Why Compare Severity Shapes?
A severity model describes the size of an individual claim payment, conditional on a payment being positive. The Gamma and inverse Gaussian distributions are both continuous, positively supported, and right-skewed, so either may look plausible when a set of claim amounts has a long upper tail. Their key distinction for ratemaking is how their variance and skewness change as the mean severity changes.
The comparison is most informative across groups with different mean severities, such as hypothetical rating cells or claim categories. A single group’s mean and variance can be matched by either family after choosing its dispersion parameter. Across groups, however, a common-dispersion assumption implies different mean-variance patterns. Skewness provides a further diagnostic.
Support, Parameters, and Shape
Both distributions assign probability only to positive severities: \(Y>0\). Neither, in this form, describes a zero or negative individual payment. Their support alone therefore does not distinguish them when the data are strictly positive. The different relationships between mean, variance, and skewness are more useful.
For a Gamma random variable with shape \(\alpha>0\) and scale \(\theta>0\), the mean is \(\alpha\theta\), the variance is \(\alpha\theta^2\), and the skewness is \(2/\sqrt{\alpha}\). Set \(\alpha=1/\phi\) and \(\theta=\phi\mu\). This gives the mean-variance form above and the following shape measures:
Thus, for Gamma severities with common \(\phi\), the coefficient of variation and skewness do not change as the mean changes. The standard deviation does change: it is \(\sqrt{\phi}\mu\), proportional to the mean.
For an inverse Gaussian variable with mean \(\mu\) and shape parameter \(\lambda\), the variance is \(\mu^3/\lambda\), and its skewness is \(3\sqrt{\mu/\lambda}\). With \(\phi=1/\lambda\), these become:
For inverse Gaussian severities with common \(\phi\), both the coefficient of variation and skewness increase with the mean. Its standard deviation is \(\sqrt{\phi}\mu^{3/2}\), which grows faster with the mean than the Gamma standard deviation does. The units of \(\phi\) differ between the families, so their numerical dispersion values should not be compared directly.
| Feature | Gamma | Inverse Gaussian |
|---|---|---|
| Support | \(Y>0\) | \(Y>0\) |
| Variance at mean \(\mu\) | \(\phi\mu^2\) | \(\phi\mu^3=\mu^3/\lambda\) |
| Coefficient of variation | \(\sqrt{\phi}\) | \(\sqrt{\phi\mu}\) |
| Skewness | \(2\sqrt{\phi}\) | \(3\sqrt{\phi\mu}\) |
| Common-dispersion pattern across means | Constant variance-to-\(\mu^2\) ratio | Constant variance-to-\(\mu^3\) ratio |
Why the Moment Ratios Distinguish the Families
The variance formulas yield a direct diagnostic. Divide each variance by the relevant power of its mean. If observations from comparable groups follow a Gamma model with common dispersion, then \(\operatorname{Var}(Y)/\mu^2\) should be approximately constant. If they follow an inverse Gaussian model with common dispersion, then \(\operatorname{Var}(Y)/\mu^3\) should be approximately constant.
This follows by substitution, not by comparing raw variances. For the Gamma case, dividing \(\phi\mu^2\) by \(\mu^2\) leaves \(\phi\); for the inverse Gaussian case, dividing \(\phi\mu^3\) by \(\mu^3\) leaves \(\phi\). The ratios are useful only when a common dispersion across the groups is a reasonable assumption and the groups are sufficiently comparable.
Skewness supplies a second check. For each group, estimate skewness from the severity data and compare it with the value implied by the fitted dispersion and mean. Sample skewness can be noisy, especially with limited data, so it should support rather than replace the variance-pattern assessment. These moment checks help select a candidate model; they do not establish that the full distribution fits every feature of the data.
Check that the observations are individual positive severities, not counts or a total outcome that may include zero.
Calculate variance divided by mean squared and variance divided by mean cubed for each group.
Compare observed skewness with the family’s implied skewness, while considering whether a shared dispersion across groups is plausible.
