本示例说明如何使用金融工具箱中的Portfolio对象执行投资组合优化。
尤其是，该示例说明了如何优化投资组合以相对于市场基准最大化信息比率。具体而言，将table中包含的财务数据读入MATLAB并进行可视化（按天和按年）。Portfolio对象是使用每个资产的交易日收益的市场数据来创建的。使用支持Portfolio对象的函数，可以直接计算有效边界。然后，解决定制的优化问题，以找到信息比率最大的资产分配。
使用MATLAB导入历史数据
导入资产范围和道琼斯工业平均指数（DJI）市场基准的历史价格。使用MATLAB的readtable函数，将数据从Microsoft®Excel电子表格导入到table。
data = readtable('dowPortfolio.xlsx');head(data, 10)ans=10×32 tableDates DJIAA AIGAXPBA C CATDD DISGE GM HD HONHPQIBMINTCJNJJPMKO MCDMMMMO MRKMSFTPFEPG T UTXVZ WMTXOM ___________ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____03-Jan- 10847 28.72 68.41 51.53 68.63 45.26 55.86 40.68 24.1833.6 17.82 39.79 36.14 28.35 80.13 24.57 59.08 37.78 38.98 32.72 75.93 52.27 30.73 26.19 22.16 56.3822.7 54.94 26.7944.9 56.6404-Jan- 10880 28.89 68.51 51.03 69.34 44.42 57.29 40.46 23.77 33.5618.3 39.05 35.99 29.18 80.0324.9 59.99 37.56 38.91 33.01 75.54 52.65 31.08 26.32 22.88 56.48 22.87 54.61 27.58 44.99 56.7405-Jan- 10882 29.1268.6 51.57 68.53 44.65 57.29 40.38 24.19 33.47 19.34 38.67 35.97 28.97 80.56 25.25 59.74 37.6739.1 33.05 74.85 52.52 31.13 26.3422.956.3 22.92 54.4127.9 44.38 56.4506-Jan- 10959 29.02 68.89 51.75 67.57 44.65 58.43 40.55 24.5233.7 19.61 38.96 36.5329.8 82.96 25.28 60.01 37.94 39.47 33.25 75.47 52.95 31.08 26.26 23.16 56.24 23.21 54.58 28.01 44.56 57.5709-Jan- 11012 29.37 68.57 53.04 67.01 44.43 59.49 40.32 24.78 33.61 21.12 39.38 36.23 30.17 81.76 25.44 60.38 38.55 39.66 33.88 75.84 53.11 31.58 26.21 23.16 56.6723.355.2 28.1244.4 57.5410-Jan- 11012 28.44 69.18 52.88 67.33 44.57 59.2540.2 25.09 33.43 20.79 40.33 36.17 30.3382.125.1 60.49 38.6139.7 33.91 75.37 53.04 31.27 26.35 22.77 56.45 23.16 55.24 28.24 44.54 57.9911-Jan- 11043 28.0569.6 52.5968.3 44.98 59.28 38.87 25.33 33.66 20.61 41.44 36.19 30.88 82.19 25.12 59.91 38.58 39.7234.5 75.22 53.31 31.39 26.63 23.06 56.65 23.34 54.41 28.58 45.23 58.3812-Jan- 10962 27.68 69.0452.667.9 45.02 60.13 38.02 25.41 33.25 19.76 41.05 35.77 30.57 81.61 24.96 59.63 37.8739.5 33.96 74.57 53.23 31.41 26.4822.9 56.02 23.2453.9 28.69 44.43 57.7713-Jan- 10960 27.81 68.8452.567.7 44.92 60.24 37.86 25.47 33.3519.2 40.43 35.85 31.43 81.22 24.78 59.26 37.84 39.37 33.65 74.38 53.2931.4 26.53 22.99 56.49 23.2754.1 28.7544.1 59.0617-Jan- 10896 27.97 67.84 52.03 66.93 44.47 60.85 37.75 25.1533.2 18.68 40.11 35.5631.2 81.05 24.52 58.74 37.64 39.11 33.77 73.99 52.85 31.16 26.34 22.63 56.25 23.13 54.41 28.12 43.66 59.61
从表格中分离资产名称，资产价格和DJI基准价格。可视化图片显示：归一化为统一的所有资产价格（即累积收益）的演变。
benchPrice = data.DJI;assetNames = data.Properties.VariableNames(3:2:end); % using half of the assets for displayassetPrice = data(:,assetNames).Variables;assetP = assetPrice./assetPrice(1, :); benchmarkP = benchPrice / benchPrice(1);figure;plot(data.Dates,assetP);hold on;plot(data.Dates,benchmarkP,'LineWidth',3,'Color','k');hold off;xlabel('Date');ylabel('Normalized Price');title('Normalized Asset Prices and Benchmark');grid on;
