Worked Comparisons Of Exponential And Gamma Severities

Why Compare Exponential and Gamma Severities?

Two severity models can have the same mean and still describe materially different loss patterns. In particular, matching a mean does not determine the probability of a large loss: that probability depends on the model’s shape as well as its scale. Exponential and gamma models make a useful comparison because the exponential distribution is a special case of the gamma family, while other gamma shapes allow different hazard patterns.

This tutorial treats $X$ as a nonnegative, continuous loss amount in dollars, with no deductible, limit, inflation adjustment, or censoring. The calculations concern severity distributions only; interest and mortality assumptions do not apply. A comparison is meaningful only after each model’s parameterization and fitted quantity—such as its mean or variance—are stated.

Definition: An exponential severity with rate $\lambda>0$ has density $f_X(x)=\lambda e^{-\lambda x}$ and survival function $S_X(x)=P(X>x)=e^{-\lambda x}$, for $x\geq 0$. Its mean is $1/\lambda$, its variance is $1/\lambda^2$, and its hazard is the constant $\lambda$.

Definition: A gamma severity with shape $\alpha>0$ and scale $\theta>0$ has density $f_X(x)=x^{\alpha-1}e^{-x/\theta}/(\Gamma(\alpha)\theta^\alpha)$, for $x>0$. Its mean is $\alpha\theta$, and its variance is $\alpha\theta^2$. The scale $\theta$ has the same units as $X$; the corresponding rate parameter is $1/\theta$.

The gamma model includes the exponential model: setting $\alpha=1$ gives an exponential distribution with rate $1/\theta$. For a positive integer shape $m$, the gamma survival function has the convenient form $S_X(x)=e^{-z}\sum_{j=0}^{m-1}z^j/j!$, where $z=x/\theta$. For non-integer shape, the survival function is generally evaluated using the incomplete gamma function or numerical software. Gamma quantiles also usually require numerical solution.

Comparing Models on a Common Basis

When both models are fitted to the same mean $\mu$, the exponential rate is $\lambda=1/\mu$, while a gamma model with chosen shape $\alpha$ must have scale $\theta=\mu/\alpha$. Their variances need not agree: the exponential variance is $\mu^2$, whereas the gamma variance is $\mu^2/\alpha$. Thus, specifying only a common mean leaves the gamma shape—and consequently its spread, hazard, and tail probabilities—undetermined.

If both a mean $\mu$ and variance $\sigma^2$ are specified for a gamma model, its parameters are determined by the moment equations:

\alpha=\frac{\mu^2}{\sigma^2}, \qquad \theta=\frac{\sigma^2}{\mu}.

A moment-matched exponential has variance $\mu^2$. Therefore, an exponential model can match both the mean and variance only when $\sigma^2=\mu^2$, which corresponds to the gamma special case $\alpha=1$. Otherwise, comparing an exponential fitted to the mean with a gamma fitted to both moments is a deliberate comparison of different spreads, not a two-moment match.

Worked Example: Equal Means, Different Medians and Tail Probabilities

A hypothetical portfolio’s severity is represented either by an exponential model with mean $1,000 or by a gamma model with shape $\alpha=2$ and scale $\theta=$500$. Compare their variances, medians, hazards at $1,000, and probabilities of exceeding $1,500. Use dollars throughout and round probabilities to four decimals and dollar amounts to the nearest cent.

Method and parameter check. The exponential rate is $\lambda=1/1000=0.001$ per dollar. The gamma mean is $\alpha\theta=2(500)=$1,000$, so the means agree. The gamma variance is $\alpha\theta^2=2(500)^2=$500,000 squared; the exponential variance is \(1/\lambda^2=$1,000,000 squared$.

Medians. The exponential median $q_{0.5}$ solves $e^{-q_{0.5}/1000}=0.5$, so $q_{0.5}=1000\ln 2=$693.15$. For a shape-2 gamma, the survival function is $e^{-z}(1+z)$, where $z=x/500$. Its median solves $e^{-z}(1+z)=0.5$, whose positive root is $z\approx1.67834699$. Thus the gamma median is $500(1.67834699)=$839.17$, rounded to the nearest cent.

Exceedance probabilities at $1,500. The exponential probability is $e^{-1500/1000}=e^{-1.5}\approx0.2231$. For the gamma model, $z=1500/500=3$, so $P(X>1500)=e^{-3}(1+3)=4e^{-3}\approx0.1991$.

