Turcotte, Roxane, Hélène Cossette, and Mathieu Pigeon. 2021. “Working with a Parametric Copula-Based Model for Individual Non-Life Loss Reserving.” Variance 14 (2).
• Figure 1. Typical development pattern of a claim in non-life insurance
• Table 1. Information available in the data set
• Table 2. Additional features of the data set
• Table 3. Descriptive statistics for total payments of closed claims
• Table 4. Incremental run-off triangle for AB (in $100,000) • Table 5. Incremental run-off triangle for BI (in$100,000)
• Table 6. Incremental run-off triangle for APD (in $100,000) • Table 7. Prediction results for collective approaches • Table 8. Final models for the time structure • Table 9. Descriptive statistics for closed claims • Table 10. Information criteria and estimated values for the selected marginal distributions • Table 11. Results for elliptical copulas • Table 12. Information criteria for the dependence structure • Figure 2. Predictive distribution based on a normal copula for AB (left) and BI (right) claims • Figure 3. Predictive distribution based on a Student copula for BI (left) and APD (right) claims • Table 13. Prediction results for the copula-based individual approach (in$)

## Abstract

In this paper, we propose a generalization of the individual loss reserving model introduced by Pigeon et al. (2013) considering a discrete time framework for claims development. We use a copula to model the potential dependence within the development structure of a claim, which allows a wide variety of marginal distributions. We also add a specific component to consider claims closed without payment. We provide a case study based on a detailed personal auto insurance data set from a North American insurance company.

Accepted: June 16, 2020 EDT

# Appendix: Distributions

The pdf of a lognormal rv is

\begin{aligned} f(x) &= \frac{1}{x \theta_2 \sqrt{2 \pi}} \exp\left( \dfrac{- \left(\ln(x) - \theta_1\right)^2}{2 \theta_2^2} \right)\mathbb{I}_{(0,\infty)}(x)\end{aligned}

where $$\theta_1 \in \mathbb{R}$$ and $$\theta_2 \in \mathbb{R^*_+}.$$

The pdf of a Pareto Type II rv is

\begin{aligned} f(x) &= k \frac{\theta_1^{\theta_2}}{(\theta_1+x)^{\theta_2+1}}\mathbb{I}_{(0,\infty)}(x)\end{aligned}

where $$\theta_1 \in \mathbb{R^*_+}$$ and $$\theta_2 \in \mathbb{R^*_+}.$$

Finally, the pdf of a Burr rv is

\begin{aligned} f(x) &= \theta_1 \theta_2 \times \frac{x^{\theta_2-1}}{ \left( 1 + x^{\theta_2} \right)^{\theta_1 + 1}}\mathbb{I}_{(0,\infty)}(x)\end{aligned}

where $$\theta_1 \in \mathbb{R^*_+}$$ and $$\theta_2 \in \mathbb{R^*_+}.$$