This paper considers the regression model y = Xβ+e with all the classical assumptions (including normality) but one, viz. it is assumed that the covariance matrix of the disturbances depends upon a finite number of unknown parameters θ1 … θm. The paper gives a method to derive simultaneously the maximum likelihood estimates of β and θ. Also the information matrix is presented. It is proved that β is unbiased if its mean exists. Conditions are given under which the maximum likelihood estimates are consistent, asymptotically normal, and asymptotically efficient. Finally, applications are given to the autocorrelated errors model and to Zellner-type regressions.

Maximum likelihood estimation of the GLS model with unknown parameters in the disturbance covariance matrix

As part of the new regulatory framework of Solvency II, introduced by the European Union, insurance companies are required to monitor their solvency by computing a key risk metric called the Solvency Capital Requirement (SCR). The official description of the SCR is not rigorous and has lead researchers to develop their own mathematical frameworks for calculation of the SCR. These frameworks are complex and are difficult to implement. Recently, Bauer et al. suggested a nested Monte Carlo (MC) simulation framework to calculate the SCR. But the proposed MC framework is computationally expensive even for a simple insurance product. In this paper, we propose incorporating a neural network approach into the nested simulation framework to significantly reduce the computational complexity in the calculation. We study the performance of our neural network approach in estimating the SCR for a large portfolio of an important class of insurance products called Variable Annuities (VAs). Our experiments show that the proposed neural network approach is both efficient and accurate.

Efficient Valuation of SCR via a Neural Network Approach

Properly and thoroughly documenting a crime scene is of utmost importance in solving a crime, yet the process is often difficult and labor intensive. 3D scanning technology allows for the rapid acquisition of highly detailed 3D models of a physical environment. In recent years, adoption of 3D scanning technology has increased amongst crime scene investigation (CSI) units across the world to aid in the investigative process. The technology allows investigators to analyze crime scenes at a later point in time with a scanned 3D model. Unfortunately, the adoption cost of this technology remains a detractor to many CSI units. The presented research seeks to better understand whether investing in this technology results in an overall benefit for CSI units, and if so, when and how? To answer these questions, the research team developed a formal cost-benefit analysis algorithm for 3D scanning technology adoption. Considerations of the algorithm were influenced by actual 3D scanning results of mock crime scenes, together with a focus group of crime scene investigators that discussed the technology. A 100,000 count Monte-Carlo simulation of the developed cost-benefit algorithm enabled researchers to seek an answer to the question of adoption. The results of the simulation are presented in this document as well as through an interactive web application.

A cost-benefit analysis of 3D scanning technology for crime scene investigation

A variable annuity (VA) is an equity-linked annuity that provides investment guarantees to its policyholder and its contributions are normally invested in multiple underlying assets (e.g., mutual funds), which exposes VA liability to significant market risks. Hedging the market risks is therefore crucial in risk managing a VA portfolio as the VA guarantees are long-dated liabilities that may span decades. In order to hedge the VA liability, the issuing insurance company would need to construct a hedging portfolio consisting of the underlying assets whose positions are often determined by the liability Greeks such as partial dollar Deltas. Usually, these quantities are calculated via nested simulation approach. For insurance companies that manage large VA portfolios (e.g., 100k+ policies), calculating those quantities is extremely time-consuming or even prohibitive due to the complexity of the guarantee payoffs and the stochastic-on-stochastic nature of the nested simulation algorithm. In this paper, we extend the surrogate model-assisted nest simulation approach in Lin and Yang [(2020) Insurance: Mathematics and Economics, 91, 85–103] to efficiently calculate the total VA liability and the partial dollar Deltas for large VA portfolios with multiple underlying assets. In our proposed algorithm, the nested simulation is run using small sets of selected representative policies and representative outer loops. As a result, the computing time is substantially reduced. The computational advantage of the proposed algorithm and the importance of dynamic hedging are further illustrated through a profit and loss (P&L) analysis for a large synthetic VA portfolio. Moreover, the robustness of the performance of the proposed algorithm is tested with multiple simulation runs. Numerical results show that the proposed algorithm is able to accurately approximate different quantities of interest and the performance is robust with respect to different sets of parameter inputs. Finally, we show how our approach could be extended to potentially incorporate stochastic interest rates and estimate other Greeks such as Rho.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/65AEE4E4D8D06B1060D3414751575F9F/S0515036120000264a.pdf/div-class-title-efficient-dynamic-hedging-for-large-variable-annuity-portfolios-with-multiple-underlying-assets-div.pdf

