portoptgads.m

该程序介绍了一种简单的遗传算法应用程序

%% Mean-variance portfolio optimization using GA and PATTERNSEARCH
%
% We seek to try out |ga| and |patternsearch| functions of the Genetic Algorithm and Direct Search Toolbox.
% We consider the unconstrained mean-variance portfolio optimization problem, handled by |portopt| and 
% |portalloc| of the Financial Toolbox - note that in absence of constraints other than |sum(w) = 1|, the 
% problem admits a simple closed-form analytic solution - and see whether |ga| and |patternsearch| succeed 
% at locating the optimal portfolio identified by |portalloc|. 
%%
% In the event, |patternsearch| delivers respectable performance, whereas |ga| does not. (Can this be improved
% via judicious choice of |ga| options?) Oddly enough, |patternsearch| solutions commonly register higher
% utility levels than their |portalloc| counterparts. 
%%
% Hopefully, this code sample will aid other Matlab users experimenting with Matlab's optimization facilities.
%% 
% *Exercise parameters*:
expRet = [0.1 0.2 0.15]; 
%%
expCov = [0.005   -0.010    0.004 
         -0.010    0.040   -0.002 
          0.004   -0.002    0.023];
%%
risklessRate  = 0.12;
%%
borrowingRate = risklessRate;
%%
% *|portalloc| solution*
%% 
% Optimal portfolio as a function of risk-aversion parameter:
[portRisk,portReturn,portWts] = portopt(expRet,expCov,20);
%%
numRAValues     = 50;
riskAversionMin = .01;
riskAversionMax = 3;
%%
riskAversion = logspace(riskAversionMin,riskAversionMax,numRAValues); 
%%
[riskyReturn,riskyFraction,overallRisk,overallReturn] = deal(nan(numRAValues,1));
for i = 1:numRAValues
    [riskyRisk,riskyReturn(i),riskyWts,riskyFraction(i),overallRisk(i),overallReturn(i)] = ...
        portalloc(portRisk,portReturn,portWts,risklessRate,borrowingRate,riskAversion(i));
end

%%
figure
plot(overallRisk,overallReturn,'k'), hold on
plot(portRisk,portReturn,'LineWidth',3,'Color',[.8 .8 .8])

xlabel('portfolio risk, \sigma_{p}')
ylabel('portfolio return, r_{p}')

xlim = [0 max(overallRisk)];
ylim = [risklessRate max(expRet)];
axis([xlim ylim])