Hazards at $1,000. The exponential hazard is $0.001$ per dollar at every loss amount. For the shape-2 gamma, the hazard is $z/[\theta(1+z)]$. At $x=$1,000$, $z=2$, giving $2/[500(3)]=0.001333$ per dollar. The gamma model has a higher hazard at this point, even though its probability of exceeding $1,500 is lower.

Check and trap avoided. Both means are $1,000, but the gamma model has smaller variance and a higher median. At $1,500 its exceedance probability, 0.1991, is below the exponential model’s 0.2231. The common mistake is to assume that equal means imply equal medians or equal tail probabilities; these calculations show that they do not.

Why the Gamma Survival and Hazard Formulas Work

For shape 2, the density is $f_X(x)=xe^{-x/\theta}/\theta^2$. Set $z=x/\theta$ and substitute $t=y/\theta$ in the survival integral. This gives $S_X(x)=\int_z^\infty te^{-t}\,dt$. Integration by parts, taking $u=t$ and $dv=e^{-t}dt$, yields:

\int_z^\infty te^{-t}\,dt = \left[-te^{-t}\right]_z^\infty+\int_z^\infty e^{-t}\,dt = ze^{-z}+e^{-z} = e^{-z}(1+z).

The boundary term at infinity is zero because $te^{-t}$ tends to zero. The hazard is the density divided by the survival function, $h_X(x)=f_X(x)/S_X(x)$, wherever $S_X(x)>0$. For shape 2, substituting the density and survival function gives:

h_X(x) = \frac{(x/\theta^2)e^{-x/\theta}} {e^{-x/\theta}(1+x/\theta)} = \frac{x}{\theta(\theta+x)}.

This hazard starts at zero for shape 2 and increases toward $1/\theta$ as $x$ grows. More generally, the exponential hazard is constant, while gamma hazards depend on shape: for shape below 1 they decrease, for shape 1 they are constant, and for shape above 1 they increase. A hazard describes the instantaneous rate of loss occurrence conditional on exceeding the current amount; it is not itself the probability of exceeding a fixed threshold.

Worked Example: Moment-Matching a Gamma Model

Suppose a hypothetical severity study gives a fitted mean of $800 and variance of $320,000 squared. Construct the gamma model matching both moments, then compare it with an exponential model matched to the same mean. Calculate the probabilities of exceeding $800 and $1,200. Round probabilities to four decimals.

Method and gamma parameters. Use the gamma moment equations: $\alpha=800^2/320000=640000/320000=2$, and $\theta=320000/800=$400$. Checking, the gamma mean is $2(400)=$800$ and its variance is $2(400)^2=$320,000 squared. The exponential rate fitted to the same mean is \(\lambda=1/800$ per dollar; its variance is \(800^2=$640,000 squared.

At $800. The exponential exceedance probability is $e^{-800/800}=e^{-1}\approx0.3679$. For the gamma model, $z=800/400=2$, so the probability is $e^{-2}(1+2)=3e^{-2}\approx0.4060$. Here the gamma probability is larger.

At $1,200. The exponential probability is $e^{-1200/800}=e^{-1.5}\approx0.2231$. The gamma value of $z$ is $1200/400=3$, so its probability is $e^{-3}(1+3)=4e^{-3}\approx0.1991$. At this threshold, the exponential probability is larger.

Check and trap avoided. The comparisons reverse between $800 and $1,200, even though the models have the same mean. This is possible because survival curves can cross. Do not conclude that one model has a larger exceedance probability at every threshold just because it is more variable or has a larger probability at one selected threshold.

Comparing Hazards and Large-Loss Probabilities

A survival probability answers a direct question: $S_X(x)=P(X>x)$. It can be compared at a threshold that matters for the application. The hazard instead compares the density at $x$ with the probability of having survived beyond $x$. For a continuous severity with density $f_X$ and survival $S_X$, its hazard is $h_X(x)=f_X(x)/S_X(x)$. It has units inverse to the units of loss and should not be reported as a probability.

The distinction matters in practice. A model can have a lower probability of exceeding a particular threshold but a higher hazard at that threshold. The first is accumulated tail probability; the second is a local conditional rate. Neither comparison alone establishes that a model’s entire tail is heavier. When making a model comparison, state the threshold, the fitted quantities, and whether the result concerns a quantile, a survival probability, or a hazard.

