5.1 Tweedie Compound Poisson-Gamma Modeling

The Compound Poisson-Gamma Process in P&C Insurance

In property and casualty insurance, pure premium (or loss cost) represents aggregate losses per unit of earned exposure. Traditional ratemaking structures pure premium via two separate Generalized Linear Models: a Poisson model for claim frequency and a Gamma model for claim severity. However, casualty actuaries often model pure premium directly in a single step using the Tweedie Compound Poisson-Gamma distribution.

Consider an insurance portfolio where an insured unit over a defined coverage period experiences an unknown number of claims $N$, each generating a loss amount $Z_i$:

$$Y = \sum_{i=1}^N Z_i$$

Where:

1. Claim frequency $N$ follows a Poisson distribution with intensity parameter $\lambda > 0$:

$$P(N = n) = \frac{e^{-\lambda} \lambda^n}{n!}, \quad n \in \{0, 1, 2, \dots\}$$

1. Individual claim severities $Z_1, Z_2, \dots, Z_N$ are independent, identically distributed Gamma random variables with shape parameter $\alpha > 0$ and scale parameter $\theta > 0$:

$$Z_i \stackrel{iid}{\sim} \text{Gamma}(\alpha, \theta), \quad \mathbb{E}[Z_i] = \alpha\theta, \quad \text{Var}(Z_i) = \alpha\theta^2$$

1. The count $N$ and the individual claim amounts $Z_i$ are mutually independent.
2. If $N = 0$, the aggregate loss is identically zero: $Y = 0$.

Mean and Variance of Aggregate Losses

Applying the Law of Total Expectation:

$$\mu = \mathbb{E}[Y] = \mathbb{E}\Big[ \mathbb{E}[Y \mid N] \Big] = \mathbb{E}[N \cdot \mathbb{E}[Z]] = \mathbb{E}[N] \cdot \mathbb{E}[Z] = \lambda \alpha \theta$$

Applying the Law of Total Variance (Eve's Law):

$$\text{Var}(Y) = \mathbb{E}\Big[ \text{Var}(Y \mid N) \Big] + \text{Var}\Big( \mathbb{E}[Y \mid N] \Big)$$ $$\text{Var}(Y) = \mathbb{E}[N \cdot \text{Var}(Z)] + \text{Var}(N \cdot \mathbb{E}[Z]) = \mathbb{E}[N] \text{Var}(Z) + (\mathbb{E}[Z])^2 \text{Var}(N)$$

Substituting the Poisson and Gamma moments:

$$\text{Var}(Y) = \lambda (\alpha \theta^2) + (\alpha \theta)^2 (\lambda) = \lambda \alpha \theta^2 (1 + \alpha)$$

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The Tweedie Exponential Dispersion Family Representation

In 1984, British statistician Maurice Tweedie demonstrated that this Compound Poisson-Gamma process belongs to the Exponential Dispersion Family (EDF) with a specific power variance function:

$$V(\mu) = \mu^p, \quad \text{where } 1 < p < 2$$ $$\text{Var}(Y) = \phi V(\mu) = \phi \mu^p$$

Where $\phi$ is the dispersion parameter and $p$ is known as the Tweedie index (power) parameter.

Spectrum of Tweedie Distributions

The Tweedie family encompasses several classical distributions depending on the value of $p$:

                               TWEEDIE INDEX SPECTRUM (p)

       p = 0             p = 1             1 < p < 2            p = 2           p = 3
  ───────┼─────────────────┼───────────────────┼──────────────────┼───────────────┼───────►
       Normal           Poisson             TWEEDIE             Gamma          Inverse
      (Linear)        (Frequency)       (Pure Premium)       (Severity)       Gaussian
                                        Compound P-G                        (Heavy Tail)
                                        [Mass at Zero!]

The Point Mass at Zero

The defining feature of the Tweedie distribution with $1 < p < 2$ is that it is not purely continuous, but rather a mixed discrete-continuous distribution. It exhibits an exact point mass at zero corresponding to policies that incurred no claims ($N = 0$):

$$P(Y = 0) = P(N = 0) = e^{-\lambda} > 0$$

For $y > 0$, the distribution is strictly continuous with a smooth, right-skewed density. In a personal auto portfolio where $90\%$ of drivers have no claims during the year, the empirical pure premium distribution has a spike of probability at $0 ($90%$ mass) followed by a continuous positive distribution for the $10\%$ who incurred losses. The Tweedie distribution matches this physical reality.