Worked Moment Comparisons
Worked Example: Variance Scaling Favors Gamma
A hypothetical insurer summarizes positive claim severities, in dollars, for two comparable segments. Segment A has mean \(100\), variance \(2{,}500\), and sample skewness \(1.00\). Segment B has mean \(200\), variance \(10{,}000\), and sample skewness \(1.05\). Compare the families under a common-dispersion assumption. There is no interest discounting or mortality assumption: the quantities are individual severities. Treat the supplied moments as descriptive values.
Method. Calculate the two variance ratios for each segment, then compare the skewness with each family’s prediction.
Gamma calculation. For Segment A, \(\phi=2{,}500/100^2=2{,}500/10{,}000=0.25\). For Segment B, \(\phi=10{,}000/200^2=10{,}000/40{,}000=0.25\). The common Gamma dispersion is therefore consistent across the two segments. Its implied skewness is \(2\sqrt{0.25}=2(0.5)=1.00\) in each segment.
Inverse Gaussian calculation. For Segment A, \(\phi=2{,}500/100^3=2{,}500/1{,}000{,}000=0.0025\) per dollar. Using this dispersion for Segment B predicts variance \(0.0025(200^3)=0.0025(8{,}000{,}000)=20{,}000\), not the supplied \(10{,}000\). Its implied skewness is \(3\sqrt{0.0025(100)}=3\sqrt{0.25}=1.50\) for A and \(3\sqrt{0.0025(200)}=3\sqrt{0.50}\approx2.1213\) for B.
The Gamma ratios and skewness agree closely with the supplied summaries; the inverse Gaussian common-dispersion pattern does not. Conclusion: the descriptive moments favor Gamma. The mistake to avoid is fitting an inverse Gaussian dispersion to Segment A and assuming that one fit also explains Segment B.
Worked Example: Variance Scaling Favors Inverse Gaussian
For a different hypothetical positive-severity portfolio, Segment C has mean \(50\) dollars, variance \(5{,}000\) dollars squared, and sample skewness \(4.3\). Segment D has mean \(100\) dollars, variance \(40{,}000\) dollars squared, and sample skewness \(5.9\). Find which family is more consistent with the supplied moments under common dispersion. No discounting or mortality is involved.
Inverse Gaussian calculation. For C, \(\phi=5{,}000/50^3=5{,}000/125{,}000=0.04\) per dollar, so \(\lambda=1/\phi=25\) dollars. For D, \(\phi=40{,}000/100^3=40{,}000/1{,}000{,}000=0.04\) per dollar. The dispersion is the same. The implied variances check: \(50^3/25=125{,}000/25=5{,}000\), and \(100^3/25=1{,}000{,}000/25=40{,}000\).
The implied skewness for C is \(3\sqrt{0.04(50)}=3\sqrt{2}\approx4.2426\). For D it is \(3\sqrt{0.04(100)}=3\sqrt{4}=6.0000\). These values are close to the supplied sample skewnesses.
Gamma comparison. Its dispersion estimates are \(5{,}000/50^2=5{,}000/2{,}500=2\) for C and \(40{,}000/100^2=40{,}000/10{,}000=4\) for D. They are not common, and the implied Gamma skewnesses would be \(2\sqrt{2}\approx2.8284\) and \(2\sqrt{4}=4.0000\). Conclusion: the inverse Gaussian assumption is more consistent. Do not judge the fit by saying that a larger raw variance automatically favors one family; use the variance-to-mean-power ratios.
Worked Example: A Single Group Gives Only a Partial Diagnosis
A hypothetical book of positive equipment-damage severities has mean \(40\) dollars, variance \(400\) dollars squared, and sample skewness \(1.08\). Compare the moment implications of both families for this one group. There is no interest or mortality component.