Worked Example: Shape-3 Gamma Tail and Local Hazard

A hypothetical severity is modeled by a gamma distribution with shape $\alpha=3$ and scale $\theta=$300$. Compare it with an exponential distribution having the same mean. Find the probability of exceeding $1,500 and the hazard at $1,500 under each model. Round probabilities to four decimals and hazards to six significant digits per dollar.

Method and model check. The gamma mean is $3(300)=$900$, so the matching exponential rate is $1/900$ per dollar. At $x=$1,500$, the gamma standardized amount is $z=1500/300=5$. For integer shape 3, its survival function is $e^{-z}(1+z+z^2/2)$.

Exceedance probabilities. The gamma probability is $e^{-5}(1+5+25/2)=18.5e^{-5}\approx0.1247$. The exponential probability is $e^{-1500/900}=e^{-5/3}\approx0.1889$. Thus the gamma model has the lower exceedance probability at this threshold.

Hazards. The exponential hazard is constant at $1/900\approx0.00111111$ per dollar. For shape 3, dividing the gamma density by its survival function gives $h_X(x)=z^2/[2\theta(1+z+z^2/2)]$. At $z=5$, this is $25/[600(18.5)]=25/11100\approx0.00225225$ per dollar.

Check and trap avoided. The gamma hazard at $1,500 is greater, although its exceedance probability is smaller. This is not a contradiction: the hazard is local, while the survival probability reflects the whole distribution above the threshold. Do not substitute a hazard for a tail probability.

Quantiles, Parameter Checks, and Exam Traps

The $p$-quantile $q_p$ is a value satisfying $F_X(q_p)=p$, or equivalently $S_X(q_p)=1-p$ for a continuous distribution. For an exponential model, $q_p=-\ln(1-p)/\lambda$. For a gamma model, solve $S_X(q_p)=1-p$; when the shape is an integer, the finite-sum survival formula is often convenient, but the resulting equation may still need a numerical root. Check the root by substituting it back into the survival equation.

The most frequent calculation errors are parameterization errors. If a problem gives gamma scale $\theta$, the exponential rate in the shape-1 case is $1/\theta$, not $\theta$. Likewise, if the gamma input is a rate $\beta$, its scale is $1/\beta$. Carry units through the calculation: a rate has units per dollar, while a scale has units dollars.

A second trap is using the wrong number of terms in the integer-shape gamma survival formula. For shape 2, the polynomial is $1+z$; for shape 3, it is $1+z+z^2/2$. For example, at $z=3$, the shape-2 survival is $e^{-3}(1+3)=4e^{-3}\approx0.1991$. Incorrectly using the shape-3 polynomial would give $8.5e^{-3}\approx0.4232$, which is not the shape-2 probability.

Key takeaway: First state each model and its parameterization. Then identify what is being matched, calculate the requested quantile, hazard, or survival probability from the corresponding formula, and check the result in context. Equal means do not imply equal variances, quantiles, or tail probabilities.

Check Your Understanding

For each hypothetical severity comparison, identify the relevant model and parameterization before calculating. Probabilities are rounded to four decimals.

An exponential severity has mean $750. Find the rate and the probability of exceeding $1,500. (Answer: $\lambda=1/750$ per dollar; $P(X>1500)=e^{-2}\approx0.1353$.)
A gamma severity has shape 2 and scale $250. Find its mean, variance, and probability of exceeding $500. (Answer: Mean $500; variance 125,000 squared dollars; probability $3e^{-2}\approx0.4060$.)
A hypothetical severity has mean $900 and variance 405,000 squared dollars. Find the gamma shape and scale, then compare its probability of exceeding $900 with that of an exponential model matched to the mean. (Answer: $\alpha=2$, $\theta=$450$; gamma probability $3e^{-2}\approx0.4060$, exponential probability $e^{-1}\approx0.3679$.)
A gamma severity has shape 3 and scale $200. Find the probability of exceeding $1,000 using the integer-shape survival formula. (Answer: $z=5$; probability $18.5e^{-5}\approx0.1247$.)
Compare an exponential severity with mean $1,200 and a gamma severity with shape 2 and scale $600 at the $1,200 threshold. Which has the larger exceedance probability? (Answer: Gamma $3e^{-2}\approx0.4060$; exponential $e^{-1}\approx0.3679$; the gamma probability is larger.)

These model comparisons are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

What You'll Learn