From LDFs to DFT, the latest issue of Variance (volume 8, number 1) offers some tools actuaries can use to enhance their work products.
Actuaries quite often have to interpolate data to obtain quantities such as loss development factors (LDFs) for maturities in between the maturities included in a loss development triangle, or increased limits factors for limits between the data points used in the increased limits analysis. “Interpolation Along a Curve” by Joseph Boor presents an approach that includes the advantages of using fitted curves for nonlinear data, avoiding the errors arising from mismatches between patterns in the data and patterns inherent to the curve family used for interpolation.
Sebastian Happ, Ramona Maier and Michael Merz, in “Multivariate Bühlmann-Straub Credibility Model Applied to Claims Reserving for Correlated Run-off Triangles,” consider the claims reserving problem in a multivariate context and apply the multivariate generalization of Bühlmann and Straub’s 1970 credibility model to claims reserving. This multivariate model allows for a simultaneous study of N correlated run-off portfolios and enables the derivation of an estimator of the conditional mean square error of prediction for the credibility predictor of the ultimate claim of the total portfolio. The authors apply multivariate credibility predictors that reflect the correlation structure between the N portfolios and are optimal in terms of a classical optimality criterion. The results are illustrated by means of an example and comparison to the results derived by the multivariate chain-ladder method and the multivariate additive loss reserving method proposed by Merz and Wüthrich in 2008.
Liang Peng and Ruodu Wang, in “Interval Estimation for Bivariate t-Copulas via Kendall’s Tau,” present a copula model for risk management. Due to the properties of asymptotic dependence and easy simulation, the t-copula has often been employed in practice. A computationally simple estimation procedure for the t-copula is to first estimate the linear correlation via Kendall’s tau estimator and then to estimate the parameter of the number of degrees of freedom by maximizing the pseudo-likelihood function. This paper derives the asymptotic limit of this two-step estimator, which results in a complicated asymptotic covariance matrix. Further, the authors propose jackknife empirical likelihood methods to construct confidence intervals/regions for the parameters and the tail dependence coefficient without estimating any additional quantities. A simulation study shows that the proposed methods perform well in finite sample.
“Estimating Insurance Attrition Using Survival Analysis,” by Luyang Fu and Hongyuan Wang, uses survival analysis to estimate attrition and retention. Compared with conventional methods, this approach has three advantages: (1) it addresses not only whether the policy will leave but also when it will leave; (2) it analyzes mid-term cancellation and end-term nonrenewal sequentially, and therefore provides a dynamic insight of retention, which improves the static view derived from snapshot data; (3) it can take into account time-varying macroeconomic variables and help researchers to understand how the broader economic environment affects insurance retention. A case study illustrates the technique, from creating the panel data required by survival analysis to interpreting the model results.
“The Discrete Fourier Transform and Cyclical Overflow” by Leigh J. Halliwell is an introduction to the discrete Fourier transform (DFT). Halliwell notes that more casualty actuaries would employ the DFT if they understood it better. In addition to being an introduction to the DFT, this paper explains how the DFT treats the probability of amounts that overflow its upper bound, a topic that others either have not noticed or have deemed of little importance. The cyclical overflow originates in the modular arithmetic whereby the DFT evaluates characteristic functions. To understand this is to attain a deeper understanding of the DFT, which may lead to its wider use.
Donna Royston is the CAS publications production coordinator.
