4.4 Gamma & Inverse Gaussian Models for Claim Severity

Severity Modeling Principles in Actuarial Practice

In property and casualty (P&C) ratemaking, pure premium (loss cost) represents the expected loss per unit of exposure. Under the classical two-stage frequency-severity framework, pure premium decomposes multiplicatively into the expected claim frequency and the expected conditional claim severity:

$\text{Pure Premium} = \mathbb{E}\left[\frac{\text{Loss}}{\text{Exposure}}\right] = \mathbb{E}\left[\frac{\text{Claims}}{\text{Exposure}}\right] \times \mathbb{E}\left[\frac{\text{Loss}}{\text{Claims}} \;\middle|\; \text{Claims} > 0\right] = \text{Frequency} \times \text{Severity}$

While frequency models analyze discrete claim counts across all exposed policies (including zero-claim policies), claim severity models operate strictly on the conditional distribution of claim sizes given that a claim has occurred ($Y_i > 0$).

                               ALL EXPOSED POLICIES
                                         │
                  ┌──────────────────────┴──────────────────────┐
                  ▼                                             ▼
           No Claims (Y = 0)                             Claim Occurred (Y > 0)
      (Used in Frequency Model)                       (Used in Severity Model)
                                                                │
                                                 ┌──────────────┴──────────────┐
                                                 ▼                             ▼
                                            Claim-Level                  Policy-Level
                                            Record Grain               Average Severity
                                         (1 row per claim)       (Y_bar = Loss / N, Weight = N)

Data Grain and Actuarial Weighting

Actuaries fit severity models at one of two distinct data grains:

1. Individual Claim Grain (Transaction Level): Each observation represents a single closed or settled claim ($y_i$). The model assigns equal prior weights ($\omega_i = 1$) unless specific inflation trend adjustments or credibility weights apply.
2. Policy or Risk-Cell Grain (Aggregated Level): Each observation represents a policy or rating cell that incurred one or more claims ($N_i \ge 1$). The target response is the average claim severity:

$\bar{Y}_i = \frac{S_i}{N_i} = \frac{\sum_{j=1}^{N_i} Z_{ij}}{N_i}$

Where $S_i$ is the aggregate loss and $Z_{ij}$ is the amount of the $j$-th individual claim on policy $i$.

If individual claims $Z_{ij}$ are independent and identically distributed with mean $\mu_i$ and variance $\text{Var}(Z_{ij}) = \phi V(\mu_i)$, the variance of the average severity is:

$\text{Var}(\bar{Y}_i) = \text{Var}\left( \frac{1}{N_i} \sum_{j=1}^{N_i} Z_{ij} \right) = \frac{1}{N_i^2} \sum_{j=1}^{N_i} \text{Var}(Z_{ij}) = \frac{N_i \phi V(\mu_i)}{N_i^2} = \frac{\phi}{N_i} V(\mu_i)$

In the Exponential Dispersion Family (EDF), the dispersion function is parameterized as $a(\phi) = \frac{\phi}{\omega_i}$. Comparing this with the variance equation demonstrates that the prior weight $\omega_i$ must strictly equal the claim count $N_i$:

$\omega_i = N_i$

An observation with $10$ claims provides ten times the information and exhibits one-tenth the sampling variance of an observation with $1$ claim. Weighting by claim count ensures that parameter estimation is unbiased and statistically efficient.

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The Gamma Distribution for Claim Severity

The Gamma distribution is the standard workhorse for modeling P&C insurance claim severity. Its support is strictly positive ($y > 0$), and its right-skewed profile naturally reflects the asymmetry of property damage and collision losses.