粗线表示道琼斯工业平均指数市场基准。
计算收益和风险调整收益
根据价格序列计算收益序列，并计算资产动量（历史收益和标准差）。可视化图片显示：所有资产的风险收益特征和DJI市场基准的散点图。
benchReturn = tick2ret(benchPrice);assetReturn = tick2ret(assetPrice);benchRetn = mean(benchReturn);benchRisk = std(benchReturn);assetRetn = mean(assetReturn);assetRisk = std(assetReturn);
计算历史统计数据并绘制年度风险收益图。请注意，该图为年度水平，因此按日收益率进行换算。
scale = 252;assetRiskR = sqrt(scale) * assetRisk;benchRiskR = sqrt(scale) * benchRisk;assetReturnR = scale * assetRetn;benchReturnR = scale * benchRetn;figure;scatter(assetRiskR, assetReturnR, 6, 'm', 'Filled');hold onscatter(benchRiskR, benchReturnR, 6, 'g', 'Filled');for k = 1:length(assetNames)text(assetRiskR(k) + 0.005, assetReturnR(k), assetNames{k}, 'FontSize', 8);endtext(benchRiskR + 0.005, benchReturnR, 'Benchmark', 'Fontsize', 8);hold off;xlabel('Risk (Std Dev of Return)');ylabel('Expected Annual Return');grid on;
建立投资组合优化
通过使用Portfolio填充对象来设置投资组合优化问题。因为目标是根据基准优化投资组合分配，所以将在Portfolio对象中计算和使用每种资产的有效收益。在此示例中，投资组合中资产的预期收益和协方差设置为其历史值。
p = Portfolio('AssetList',assetNames);
设置默认投资组合约束（所有权重之和为1，不做空，对风险资产的投资为100％）。
p = setDefaultConstraints(p);
将资产收益和协方差添加到Portfolio对象。
activReturn = assetReturn - benchReturn;pAct = estimateAssetMoments(p,activReturn,'missingdata',false)pAct = Portfolio with properties:BuyCost: []SellCost: []RiskFreeRate: []AssetMean: [15x1 double]AssetCovar: [15x15 double]TrackingError: []TrackingPort: []Turnover: []BuyTurnover: []SellTurnover: []Name: []NumAssets: 15AssetList: {1x15 cell}InitPort: []AInequality: []bInequality: []AEquality: []bEquality: []LowerBound: [15x1 double]UpperBound: []LowerBudget: 1UpperBudget: 1GroupMatrix: []LowerGroup: []UpperGroup: []GroupA: []GroupB: []LowerRatio: []UpperRatio: []MinNumAssets: []MaxNumAssets: []BoundType: [15x1 categorical]
使用Portfolio对象计算有效边界
计算20个最优投资组合的均值方差有效边界。可视化单个资产的风险收益特征的边界。此外，计算和可视化沿边界的每个投资组合的信息比率。
pwgtAct = estimateFrontier(pAct, 20); % Estimate the weights.[portRiskAct, portRetnAct] = estimatePortMoments(pAct, pwgtAct); % Get the risk and return.% Extract the asset moments and names.[assetActRetnDaily, assetActCovarDaily] = getAssetMoments(pAct);assetActRiskDaily = sqrt(diag(assetActCovarDaily));assetNames = pAct.AssetList;% Rescale.assetActRiskAnnual = sqrt(scale) * assetActRiskDaily;portRiskAnnual = sqrt(scale) * portRiskAct;assetActRetnAnnual = scale * assetActRetnDaily;portRetnAnnual = scale * portRetnAct;figure;subplot(2,1,1);plot(portRiskAnnual, portRetnAnnual, 'bo-', 'MarkerFaceColor', 'b');hold on;scatter(assetActRiskAnnual, assetActRetnAnnual, 12, 'm', 'Filled');hold on;for k = 1:length(assetNames)text(assetActRiskAnnual(k) + 0.005, assetActRetnAnnual(k), assetNames{k}, 'FontSize', 8);endhold off;xlabel('Risk (Std Dev of Active Return)');ylabel('Expected Active Return');grid on;subplot(2,1,2);plot(portRiskAnnual, portRetnAnnual./portRiskAnnual, 'bo-', 'MarkerFaceColor', 'b');xlabel('Risk (Std Dev of Active Return)');ylabel('Information Ratio');grid on;
