Efficient Portfolio Optimization with Conditional Value at Risk

904 PROCEEDINGS OF THE IMCSIT. VOLUME 5, 2010with the inner optimization problemD(u) = maxx j{T∑ ∑nu t r jt x j :t=1j=1n∑µ j x j ≥ µ 0 ,j=1n∑x j = 1,j=1x j ≥ 0, j = 1,...,n}n∑ T∑= max { ( r jt u t )x j :x jj=1t=1n∑µ j x j ≥ µ 0 ,j=1n∑x j = 1,j=1x j ≥ 0, j = 1,...,n}.(11)Again, we may take advantages of the LP dual to the innerproblem. Indeed, introducing dual variable q corresponding tothe equation ∑ nj=1 x j = 1 and variable u 0 corresponding tothe inequality ∑ nj=1 µ jx j ≥ µ 0 we get the LP dualD(u) = minq,u 0{q −µ 0 u 0 :q −µ j u 0 −T∑r jt u t ≥ 0 j = 1,...,n}.t=1Hence, an alternative model for the CVaR portfolio optimization(10) can be expressed as the following LP:min q −µ 0 u 0s.t. q −µ j u 0 −T∑r jt u t ≥ 0, j = 1,...,nt=1T∑(12)u t = 1t=10 ≤ u t ≤ p tβ , t = 1,...,TLP model (12) contains T + 1 variables u t , but the Tconstraints corresponding to variables d t from (6) take theform of simple upper bounds on u t (for t = 1,...,T)thus not affecting the problem complexity. The number ofconstraints in (12) is proportional to the total of portfoliosize n, thus it is independent from the number of scenarios.Exactly, there are T +1 variables and n+1 constraints. Thisguarantees a high computational efficiency of the model evenfor very large number of scenarios. Similarly, other portfoliostructure requirements are modeled with rather small numberof constraints thus generating small number of additionalvariables in the model. Actually, the model (12) is the LPdual to the model (6), thus similar to that introduced in [23].Obviously, the optimal portfolio shares x j are not directlyrepresentedwithinthe solutionvectorofproblem(12)buttheyare easily available as the dual variables (shadow prices) forinequalities q −µ j u 0 − ∑ Tt=1 r jtu t ≥ 0.The Minimax portfolio optimization model can be writtenas the following LP problem:max ηn∑s.t. µ j x j ≥ µ 0 ,j=1n∑x j = 1j=1x j ≥ 0, j = 1,...,nn∑−η + r jt x j ≥ 0, t = 1,...,Tj=1(13)which is simpler than the standard CVaR optimization model(6). Except from the portfolio weights x j , the model containsonly one additional variable η. Nevertheless, it still contains Tlinear inequalities in addition to the core constraints. Hence,its dimensionality is (T +2)×(n+1).The Minimax portfolio optimization model representing alimiting case of the CVaR model for β tending to 0. Actually,for any β ≤ min t=1,...,T p t we gets M β (x) = M(x) thusallowing to represent the Minimax portfolio optimization bythe CVaR optimization model (6) and to take advantagesof itsdual form (12). Due to β ≤ p t for all t = 1,...,T, the upperbounds on variables u t becomes redundant and we get thefollowing dual form of the Minimax portfolio optimization:min q −µ 0 u 0s.t. q −µ j u 0 −T∑u t = 1t=1T∑r jt u t ≥ 0, j = 1,...,nt=1u t ≥ 0, t = 1,...,T(14)The model dimensionality is only (n + 1) × (T + 2) thusguaranteeing a high computational efficiency even for verylarge number of scenarios.The Mean Absolute Deviation (MAD) risk measure isdirectly given by the value of the second order cdf F x(2) atthe mean ¯δ(x) = E{max{µ(x)−R x ,0}} = F x (2) (µ(x)) [19].Therefore,its leadstoan LP portfoliooptimizationmodelverysimilar to that for the CVaR optimization (6). Indeed, we get:max −s.t.T∑p t d tt=1n∑µ j x j ≥ µ 0 ,j=1d t ≥n∑x j = 1j=1n∑(µ j −r jt )x j , d t ≥ 0, t = 1,...,Tj=1x j ≥ 0, j = 1,...,n(15)with T+n variablesand T+2 constraints. The LP dual model