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By Greg N. Gregoriou (eds.)

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Closed under multiplication by γ ≥ 0. 2. Convexity. Proof: First, it must be shown that if η ∈ Aη , then γη ∈ Aη for γ ≥ 0. This property follows from η ∈ Aη being equivalent to Pη ≥ 0 and the property P[γη] = γPη which is non-negative since both γ and Pη are non-negative. Second, if η1 , η2 ∈ Aη , implying Pη1 ≥ 0 and Pη2 ≥ 0, then γη1 + (1 − γ)η2 ∈ Aη for 0 ≤ γ ≤ 1 since P[γη1 + (1 − γ)η2 ] = γPη1 + (1 − γ)Pη2 ≥ 0. Therefore, as in the ADEH framework, unless each element of Pη is non-negative, the portfolio η is unacceptable.

Jordan, J. and Rosengren, E. (2003) “Using Loss Data to Quantify Operational Risk”, Working Paper, Federal Reserve Bank of Boston. , Rosengren, E. and Jordan, J. (2004) “Implications of Alternative Operational Risk Modeling Techniques”, Working Paper, Federal Reserve Bank of Boston. , Georges, P. and Roncalli, T. (2001) “Loss Distribution Approach for Operational Risk”, Working Paper, Groupe de Recherche Opérationnelle, Crédit Lyonnais. , Moudoulaud, O. and Roncalli, T. (2003) “Loss Distribution Approach in Practice”, Working Paper, Groupe de Recherche Opérationnelle, Crédit Lyonnais.

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). 20 DETERMINATION OF THE CAPITAL CHARGE FOR OPERATIONAL RISK 3. Consistent means that for large n, the estimates converge to the true value of the parameters, Unbiased means that for all sample sizes the parameter of interest is calculated correctly, Efficient means that the ML estimate is the estimate with the smallest variance while Sufficient indicates that is uses all the information in the observations.

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