使用优化工具箱执行信息比率最大化
运行混合优化，以沿着边境找到具有最大信息比率的投资组合。信息比率是相对回报与相对风险的比率（称为“跟踪误差”）。夏普（Sharpe）比率着眼于相对于无风险资产的收益，而信息比率是基于相对于风险基准（在本例中为DJI基准）的收益。这是通过运行优化来完成的，该优化找到最佳收益约束，对于该最优收益约束，投资组合优化问题将返回最大信息比率投资组合。从目标函数infoRatioTargetReturn调用投资组合优化函数，该目标函数是通过优化工具箱函数fminbnd来优化的。给定目标有效收益率，局部函数infoRatioTargetReturn计算最小（有效）风险组合。
本地函数infoRatioTargetReturn被调用为优化程序（fminbnd）的目标函数，旨在寻找最大化的信息比率和负面信息的比例最小化的目标回报率。
objFun = @(targetReturn) -infoRatioTargetReturn(targetReturn,pAct);options = optimset('TolX',1.0e-8);[optPortRetn, ~, exitflag] = fminbnd(objFun,min(portRetnAct),max(portRetnAct),options);
获取权重，信息比率和最佳投资组合的风险回报。
[optInfoRatio,optWts] = infoRatioTargetReturn(optPortRetn,pAct);optPortRisk = estimatePortRisk(pAct,optWts) optPortRisk = 0.0040
绘制最佳投资组合
验证找到的投资组合确实是最大信息比率的投资组合。
% Rescale.optPortRiskAnnual = sqrt(scale) * optPortRisk;optPortReturnAnnual = scale * optPortRetn;figure;subplot(2,1,1);scatter(assetActRiskAnnual, assetActRetnAnnual, 6, 'm', 'Filled');hold onfor k = 1:length(assetNames)text(assetActRiskAnnual(k) + 0.005,assetActRetnAnnual(k),assetNames{k},'FontSize',8);endplot(portRiskAnnual,portRetnAnnual,'bo-','MarkerSize',4,'MarkerFaceColor','b');plot(optPortRiskAnnual,optPortReturnAnnual,'ro-','MarkerFaceColor','r');hold off;xlabel('Risk (Std Dev of Active Return)');ylabel('Expected Active Return');grid on;subplot(2,1,2);plot(portRiskAnnual,portRetnAnnual./portRiskAnnual,'bo-','MarkerSize',4,'MarkerFaceColor','b');hold onplot(optPortRiskAnnual,optPortReturnAnnual./optPortRiskAnnual,'ro-','MarkerFaceColor','r');hold off;xlabel('Risk (Std Dev of Active Return)');ylabel('Information Ratio');title('Information Ratio with Optimal Portfolio');grid on;
显示投资组合优化解决方案
显示投资组合优化解决方案。
assetIndx = optWts > .001;results = table(assetNames(assetIndx)', optWts(assetIndx)*100, 'VariableNames',{'Asset', 'Weight'});disp('Maximum Information Ratio Portfolio:')Maximum Information Ratio Portfolio:disp(results)AssetWeight ________ _______{'AA' }1.539{'AXP' } 0.35551{'C' }9.6533{'DD' }4.0684{'HPQ' }17.698{'JPM' }21.565{'MCD' }26.736{'MO' }13.648{'MSFT'}2.6858{'UTX' }2.0509fprintf('Active return for Max. Info Ratio portfolio is %0.2f%%n', optPortRetn*25200);Active return for Max. Info Ratio portfolio is 12.14%fprintf('Tracking error for Max. Info Ratio portfolio is %0.2f%%n', optPortRisk*sqrt(252)*100);Tracking error for Max. Info Ratio portfolio is 6.32%
局部函数
function [infoRatio,wts] = infoRatioTargetReturn(targetReturn,portObj)% Calculate information ratio for a target-return portfolio along the% efficient frontier.wts = estimateFrontierByReturn(portObj,targetReturn);portRiskAct = estimatePortRisk(portObj,wts);infoRatio = targetReturn/portRiskAct;end
注：本文根据MATLAB官网内容修改而成。
前往索引：
吃小羊：MATLAB金融工具箱：00：索引
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