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Mathematical Mapping: Tweedie vs. Poisson-Gamma Parameters

There is an exact, invertible algebraic mapping between the Tweedie GLM parameters $(\mu, \phi, p)$ and the underlying compound Poisson-Gamma parameters $(\lambda, \alpha, \theta)$:

Derivation of the Transformation

We equate the Tweedie mean and variance to the Poisson-Gamma mean and variance:

$$\mu = \lambda \alpha \theta$$ $$\text{Var}(Y) = \phi \mu^p = \lambda \alpha \theta^2 (1 + \alpha)$$

Dividing the variance by the mean:

$$\frac{\text{Var}(Y)}{\mu} = \frac{\phi \mu^p}{\mu} = \phi \mu^{p-1} = \frac{\lambda \alpha \theta^2 (1 + \alpha)}{\lambda \alpha \theta} = \theta (1 + \alpha)$$ $$\theta = \frac{\phi \mu^{p-1}}{1 + \alpha}$$

Substituting $\theta$ back into the mean expression $\mu = \lambda \alpha \theta$:

$$\mu = \lambda \alpha \left( \frac{\phi \mu^{p-1}}{1 + \alpha} \right) \implies \lambda = \frac{\mu^{2-p}}{\phi} \left( \frac{1 + \alpha}{\alpha} \right)$$

By matching the power dependencies of the cumulant function within the Exponential Dispersion Family, the relationship between $p$ and the Gamma shape $\alpha$ is established as:

$$p = \frac{\alpha + 2}{\alpha + 1}$$

Solving for the Gamma shape parameter $\alpha$:

$$\alpha(p - 1) = 2 - p \implies \alpha = \frac{2 - p}{p - 1}$$

Because $1 < p < 2$, both $(2 - p) > 0$ and $(p - 1) > 0$, ensuring $\alpha > 0$.

Substituting $\alpha$ into the expressions for $\lambda$ and $\theta$ yields the fundamental actuarial bridge formulas:

$$\textbf{Gamma Shape:} \quad \alpha = \frac{2 - p}{p - 1}$$ $$\mathbf{Poisson\;Mean:} \quad \lambda = \frac{\mu^{2-p}}{\phi (2 - p)}$$ $$\textbf{Gamma Scale:} \quad \theta = \phi (p - 1) \mu^{p-1}$$

Actuarial Implication for Zero-Claim Probability

Substituting $\lambda$ into the point mass at zero yields:

$$P(Y = 0) = \exp(-\lambda) = \exp\left( -\frac{\mu^{2-p}}{\phi (2 - p)} \right)$$

Because $2 - p > 0$, as expected pure premium $\mu$ increases, the exponent becomes more negative, causing $P(Y = 0)$ to decrease. Higher-risk policyholder segments have a substantially lower probability of remaining claim-free, precisely matching actuarial observations.

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Direct Pure Premium vs. Two-Stage Frequency-Severity Modeling

When pricing personal or commercial lines, casualty actuaries must decide between fitting a single direct Tweedie GLM or building a two-stage frequency-severity architecture.

                         PURE PREMIUM MODELING PARADIGMS
                                        │
                ┌───────────────────────┴───────────────────────┐
                ▼                                               ▼
      DIRECT TWEEDIE MODEL                         TWO-STAGE ARCHITECTURE
      Response: Loss / Exposure                    Stage 1: Frequency GLM (Poisson)
      Prior Weights: Exposure                      Stage 2: Severity GLM (Gamma/IG)
      Link: Log                                    Pure Premium = Frequency * Severity
      Single Objective Optimization                Multiplicative Combination

Comprehensive Actuarial Comparison

Strategic Guidance for Exam and Practice

• Use Direct Tweedie when: Speed to market, operational simplicity, and direct loss cost optimization are paramount, particularly in high-volume personal lines (personal auto comprehensive, renters) with uniform policy limits.
• Use Two-Stage Models when: Analyzing lines with high limit variability, significant deductible options (commercial property), or when underwriters require separate frequency and severity relativities to design underwriting guidelines.

