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Using Bayesian Approach for Modeling Operational Risk
Ursulenko A.
Introduction
Operational Risk is an important quantitative topic in the banking world as a result of the Basel II regulatory requirements. Through the Advanced Measurement Approach, banks are permitted significant flexibility over the approaches that may be used in the development of operational risk models. Such models incorporate internal and external loss data observations in combination with expert opinion surveyed from business subject matter experts. Accordingly the Bayesian approach provides a natural, probabilistic framework in which to evaluate risk models.
Of the methods developed to model operational risk, the majority follow the Loss Distributional Approach (LDA). The idea of LDA is to fit severity and frequency distributions over a predetermined time horizon, typically annual. Popular choices include exponential, weibull, lognormal, generalised Pareto, and g-and-h distributions . The best fitting models are then used to produce compound processes for the annual loss distribution, from which VaR and capital estimates may be derived.
Under the compound process:
r = ZfL iXt,
where Xi~f(x) — the random variable that follows the fitted severity distribution;
*i~/M — the random variable that, the fitted frequency distribution, is commonly modeled by Poisson, binomial and negative binomial distributions.
There are a number of pertinent issues in fitting models to operational risk data: the combination of data sources from expert opinions and observed loss data; the elicitation of information from subject matter experts, which incorporates survey design considerations; sample biases in loss data collection, such as survival bias, censoring, incomplete data sets, truncation and, since rare events are especially important, small data sets.
For the posterior simulation possible methods include Markov chain Monte Carlo (MCMC), importance sampling, annealed MCMC, sequential Monte Carlo (SMC) and approximate Bayesian computation (ABC). In this article we examine the MCMC, annealed MCMC and ABC.
Simulation Technique 1: Markov chain Monte Carlo
Markov chain Monte Carlo (MCMC) constructs an ergodic Markov chain, {Si.-.e*}, taking values st in a measurable space. Under the general framework established by Metropolis and Hastings , transitions from one element in the chain to the next are determined via a transition kernel k which satisfies the detailed balance condition: lyi:»)K (Si-1. 6/) = Pto \yi:n)K («1, 0i-l), where p(Si-i\y1:n) — the desired stationary distribution. The transition kernel contains in its definition a proposal density q, from which the proposed next value in the chain is drawn, and an acceptance probability a, which determines whether the proposed value is accepted or if the chain remains in its present state. The acceptance probability is crucial as it
ensures that the Markov chain has the required stationary distribution. .
Metropolis-Hastings Algorithm:
Initialise i = 0 and 00 = [a, b, p, q] randomly sampled from the support of the posterior.
Draw proposal 0-+1 from proposal distribution q(0t). Evaluate the acceptance probability:
a(.6i,8*+1) = min{l,
P(9i+l|yi:n)g(gj+l.gt)-
P(0i|yi:n)?(Si.8i*+l)
Sample random variate U~U[0,1].
If U < a(8i, 0*+1) then set 9i+1 = 0*+1; else set 8i+1 = 9t.
Increment i = i + 1.
If i<N go to 2.
Note that this sampling strategy only requires the posterior distribution to be known up to its normalization constant. This is important as solving the integral for the normalizing constant in this model is not straightforward.
Simulation Technique 2: Approximate Bayesian Computation It is not possible to apply standard MCMC techniques for Model 2, as the likelihood function, and therefore the acceptance probability, may not be written down analytically or evaluated. A recent advancement in computational statistics, approximate Bayesian computation methods are ideally suited to situations in which the likelihood is either computationally prohibitive to evaluate or cannot be expressed analytically .
Approximate Bayesian computation algorithm:
Initialise i = 0 and 0O = [A, B, p, q] randomly sampled from the support of the posterior.
Draw proposal 0*+1 from proposal distribution q(9i). Generate a simulated data set yl;n from the likelihood conditional on the proposal parameters 0*+1.
Evaluate the acceptance probability:
a(0j, 0*+i) = min {1
p(9*+l)1l(9*+l-ei)
I[p(S(yI:n).S(yi:„)) < £]}.
p(8i)q(0i.0,'+i)
Sample random variate U~U[0,1].
If U < a(0i, 0,*+1) then set 0i+1 = 0*+1; else set ei+1 = 0j. Increment i = i + 1.
If i < N go to 2.
Various summary statistics could be used, such as mean, variance, skewness, kurtosis and quantiles.
Simulation Technique 3: Simulated Annealing Developed in statistical physics , simulated annealing is a form of probabilistic optimization which may be used to obtain the maximum a posteriori parameter estimates. It implements an MCMC sampler which traverses a sequence of distributions in such a way that one may quickly find the mode of the distribution with the largest mass.
Simulated Annealing algorithm:
Initialise i = 0 and 0O = [a, b, p, q] randomly sampled from the support of the posterior.
Draw 0,*+i proposal from proposal distribution q(0i). Evaluate the acceptance probability:
p(ei\y1nyVq(,e,.e;+1)
Sample random variate U~U[0,1].
If U < a(0j, 0*+1) then set 01+1 = 0*+1; else set 0i+1 = 0j.
Increment i = i +1.
If i < N go to 2.
Conclusion
In this article, by introducing existing and novel simulation procedures we have substantially extended the range of models admissible for Bayesian inference under the LDA operational risk modeling framework. We strongly advocate that Bayesian approaches to operational risk modeling should be considered as a serious alternative for practitioners in banks and financial institutions, as it provides a mathematically rigorous paradigm in which to combine observed data and expert opinion. We hope that the presented class of algorithms provides an attractive and feasible approach in which to realize these models.
References
1. Dutta K. and J. Perry. An empirical analysis of loss distribution models for estimating operational risk capital. Federal Reserve Bank of Boston, 2006. Working Papers № 06-13.
2. Tavare S., P. Marjoram, J. Molitor and V. Plagnol. Markov Chain Monte Carlo without Likelihoods, Proceedings of the National Academy of Science of USA, 20O2, № 100, P. 24-28.
Author
• Ursulenko A., post-graduate student, Taras Shevchenko National University of Kiev
Всероссийский журнал научных публикаций, август 2011
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