Advances in Risk Management

YVES CRAMA, GEORGES HÜBNER AND JEAN-PHILIPPE PETERS 19

1,500,000

1,300,000

1,100,000

900,000

700,000

500,000

0,25 1 5 10

Collection threshold

Capital charge

20 50

BL1 BL2

Figure 1.5 Regulatory capital charge for BL1 and BL2 with various

collection thresholds

APPENDIX: FINDING THE EVT THRESHOLD

The algorithm performs the following steps:

1 Let (x 1 ,...,xn) be the ordered sample of observations. Considermcandidate thresh-

oldsU 1 ,...,Umsuch thatxn−i,...,xn>Uifori=1,...,m.

2 For each thresholdUi, use the weighted average of Hill estimators proposed in Huis-

manet al.(2001) to estimate the tail indexξiof the GPD distribution. This method

corrects for the small-sample bias of the original Hill estimator.

3 Then compute the maximum likelihood estimator of the scale parameterβiof the GPD,

with the tail indexξifixed to the value obtained in step 2.

4 For each thresholdUi, compute the Mean Squared Error statistic^11 MSE(Ui)=n^1 i

∑ni

k= 1

(Fk−Fˆk)^2 , whereniis the number of losses above thresholdUi,Fkis the cdf of the

GPD(ξk,βk,μk) andFˆkis the empirical cdf.

5 IdentifyMSE(Uopt)=min(MSE(U 1 ),...,MSE(Um));Uoptis retained as estimator of the

cut-off threshold and the fitted distribution is the GPD(ξopt,βopt,Uopt).

NOTES

1. The Basel Committee thus assumes perfect positive dependence between operational
   risks; alternatively, it also allows banks to use internally defined correlations. See
   paragraph 669 of BCBS (2004).


2. Basel II states: “Supervisors will require the bank to calculate its regulatory capital
   requirement as the sum of expected loss (EL) and unexpected loss (UL), unless the
   bank can demonstrate that it is adequately capturing EL in its internal business
   practices” (BCBS, 2004, §669).