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                           حسابداری و مالی:: 
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                        , when all PDs, LGDs and EADs are assumed to be known but not the dependence), some explicit bounds were found by Ru¨schendorf (1982) for the two- dimensional  case  and  by  Puccetti  and  Ru¨schendorf  (2012b)  for  homogeneous  portfolios  in higher dimensions.
The following proposition provides a sharp upper VaR bound for a homogeneous portfolio.
Proposition 5.3 (Sharp moment-constrained bounds for a homogeneous portfolio).
Assuming  a  homogeneous  portfolio  (vi  =  v, ρij  =  ρ)  and  asset  returns  that  are  driven  by one single factor M only, then when the number of loans n → ∞, we find  that
6.2.1      Homogeneous portfolio (Beta model)
 
                        
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