Concept

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.

How To Use This Atlas

Use this page as a map, not as a substitute for the official exam tables. The official FAM tables state the examination parameterizations and the ASTAM formula sheet states the generating-function notation. ActuaryPath adds the interpretation layer: what each formula is for, when moments exist, and where each distribution fits in FAM or ASTAM.

For a candidate, the key split is count versus amount. Frequency distributions use PMFs and probability generating functions. Severity distributions use PDFs, CDFs, survival functions, moments, limited expected values, hazards, and tail diagnostics.

Frequency Models

Frequency models count claims or events. FAM and ASTAM use Poisson, Binomial, Negative Binomial, and Geometric as the core (a,b,0) families, with zero-modified and zero-truncated variants appearing in the (a,b,1) setting.

Probability generating functions are the clean way to combine counts with discrete severities. The count PGF P_N(z) becomes the aggregate PGF after substituting the severity PGF into it.

Poisson PMF, PGF, and moments

P(N=n)=e^{-\lambda}\frac{\lambda^n}{n!},\quad P_N(z)=e^{\lambda(z-1)},\quad E[N]=\operatorname{Var}(N)=\lambda

Binomial PMF and PGF

P(N=n)=\binom{m}{n}q^n(1-q)^{m-n},\quad P_N(z)=(1-q+qz)^m

Negative binomial PGF in SOA form

P_N(z)=\left(1+\beta(1-z)\right)^{-r},\quad E[N]=r\beta,\quad \operatorname{Var}(N)=r\beta(1+\beta)

Pareto Type II

The SOA tables use Pareto Type II, also called Lomax, with shape alpha and scale theta. It is the standard heavy-tail severity family and the main bridge into mean-excess functions, GPD threshold thinking, and high-layer reinsurance.

Moments exist only up to the shape parameter. That existence condition is not fine print. If alpha is no larger than 1, the mean is infinite; if alpha is no larger than 2, the variance is infinite.

PDF and CDF

f(x)=\frac{\alpha\theta^{\alpha}}{(x+\theta)^{\alpha+1}},\quad F(x)=1-\left(\frac{\theta}{x+\theta}\right)^{\alpha},\quad x>0

Mean and variance

E[X]=\frac{\theta}{\alpha-1}\quad(\alpha>1),\qquad \operatorname{Var}(X)=\frac{\alpha\theta^2}{(\alpha-1)^2(\alpha-2)}\quad(\alpha>2)

Limited expected value

E[X\wedge u]=\frac{\theta}{\alpha-1}\left[1-\left(\frac{\theta}{u+\theta}\right)^{\alpha-1}\right],\quad \alpha\ne 1

Lognormal

A lognormal severity is X = exp(Y) where Y is normal with mean mu and standard deviation sigma. The parameters live on the log scale. This makes MLE simple because the log-data are normal, but it also makes the original-scale mean easy to misread.

Lognormal has all positive moments but no finite moment generating function for positive t. That is why it behaves like a heavy-tailed severity model in aggregate-loss and ruin settings even though its moments exist.

PDF

f(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(\ln x-\mu)^2}{2\sigma^2}\right),\quad x>0

Mean and variance

E[X]=e^{\mu+\sigma^2/2},\quad \operatorname{Var}(X)=e^{2\mu+\sigma^2}\left(e^{\sigma^2}-1\right)

Limited expected value

E[X\wedge u]=e^{\mu+\sigma^2/2}\Phi\!\left(\frac{\ln u-\mu-\sigma^2}{\sigma}\right)+u\left[1-\Phi\!\left(\frac{\ln u-\mu}{\sigma}\right)\right]

Gamma And Chi-Squared

The SOA gamma uses shape alpha and scale theta. Exponential is the alpha = 1 special case. Chi-squared with nu degrees of freedom is Gamma(nu/2, 2), which is why chi-squared tests and gamma severity models share the same functional shape.

Gamma is light-tailed relative to lognormal and Pareto. It is useful when severity is positive, skewed, and not dominated by extremely high losses. It also supplies the conjugate prior for Poisson frequency and the severity component of compound-Poisson-Gamma Tweedie models.

Gamma PDF and MGF

f(x)=\frac{x^{\alpha-1}e^{-x/\theta}}{\theta^{\alpha}\Gamma(\alpha)},\quad M_X(t)=(1-\theta t)^{-\alpha},\quad t<1/\theta

Gamma moments

E[X]=\alpha\theta,\qquad \operatorname{Var}(X)=\alpha\theta^2

Chi-squared as gamma

X\sim\chi^2_{\nu}\iff X\sim\operatorname{Gamma}(\nu/2,2),\quad E[X]=\nu,\quad \operatorname{Var}(X)=2\nu

Beta And Weibull

Beta is the bounded distribution on (0,1). Exam P uses it as a standard continuous distribution; actuarial severity work often uses transformed beta forms for bounded losses. Weibull is positive and uses shape tau with scale theta; its main actuarial use is hazard-rate flexibility.

Weibull with tau = 1 is exponential. Tau less than 1 gives decreasing hazard. Tau greater than 1 gives increasing hazard. That hazard interpretation is more important than the density shape by itself.

Beta PDF, mean, and variance

f(x)=\frac{x^{a-1}(1-x)^{b-1}}{B(a,b)},\quad E[X]=\frac{a}{a+b},\quad \operatorname{Var}(X)=\frac{ab}{(a+b)^2(a+b+1)}

Weibull CDF and survival

F(x)=1-e^{-(x/\theta)^{\tau}},\quad \bar F(x)=e^{-(x/\theta)^{\tau}}

Weibull mean, variance, and hazard

E[X]=\theta\Gamma(1+1/\tau),\quad \operatorname{Var}(X)=\theta^2\left[\Gamma(1+2/\tau)-\Gamma(1+1/\tau)^2\right],\quad h(x)=\frac{\tau}{\theta}\left(\frac{x}{\theta}\right)^{\tau-1}

Worked Example: Pick The Formula Layer

A question asks for the expected payment under an ordinary deductible. That is not a PDF problem first; it is a payment-variable problem. Use limited expected value or stop-loss formulas before choosing a computational route.

A question asks for the distribution of aggregate losses with a discrete severity and a Poisson frequency. That is a PGF or Panjer problem, not a direct CDF lookup. Use P_S(z) = exp(lambda(P_X(z) - 1)) or the Panjer recursion depending on whether the question asks for a formula or individual probabilities.