Chain Ladder, Flexible Factor Chain Ladder and Linear Regression Models for General Insurance Reserving

In General and Property-Casualty Insurance business, there is a time delay between the event giving rise to the claim and the final settlement of the claim. This time delay can be conspicuous: from few months to many years. For both statutory and internal purposes, the insurer must be able to quantify these future liabilities in order to set up an opportune capital reserve. The aim of reserving regulations is to protect the policyholders against insurer’s insolvency and enable the insurer to assess its financial position correctly.

Actuarial practitioners estimate the reserve by arranging past claim data in the so-called run-off triangle. The oldest and still most popular method of the Chain-Ladder (CL) is a special case of weighted linear regression. Because of the complex longitudinal nature of the data, judgmental intervention is also common. The final estimate is usually a corrected-CL, after "manual" adjustments of some opportune data summaries (called link ratios).

From the nonparametric model of Mack (1993) to the exponential family parametric extension (Taylor (2011)), the literature provided many stochastic models that produces the CL as the best estimator, thus enabling variance and error-interval calculations in many specific circumstances. If those results seem to legitimate the ubiquity of the CL, there are, however, also strong criticisms against. Zenwirth, B. and Barnett, G. (2000) introduce a comprehensive set of regression diagnostics for the CL. The CL is, in fact, a weighted linear regression model with no intercept and a purely interactive effect between explanatory variable and the maturity of the claims aggregate (the so called development period). The authors show that the limitations of the CL are severe even on classical CL-literature datasets. The fit is often inadequate: data show trends, strong pattern of variations between development periods and, often, alternative regression models are preferable to the CL.

Those limitations can at least partially explain the industrial practice of judgmentally adjusting the link ratios. As a result, however, those practices cannot refer to classical CL models for assessing uncertainty as the corresponding prediction intervals would be generally biased.

Murphy (2009) try to solve the impasse by directly justifying the corrected CL practice. In fact, their Flexible Factor Chain Ladder (FFCL) model produces estimates which are consistent with the chain ladder estimates based on judgmentally selected link ratios. In order to achieve this, they generalize the volatility pattern and show a heuristic procedure for trend adjustment and model validation. The validation procedure, however, is less cogent than the one proposed by Zenwirth and Barnett and, for this reason, less persuasive.
More recently, Margetts and Clark (2016) reconsider the extended volatility structure proposed by Bardis, Majidi and Murphy (2009) and derive the maximum likelihood equations for the case in which the FFCL model is not constrained to match the selected link ratios. If the data support the judgmental link ratios, the two procedures should coincide, that is is a way to validate the actuary’s choice.

The main motivation of the current proposal is that the industrial practice has fundamental need of evaluating the stochastic performance of actuarial practices. Hence, our aim is to construct specific methods for model identification and validation and to derive empirical and analytical measures of uncertainty. For this, we will investigate the potentiality of the FFCL in justifying the actuarial practice. In particular, we will use empirical data to derive volatility structures of claim development per line of business, and identify the cases in which the judgmentally corrected CL can be validated by the Margetts and Clark approach. The technical objective is to derive prediction intervals which realistically reflects both sampling error and model error. On the footsteps of Zenwirth and Barnett, moreover, we will also use their extended regression diagnostics for assuring that no simpler model will be excluded from consideration.