Effectiveness of Stochastic Neural Network for Prediction of Fall or Rise of TOPIX

Stochastic neural network is a hierarchical network of stochastic neurons which emit 0 or 1 with the probability determined by the values of inputs. We have developed an efficient training algorithm so as to maximize the likelihood of such a neural network. This algorithm enables us to apply the stochastic neural network to a practical problem like prediction of fall or rise of Tokyo Stock Price Index (TOPIX). We trained it with the data from 1994 to 1996 and predicted the fall or rise of 1 day ahead of TOPIX for the period from 1997 to 2000. The result is quite promising. The accuracy of the prediction of the stochastic network is the 60.28%, although those of non-stochastic neural network, autoregressive model and GARCH model are 50.02, 51.38 and 57.21%, respectively. However, the stochastic neural network is not so advantageous over other networks or models for prediction of the TOPIX used for training. This means that the stochastic neural network is less over fitting to the training data than others, and results in the best prediction. We will demonstrate how the stochastic neural network learns well non-linear structure behind of the data in comparison to other models or networks, including Generalized Linear model (GLM).JEL codes: D24, L60, 047     

Spatial statistical modelling of insurance risk: a spatial epidemiological approach to car insurance

Spatial models, such as the Besag, York and Mollie (BYM) model, have long been used in epidemiology and disease mapping. A common research question in these subjects is modelling the number of disease events per region; here the BYM models provides a holistic framework for both covariates and dependencies between regions. We use these tools to assess the relative insurance risk associated with the policyholders geographical location. A Bayesian modelling approach is presented and an elastic net is used to reduce the large number of possible geographic covariates. The final inference is performed using Integrated Nested Laplace Approximation. The model is applied to car insurance data from If P&C Insurance together with spatially referenced covariate data of high resolution, provided by Insightone. The entire analysis is performed using freely available R-packages. Including spatial dependence when modelling the number of claims significantly improves on the result obtained using ordinary generalised linear models. However, the support for adding a spatial component to the model for claims cost is weaker.     

On Stochastic Approximation and Credibility

Stochastic approximation is a powerful tool for sequential estimation of zero points of a function. This methodology is defined and is shown to be related to a broad class of credibility formulae derived for the Exponential Dispersion Family (EDF). We further consider a Location Dispersion Family (LDF) which is rich enough and for which no simple credibility formula exists. For this case, a Generalized Sequential Credibility Formula is suggested and an optimal stepwise gain sequence is derived.     

Credibility ratemaking using collateral information

Credibility ratemaking is a technique used in pricing health care, property and casualty, workers’ compensation, and group life coverages. It has been a part of actuarial practice since the time of Mowbray's (1914) contribution. In earlier work, we showed how many types of credibility models could be expressed as special cases of mixed linear models. This article extends this approach to credibility by formally introducing collateral information through the use of Bayesian methods.

Specifically, we derive credibility estimators and mean square errors for normal hierarchical linear models. We provide intuition for the credibility estimators by establishing the link between these estimators and homogeneous and inhomogeneous estimators that appear in non-Bayesian credibility theory.     

Recurrences formulae for the semi-invariants of some discontinuous frequency functions of n variables

Abstract

Professor Ragnar Frisch has as is well-known given a recurrence formula for the semi-invariants of the Bernouillian frequency function of one variable. Mr. Paul Quale has later generalised his result to certain frequency functions (Pascal and Poisson laws).     

Improved approximations for the aggregate claims distribution of a life insurance portfolio

Abstract

In an earlier paper the author derived a recursion formula which permits the exact computation of the aggregrate claims distribution in the individual life model. This exact procedure requires of course more computing time than approximative methods such as Kornya's algorithm, which seemed to be the best compromise between accuracy and computational effort. In the present paper it is shown that, to save time, the exact formula can be used in an approximative way and that the corresponding error bound is smaller than the one of the Kornya-type approximations.     