Probability Density Function and EDF Parameterization

In classical statistics, the Gamma distribution is parameterized by shape parameter $\alpha > 0$ and scale parameter $\theta > 0$. In Generalized Linear Models, actuaries reparameterize the distribution in terms of its mean $\mu = \mathbb{E}[Y] = \alpha \theta$ and dispersion parameter $\phi = \frac{1}{\alpha} > 0$:

$f(y; \mu, \phi) = \frac{1}{\Gamma(1/\phi)} \left( \frac{1}{\phi \mu} \right)^{1/\phi} y^{1/\phi - 1} \exp\left( -\frac{y}{\phi \mu} \right), \quad y > 0$

To express this in canonical Exponential Dispersion Family form:

$f(y; \theta_c, \phi) = \exp\left( \frac{y \theta_c - b(\theta_c)}{\phi} + c(y, \phi) \right)$

Where:

• Natural (canonical) parameter: $\theta_c = -\frac{1}{\mu} \implies \mu = -\frac{1}{\theta_c}$
• Cumulant function: $b(\theta_c) = -\ln(-\theta_c) = \ln(\mu)$
• Dispersion parameter: $a(\phi) = \frac{\phi}{\omega_i}$
• Normalizing function: $c(y, \phi) = \frac{1 - \phi}{\phi} \ln(y) - \frac{\ln(\phi)}{\phi} - \ln\Gamma\left(\frac{1}{\phi}\right)$

Derivation of the Variance Function

The mean and variance functions derive directly from the cumulant function $b(\theta_c)$:

$\mu = b'(\theta_c) = -\frac{1}{-\theta_c}(-1) = -\frac{1}{\theta_c}$

$V(\mu) = b''(\theta_c) = \frac{1}{\theta_c^2} = \left( -\frac{1}{\mu} \right)^{-2} = \mu^2$

Thus, the variance of a Gamma random variable with prior weight $\omega_i$ is:

$\text{Var}(Y_i) = \frac{\phi}{\omega_i} V(\mu_i) = \frac{\phi}{\omega_i} \mu_i^2$

The Constant Coefficient of Variation (CV) Property

A critical mathematical property of the Gamma GLM is that its Coefficient of Variation ($CV$) is strictly constant across all risk profiles:

$CV = \frac{\sigma}{\mu} = \frac{\sqrt{\text{Var}(Y)}}{\mu} = \frac{\sqrt{\phi \mu^2}}{\mu} = \sqrt{\phi} = \frac{1}{\sqrt{\alpha}}$

This constant $CV$ property has profound actuarial meaning:

1. Proportional uncertainty is identical across rating tiers: A risk group with an expected claim severity of $1,000 (e.g., minor auto bumper repairs) has a standard deviation of $\sqrt{\phi} \times 1,000$. A risk group with an expected severity of $20,000 (e.g., luxury sports car repairs) has a standard deviation of $\sqrt{\phi} \times 20,000$.
2. If $\phi = 0.25$, then $CV = \sqrt{0.25} = 0.50$ ($50%$) for both risk classes.
3. This reflects property lines where loss volatility scales quadratically with the value of the underlying asset.

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Link Functions for Severity: Log Link vs. Canonical Link

The canonical link function for the Gamma distribution equates the linear predictor $\eta_i = \mathbf{x}_i^T \boldsymbol{\beta}$ to the natural parameter $\theta_c = -1/\mu_i$:

$g(\mu_i) = -\frac{1}{\mu_i} \quad \left(\text{or operational inverse link } g(\mu_i) = \frac{1}{\mu_i}\right)$

Why Actuaries Universally Reject the Canonical Link for Severity

Despite its mathematical appeal (such as minimal sufficient statistics and Newton-Raphson equivalence), the canonical reciprocal link is virtually never used in commercial insurance pricing:

1. Risk of Negative or Undefined Fitted Values: Under the inverse link, $\hat{\mu}_i = \frac{1}{\mathbf{x}_i^T \hat{\boldsymbol{\beta}}}$. If the linear predictor evaluates to zero or a negative number for any policyholder profile, the model predicts an infinite or negative expected claim severity. In insurance, negative claim severities are a physical impossibility that will crash automated rating engines.
2. Non-Multiplicative Rating Structure: The inverse link produces an additive model on the reciprocal scale, which cannot be incorporated into standard multiplicative insurance rating manuals.

