Concepts
The concept atlas is the public map of ActuaryPath: probability foundations, distribution models, reserving, credibility, finance, life contingencies, and practice-linked actuarial modeling topics.
60 live concept pages organized by exam use.
Use the atlas as the public internal-link spine: prelim probability, distribution models, statistics, reserving, credibility, life contingencies, and property-casualty methods.
Probability and distributions
29 pagesBayes Theorem
Bayes theorem updates a prior probability after a signal arrives. In actuarial work, it explains why base rates matter in underwriting flags, claim triage, credibility, and classification models.
Binomial Distribution
The binomial distribution counts successes in a fixed number of independent Bernoulli trials. It sits inside the (a, b, 0) frequency class and is the right model whenever the trial count is fixed and the trial probability is constant.
Central Limit Theorem
The Central Limit Theorem explains why sums and averages of many iid random variables often behave approximately normally. On Exam P, it is the main approximation tool for large aggregated random quantities.
Compound Poisson and Tweedie
An aggregate loss S = X_1 + ... + X_N with N a Poisson frequency and the X_i independent severities is a compound Poisson. When the severities are gamma, S has a Tweedie distribution with a positive mass at zero and a continuous mass on the positives, which makes it the standard pure-premium GLM target.
Compound Poisson Distribution
A compound Poisson model adds independent claim severities over a Poisson claim count. It is the basic aggregate-loss model behind FAM, ASTAM, Panjer recursion, and many ruin-theory examples.
Conditional Probability
Conditional probability changes the denominator to the information you are given. It is the language behind underwriting signals, claim flags, diagnostic tests, and Bayes theorem.
Correlation
Correlation rescales covariance so you can read direction and strength without being trapped by the original units of the variables. On Exam P, it is the cleaner way to compare dependence across different settings.
Covariance
Covariance measures whether two random variables tend to move together or in opposite directions. On Exam P, it matters because dependence changes the variance of sums and linear combinations.
De Moivre-Laplace and Continuity Correction
The De Moivre-Laplace theorem is the binomial-to-normal special case of the central limit theorem. It is also the place where Exam P candidates first encounter the continuity correction, the half-integer shift that recovers about two decimal places of accuracy in normal approximations to discrete distributions.
Distributions Atlas: Actuarial View
Actuarial work uses a small fixed set of distributions, used in two roles: frequency models for counts and severity models for amounts. The atlas organizes them by role and by tail weight so the right model is easy to pick before any algebra starts.
Expected Value
Expected value is the probability-weighted center of a random variable. For actuaries, it is the first version of expected claim cost, expected payment, and model output.
Exponential Distribution
The exponential distribution models waiting time to the next event in a Poisson process. It is the cleanest entry point into survival functions, hazard rates, and memorylessness.
Gamma Distribution
The gamma distribution is the default positive-continuous severity model on FAM and ASTAM. It extends the exponential to a two-parameter family, supplies the mixing distribution that turns Poisson into negative binomial, and supplies the conjugate prior for the Poisson rate.
Geometric and Negative Binomial Distributions
The geometric distribution counts failures before the first success; the negative binomial counts failures before the r-th success. The SOA Loss Models parameterization (r, β) is convenient for the gamma-mixed-Poisson identity, which is the cleanest route into overdispersed count models.
Joint Distributions
A joint distribution describes two random variables together. Once you understand the joint object, marginals, conditionals, moments, and dependence questions all become controlled rewrites of the same information.
Law of Large Numbers
The law of large numbers says the sample mean of many iid observations settles at the population mean. It is the formal reason credibility theory works, why empirical claim-rate estimates make sense, and why CLT statements about averages have something to converge to in the first place.
Law of Total Probability
The law of total probability calculates an overall probability by splitting the population into clean groups, calculating the probability within each group, and weighting by group size.
Limited Expected Value and Loss Caps
Limited expected value is the expected capped loss. It is the notation bridge from FAM severity models into ASTAM deductibles, policy limits, increased limits factors, stop-loss premiums, and severity discretization.
Linear Combinations Of Random Variables
Linear combinations are where multivariate probability becomes practical. Exam P uses them to test whether you can aggregate means, handle covariance terms correctly, and recognize when a normal or CLT approximation becomes available.
Lognormal Distribution
The lognormal distribution is the distribution of exp(Y) when Y is normal. It is the default heavy-tailed severity model on FAM and ASTAM and inherits closed-form MLEs from the normal distribution by taking logs of the data.
Loss Distribution Formula Atlas
FAM and ASTAM use a small distribution library. The fastest way to make it usable is to separate frequency formulas from severity formulas, keep the SOA scale conventions visible, and attach each formula to the actuarial question it answers.
