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<html xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <!--This HTML is auto-generated from an M-file.To make changes, update the M-file and republish this document. --> <title>Mean-variance portfolio optimization using GA and PATTERNSEARCH</title> <meta name="generator" content="MATLAB 7.2"> <meta name="date" content="2007-10-15"> <meta name="m-file" content="portoptgads"><style>body { background-color: white; margin:10px;}h1 { color: #990000; font-size: x-large;}h2 { color: #990000; font-size: medium;}/* Make the text shrink to fit narrow windows, but not stretch too far in wide windows. On Gecko-based browsers, the shrink-to-fit doesn't work. */ p,h1,h2,div.content div { /* for MATLAB's browser */ width: 600px; /* for Mozilla, but the "width" tag overrides it anyway */ max-width: 600px; /* for IE */ width:expression(document.body.clientWidth > 620 ? "600px": "auto" );}pre.codeinput { background: #EEEEEE; padding: 10px;}@media print { pre.codeinput {word-wrap:break-word; width:100%;}} span.keyword {color: #0000FF}span.comment {color: #228B22}span.string {color: #A020F0}span.untermstring {color: #B20000}span.syscmd {color: #B28C00}pre.codeoutput { color: #666666; padding: 10px;}pre.error { color: red;}p.footer { text-align: right; font-size: xx-small; font-weight: lighter; font-style: italic; color: gray;} </style></head> <body> <div class="content"> <h1>Mean-variance portfolio optimization using GA and PATTERNSEARCH</h1> <introduction> <p>We seek to try out <tt>ga</tt> and <tt>patternsearch</tt> functions of the Genetic Algorithm and Direct Search Toolbox. We consider the unconstrained mean-variance portfolio optimization problem, handled by <tt>portopt</tt> and <tt>portalloc</tt> of the Financial Toolbox - note that in absence of constraints other than <tt>sum(w) = 1</tt>, the problem admits a simple closed-form analytic solution - and see whether <tt>ga</tt> and <tt>patternsearch</tt> succeed at locating the optimal portfolio identified by <tt>portalloc</tt>. </p> </introduction> <p>In the event, <tt>patternsearch</tt> delivers respectable performance, whereas <tt>ga</tt> does not. (Can this be improved via judicious choice of <tt>ga</tt> options?) Oddly enough, <tt>patternsearch</tt> solutions commonly register higher utility levels than their <tt>portalloc</tt> counterparts. </p> <p>Hopefully, this code sample will aid other Matlab users experimenting with Matlab's optimization facilities.</p> <p><b>Exercise parameters</b>: </p><pre class="codeinput">expRet = [0.1 0.2 0.15];</pre><pre class="codeinput">expCov = [0.005 -0.010 0.004 -0.010 0.040 -0.002 0.004 -0.002 0.023];</pre><pre class="codeinput">risklessRate = 0.12;</pre><pre class="codeinput">borrowingRate = risklessRate;</pre><p><b><tt>portalloc</tt> solution</b></p> <p>Optimal portfolio as a function of risk-aversion parameter:</p><pre class="codeinput">[portRisk,portReturn,portWts] = portopt(expRet,expCov,20);</pre><pre class="codeinput">numRAValues = 50;riskAversionMin = .01;riskAversionMax = 3;</pre><pre class="codeinput">riskAversion = logspace(riskAversionMin,riskAversionMax,numRAValues);</pre><pre class="codeinput">[riskyReturn,riskyFraction,overallRisk,overallReturn] = deal(nan(numRAValues,1));<span class="keyword">for</span> i = 1:numRAValues [riskyRisk,riskyReturn(i),riskyWts,riskyFraction(i),overallRisk(i),overallReturn(i)] = <span class="keyword">...</span> portalloc(portRisk,portReturn,portWts,risklessRate,borrowingRate,riskAversion(i));<span class="keyword">end</span></pre><pre class="codeinput">figureplot(overallRisk,overallReturn,<span class="string">'k'</span>), hold <span class="string">on</span>plot(portRisk,portReturn,<span class="string">'LineWidth'</span>,3,<span class="string">'Color'</span>,[.8 .8 .8])xlabel(<span class="string">'portfolio risk, \sigma_{p}'</span>)ylabel(<span class="string">'portfolio return, r_{p}'</span>)xlim = [0 max(overallRisk)];ylim = [risklessRate max(expRet)];axis([xlim ylim])title(<span class="string">'Capital allocation line'</span>)set(gcf,<span class="string">'color'</span>,<span class="string">'white'</span>)</pre><img vspace="5" hspace="5" src="portoptgads_01.png"> <p>Optimal share of risky assets and portfolio return as functions of the risk-aversion parameter:</p><pre class="codeinput">figure[a,h1,h2] = plotyy(riskAversion,riskyFraction,riskAversion,overallReturn,<span class="string">'semilogx'</span>);set(a(1),<span class="string">'XColor'</span>,<span class="string">'k'</span>,<span class="string">'YColor'</span>,<span class="string">'k'</span>), set(h1,<span class="string">'Color'</span>,<span class="string">'k'</span>), ylabel(a(1),<span class="string">'Risky-asset fraction'</span>)set(a(2),<span class="string">'XColor'</span>,<span class="string">'k'</span>,<span class="string">'YColor'</span>,<span class="string">'r'</span>), set(h2,<span class="string">'Color'</span>,<span class="string">'r'</span>), ylabel(a(2),<span class="string">'Portfolio return, r_{p}'</span>)xlabel(<span class="string">'Risk-aversion parameter, \lambda'</span>)title(<span class="string">'Optimal risky-assets share and portfolio return as functions of the risk-aversion parameter'</span>)set(gcf,<span class="string">'color'</span>,<span class="string">'white'</span>)</pre><img vspace="5" hspace="5" src="portoptgads_02.png"> <p><b>Genetic-algorithm and direct-search optimization</b></p><pre class="codeinput">optMethods = {<span class="string">'Benchmark: portalloc'</span>, <span class="keyword">...