SOA FAMFundamentals of Actuarial Mathematics

Associateship (ASA)

3.5 hours (1.75h per half if split)·34 multiple-choice (17 + 17)·≈350 study hours

Syllabus learning objectives

Paraphrased from the SOA syllabus so the study-plan builder and practice sets track the topics you will actually be examined on. Weights are the official topic ranges.

FAM-S: Insurance & reinsurance coverages

Describe common short-term insurance and reinsurance contract types, including quota-share, surplus-share, excess-of-loss and facultative arrangements.
Apply policy limits, deductibles, coinsurance and coverage modifications to loss payments for a single claim.
Determine insurer vs. reinsurer retained losses under proportional and non-proportional treaties.
Identify the effect of policy provisions (aggregate limits, per-occurrence limits) on ceded and retained loss distributions.

FAM-S: Severity, frequency & aggregate models

20%

Compute moments, percentiles and limited expected values for common severity distributions (Pareto, gamma, lognormal, Weibull, exponential).
Apply deductibles, limits, coinsurance and inflation to severity distributions and derive the resulting payment-per-loss and payment-per-payment distributions.
Compute the distribution, mean and variance of aggregate claims via the compound (Poisson, negative binomial, binomial) frequency-severity model.
Use the recursive (Panjer) method and moment-based approximations (normal, lognormal, translated gamma) to approximate aggregate loss distributions.
Adjust frequency distributions for exposure changes and coverage modifications (deductibles affecting claim counts).

FAM-S: Parametric estimation

Fit parametric loss models by the method of moments and maximum likelihood, including for grouped, censored and truncated data.
Construct and interpret the likelihood function for data subject to policy limits and deductibles.
Apply the delta method and observed/expected Fisher information to estimate variances of MLE parameter estimates.
Use graphical and hypothesis-based goodness-of-fit tools (Q-Q plots, chi-square, Kolmogorov-Smirnov) to select among competing models.

FAM-S: Introductory credibility

Apply limited fluctuation (classical) credibility to determine the standard for full credibility and partial credibility factors.
Compute Bühlmann and Bühlmann-Straub credibility premiums from expected value of process variance and variance of hypothetical means.
Estimate credibility parameters (k, Z) from sample data using nonparametric and semiparametric methods.
Interpret credibility-weighted estimates as a blend between individual experience and a manual/collective rate.

FAM-S: Pricing & reserving for short-term insurance

Adjust historical premium to current rate level and trend losses to the future policy period.
Estimate ultimate losses using the chain-ladder, Bornhuetter-Ferguson and expected-loss-ratio reserving methods.
Calculate the indicated overall rate change combining loss ratio and pure premium approaches.
Evaluate the impact of reinsurance and loss development assumptions on indicated reserves and rates.

FAM-S: Option pricing fundamentals

Price European put and call options using put-call parity and the binomial lattice model.
Apply risk-neutral valuation and the Black-Scholes formula to value simple derivatives.
Describe how embedded options in insurance products (guarantees, caps, floors) can be decomposed into vanilla option positions.
Compute the delta of an option position and construct a simple replicating/hedging portfolio.

FAM-L: Long-term insurance coverages

Describe the benefit structure of whole life, term, endowment, and deferred insurance and annuity contracts.
Translate verbal descriptions of long-term coverage provisions into actuarial present value expressions.
Distinguish level, increasing and decreasing benefit patterns and their effect on reserves and premiums.
Identify how riders (e.g., waiver of premium, accidental death) modify the basic contract's cash flows.

FAM-L: Survival models

12%

Define the survival function, force of mortality, and curtate/complete expectation of life, and relate them algebraically.
Compute probabilities and moments of time-to-death and time-to-failure under standard parametric mortality laws (constant force, De Moivre, Gompertz, Makeham) and life tables.
Apply select-and-ultimate mortality tables and fractional-age assumptions (uniform distribution of deaths, constant force) to interpolate probabilities.
Compute probabilities for multiple lives (joint life, last survivor) under independence assumptions.

FAM-L: Present value random variables

10%

Derive the distribution and moments of the present value random variable for insurances and annuities payable continuously, annually or mthly.
Apply the relationships between insurance and annuity APVs (e.g., $A=1-d\ddot a$) to simplify calculations.
Compute actuarial present values under a fully discrete, fully continuous, or semicontinuous model.
Use recursive relationships to compute actuarial present values for benefits deferred or paid at the end/beginning of the year of death.

FAM-L: Premium calculation

10%

Apply the equivalence principle to determine net premiums for insurances and annuities under various benefit and premium timing patterns.
Incorporate expenses (percent of premium, per-policy, per-unit) to compute gross premiums.
Compute the variance of loss-at-issue random variables for premium-paying contracts.
Apply premium refund/return-of-premium and percentile premium principles to alternative premium calculations.