Efficient dynamic hedging for large variable annuity portfolios with multiple underlying assets

Under the Solvency II regime, life insurance companies are asked to derive their solvency capital requirements from the full loss distributions over the coming year. Since the industry is currently far from being endowed with sufficient computational capacities to fully simulate these distributions, the insurers have to rely on suitable approximation techniques such as the least-squares Monte Carlo (LSMC) method. The key idea of LSMC is to run only a few wisely selected simulations and to process their output further to obtain a risk-dependent proxy function of the loss. In this paper, we present and analyze various adaptive machine learning approaches that can take over the proxy modeling task. The studied approaches range from ordinary and generalized least-squares regression variants over generalized linear model (GLM) and generalized additive model (GAM) methods to multivariate adaptive regression splines (MARS) and kernel regression routines. We justify the combinability of their regression ingredients in a theoretical discourse. Further, we illustrate the approaches in slightly disguised real-world experiments and perform comprehensive out-of-sample tests.

https://www.mdpi.com/2227-9091/8/1/21/pdf

Machine Learning in Least-Squares Monte Carlo Proxy Modeling of Life Insurance Companies

The Solvency II directive asks insurance companies to derive their solvency capital requirement from the full loss distribution over the coming year. While this is in general computationally infeasible in the life insurance business, an application of the Least-Squares Monte Carlo (LSMC) method offers a possibility to overcome this computational challenge. We outline in detail the challenges a life insurer faces, the theoretical basis of the LSMC method and the necessary steps on the way to a reliable proxy modeling in the life insurance business. Further, we illustrate the advantages of the LSMC approach via presenting (slightly disguised) real-world applications.

/pdf/a-least-squares-monte-carlo-framework-in-proxy-modeling-of-485cohjtrp.pdf

A Least-Squares Monte Carlo Framework in Proxy Modeling of Life Insurance Companies

Under the Solvency II regime, life insurance companies are asked to derive their solvency capital requirements from the full loss distributions over the coming year. Since the industry is currently far from being endowed with sufficient computational capacities to fully simulate these distributions, the insurers have to rely on suitable approximation techniques such as the least-squares Monte Carlo (LSMC) method. The key idea of LSMC is to run only a few wisely selected simulations and to process their output further to obtain a risk-dependent proxy function of the loss. In this paper, we present and analyze various adaptive machine learning approaches that can take over the proxy modeling task. The studied approaches range from ordinary and generalized least-squares regression variants over GLM and GAM methods to MARS and kernel regression routines. We justify the combinability of their regression ingredients in a theoretical discourse. Further, we illustrate the approaches in slightly disguised real-world experiments and perform comprehensive out-of-sample tests.

The least-squares Monte Carlo method has proved to be a suitable approximation technique for the calculation of a life insurer’s solvency capital requirements. We suggest to enhance it by the use of a neural network based approach to construct the proxy function that models the insurer’s loss with respect to the risk factors the insurance business is exposed to. After giving a mathematical introduction to feed forward neural networks and describing the involved hyperparameters, we apply this popular form of neural networks to a slightly disguised data set from a German life insurer. Thereby, we demonstrate all practical aspects, such as the hyperparameter choice, to obtain our candidate neural networks by bruteforce, the calibration (“training”) and validation (“testing”) of the neural networks and judging their approximation performance. Compared to adaptive OLS, GLM, GAM and FGLS regression approaches, an ensemble built of the 10 best derived neural networks shows an excellent performance. Through a comparison with the results obtained by every single neural network, we point out the significance of the ensemble-based approach. Lastly, we comment on the interpretability of neural networks compared to polynomials for sensitivity analyses.

https://www.mdpi.com/2227-9091/8/4/116/pdf

Anne-Sophie Krah

Papers

A Least-Squares Monte Carlo Framework in Proxy Modeling of Life Insurance Companies

Machine Learning in Least-Squares Monte Carlo Proxy Modeling of Life Insurance Companies

Machine Learning in Least-Squares Monte Carlo Proxy Modeling of Life Insurance Companies

Least-Squares Monte Carlo for Proxy Modeling in Life Insurance: Neural Networks