Comparing Severity Tails And Moment Existence

Why Tail Behavior and Moments Must Be Considered Together

A severity model describes the size of a loss \(X\), which is assumed here to be nonnegative. Two useful questions are related but not identical: how quickly does the probability of an extreme loss decrease, and which averages such as \(E[X]\) or \(E[X^2]\) exist? A model may assign relatively substantial probability to large losses and still have a finite mean and variance. Conversely, a tail that decreases as a power of the loss can make higher moments infinite.

This comparison is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The models below use their standard parameterizations. In particular, a Weibull model is given by its scale \(\theta>0\) and shape \(k>0\); a lognormal model is specified by the distribution of \(\ln X\); and the Pareto models use a positive shape parameter \(\alpha\).

A Tail-Integral Justification

For a nonnegative loss, a moment can be expressed directly in terms of the survival function \(S_X(x)=P(X>x)\). For \(r>0\), the identity is

provided the integral is interpreted as possibly infinite. To see why it holds, write \(X^r=\int_0^X r x^{r-1}\,dx\). Equivalently, for each outcome, the indicator \(1_{\{X>x\}}\) is one exactly for \(0\leq x<X\), so

Taking expectations and interchanging the nonnegative integral and expectation gives \(E[X^r]=\int_0^\infty r x^{r-1}P(X>x)\,dx\). This formula explains the role of the far tail: finiteness is determined by whether the integrand remains integrable as \(x\) becomes large. It also applies to continuous and discrete nonnegative losses.

Exponential and Weibull Tails

An exponential severity with rate \(\lambda>0\) has survival function \(S_X(x)=e^{-\lambda x}\) for \(x\geq 0\). Its \(r\)th raw moment is \(r!/\lambda^r\) for a positive integer \(r\); more generally, \(E[X^r]=\Gamma(r+1)/\lambda^r\) for real \(r>0\). Every positive moment is finite because the exponential tail decreases quickly enough to make the tail integral converge.

A Weibull severity with scale \(\theta>0\) and shape \(k>0\) has survival function \(S_X(x)=\exp[-(x/\theta)^k]\) for \(x\geq0\). Its raw moments are

Every positive moment is finite for every \(k>0\). The shape changes the rate of tail decay relative to an exponential, but does not change this moment-existence conclusion. If \(k=1\), the Weibull is exponential with rate \(1/\theta\). If \(0<k<1\), its survival tail eventually exceeds that of any fixed-rate exponential. If \(k>1\), it eventually falls below that exponential tail. For all \(k>0\), however, the Weibull tail decreases faster than any power \(x^{-a}\), where \(a>0\).

Lognormal Tail and Moments

A lognormal severity has \(X>0\) and \(\ln X\sim N(\mu,\sigma^2)\), where \(\sigma>0\) and the second normal parameter is the variance. For every real \(r>0\),

Consequently, all positive raw moments of a lognormal severity are finite. Its tail is nevertheless much larger, for sufficiently large \(x\), than the tail of any exponential model. It decreases faster than any fixed power tail \(x^{-a}\), \(a>0\), as \(x\) tends to infinity. “Heavy-tailed” is used in different ways; in comparing these models, state the comparison being made rather than treating the label as a moment-existence rule.

A related distinction is important: all positive raw moments being finite does not imply that the moment-generating function is finite for positive arguments. For a nondegenerate lognormal severity, \(E[e^{tX}]\) is infinite for every \(t>0\). The raw moments and the moment-generating function answer different questions.

Pareto Tails and the Moment Cutoff

For a Pareto Type I severity with minimum value \(x_m>0\) and shape \(\alpha>0\), the survival function is \(S_X(x)=(x_m/x)^\alpha\) for \(x\geq x_m\). Its \(r\)th raw moment is

For \(r\geq\alpha\), the moment is infinite. The cutoff follows from the tail integral: beyond \(x_m\), its integrand is proportional to \(x^{r-1-\alpha}\). The integral to infinity converges exactly when \(r-1-\alpha<-1\), or \(r<\alpha\). Equality does not give a finite moment; it produces a divergent logarithmic integral.

For a Pareto Type II (Lomax) severity with scale \(\theta>0\) and shape \(\alpha>0\), the support is \(x\geq0\) and \(S_X(x)=(1+x/\theta)^{-\alpha}\). Its raw moments are

They are infinite for \(r\geq\alpha\). Although Type I and Type II have different supports and moment formulas, both have the same shape-based condition for finiteness. Their tails decrease at a power rate proportional to \(x^{-\alpha}\) for large \(x\). In either form, the mean is finite only if \(\alpha>1\), and the second raw moment is finite only if \(\alpha>2\).

Comparing the Models and Avoiding Common Traps

A common exam trap is to conclude that any model with a tail heavier than an exponential must have an infinite mean or variance. That conclusion is false. For example, a Weibull model with \(0<k<1\) has a tail that eventually exceeds a fixed-rate exponential tail, but all its positive raw moments remain finite. A lognormal also has all positive raw moments finite. By contrast, a Pareto model with \(1<\alpha\leq2\) has a finite mean but an infinite second raw moment.

The incorrect reasoning might be: “The tail is heavier than exponential, so \(E[X^2]=\infty\).” The correction is to identify the model and apply its moment condition. For a Weibull severity, \(E[X^2]=\theta^2\Gamma(1+2/k)\), which is finite for every \(k>0\). For a Pareto severity, check whether \(2<\alpha\); if not, the second raw moment is infinite. Tail comparisons alone do not replace these checks.