### Expected Shortfall Estimation For Apparently Infinite-Mean Models Of Operational Risk

Delft University of Technology; Delft University of Technology – Delft Institute of Applied Mathematics (DIAM)

Independent Researcher; NYU-Poly Institute; New England Complex Systems Institute

October 27, 2015

**Abstract: **

Statistical analyses on actual data depict operational risk as an extremely heavy-tailed phenomenon, able to generate losses so extreme as to suggest the use of infinite-mean models. But no loss can actually destroy more than the entire value of a bank or of a company, and this upper bound should be taken into consideration when dealing with tail-risk assessment. In financial risk management no risk can really be infinite, given that it always exists an upper bound, no matter how big.

Introducing what we call the dual distribution, we show how to deal with heavy-tailed phenomena with a remote yet finite upper bound. We show how to compute relevant tail quantities such as the Expected Shortfall (ES).

Our methodology is useful with apparently infinite-mean phenomena, as in the case of operational risk, but it can be applied in all those situations involving fat-tails and bounded supports.

### Expected Shortfall Estimation For Apparently Infinite-Mean Models Of Operational Risk – Introduction

According to the Basel Committee on Banking Supervision: “Operational risk is defined as the risk of loss resulting from inadequate or failed internal processes, people and systems or from external events. This definition includes legal risk, but excludes strategic and reputational risk” (BCBS 2011a, 2014). Operational risk is one of the main risks banks (and insurance companies) have to deal with, together with market, credit and liquidity risk (Hull 2015, McNeil et al. 2015).

As shown in Moscadelli (2004), de Fontnouvelle (2005), de Fontnouvelle et al. (2003, 2005), and further discussed in Fiordelisi et al. (2014), Ne?slehová et al. (2006) and Peters and Shevchenko (2015), a peculiar characteristic of operational risk is that the distribution of losses is extremely heavy-tailed, showing a clear Paretian behavior for the upper tail, when we consider losses as positive amounts. Following the standard division of banks’ activities into business lines, as required by the so-called standardized and advanced measurement approaches (BCBS 2011b, 2014), Moscadelli (2004) has even shown that for corporate finance, trading and sales, and payment and settlement, the loss distribution has such a fat right tail not to have a finite mean (with a shape parameter > 1, see Subsection 2.2 for more details). No need to say that an infinite mean implies that also the expected shortfall is infinite, while value-at-risk tends to assume large values (Neslehova et al.2006, Puccetti and Ruschendorf 2014), especially for very large confidence levels, as the 99% and the 99.9% prescribed by regulations (BCBS 2014). And, since the distribution with the heaviest tail tends to dominate, when loss distributions are aggregated (de Haan and Ferreira 2006, Embrechts et al. 2003, McNeil et al. 2015), one single business line with infinite mean is sufficient to have an infinite mean for the whole bank’s distribution of operational losses.

The basic arithmetic rules of the Basel Accords (BCBS 2011a,b) look inappropriate to really deal with losses like the 1.4 billion dollars Daiwa lost for fraudulent trading, the 250 million dollars Merrill Lynch paid for a legal settlement related to gender discrimination, the 225 million dollars Bank of America lost for systems integration failures, or the 140 million dollars Merrill Lynch lost because of damages to is facilities after the 9/11 events. For this reason, under the Advanced Measurement Approach (BCBS 2011b), many solutions have been proposed in the literature to better assess operational risk, and to deal with its extremely heavy-tailed behavior, e.g. Bocker and Kluppelberg (2010), Chavez-Demoulin et al. (2006, 2015), Moscadelli (2004), Puccetti and Rüschendorf (2014), and Tursunalieva and Silvapulle (2014). All these contributions seem to agree on the use of extreme value theory, and in particular of the Generalized Pareto approximation of the right tail (de Haan and Ferreira 2006, Falk et al. 2004), to study the behavior of large operational losses (see Subsection 2.2 for more details.). The tail of the distribution is indeed what really matters when we are interested in quantities like value-at-risk and expected shortfall, and in the related minimum capital requirements.

If we take for granted the infiniteness of the mean for operational losses, we find ourselves in what Ne?slehová et al. (2006) define the “one loss causes ruin problem”. If the mean is infinite (and so necessarily are the variance and all the higher moments), one single loss can be so large as to deplete the minimum capital of a bank, causing a technical default. Even worse: it can be so large as to destroy more value than the entire capitalization of the bank under consideration.

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