Gamma Severity Parameterizations

Introduction

The gamma distribution is a flexible model for positive claim severity. Its formulas appear in two common forms: shape and scale, and shape and rate. The formulas describe the same family of distributions, but the second parameter means something different in each convention. Confusing scale with rate can change a calculated mean by a large factor, even when the shape parameter is correct.

This tutorial sets out both parameterizations, shows how to translate between them, and applies them to densities, moments, and quantiles. The examples use hypothetical claim severities measured in dollars. No interest or mortality assumptions are needed for these severity calculations.

Definition: A positive random variable $X$ has a gamma distribution with shape $\alpha>0$ and scale $\theta>0$, written $X\sim\operatorname{Gamma}(\alpha,\theta)$, if its density is $$ f_X(x)=\frac{x^{\alpha-1}e^{-x/\theta}}{\Gamma(\alpha)\theta^\alpha},\qquad x>0, $$ and $f_X(x)=0$ for $x\leq 0$. Here $\Gamma(\alpha)=\int_0^\infty t^{\alpha-1}e^{-t}\,dt$ is the gamma function. The scale $\theta$ has the same units as $X$.

Scale and Rate Are Reciprocals

The scale $\theta$ controls the horizontal size of the distribution: multiplying $\theta$ by a constant multiplies the severity scale by that constant. The corresponding rate parameter is $\beta=1/\theta$. A rate has inverse units, such as dollars$^{-1}$ when severity is measured in dollars.

Using $\beta=1/\theta$, the same density can be written in shape-rate form. This follows by replacing each occurrence of $\theta$ in the scale density with $1/\beta$:

f_X(x)=\frac{x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)(1/\beta)^\alpha} =\frac{\beta^\alpha x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)},\qquad x>0.

Thus $X\sim\operatorname{Gamma}(\alpha,\beta)$ in shape-rate notation has the density on the right, not the scale-form density with $\beta$ substituted for $\theta$. Because the symbol $\operatorname{Gamma}(\alpha,\cdot)$ is used with both conventions in different references, identify the convention whenever quoting parameters.

Parameter conversion: Shape stays the same when changing conventions. The scale and rate are reciprocals: $$ \beta=\frac{1}{\theta},\qquad \theta=\frac{1}{\beta}. $$ In scale form, $E[X]=\alpha\theta$ and $\operatorname{Var}(X)=\alpha\theta^2$. In rate form, $E[X]=\alpha/\beta$ and $\operatorname{Var}(X)=\alpha/\beta^2$. These moment formulas assume a gamma distribution with positive shape and scale, or equivalently positive shape and rate.

The two moment expressions agree under conversion. For example, $\alpha\theta=\alpha/\beta$ because $\theta=1/\beta$, and $\alpha\theta^2=\alpha/\beta^2$. The variance has squared severity units; the mean has severity units. This is a useful dimensional check when calculating with either convention.

Worked Calculations

Worked Example: Convert Parameters and Evaluate a Density

A hypothetical insurer models an individual claim amount as gamma with shape $\alpha=3$ and scale $\theta=\$400$. Find the equivalent rate, mean, variance, and density at $x=\$600$. There is no discounting or mortality in this severity model.

Method. First convert the scale to a rate, then use the matching rate-form density and moment formulas. This keeps the parameter meanings and units explicit.

The rate is $ \beta=1/\theta=1/400=0.0025 $ per dollar. The mean is $ E[X]=\alpha\theta=3(400)=\$1{,}200 $. The variance is $ \operatorname{Var}(X)=\alpha\theta^2=3(400^2)=3(160{,}000)=480{,}000 $ dollars squared.

Since $\Gamma(3)=2!=2$, the rate-form density at 600 dollars is

f_X(600)=\frac{(0.0025)^3(600)^{3-1}e^{-(0.0025)(600)}}{\Gamma(3)} =\frac{(1.5625\times10^{-8})(360{,}000)e^{-1.5}}{2}.

Now $(1.5625\times10^{-8})(360{,}000)=0.005625$, and $e^{-1.5}\approx0.2231302$. Therefore $f_X(600)\approx(0.005625)(0.2231302)/2=0.00062755$ per dollar.

Check. The exponent is $-\beta x=-1.5$, matching $-x/\theta=-600/400=-1.5$. The density is positive, and its units are inverse dollars. The common mistake avoided here is using $0.0025$ as a scale; it is the rate.

Worked Example: Use the Shape-Rate Convention

For a hypothetical portfolio severity model, $X$ has shape $\alpha=4$ and rate $\beta=0.002$ per dollar. Calculate the equivalent scale, mean, variance, and density at $x=\$1{,}000$. No interest or mortality assumption is relevant.

Method. Work directly from the rate form, converting to scale only to check the moments. The scale is $ \theta=1/\beta=1/0.002=\$500 $.

The rate-form mean and variance are $ E[X]=\alpha/\beta=4/0.002=\$2{,}000 $ and $ \operatorname{Var}(X)=\alpha/\beta^2=4/(0.002)^2=4/0.000004=1{,}000{,}000 $ dollars squared. Using scale form gives the same results: $4(500)=2{,}000$ and $4(500^2)=4(250{,}000)=1{,}000{,}000$.

Because $\Gamma(4)=3!=6$, the density is

f_X(1{,}000)=\frac{(0.002)^4(1{,}000)^3e^{-(0.002)(1{,}000)}}{6} =\frac{(1.6\times10^{-11})(1{,}000{,}000{,}000)e^{-2}}{6}.

The product before the exponential is $0.016$, and $e^{-2}\approx0.1353353$. Thus $f_X(1{,}000)\approx0.016(0.1353353)/6=0.00036089$ per dollar.