</span> <span class="string">'Genetic algorithm'</span>, <span class="keyword">...</span> <span class="string">'Direct search'</span>};numMethods = length(optMethods);</pre><p>Means and covariances of returns, including the riskless asset:</p><pre class="codeinput">expRetExt = [risklessRate expRet]; numAssets = length(expRetExt); <span class="comment">%#ok</span></pre><pre class="codeinput">expCovExt = zeros(numAssets); expCovExt(2:end,2:end) = expCov; <span class="comment">%#ok</span></pre><p>Values of the risk-aversion parameter, resulting in a long-only optimal allocation:</p> <p>(We make things easier for <tt>ga</tt> and <tt>patternsearch</tt>, restricting optima to the unit hypercube) </p><pre class="codeinput">riskAversion = riskAversion(riskyFraction < 1);numRAValues = 10;</pre><p>Objective-function definition used by <tt>ga</tt> and <tt>patternsearch</tt>: </p> <p><tt>util = -(expRetExt*(Wts') - A*Wts*expCovExt*(Wts')/2)</tt></p><pre class="codeinput">objfun = @(Wts) util(Wts);</pre><p>Objective function is the negative of utility, since <tt>ga</tt> and <tt>patternsearch</tt> are minimizers. Refer to the Financial Toolbox documentation page entitled 'Portfolio selection and risk aversion' for a (rather misleading) definition of the utility function. </p> <p>Constraint definitions used by <tt>ga</tt> and <tt>patternsearch</tt>: </p><pre class="codeinput">Aeq = ones(1,4); beq = 1; <span class="comment">% weights sum to 1</span>lb = zeros(1,4); <span class="comment">% weights are positive (see above)</span>ub = ones(1,4); <span class="comment">% weights are below one (see above)</span></pre><p>Options used by <tt>ga</tt> and <tt>patternsearch</tt>: </p><pre class="codeinput">gaoptions = gaoptimset(<span class="string">'Display'</span>,<span class="string">'off'</span>,<span class="string">'MutationFcn'</span>,@mutationadaptfeasible);psoptions = psoptimset(<span class="string">'Display'</span>,<span class="string">'off'</span>);</pre><p>We suppress the functions' interim command-window output, and modify <tt>MutationFcn</tt> property of <tt>ga</tt>, as suggested by the <tt>ga</tt> warning produced with the option's default setting 'mutationgaussian'. </p> <p>Arrays storing optimization results:</p><pre class="codeinput">optFun = nan(numRAValues,1,numMethods);optWts = nan(numRAValues,numAssets,numMethods);</pre><p>Optimization:</p> <p><tt>portalloc</tt> is invoked once for each risk-aversion case, whereas <tt>ga</tt> and <tt>patternsearch</tt> are allowed <tt>numAttempts</tt> = 10 attempts. Only successful attempts, exiting with an 'appropriate' value of <tt>exittext</tt> or <tt>exitflag</tt> (see the functions' documentation), count towards the limit. </p><pre class="codeinput">numAttempts = 10;</pre><pre class="codeinput">warning <span class="string">off</span> <span class="comment">%#ok</span><span class="keyword">global</span> expRetExt expCovExt A</pre><p>An unappealing temporary fix. Documentation-suggested ways to parameterize the objective function resulted in errors and irregularities.</p><pre class="codeinput">tic<span class="keyword">for</span> i = 1:numRAValues <span class="keyword">for</span> j = 1:numMethods <span class="keyword">if</span> 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; <span class="keyword">else</span> fmax = -Inf; <span class="keyword">for</span> k = 1:numAttempts goOn = true; <span class="keyword">while</span> goOn <span class="keyword">if</span> j == 2 [x f exittext] = ga(objfun,4,[],[],Aeq,beq,lb,ub,[],gaoptions); goOn = isempty(strfind(exittext,<span class="string">'change in the fitness value less than options.TolFun'</span>)); <span class="keyword">else</span> x0 = rand(1,numAssets); [x f exitflag] = patternsearch(objfun,x0,[],[],Aeq,beq,lb,ub,psoptions); goOn = exitflag < 1; <span class="keyword">end</span> <span class="keyword">end</span> <span class="keyword">if</span> -f > fmax optWts(i,:,j) = x; fmax = -f; <span class="keyword">end</span> <span class="keyword">end</span> optFun(i,:,j) = fmax; <span class="keyword">end</span> <span class="keyword">end</span><span class="keyword">end</span>toc</pre><pre class="codeoutput">Elapsed time is 363.962409 seconds.</pre><p>Selected outputs of alternative optimizers:</p><pre class="codeinput">metricNames = {<span class="string">'Maximized value of objective function'</span>, <span class="keyword">...</span> <span class="string">'Optimal portfolio share of the riskless asset'</span>, <span class="keyword">...</span> <span class="string">'Optimal share of asset 1 relative to the risky-asset portfolio'</span>, <span class="keyword">...</span> <span class="string">'Optimal share of asset 2 relative to the risky-asset portfolio'</span>};metricValues = {optFun, <span class="keyword">...</span> optWts(:,1,:), <span class="keyword">...</span> optWts(:,2,:)./(1 - optWts(:,1,:)), <span class="keyword">...</span> optWts(:,3,:)./(1 - optWts(:,1,:))};</pre><pre class="codeinput"><span class="keyword">for</span> i = 1:length(metricValues)⌨️ 快捷键说明
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