Author’s Reply: Skewness and Stock Option Prices - Discussion by Terence Chan,Michel Jacques,Heinz H. Mu¨ller,Geérard Pafumi,Elias S.W. Shiu,and Serena Tiong,July 1997

Abstract

In recent years, the combined effects of deregulation in financial services, along with advances in telecommunications and information technology, are forcing far-reaching changes upon the insurance industry. The result is the industry is becoming more competitive. The emerging role of electronic commerce (e-commerce) is particularly important and interesting to study.

I offer a brief survey of the role of e-commerce in the insurance industry. The paper is organized in the following manner: Section 1 summarizes Internet trends and discusses various related public policy issues; Section 2 addresses online insurance supply and demand; Section 3 discusses the economics of disintermediation and reintermediation and explains how this applies to e-commerce in the insurance industry. Finally, Section 4 offers a set of concluding remarks.     

Comparison of Some Bayesian Analyses of Heterogeneity in Group Life Insurance

Norberg (1989) analyses the heterogeneity in a portfolio of group life insurances using a parametric empirical Bayesian approach. In the present paper the model of Norberg is compared to a parametric fully Bayesian model and to a non-parametric fully Bayesian model.     

A Note on Compound Generalized Distributions: A Comment on the Article by B. Kling and M. Goovaerts

Abstract

In this paper, the recursion for Generalized Power Series Distributions (GPSD) recently developed by Kling & Goovaerts (1993) is reviewed. Some errors are discovered in their results. Corrected and slightly more general results are proposed here.     

Bonus-Malus Systems

Abstract

Bonus-malus is a merit-rating technique used in most of Europe and Asia, and some Latin American and African countries. Policyholders from a given risk cell are subdivided into bonus-malus classes. Their claims histories then modify the class upon each renewal. Markov chain theory provides the tools for the design, evaluation, and comparison of these systems. In this article, definitions and examples of bonus-malus systems are provided (Section 2). The main actuarial tools for the study and design of bonus-malus systems are reviewed (Section 3). In the discussions that follow, Krupa Subramanian outlines a model for analyzing market shares in a competitive environment, a crucial research topic given current deregulation trends, and Pierre Lemaire compares actuarial with regulatory approaches to bonus-malus.     

Bayesian Risk Measures for Derivatives via Random Esscher Transform

Abstract

This paper proposes a model for measuring risks for derivatives that is easy to implement and satisfies a set of four coherent properties introduced in Artzner et al. (1999). We construct our model within the context of Gerber-Shiu’s option-pricing framework. A new concept, namely Bayesian Esscher scenarios, which extends the concept of generalized scenarios, is introduced via a random Esscher transform. Our risk measure involves the use of the risk-neutral Bayesian Esscher scenario for pricing and a family of real-world Bayesian Esscher scenarios for risk measurement. Closed-form expressions for our risk measure can be obtained in some special cases.     

Recursions for Distribution Functions and Stop-Loss Transforms

For any function f on the non-negative integers, we can evaluate the cumulative function o f given by o f ( s )= ~ s x=0 f ( x ) from the values of f by the recursion o f ( s )= o f ( s -1)+ f ( s ). Analogously we can use this procedure t times to evaluate the t -th order cumulative function o t f . As an alternative, in the present paper we shall derive recursions for direct evaluation of o t f when f itself satisfies a certain sort of recursion. We shall also derive recursions for the t -th order tails v t f where v f ( s )= ~ X x=s+1 f ( x ). The recursions can be applied for exact and approximate evaluation of distribution functions and stop-loss transforms of probability distributions. The class of recursions for f includes the classes discussed by Sundt (1992), incorporating the class studied by Panjer's (1981). We discuss in particular convolutions and compound functions.     

