Scale and Shape Parameters in Continuous Models

Two actuaries fit a Pareto with the same $\theta =10,000$. Actuary A uses $\alpha =2.5$. Actuary B uses $\alpha =4.0$. Same scale parameter, but A's expected excess loss at $50,000 is more than three times larger. The difference comes entirely from the shape parameter.

A scale parameter stretches or compresses a distribution along the horizontal axis. If $Y=cX$, the new distribution has scale parameter $c\theta$. The functional form is unchanged.

HIGH-FREQUENCY: Scaling effects on moments: $E[cX]=cE[X]$, $\text{Var}(cX)=c^2\text{Var}(X)$. The coefficient of variation is unchanged.

Percentiles scale linearly. Scale parameters have the same units as the random variable.

KEY: Scaling a distribution multiplies the mean by $c$ and the variance by $c^2$, but the coefficient of variation stays the same. CV depends only on the shape parameter.

A shape parameter changes the distribution's fundamental form, tail weight, skewness, modality, hazard rate behavior. Shape parameters are dimensionless. Scaling $X$ by a constant does not change the shape parameter.

Common mistakes

• Changing shape under inflation. Inflation only changes the scale parameter. Adjusting $\alpha$ or $\tau$ produces a fundamentally different distribution.
• Treating lognormal $\sigma$ as scale. Scaling $X\to cX$ adds $\ln c$ to $\mu$ and leaves $\sigma$ alone. A trap asks for the new CV after inflation, answer is unchanged.
• Forgetting the leverage effect. If ground-up losses inflate by $r$, excess costs do not inflate by $r$. Computing $(1+r)\times \text{old excess}$ ignores leverage. Trap: $1.08\times 278=300$ vs. correct 332.

Bottom line

• Scale parameter stretches the distribution without changing its shape. $E[cX]=cE[X]$, $\text{Var}(cX)=c^2\text{Var}(X)$, CV unchanged
• Shape parameter changes tail weight, skewness, and hazard rate. CV depends only on shape
• Inflation is a scale change: adjust only the scale parameter; shape stays fixed
• Lognormal exception: $\mu$ is scale (additive on the log scale), $\sigma$ is shape

Exam shortcut

When you see inflation, immediately identify the scale parameter and multiply by $(1+r)$. Do not touch the shape parameter. If the problem also has a fixed attachment, expect the answer to show leverage. "Scale stretches, shape sculpts." "CV is shape-only." "Lognormal sigma is shape", scaling adds to $\mu$, never touches $\sigma$.

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Learning objectives

• 2b

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