title('Capital allocation line')
set(gcf,'color','white')
%% 
% Optimal share of risky assets and portfolio return as functions of the risk-aversion parameter:
figure
[a,h1,h2] = plotyy(riskAversion,riskyFraction,riskAversion,overallReturn,'semilogx');
set(a(1),'XColor','k','YColor','k'), set(h1,'Color','k'), ylabel(a(1),'Risky-asset fraction')  
set(a(2),'XColor','k','YColor','r'), set(h2,'Color','r'), ylabel(a(2),'Portfolio return, r_{p}') 
xlabel('Risk-aversion parameter, \lambda')
title('Optimal risky-assets share and portfolio return as functions of the risk-aversion parameter')
set(gcf,'color','white')
%%
% *Genetic-algorithm and direct-search optimization*
optMethods = {'Benchmark: portalloc', ...
              'Genetic algorithm', ...
              'Direct search'};
numMethods = length(optMethods);
%%
% Means and covariances of returns, including the riskless asset:
expRetExt = [risklessRate expRet]; numAssets = length(expRetExt);    %#ok
%%
expCovExt = zeros(numAssets); expCovExt(2:end,2:end) = expCov;       %#ok
%%
% Values of the risk-aversion parameter, resulting in a long-only optimal allocation:
%%
% (We make things easier for |ga| and |patternsearch|, restricting optima to the unit hypercube)
riskAversion = riskAversion(riskyFraction < 1);
numRAValues  = 10; 
%%
% Objective-function definition used by |ga| and |patternsearch|:
%%
% |util = -(expRetExt*(Wts') - A*Wts*expCovExt*(Wts')/2)|
objfun = @(Wts) util(Wts);
%%
% Objective function is the negative of utility, since |ga| and |patternsearch| are minimizers.
% Refer to the Financial Toolbox documentation page entitled 'Portfolio selection and risk aversion' for 
% a (rather misleading) definition of the utility function.
%%
% Constraint definitions used by |ga| and |patternsearch|:
Aeq = ones(1,4); beq = 1;     % weights sum to 1
lb  = zeros(1,4);             % weights are positive  (see above)
ub  = ones(1,4);              % weights are below one (see above)
%% 
% Options used by |ga| and |patternsearch|:
gaoptions = gaoptimset('Display','off','MutationFcn',@mutationadaptfeasible);
psoptions = psoptimset('Display','off');
%%
% We suppress the functions' interim command-window output, and modify |MutationFcn| property of |ga|,
% as suggested by the |ga| warning produced with the option's default setting 'mutationgaussian'. 
%%
% Arrays storing optimization results:
optFun = nan(numRAValues,1,numMethods);
optWts = nan(numRAValues,numAssets,numMethods);
%%
% Optimization:
%%
% |portalloc| is invoked once for each risk-aversion case, whereas |ga| and |patternsearch| are allowed |numAttempts| = 10 attempts. 
% Only successful attempts, exiting with an 'appropriate' value of |exittext| or |exitflag| (see the functions' documentation),
% count towards the limit.
%%
numAttempts = 10;
%%
warning off                                                          %#ok
global expRetExt expCovExt A
%%
% An unappealing temporary fix. Documentation-suggested ways to parameterize the objective function resulted in errors 
% and irregularities.
%%
tic
for i = 1:numRAValues
    for j = 1:numMethods
        if j == 1
           A = riskAversion(i);
           [a,c,riskyWts,riskyFraction,d,e] = portalloc(portRisk,portReturn,portWts,risklessRate,borrowingRate,A);
           Wts = [(1 - riskyFraction) riskyFraction*riskyWts];
           optWts(i,:,j) = Wts;
           optFun(i,:,j) = expRetExt*Wts' - A*Wts*expCovExt*(Wts')/2;
        else
           fmax = -Inf;
           for k = 1:numAttempts
               goOn = true;  
               while goOn
                  if j == 2
                     [x f exittext] = ga(objfun,4,[],[],Aeq,beq,lb,ub,[],gaoptions);
                     goOn = isempty(strfind(exittext,'change in the fitness value less than options.TolFun'));
                  else
                     x0 = rand(1,numAssets); 
                     [x f exitflag] = patternsearch(objfun,x0,[],[],Aeq,beq,lb,ub,psoptions);
                     goOn = exitflag < 1; 
                  end   
               end
               if -f > fmax
                  optWts(i,:,j) = x; 
                  fmax = -f;
               end   
           end
           optFun(i,:,j) = fmax;
        end
    end
end
toc
%%
% Selected outputs of alternative optimizers:
metricNames  = {'Maximized value of objective function', ...
                'Optimal portfolio share of the riskless asset', ...
                'Optimal share of asset 1 relative to the risky-asset portfolio', ...
                'Optimal share of asset 2 relative to the risky-asset portfolio'};
metricValues = {optFun, ...
                optWts(:,1,:), ...
                optWts(:,2,:)./(1 - optWts(:,1,:)), ...
                optWts(:,3,:)./(1 - optWts(:,1,:))};
%%            
for i = 1:length(metricValues)
    figure
    plot(riskAversion(1:numRAValues),squeeze(metricValues{i}))
    legend(optMethods,'Location','SouthEast')
    xlabel('Risk-aversion parameter, \lambda')
    title(metricNames{i})
    set(gcf,'color','white')
end    
%%
close all
% Example: portoptgads (Oh, FEX code metrics..)