FAM-L: Reserves (policy values)

10%

Compute prospective and retrospective policy values (reserves) for standard insurance and annuity contracts.
Apply the recursive reserve relationship to project reserves forward one year and decompose gains by source.
Compute Fackler's accumulation formula and full/modified preliminary term reserves.
Interpret the relationship between reserves, premiums and expected mortality/interest gains.

FAM-L: Interest-rate risk & pension intro

Describe key duration and convexity measures and apply them to assess a portfolio's exposure to interest-rate movements.
Apply immunization (Redington and full immunization) techniques to match asset and liability cash flows.
Describe the basic structure of defined-benefit and defined-contribution pension plans and their funding objectives.
Compute simple pension actuarial liabilities under a career-average or final-salary benefit formula.

Lecture videos for this exam

Open the full video library →

[MATH 5639 Actuarial Loss Models] Lecture 1: Probability Exercise 1

Bin Z · Loss Models

[MATH 5639 Actuarial Loss Models] Lecture 17: Ch2.5 Deductible

Bin Z · Loss Models

[MATH 5639 Actuarial Loss Models] Lecture 21: Ch3 Individual Risk Model

Bin Z · Loss Models

Introduction to Contingencies- Lecture 1

H&J Online Academy · Life Contingencies

The Life Table (Contingencies: Actuarial Mathematics)

inTuition · Life Contingencies

Life Assurance Contracts Part I (Contingencies: Actuarial Mathematics)

inTuition · Life Contingencies

Whole Life and Temporary Annuities

Mike, the Mathematician · Life Contingencies

Overview

FAM merges the old LTAM and STAM cores. FAM-L covers survival models, insurance and annuity present values, premiums and policy values; FAM-S covers loss models, coverage modifications, estimation, credibility and short-term pricing/reserving. Candidates may sit the two halves separately.

Duration: 3.5 hours (1.75h per half if split)
Questions: 34 multiple-choice (17 + 17)
Style: Computer-based
Passing: Scaled 6 of 10

Syllabus map

FAM-S: Insurance & reinsurance coverages

Reinsurance Structures and Pricing

FAM-S: Severity, frequency & aggregate models

20%

Aggregate Loss Models Probability and Distributions for Insurance

FAM-S: Parametric estimation

Statistics and Estimation for Actuaries

FAM-S: Introductory credibility

Credibility Theory: Limited Fluctuation, Bühlmann, and Bayes

FAM-S: Pricing & reserving for short-term insurance

Ratemaking Fundamentals Reserving Methods: Chain Ladder, Bornhuetter–Ferguson, Benktander, and Cape Cod

FAM-S: Option pricing fundamentals

Interest Theory and Time Value of Money

FAM-L: Long-term insurance coverages

Life Contingencies Primer

FAM-L: Survival models

12%

Life Contingencies Primer

FAM-L: Present value random variables

10%

Life Contingencies Primer

FAM-L: Premium calculation

10%

Life Contingencies Primer

FAM-L: Reserves (policy values)

10%

Life Contingencies Primer

FAM-L: Interest-rate risk & pension intro

Interest Theory and Time Value of Money

Key formulas

Constant force of mortality $\;{}_tp_x=e^{-\mu t}$ , $\bar A_x=\dfrac{\mu}{\mu+\delta}$ , $\bar a_x=\dfrac{1}{\mu+\delta}$

Equivalence-principle net premium (whole life, annual): $\;P=\dfrac{A_x}{\ddot a_x}=\dfrac{1}{\ddot a_x}-d$

Policy value recursion $\;({}_tV+P)(1+i)=q_{x+t}\,b_{t+1}+p_{x+t}\,{}_{t+1}V$

Compound Poisson aggregate $\;E[S]=\lambda E[X]$ , $\operatorname{Var}(S)=\lambda E[X^2]$

Limited expected value, Pareto( $\alpha,\theta$ ): $\;E[X\wedge u]=\dfrac{\theta}{\alpha-1}\left[1-\left(\dfrac{\theta}{u+\theta}\right)^{\alpha-1}\right]$

Bühlmann $\;Z=\dfrac{n}{n+EPV/VHM}$

Study strategy

Learn the FAM tables (Standard Ultimate Life Table, loss-distribution formula sheet) inside out — you get them on the exam, so speed of lookup is the edge.
For FAM-L, derive every insurance/annuity relationship from first principles once, then rely on the $A=1-d\ddot a$ identities.
For FAM-S, memorize the coverage-modification recipe: deductible, limit, coinsurance, inflation, in that order.
Do mixed timed sets; the two halves reward different rhythms.

Common traps

Confusing continuous ( $\bar A$ ), discrete ( $A$ ) and mthly ( $A^{(m)}$ ) benefit timing.
Applying the deductible after the policy limit in a per-loss calculation.
Using $\operatorname{Var}(S)=\lambda\operatorname{Var}(X)$ instead of $\lambda E[X^2]$ for compound Poisson.
Forgetting that per-payment quantities condition on the loss exceeding the deductible.

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