The Operational Dominance of the Log Link

Actuaries universally pair the Gamma distribution with the log link function:

$g(\mu_i) = \ln(\mu_i) = \mathbf{x}_i^T \boldsymbol{\beta} \iff \mu_i = \exp(\mathbf{x}_i^T \boldsymbol{\beta})$

This specification provides two indispensable benefits:

1. Strict Positivity: Because $\exp(\eta) > 0$ for all $\eta \in (-\infty, \infty)$, predicted severity is mathematically guaranteed to be strictly positive regardless of predictor combinations.
2. Multiplicative Relativities: The model decomposes directly into standard actuarial rating relativities:

$\mu_i = \exp(\beta_0) \times \exp(\beta_1 x_{i1}) \times \exp(\beta_2 x_{i2}) \times \dots \times \exp(\beta_k x_{ik})$

• The base severity is $\text{Base Rate} = \exp(\beta_0)$.
• Each categorical attribute contributes a multiplicative relativity $\text{Relativity}_j = \exp(\beta_j)$.

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The Inverse Gaussian Distribution for Heavy-Tailed Severity

While the Gamma distribution fits well for well-behaved property damage claims, it frequently underestimates loss costs in commercial lines characterized by severe shock losses, long-tail casualty litigation, and catastrophic personal injury. For these lines, the Inverse Gaussian (IG) distribution provides a superior alternative.

Probability Density Function and EDF Parameterization

The Inverse Gaussian distribution (also known as the Wald distribution) has probability density function:

$f(y; \mu, \phi) = \frac{1}{\sqrt{2\pi \phi y^3}} \exp\left( -\frac{(y - \mu)^2}{2 \phi \mu^2 y} \right), \quad y > 0$

In Exponential Dispersion Family notation:

$f(y; \theta_c, \phi) = \exp\left( \frac{y \theta_c - b(\theta_c)}{\phi} + c(y, \phi) \right)$

Where:

• Natural parameter: $\theta_c = -\frac{1}{2\mu^2} \implies \mu = (-2\theta_c)^{-1/2}$
• Cumulant function: $b(\theta_c) = -\sqrt{-2\theta_c} = -\frac{1}{\mu}$
• Dispersion parameter: $a(\phi) = \frac{\phi}{\omega_i}$
• Normalizing function: $c(y, \phi) = -\frac{1}{2}\ln(2\pi \phi y^3) - \frac{1}{2\phi y}$

Derivation of the Variance Function

Differentiating the cumulant function yields the mean and variance:

$\mu = b'(\theta_c) = -\frac{1}{2}(-2\theta_c)^{-1/2}(-2) = (-2\theta_c)^{-1/2} = \mu$

$V(\mu) = b''(\theta_c) = -\frac{1}{2}(-2\theta_c)^{-3/2}(-2) = (-2\theta_c)^{-3/2} = \left(\frac{1}{\mu^2}\right)^{-3/2} = \mu^3$

Thus, the variance of an Inverse Gaussian random variable is cubic in the mean:

$\text{Var}(Y_i) = \frac{\phi}{\omega_i} V(\mu_i) = \frac{\phi}{\omega_i} \mu_i^3$

Non-Constant Coefficient of Variation in Inverse Gaussian

Because the variance function is $V(\mu) = \mu^3$, the Coefficient of Variation is not constant; it increases with the square root of the mean:

$CV = \frac{\sigma}{\mu} = \frac{\sqrt{\phi \mu^3}}{\mu} = \sqrt{\phi \mu}$

This behavior matches long-tail liability insurance: high-severity risk classes (such as heavy commercial trucking or hazardous industrial manufacturing) exhibit not only higher absolute dollar variances, but also proportionately greater relative volatility than low-severity risk classes.

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Tail Behavior Comparison: Gamma vs. Inverse Gaussian

The fundamental actuarial distinction between the Gamma and Inverse Gaussian distributions lies in their asymptotic tail decay. Consider the right tail behavior as claim size $y \to \infty$:

$\text{Gamma Density Decay:} \quad f_{\text{Gamma}}(y) \propto y^{1/\phi - 1} \exp\left(-\frac{y}{\phi \mu}\right)$

$\text{Inverse Gaussian Density Decay:} \quad f_{\text{IG}}(y) \propto y^{-3/2} \exp\left(-\frac{y}{2\phi \mu^2}\right)$

Density f(y)
    │
    │      ▲
    │     ╱ ╲
    │    ╱   ╲       Gamma (Decays faster in right tail: V(mu) = mu^2)
    │   ╱     ╲─────────────────────────┐
    │  ╱                                └───┐
    │ ╱      Inverse Gaussian                └──────────────────
    │╱       (Heavy right tail: V(mu) = mu^3)    ┌──────────────
    └────────────────────────────────────────────┴───────────────► Claim Size (y)

