Many banks adopt the Loss Distribution Approach to quantify the operational risk capital charge under Basel II requirements. It is common practice to estimate the capital charge using the 0.999 quantile of the annual loss distribution, calculated using point estimators of the frequency and severity distribution parameters. The uncertainty of the parameter estimates is typically ignored. One of the unpleasant consequences for the banks accounting for parameter uncertainty is an increase in the capital requirement. This paper demonstrates how the parameter uncertainty can be taken into account using a Bayesian framework that also allows for incorporation of expert opinions and external data into the estimation procedure.
Deep Dive into Estimation of Operational Risk Capital Charge under Parameter Uncertainty.
Many banks adopt the Loss Distribution Approach to quantify the operational risk capital charge under Basel II requirements. It is common practice to estimate the capital charge using the 0.999 quantile of the annual loss distribution, calculated using point estimators of the frequency and severity distribution parameters. The uncertainty of the parameter estimates is typically ignored. One of the unpleasant consequences for the banks accounting for parameter uncertainty is an increase in the capital requirement. This paper demonstrates how the parameter uncertainty can be taken into account using a Bayesian framework that also allows for incorporation of expert opinions and external data into the estimation procedure.
Under the Basel II requirements BIS (2006) banks are required to quantify the capital charge for operational risk. Many banks have adopted the Loss Distribution Approach (LDA), where frequency and severity of operational losses for each risk cell (in the matrix of eight business lines times seven event types) are estimated over a one year period. Then the capital is estimated using the 0.999 quantile of the distribution for the total annual loss in the bank. Under the LDA, the annual loss in a risk cell is
where N is the annual number of events, modelled as a random variable from some discrete distribution with a density ) | (. α p , and N i X i ,…, 1 , =
, are the severities of the events modelled as independent random variables from a continuous distribution with a density ) | (. β f . Here, α and β are distribution parameters. The frequency N and severities i X are assumed to be independent.
Estimation of the frequency and severity distributions is a challenging task but it will not be discussed here. There is an extensive literature on modelling the risk process (1), see e.g. McNeil, Frey and Embrechts (2005). In this paper, we assume that the distribution type is known and address the issue of the parameter uncertainty (parameter risk). The parameters α and β are unknown and statistical estimation of the parameters would involve some uncertainty due to the finite sample size of the observed data. Typically, the capital is estimated using point estimators for α and β , e.g. Maximum Likelihood Estimators (MLEs). Although the parameter estimates are uncertain, in practice this uncertainty is commonly ignored in the estimation of the operational risk capital charge. Taking parameter uncertainty into account is typically regarded as very difficult and time consuming calculation. One unpleasant consequence for a bank accounting for parameter uncertainty is an increase in the capital charge estimate when compared to the capital based on the point estimators for model parameters.
In our opinion, this subject deserves more attention. Only a few papers mention this issue. Frachot, Moudoulaud and Roncalli (2004) discussed the accuracy of the capital charge by constructing a confidence interval for the 0.999 quantile of the annual loss distribution. Mignola and Ugoccioni (2006) assessed the standard deviation of the 0.999 quantile using a first order Taylor expansion approximation. They demonstrated that even small uncertainty in the parameters can lead to profound uncertainty in the 0.999 quantile. It is critical to account for parameter uncertainty, otherwise capital charge could be underestimated significantly. The official position of BIS (2006) regarding accounting for parameter uncertainty is not clear. However, in practice, we experienced that often local regulators have concerns about the correctness of the parameter point estimators used by a bank. Appropriate accounting for parameter uncertainty would help a bank to resolve these concerns. In this paper, we present a convenient and practical approach (not difficult to implement) to account for parameter uncertainty using Bayesian inference.
Bayesian inference is a statistical technique well suited to model parameter uncertainty. There is a broad literature covering Bayesian inference and its applications for the insurance industry as well as other areas; see e.g. Berger (1985). The method allows for expert opinions or external data to be incorporated into the analysis via specifying so-called prior distributions for model parameters. In the application to operational risk, the method is relatively new, see Shevchenko and Wüthrich (2006). Below we focus on the application of the method to the estimation of the operational risk capital, taking into account the parameter uncertainty. Accounting for parameter uncertainty is critical for operational risk due to the limited internal data for some risks, very high quantile level and long time horizon used for capital estimation.
For simplicity, consider one risk cell only.
, is used for the capital charge calculation.
However, the parameters θ are unknown and can be modeled as random variable with the density ) | ( Y θ π . Here, Y is a vector of all loss events (frequencies and severities) used in the estimation procedure. Then, the marginal (predictive) distribution of the annual loss Z, taking into account the parameter uncertainty, is
(2)
The capital charge, accounting for the parameter uncertainty, should be based on the 0.999 quantile, B Q 999 . 0
, where
. Then according to Bayes’ theorem
is the density of parameters, a so-called prior distribution.
) (θ π depends on a set of further parameters that are called hyper-parameters, omitted hereafter for simplicity of notation;
is the joint density of data and parameters;
is a likelihood function, density of Y given parameters θ (here, we use the assumption that frequencies and severities are independent given θ );
Hereafter, we consider con
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