Normal Distribution
The normal distribution is the main approximation tool on Exam P: it turns means and variances into z-scores and helps model sums, averages, and bell-shaped uncertainty.
Order Statistics
Order statistics are the sorted values from a sample. On Exam P, the highest-yield cases are the minimum and maximum, because they force you to reason carefully about joint events and complements.
Pareto Distribution
The SOA Pareto is a two-parameter heavy-tail severity model with shape α and scale θ. Its mean exists only for α > 1 and its variance only for α > 2, which is exactly the tail-weight signal that makes Pareto useful for excess-of-loss layers and reinsurance pricing.
Poisson Approximation to Binomial
When n is large and p is small, the binomial Bin(n,p) is very close to the Poisson with rate lambda = np. This is the right approximation for rare-event claim counts where the normal approximation would be miscalibrated, and the Le Cam bound says exactly how close.
Poisson Distribution
The Poisson distribution models event counts over a fixed exposure when events occur at a stable average rate. It is the first actuarial model for claim frequency, arrivals, and rare-event approximations.
Probability Generating Functions
A probability generating function packages a nonnegative-integer distribution into one function. In actuarial aggregate models, PGFs explain why compound distributions compose cleanly and why Poisson, Binomial, and Negative Binomial fit the Panjer framework.
Variance
Variance measures how far a random outcome tends to sit from its mean. For actuaries, it is the first step from expected cost to volatility, capital, and aggregate risk.
Weibull Distribution
The Weibull distribution generalizes the exponential by adding a shape parameter that controls hazard-rate behavior. Decreasing hazard (early-claim concentration), constant hazard (exponential), and increasing hazard (aging-style risks) all fit naturally inside a single Weibull family.
Statistics and model selection
4 pagesChi-Squared Goodness-of-Fit Test
The Pearson chi-squared goodness-of-fit test compares observed counts in grouped cells against counts predicted by a fitted distribution. It is the simplest test to set up, the easiest to misuse on small expected cells, and the test whose degrees of freedom must be adjusted for estimated parameters.
Kolmogorov-Smirnov and Anderson-Darling Tests
Kolmogorov-Smirnov measures the largest gap between empirical and fitted CDFs. Anderson-Darling weights the gap by an inverse-variance factor that emphasizes the tails. The two tests can reach different conclusions on the same data, and for actuarial severity fits that disagreement is informative.
Maximum Likelihood Estimation
Maximum likelihood estimation chooses the parameter values that make the observed data look most plausible under a model. It is one of the main bridges from actuarial exam statistics into modern inference and machine learning.
Model Selection: LRT, AIC, and BIC
Use the likelihood ratio test when one model is nested inside another. Use AIC or BIC when comparing models that are not nested. The three criteria can disagree, and understanding when they disagree is the actuarial point.
ASTAM and short-term risk
10 pagesAggregate Loss Models and the Panjer Recursion
An aggregate loss S = X_1 + ... + X_N is a sum of a random number of independent severities. When N belongs to the Panjer (a, b, 0) class, the PMF of an integer-valued S satisfies a one-step recursion that replaces the infinite convolution with a feasible spreadsheet computation. This page covers the moments, MGF, classification of (a, b, 0) and (a, b, 1), severity discretization, and three end-to-end recursions of the kind ASTAM and CAS exams test.
Buhlmann Credibility
Buhlmann credibility estimates a risk by blending its own observed experience with the collective mean. The credibility weight increases when the risk has more exposure or less process noise.
Buhlmann-Straub Credibility
Buhlmann-Straub is Buhlmann credibility with unequal exposure. It is the right model when different years, classes, or risk cells have different volumes, so each observation does not deserve the same weight.
Copulas and Multivariate Dependence
A copula is the joint distribution of variables transformed to uniform marginals. Sklar’s theorem says every joint distribution decomposes into marginals plus a copula, which lets dependence be modeled separately from one-dimensional behavior. Gaussian, t, and Archimedean families (Clayton, Gumbel, Frank) span standard actuarial uses, distinguished by their tail-dependence coefficients. Note: copulas have been removed from the ASTAM 2026 syllabus; this page remains relevant for CFE 101, CERA, and CP 351.
Credibility Theory
Credibility theory blends an individual risk's own experience with broader collective experience so an actuarial estimate can use noisy data without overreacting to it.
Empirical Bayes
Empirical Bayes estimates the prior distribution from the portfolio, then uses that fitted prior to shrink individual risks toward the collective pattern.
