Annals of Actuarial Science

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Annals of Actuarial Science is a well researched Project Topic, it can be used as a guide or framework for your Academic Research.

Abstract

Risk aggregation is a popular method used to estimate the sum of a collection of financial assets or events, where each asset or event is modeled as a random variable. Applications include insurance, operational risk, stress testing, and sensitivity analysis.

In practice, the sum of a set of random variables involves the use of two well-known mathematical operations: n-fold convolution (for a fixed number n) and N-fold convolution, defined as the compound sum of a frequency distribution November and a severity distribution, where the number of constant n-fold convolutions is determined by N, where the severity and frequency variables are independent and continuous, currently numerical solutions such as Panjer’s recursion, fast Fourier transforms and Monte Carlo simulation produces acceptable results.

However, they have not been designed to cope with new modeling challenges that require hybrid models containing discrete explanatory (regime-switching) variables or where discrete and continuous variables are inter-dependent and may influence the severity and frequency in complex, non-linear, ways. This paper describes a Bayesian Factorisation and Elimination (BFE) algorithm that performs convolution on the hybrid models required to aggregate risk in the presence of causal dependencies.

This algorithm exploits a number of advances from the field of Bayes in Networks, covering methods to approximate statistical and conditionally deterministic functions to factorize multivariate distributions for efficient computation. Experiments show that BFE is as accurate on conventional problems as competing methods.

For more difficult hybrid problems BSE can provide a more general solution that the others cannot offer. In addition, the BFE approach can be easily extended to perform deconvolution for the purposes of stress testing and sensitivity analysis in a way that competing methods do not.

Introduction

Risk aggregation is a popular method used to estimate the sum of a collection of financial assets or events, where each asset or event is modeled as a random variable. Existing techniques make a number of assumptions about these random variables. First, they are almost always continuous. Second, if they are independent then they are identically distributed.

Third, should they be dependent, these dependencies are best represented by correlation functions, such as copulas (Nelsen, 2007; Embrechts, 2009), where marginal distribution functions are linked by some dependence structure. These statistical methods have tended to model associations between variables as a purely phenomenological artifact extant in historical statistical data.

Recent experience, at least since the beginning of the financial crisis in 2007, has amply demonstrated the inability of these assumptions to handle non-linear effects or “shocks” on financial assets and events, resulting in models that are inadequate for prediction, stress testing, and model comprehension (Laeven & Valencia, 2008; IMF, 2009).

It has been extensively argued that modeling dependence as correlation is insufficient, as it ignores any views that the analyst may, quite properly, hold about those causal influences that help generate and explain the statistical data observed (Meucci, 2008; Rebonato, 2010). Such causal influences are commonplace and permeate all levels of economic and financial discourse.

For example, does a dramatic fall in equity prices cause an increase in equity implied volatilities, or is it an increase in implied volatility that causes a fall in equity prices? The answer is trivial in this case since a fall in equity prices is well known to affect implied volatility, but correlation alone contains no information about the direction of causation.

To incorporate causation we need to involve the analyst or expert and “fold into” the model views of how discrete events interact and the effects of this interaction on the aggregation of risk. This approach extends the methodological boundaries last pushed back by the celebrated Black–Litterman model (Black & Litterman, 1991).

In that approach, a risk manager’s role is as an active participant in the risk modeling, and the role of the model is to accommodate their subjective “views”, expressed as Bayesian priors of expectations and variances of asset returns.
In this paper, we aim to represent these Bayesian “views” in an explicit causal structure, whilst providing the computational framework for solutions. Such causal models would involve discrete explanatory (regime-switching) variables and hybrid mixtures of inter-dependent discrete and continuous variables.

A causal risk aggregation model might incorporate expert derived views about macro-economic, behavioral, operational or strategic factors that might influence the assets or events under “normal” or “abnormal” conditions. Applications of the approach include insurance, stress testing, operational risk, and sensitivity analysis.

At its heart risk aggregation requires the sum of n random variables. In practice, this involves the use of two well-known mathematical operations: n-fold convolution (for a fixed value of n) and N-fold convolution (Heckman & Meyers, 1983), defined as the compound sum of a frequency distribution N, and a severity distribution S, where the number of constant n-fold convolutions is determined by N, stochastically.

Currently, popular methods such as Panjer’s recursion (Panjer, 1981), fast Fourier transforms (FFT, Heckman & Meyers, 1983) and Monte Carlo (MC) simulation (Fishman, 1996) perform risk aggregation numerically using parameters derived from historical data to estimate the distributions for both S and N.

Where S and N are independent and continuous, these approaches produce acceptable results. However, they have not been designed to cope with the new modeling challenges outlined above. In the context of modelling general dependencies among severity variables, a popular approach is to use copulas, both to model the dependent variables and to perform risk aggregation.

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Additional information

Type

Project Topic and Material

Category

Actuarial Science

No of Chapters

5

Reference

Yes

Format

PDF

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