02 Portfolio Optimization June 2012

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1、FundamentalsofOptimizationandApplicationtoPortfolioSelectionFundamentalsofOptimizationandApplicationtoPortfolioSelectionCQF1/122FundamentalsofOptimizationandApplicationtoPortfolioSelectionInthislecture...FundamentalsofOptimization:Ihowtoformulateanoptim

2、izationproblem;Ihowtousecalculustosolveunconstrainedoptimizationproblems;IapplicationtolinearregressionIconstrainedoptimizationproblemsandthemethodofLagrange;ItheKuhn-Tuckerconditions;2/122FundamentalsofOptimizationandApplicationtoPortfolioSelectionAppl

3、icationtoportfolioselection:IhowtoselectaportfoliointheMarkowitzworld;Itheminimumvarianceportfolio;Iportfolioselectionwitharisk-freeasset;IhowtoderivetheCAPM.3/122FundamentalsofOptimizationandApplicationtoPortfolioSelectionPartI:FundamentalsofOptimizati

4、on4/122FundamentalsofOptimizationandApplicationtoPortfolioSelection1-IntroductionAnoptimizationproblemisoneinwhichyouaretryingtondthebestpossiblevalue"thatafunction,sayf,cantakesubjecttoanumberofconstraints.Thisgenerallyinvolvesndingtheminimumormaxim

5、umofforofafunctionbuiltaroundf.Optimizationtechniquesareveryoftenusedinnancetosolveawiderangeofproblemsrangingfromcalculatingthevalueofabondyield,tosolvingaportfolioselectionproblemindiscreteandcontinuoustimeandtovaluingderivativessuchasAmericanandpass

6、portoptions.5/122FundamentalsofOptimizationandApplicationtoPortfolioSelectionAsoptimizationproblemscomeinallshapeandforms,youshouldexpectsometobeincrediblyeasytosolveandsomeothertobeincrediblyhard.Inanycase,formulatingtheproblemwellwillspareyoumanyheada

7、ches.6/122FundamentalsofOptimizationandApplicationtoPortfolioSelection2-FormulatinganoptimizationproblemOptimizationproblemsaremostoftendenedasminf(x1;:::;xn)(1)x1;:::;xnsubjectto:98g1(x1;:::;xn)>=>;>:gm(x1;:::;xn)bm7/122FundamentalsofOpti

8、mizationandApplicationtoPortfolioSelectionThefunctionfiscalledtheobjectivefunction.Itisthefunctionwewanttooptimize(hereminimize).Thevariablesx1;:::;xnarethedecisionvariableswithrespecttowhichwewanttooptimizethefunction.Thefunctio