Why Inverse Gaussian Has a Heavier Right Tail

1. Variance Growth Rate: Because $V(\mu) = \mu^3$ for Inverse Gaussian versus $V(\mu) = \mu^2$ for Gamma, the conditional variance grows at a much faster rate as expected losses increase.
2. Shock Loss Sensitivity: The Inverse Gaussian distribution allocates substantially higher probability mass to extreme positive values ($y \gg \mu$).
3. Actuarial Suitability by Line of Business:

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Model Selection and Actuarial Evaluation Framework

When evaluating competing severity distributions (Gamma vs. Inverse Gaussian), actuaries apply a multi-faceted statistical and operational framework:

1. Unit Deviance Comparison

The unscaled unit deviance $d(y_i, \mu_i)$ reflects the discrepancy between the observed claim size and the fitted mean:

• Gamma Unit Deviance:

$d_{\text{Gamma}}(y, \mu) = 2 \left[ -\ln\left(\frac{y}{\mu}\right) + \frac{y - \mu}{\mu} \right]$

• Inverse Gaussian Unit Deviance:

$d_{\text{IG}}(y, \mu) = \frac{(y - \mu)^2}{\mu^2 y}$

The total deviance $D(\mathbf{y}, \hat{\boldsymbol{\mu}}) = \sum_{i=1}^n \omega_i d(y_i, \hat{\mu}_i)$ summarizes overall discrepancy.

2. Information Criteria (AIC and BIC)

Because Gamma and Inverse Gaussian models have identical parameter counts for the same predictor set, comparing log-likelihoods directly indicates relative goodness-of-fit:

$\text{AIC} = -2\ln(L) + 2k$ $\text{BIC} = -2\ln(L) + k\ln(n)$

The model achieving lower AIC/BIC is preferred. In severe liability lines, Inverse Gaussian frequently achieves an AIC improvement of hundreds or thousands of points over Gamma due to superior tail fit.

3. Normal Q-Q Plots of Deviance Residuals

Deviance residuals for observation $i$ are defined as:

$r_{D, i} = \text{sign}(y_i - \hat{\mu}_i) \sqrt{\omega_i d(y_i, \hat{\mu}_i)}$

Actuaries plot the ordered deviance residuals against standard normal theoretical quantiles. If the Gamma model's residuals curve sharply upward in the upper right quadrant, the observed data possesses a heavier tail than the Gamma distribution can accommodate. Switching to an Inverse Gaussian model often straightens the upper tail along the 45-degree reference line.

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Actuarial Exam Traps & Pitfalls

[!WARNING] Trap 1: Including Zero-Claim Policies in Severity Models Both Gamma and Inverse Gaussian distributions have strictly positive support ($y > 0$). If an actuary mistakenly includes policies with zero claims ($y_i = 0$), evaluating the density or deviance requires computing $\ln(0)$ or division by zero, causing the estimation routine to crash. Severity models must strictly filter data to $y_i > 0$.

[!WARNING] Trap 2: Omitting Prior Weights on Aggregated Severity Data When modeling average claim severity at the rating cell level ($\bar{Y}_i = S_i / N_i$), failure to set prior weights weights = N treats a cell with 1,000 claims as having the same sampling reliability as a cell with 1 claim. This inflates the parameter estimation variance and distorts rating relativities.

[!WARNING] Trap 3: Assuming Inverse Gaussian Has a Constant Coefficient of Variation On exam questions, candidates frequently assume all severity GLMs have constant $CV$. Remember: only the Gamma distribution has constant $CV = \sqrt{\phi}$. The Inverse Gaussian distribution has $CV = \sqrt{\phi \mu}$, which scales directly with the square root of the expected claim size.

[!WARNING] Trap 4: Using the Canonical Inverse Link for Gamma Severity Never specify the canonical link $g(\mu) = -1/\mu$ (or $1/\mu$) for ratemaking. If the linear predictor evaluates to $\le 0$ for favorable rating tiers, it produces negative or infinite expected severities. Always use the log link $g(\mu) = \ln(\mu)$.