Check. The exponent is $-2$, also obtained from $-1{,}000/500$. The scale-form moment check agrees with the rate-form calculations. A frequent error is to calculate the mean as $\alpha\beta$; the rate-form mean is $\alpha/\beta$, not $\alpha\beta$.

Gamma Quantiles

A quantile is often needed to set a severity threshold or describe a percentile of the modeled loss distribution. For general shape $\alpha$, the gamma cumulative distribution function does not usually have an elementary inverse. Define the lower incomplete gamma function by $ \gamma(\alpha,z)=\int_0^z t^{\alpha-1}e^{-t}\,dt $. Then the gamma CDF is $F_X(x)=\gamma(\alpha,x/\theta)/\Gamma(\alpha)$ for $x>0$.

Quantile formula: For $0<p<1$, the $p$-quantile $Q_p$ is the value satisfying $F_X(Q_p)=p$. In scale form it satisfies $$ \frac{\gamma(\alpha,Q_p/\theta)}{\Gamma(\alpha)}=p. $$ Equivalently, find $z_p$ such that $\gamma(\alpha,z_p)/\Gamma(\alpha)=p$, then $Q_p=\theta z_p$. In rate form $Q_p=z_p/\beta$. Except for special cases, $z_p$ is obtained numerically.

For shape $\alpha=1$, the gamma model is exponential and the quantile has the closed form $Q_p=-\theta\ln(1-p)$. For other shapes, statistical software or a numerical root finder can solve the CDF equation. The resulting quantile must then be rescaled by $\theta$, or divided by $\beta$, depending on the convention.

Worked Example: Find a Gamma Severity Percentile

A hypothetical equipment-loss amount follows a gamma distribution with shape $\alpha=2$ and scale $\theta=\$1{,}500$. Find its 75th percentile. No interest or mortality assumptions apply.

Method. For shape 2, $\Gamma(2)=1$, and integration of the density gives $F_X(x)=1-e^{-x/\theta}(1+x/\theta)$, for $x>0$. Set $z=x/\theta$. The 75th percentile solves $1-e^{-z}(1+z)=0.75$, or $e^{-z}(1+z)=0.25$.

This equation has no elementary solution, so solve it numerically. At $z=2.6926$, $e^{-z}\approx0.067705$ and $1+z=3.6926$, giving $e^{-z}(1+z)\approx(0.067705)(3.6926)\approx0.2500$. Thus $F_X\approx1-0.2500=0.7500$, to four decimal places. The dimensionless root is approximately $z_{0.75}=2.6926$.

Rescale to dollars: $Q_{0.75}=\theta z_{0.75}=1{,}500(2.6926)=\$4{,}038.90$, or $\$4{,}039$ to the nearest dollar.

Check. The calculated CDF is approximately 0.75, and the percentile exceeds the mean $\alpha\theta=2(1{,}500)=\$3{,}000$, as expected for the 75th percentile. The common mistake avoided is reporting $2.6926$ dollars: it is the quantile on the unit-scale $z$ axis and must be multiplied by $\theta$.

Parameter Checks and Exam Traps

A reliable calculation starts by writing the convention next to the parameters. If the model is stated as shape-scale, record $\alpha$ and $\theta$; if it is shape-rate, record $\alpha$ and $\beta$. Then check that the scale and rate are reciprocals and that the exponent in the density is $-x/\theta$ or $-\beta x$, respectively.

For example, in the first worked example, mistakenly treating $\beta=0.0025$ as a scale would give the incorrect mean $3(0.0025)=0.0075$ dollars. But $0.0025$ has units of dollars$^{-1}$, so that product does not even have units of dollars. The correct scale is $1/0.0025=400$ dollars, giving mean $\$1{,}200$. The unit check exposes the error before it propagates into a density or percentile.

Do not interchange the formulas. Scale form uses $E[X]=\alpha\theta$; rate form uses $E[X]=\alpha/\beta$.
Check the variance units. It must be in squared severity units, so the scale or rate is squared in the variance formula.
Do not omit the gamma function. The density normalizer includes $\Gamma(\alpha)$; for positive integers $n$, $\Gamma(n)=(n-1)!$.
Rescale a numerical quantile. A root for $z=x/\theta$ is dimensionless; the severity quantile is $\theta z$.

Key takeaway: Gamma shape is unchanged across conventions, while scale and rate are reciprocals. Label the convention, preserve units, and use the matching density and moment formulas. For a general gamma quantile, solve the dimensionless CDF equation and then convert back to severity units.

Check Your Understanding

For each hypothetical severity model, identify the parameterization before calculating. No interest or mortality assumptions are involved.

A gamma severity has shape 2 and scale $\$300$. Give its equivalent rate, mean, and variance. (Answer: rate $1/300$ per dollar; mean $\$600$; variance $180{,}000$ dollars squared.)
A gamma model has shape 5 and rate $0.001$ per dollar. Give its scale, mean, and variance. (Answer: scale $\$1{,}000$; mean $\$5{,}000$; variance $5{,}000{,}000$ dollars squared.)
For shape 3 and scale 2 dollars, calculate the density at $x=2$. (Answer: $1/(4e)\approx0.0920$ per dollar.)
A gamma severity has shape 2 and scale $\$1{,}000$. Use the unit-scale 50th percentile $z_{0.50}\approx1.67835$ to find the severity median. (Answer: approximately $\$1{,}678.35$.)
A gamma severity has shape 4 and mean $\$2{,}400$. Find its scale, variance, and density at $\$1{,}200$. (Answer: scale $\$600$; variance $1{,}440{,}000$ dollars squared; density approximately $0.0003007$ per dollar.)

What You'll Learn