Gamma fit. The dispersion is \(\phi=400/40^2=400/1{,}600=0.25\). The corresponding shape is \(\alpha=1/\phi=4\), and the scale is \(\theta=\phi\mu=0.25(40)=10\) dollars. The variance checks as \(\alpha\theta^2=4(10^2)=400\). The implied skewness is \(2\sqrt{0.25}=1.00\).
Inverse Gaussian fit. The dispersion is \(\phi=400/40^3=400/64{,}000=0.00625\) per dollar, so \(\lambda=1/0.00625=160\) dollars. The variance checks as \(40^3/160=64{,}000/160=400\). Its implied skewness is \(3\sqrt{40/160}=3\sqrt{0.25}=1.50\).
The observed skewness \(1.08\) is closer to the Gamma prediction \(1.00\) than to \(1.50\), so these moments favor Gamma. But both models reproduce the one group’s mean and variance by construction. A single mean-variance pair cannot distinguish them; evidence across multiple mean levels would make the variance-scaling comparison possible.
Common Exam Traps and Practical Interpretation
A frequent error is to treat both families as having variance proportional to \(\mu^2\). That is the Gamma relationship; the inverse Gaussian relationship is proportional to \(\mu^3\). Another error is to interpret a fitted dispersion number without its family and units. Gamma \(\phi\) is dimensionless, whereas inverse Gaussian \(\phi\) has inverse-severity units. Their numerical values are not directly comparable.
- Do not use raw variance alone. A group with a higher mean can naturally have a higher variance under either family. Normalize by the appropriate power of the mean.
- Do not overstate evidence from one group. Either family can match a positive mean and variance by selecting its dispersion parameter. Use skewness or additional groups to compare shape.
- Check the common-dispersion condition. If severity mix or other group characteristics change, dispersion may not be common. A failed ratio pattern need not mean the family is inherently impossible.
- Keep the modeled outcome straight. These formulas describe positive individual severities. Do not apply them without adjustment to a response that includes zero outcomes.
In practice, descriptive moments are a screening tool. A selected candidate still requires assessment against the data and the intended modeling purpose. The key exam skill is to connect a stated assumption to its consequence: constant \(\operatorname{Var}(Y)/\mu^2\) and constant skewness suggest the Gamma pattern, while constant \(\operatorname{Var}(Y)/\mu^3\) with skewness increasing with the mean suggests the inverse Gaussian pattern. This distribution comparison is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each question, treat the moments as descriptive of positive individual severities and state the family implication requested.
- A Gamma severity has mean \(80\) dollars and variance \(3{,}200\) dollars squared. Find \(\phi\), the Gamma shape \(\alpha\), and implied skewness. (Answer: \(\phi=0.5\), \(\alpha=2\), skewness \(\approx1.4142\).)
- An inverse Gaussian severity has mean \(40\) dollars and variance \(1{,}280\) dollars squared. Find \(\phi\), \(\lambda\), and implied skewness. (Answer: \(\phi=0.02\) per dollar, \(\lambda=50\) dollars, skewness \(\approx2.6833\).)
- Two comparable groups have means \(20\) and \(40\), and variances \(200\) and \(800\). Which family’s common-dispersion variance pattern do these moments match? Show the ratios. (Answer: Gamma; \(200/20^2=0.5\) and \(800/40^2=0.5\).)
- Two comparable groups have means \(25\) and \(50\), and variances \(625\) and \(5{,}000\). Which family’s common-dispersion variance pattern do these moments match, and what is the common \(\phi\)? (Answer: Inverse Gaussian; \(625/25^3=0.04\) and \(5{,}000/50^3=0.04\) per dollar.)
- A single group has mean \(60\), variance \(7{,}200\), and sample skewness \(1.2\). Calculate the skewness implied by each family when its dispersion is fitted to the mean and variance. Which is closer, and can this single group establish the family? (Answer: Gamma skewness \(2\sqrt{2}\approx2.8284\); inverse Gaussian skewness \(3\sqrt{2}\approx4.2426\). Gamma is closer, but one group cannot establish the family.)