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Ludkovski, Michael, and Howard Zail. 2022. “Gaussian Process Models for Incremental Loss Ratios.” Variance 15 (1).
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  • 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
  • B. Step-Ahead RMSE
  • Figure 8. RMSE of cumulative loss ratios as a function of step-ahead \(n\) across the six business lines
  • C. Typical ILR Triangle

    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 \(L_{p,q}\)


We develop Gaussian process (GP) models for incremental loss ratios in loss development triangles. Our approach brings a machine learning, spatial-based perspective to stochastic loss modeling. GP regression offers a nonparametric probabilistic distribution regarding future losses, capturing uncertainty quantification across three distinct layers—model risk, correlation risk, and extrinsic uncertainty due to randomness in observed losses. To handle statistical features of loss development analysis—namely, spatial nonstationarity, convergence to ultimate claims, and heteroskedasticity—we develop several novel implementations of fully Bayesian GP models. We perform extensive empirical analyses over the NAIC loss development database across six business lines, comparing and demonstrating the strong performance of our models. Our computational work is performed using the R and Stan programming environments and is publicly shareable.

Accepted: June 16, 2020 EDT