\,\ddot{a}_x
Aˉx=1−δ aˉx\bar{A}_x=1-\delta\,\bar{a}_x (continuous)
Net premium (benefit premium)
P=Axa¨xP=\dfrac{A_x}{\ddot{a}_x}
(equivalence principle: APV premiums = APV benefits)
Prospective policy value
tV=Ax+t−P a¨x+t{}_t V = A_{x+t} - P\,\ddot{a}_{x+t}
PV future benefits minus PV future premiums

Frequently Asked Questions

Is the Exam FAM formula sheet free?
Yes. The full Exam FAM formula sheet is free, with no signup, no email, and no credit card required. 23 formulas across 9 topics, all rendered with the same KaTeX math notation used in the FreeFellow study app.
Can I download the Exam FAM 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 FAM 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.
Is this formula sheet affiliated 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 FAM candidates?
The full question bank with detailed solutions, mixed practice, readiness tracking, lessons (where available), and the formula sheet are all free forever. Fellow ($59/quarter or $149/year per track) unlocks timed mock exams, spaced-repetition flashcards, performance analytics, AI essay grading, and a personalized study plan.
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