Failure Probability Under Parameter Uncertainty

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent repository link: City Research OnlineElectronic copy available at: http://ssrn.com/abstract=1486317Electronic copy available at: http://ssrn.com/abstract=1486317Failure probability under parameter uncertainty * R. Gerrard A. Tsanakas † ‡ Cass Business School, City University LondonAbstract: In many problems of risk analysis, failure is equivalent to the event of a random risk factor exceeding a given threshold. Failure probabilities can be controlled if a decision maker is able to set the threshold at an appropriate level. This abstract situation applies for example to environmental risks with infrastructure controls; to supply chain risks with inventory controls; and to insurance solvency risks with capital controls. However, uncertainty around the distribution of the risk factor implies that parameter error will be present and the measures taken to control failure probabilities may not be effective. We show that parameter uncertainty increases the probability (understood as expected frequency) of failures. For a large class of loss distributions, arising from increasing transformations of location-scale families (including the Log-Normal, Weibull and Pareto distributions), the paper shows that failure probabilities can be exactly calculated, as they are independent of the true (but unknown) parameters. Hence it is possible to obtain an explicit measure of the effect of parameter uncertainty on failure probability. Failure probability can be controlled in two different ways: (a) by reducing the nominal required failure probability, depending on the size of the available data set and (b) by modifying of the distribution itself that is used to calculate the risk control. Approach (a) corresponds to a frequentist/regulatory view of probability, while approach (b) is consistent with a Bayesian/personalistic view. We furthermore show that the two approaches are consistent in achieving the required failure probability. Finally, we briefly discuss the effects of data pooling and its systemic risk implications.

Model uncertainty in risk capital measurement

“…The proof follows directly from the results in Gerrard and Tsanakas (2011), who show that P(Y ≤ VaR p [F µ,σ (·|X)]) = p, wherefrom the stated result is derived.…”

“…The estimated capital is considered a random variable via its dependence on a random sample. Gerrard and Tsanakas (2011) investigate the changes in the probability of future losses exceeding capital, when the estimated capital is random due to parameter uncertainty. Bignozzi and Tsanakas (2015) quantify the impact of parameter uncertainty on risk measures beyond VaR.…”

“…Approaches in i) focus on the estimation of the parameters assuming that a model for Y is known; see Gerrard and Tsanakas (2011), Bignozzi and Tsanakas (2015). Method ii) has the advantage that few assumptions on the model have to be made, however it requires a large sample of data to give accurate results and hence is often not practical.…”

Modelling parameter uncertainty for risk capital calculation

“…This question has already been raised in [11], where the authors showed that the probability of insolvency can be substantially higher than 0:5 % if the undertaking calculates its solvency risk capital without taking parameter uncertainty into account. As a consequence, the confidence level of 99.5 % required by Solvency II can only be achieved by a risk capital reflecting the parameter uncertainty.…”

“…Gerrard and Tsanakas [11] have applied this approach to parameter uncertainty for VaR-based solvency capital calculations. We discuss their results later in the paper.…”