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Estimating and Profiling the Index Parameter $p$

In standard Generalized Linear Models (such as Poisson or Gamma), the variance power $p$ is known ($p = 1$ or $p = 2$), allowing the parameters $\boldsymbol{\beta}$ to be solved directly via Iteratively Reweighted Least Squares (IRLS). In Tweedie models, however, the power parameter $p$ is unknown and enters the variance function nonlinearly ($V(\mu) = \mu^p$).

Because $p$ determines the Exponential Dispersion Family density itself, $p$ cannot be estimated simultaneously with $\boldsymbol{\beta}$ using standard IRLS. Instead, actuaries employ Profile Log-Likelihood Optimization.

The Profile Log-Likelihood Algorithm

                               PROFILE LIKELIHOOD ALGORITHM
                                            │
               ┌────────────────────────────┴────────────────────────────┐
               ▼                                                         ▼
      1. Define Grid for p                                      2. Loop over each p_k
         p in [1.05, 1.95], step = 0.05                            Fit Tweedie GLM via IRLS
                                                                   Obtain beta_hat(p_k), phi_hat(p_k)
                                                                         │
               ┌─────────────────────────────────────────────────────────┘
               ▼
      3. Calculate Exact Log-Likelihood
         Evaluate l_profile(p_k) using Dunn & Smyth series expansion
               │
               ▼
      4. Identify Optimal p*
         p* = argmax l_profile(p_k)
         Refine via golden-section search near p*

1. Define Search Interval: Construct a discrete grid over the theoretical domain $p \in (1.01, 1.99)$ (e.g., $p \in \{1.1, 1.2, \dots, 1.9\}$, refined to step size $0.01$).
2. Conditional IRLS Fit: For each fixed candidate value $p_k$:
   • Fit the GLM for $\boldsymbol{\beta}$ using standard IRLS with variance function $V(\mu) = \mu^{p_k}$.
   • Estimate the dispersion parameter $\hat{\phi}(p_k)$ from the Pearson statistic.
3. Evaluate Exact Log-Likelihood: Evaluate the full log-likelihood $\ell(\hat{\boldsymbol{\beta}}(p_k), \hat{\phi}(p_k); p_k)$. Because the Tweedie normalizing function $c(y, \phi)$ has no closed-form analytic expression for $1 < p < 2$, software evaluates the infinite series representation using the Dunn and Smyth (2005) Fourier inversion algorithm.
4. Maximize Profile Curve: Plot the profile log-likelihood curve against $p$ and select the optimal value:

$$p^* = \arg\max_{p \in (1, 2)} \ell_{\text{profile}}(p)$$

Actuarial Interpretation of the Optimal $p$

• Lower values ($p \to 1.1 - 1.3$): Indicate that pure premium volatility is dominated by claim frequency. Claims occur with Poisson-like variation, while individual claim severities are relatively constant (low variance). Typical for auto glass or roadside assistance.
• Higher values ($p \to 1.7 - 1.9$): Indicate that pure premium volatility is dominated by claim severity. Claim frequency is low, but loss sizes exhibit extreme right-skewness and high variance. Typical for commercial liability and excess property.

Software Implementation Reference

• R (tweedie and cplm packages):

  library(tweedie)
  # Step 1: Profile p over a grid
  p_prof <- tweedie.profile(pure_premium ~ age + territory, 
                            weights = exposure, data = df, 
                            p.vec = seq(1.1, 1.9, by = 0.05))
  best_p <- p_prof$p.max
  
  # Step 2: Fit model with optimal p
  fit <- glm(pure_premium ~ age + territory, 
             family = tweedie(var.power = best_p, link.power = 0), 
             weights = exposure, data = df)
  

• Python (statsmodels and scikit-learn):

  # scikit-learn TweedieRegressor with log link
  from sklearn.linear_model import TweedieRegressor
  reg = TweedieRegressor(power=1.5, link='log')
  reg.fit(X, y_pure_premium, sample_weight=df['exposure'])
  

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