Exchange Rates and Fundamentals: A General Equilibrium Exploration

Engel and West (2005) show that the observed near random‐walk behavior of nominal exchange rates is an equilibrium outcome of a partial equilibrium asset approach when economic fundamentals follow exogenous first‐order integrated processes and the discount factor approaches one. In this paper, I argue that the unit market discount factor creates a theoretical trade‐off within a two‐country general equilibrium model. The unit discount factor generates near random‐walk nominal exchange rates, but it counterfactually implies perfect consumption risk sharing and flat money demand. Bayesian posterior simulation exercises, based on post‐Bretton Woods data from Canada and the United States, reveal difficulties in reconciling the equilibrium random‐walk proposition within the canonical model; in particular, the market discount factor is identified as being much smaller than one. A relative money demand shock is identified as the main driver of nominal exchange rates.     

Gradual Retirement: An Additional Option in Work and Retirement

Abstract

Aging of the population raises many questions and issues for individuals, families, and society: economic, political, social, psychological, medical, ethical, moral, religious, and legal, all of which bear on the quality of life in its many dimensions. Economic security in old age is one rubric under which many of the problems and their possible solutions may be discussed. How a society arranges for its members to work and retire is an important facet in the provision for old-age economic security.

This article is concerned with the implications of demographic and labor force changes for work and retirement. It discusses the role of gradual (or phased) retirement in introducing flexibility into the range of choices between work and retirement.

Section 1 explains the rationale for gradual retirement. Section 2 spells out the barriers to implementing gradual retirement programs, including legal barriers and barriers relating to pension plan objectives. Section 3 discusses some possible solutions for implementing gradual retirement programs. Section 4 describes some selected examples of gradual retirement programs, and Section 5 contains concluding remarks.     

Probability of ruin with variable premium rate

Abstract

The probability of ruin is investigated under the influence of a premium rate which varies with the level of free reserves. Section 4 develops a number of inequalities for the ruin probability, establishing upper and lower bounds for it in Theorem 4. Theorem 5 gives an expression for the ruin probability, and it is seen in Section 5 that this amounts to a generalization of the ruin probability given by Gerber for the special case of a negative exponential claim size distribution. In that same section it is shown the Lundberg's inequality is not derivable from the generalized theory of Section 4, and this is seen as a drawback of the methods used there. Sections 6 and 7 deal with some special cases, including claim size distributions with monotone failure rates. Section 8 shows that, in contrast with the result for a constant premium that the probability of ruin for zero initial reserve is independent of the claim size distribution, the same result does not hold when the premium rate is allowed to vary. Section 9 gives some comments on the possible effect of “dangerousness” of a claim size distribution on ruin probability.     

Remarks on “A note on compound generalized distributions”

Abstract

1. INTRODUCTION

In [1] the authors derive recursion formulae for computing total claim probabilities for a general class of modified power series distributions. Such formulae provide an important tool for computing total claim size probabilities in risk-theory. As it turns out, two of their recursions (Theorem 3.2 and Theorem 3.4) need modifications. Unfortunately, these modifications have the effect that the recursions break down. In the following, we will state the modified theorems and show how these obstacles can be overcome.     

Pricing Discrete Dynamic Fund Protections

Abstract

The authors investigate the pricing of discretely monitored dynamic fund protections when the fund price follows a lognormal process or a constant elasticity of variance (CEV) process. A backward recursive pricing formula is derived. By employing a numerical technique that combines function approximation and numerical quadrature, the authors demonstrate how to complete each recursion level efficiently. Numerical experiments show that the results compare favorably with those obtained by other pricing methods.     

Actuarial Modeling with MCMC and BUGs

Abstract

In this paper, the author reviews some aspects of Bayesian data analysis and discusses how a variety of actuarial models can be implemented and analyzed in accordance with the Bayesian paradigm using Markov chain Monte Carlo techniques via the BUGS (Bayesian inference Using Gibbs Sampling) suite of software packages. The emphasis is placed on actuarial loss models, but other applications are referenced, and directions are given for obtaining documentation for additional worked examples on the World Wide Web.