Ludkovski, Michael, and Howard Zail. 2022. “Gaussian Process Models for Incremental Loss Ratios.” Variance 15 (1).
• Figure 1. Distribution of ILRs for each of the six business lines
• Figure 2. Training a GP model for a toy one-dimensional example
• Figure 3. Three-dimensional view of the loss square for a representative comauto data set with red dots indicating the training $$L_{p,q}$$’s and black dots indicating the bottom triangle to be completed
• Figure 4. Predictive distribution of the incremental loss ratios $$L_{p,q}$$ for three representative accident years
• Figure 5. Left: Percentile rank of realized ultimate losses in terms of the predictive distribution of the ILR-Hurdle+Virt model across 57 wkcomp triangles. Right: Kolmogorov-Smirnov test across three models for wkcomp
• Figure 6. Lengthscales $$\rho_{AY}$$ and $$\rho_{DL}$$ (left panel), observation variance $$\sigma_q$$ (middle panel), and hurdle probability $$h_q$$ (right panel) for three representative business lines
• Table 1. Results for wkcomp
• Figure 7. Top row: 1,000 conditional simulations of future cumulative losses $$CC_{p,q}$$ for AY $$= 1995, q=1,\ldots,10$$ and a representative comauto triangle. The solid cyan line is the predictive mean of $$CC_{p,q}$$, and the dashed red line represents the actual realized losses. (Left: ILR-Plain model; Right: ILR-Hurdle+Virt model.) Bottom row: Predictive density of $$R_{ult} = \sum_p CC_{p,Q}$$, together with the realized ultimate losses (vertical line) for the same triangle. (Left: GP ILR-Plain model; Right: Partial Bayesian versus Full Bayesian for ILR-Hurdle+Virt)
• A. Results for All Business Lines
• Figure 8. RMSE of cumulative loss ratios as a function of step-ahead $$n$$ across the six business lines
Table 2a. Sample run-off triangle from comauto with accident years $$p$$ in rows and development lags $$q$$ in columns: Cumulative losses $$CC_{p,q}$$
• Table 2b. Sample run-off triangle from comauto with accident years $$p$$ in rows and development lags $$q$$ in columns: Incremental loss ratios $$L_{p,q}$$