Extreme Value Theory for Heavy Tails
Extreme value theory has two limit theorems. Fisher-Tippett-Gnedenko says block maxima of i.i.d. samples converge to the generalized extreme value distribution (Gumbel, Frechet, or Weibull depending on tail behavior). Pickands-Balkema-de Haan says excesses above a high threshold converge to the generalized Pareto distribution. The mean excess function distinguishes light from heavy tails by visual diagnostic; the Hill estimator gives a numerical tail index. Together these tools drive catastrophe modeling, capital adequacy, and operational risk in CERA and CFE 101.
Reserving Diagnostics for Development Factors
Reserving diagnostics ask whether the triangle is fit for the method being used. Development-factor correlation, calendar-year effects, immature accident years, and residual patterns all test whether a chain-ladder, Mack, ODP, or credibility estimate deserves trust.
Ruin Theory and the Lundberg Bound
The Cramer-Lundberg surplus process U(t) = u + ct - S(t) tracks an insurer’s capital under continuous premium income and compound Poisson claim outflow. The probability of ultimate ruin ψ(u) is bounded by the Lundberg inequality ψ(u) ≤ e^{-Ru}, where the adjustment coefficient R is the unique positive root of λ(M_X(r) - 1) = cr. For exponential severity, an exact closed form exists; for general severity, R is found numerically.
Stochastic Processes and Martingales
A stochastic process is a collection of random variables indexed by time. A martingale is a process whose conditional expectation of the next value, given the past, equals the current value, so it models a fair game. The optional stopping theorem says that under stated conditions the expected value at a stopping time equals the starting value, and the exponential martingale built from the surplus process is the mechanism behind the Lundberg ruin bound.
ALTAM and life models
4 pagesLife Contingencies and Survival Models
A survival model describes the random future lifetime of a life aged x through its survival function and its force of mortality. The life table tabulates survival on an integer-age grid. The expected present value combines the time value of money with survival probability, giving the price of insurances and annuities, and the equivalence principle sets the level premium so that the EPV of premiums equals the EPV of benefits.
Mortality Projection: Lee-Carter and CBD
Lee-Carter and Cairns-Blake-Dowd are the two reference stochastic mortality projection models on ALTAM, ILA 201, and the longevity-risk content within CERA. Lee-Carter is a log-bilinear model in age and time; CBD is a two-factor logit-survival model. Both project a time index forward as a random walk and feed projected mortality back into life and annuity valuation.
Multi-State Markov Models for Insurance
A multi-state Markov model treats a policyholder as moving between insurance states (healthy, disabled, dead, lapsed) according to age-dependent transition forces. The generator matrix packs those forces, the Kolmogorov forward equations turn the generator into transition probabilities, and the resulting framework drives disability income, long-term care, and critical illness pricing on ALTAM and ILA 201.
Profit Testing: Profit Signature, NPV, IRR, Embedded Value
Profit testing projects the cashflows of a life insurance policy year by year, takes expected profit per policy in force, weights by in-force survival to get the profit signature per policy issued, and reports NPV, IRR, break-even period, and the value of new business. Embedded value extends this from one new policy to the entire in-force book and the company's free surplus.
CAS ratemaking and reserving
6 pagesBornhuetter-Ferguson Method
The Bornhuetter-Ferguson method blends an a priori expected ultimate loss with the observed amount already emerged, making it less sensitive than chain-ladder when early data is thin or volatile.
Cape Cod Method
The Cape Cod method is a reserving approach that estimates an expected loss ratio from the experience itself, then uses that expected-loss view to complete the unreported portion of each year.
Chain-Ladder Method
The chain-ladder method estimates ultimate losses by projecting observed claim development forward using development factors derived from historical triangles.
Loss Ratio Method
The loss ratio method builds an overall rate indication by comparing projected loss needs and fixed expenses against the premium that remains available after variable expenses and profit provisions.
On-Leveling And Trend
On-leveling adjusts historical premium to a common rate level, while trend moves losses, premium, or exposures from an old period to the future period that the selected rate is meant to cover.
Pure Premium Method
The pure premium method is a core ratemaking framework that projects loss and expense cost per exposure, then translates that projected cost into an indicated rate level.
Stochastic reserving
1 pagesPredictive analytics
1 pagesInvestment and life products
1 pagesPension valuation
1 pagesSimulation
1 pagesFam
1 pagesMas I
1 pagesHow to use the atlas
Start with the early probability and financial-math spine, then move into distributions, statistics, reserving, credibility, and life or property-casualty modeling depending on the exam path you are studying.
Each concept page is meant to carry a clear explanation, source context, worked examples or decision logic, internal links, and practice when the topic benefits from drills.
What belongs here
The atlas is intentionally broader than an Exam P formula list. It connects preliminary exams to FAM, ASTAM, ALTAM, CAS MAS, ratemaking, reserving, and FSA-level modeling topics.