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The following example sets up a basic asset allocation problem to use mean-variance portfolio optimization to estimate efficient portfolios. Suppose you want to manage an asset allocation fund with four asset classes: bonds, large-cap equities, small-cap equities, and emerging equities. The fund is long-only with no borrowing or leverage, should have no more than 85% of the portfolio in equities, and no more than 35% of the portfolio in emerging equities.
The cost to trade the first three assets is 10 basis points annualized and the cost to trade emerging equities is four times higher. Finally, you want to ensure that average turnover is no more than 15%. To solve this problem, you will set up a basic mean-variance portfolio optimization problem and then slowly introduce the various constraints on the problem to get to a solution.
For information on the workflow when using Portfolio objects, see Portfolio Object Workflow.
To set up the portfolio optimization problem, start with basic definitions of known quantities associated with the structure of this problem. Each asset class is assumed to have a tradeable asset with a real-time price. Such assets can be, for example, exchange-traded funds (ETFs). The initial portfolio with holdings in each asset that has a total of $7.5 million along with an additional cash position of $60,000. These basic quantities and the costs to trade are set up in the following variables with asset names in the cell array Asset, current prices in the vector Price, current portfolio holdings in the vector Holding, and transaction costs in the vector UnitCost.
Asset = { 'Bonds', 'Large-Cap Equities', 'Small-Cap Equities', 'Emerging Equities' }; Price = [ 52.4; 122.7; 35.2; 46.9 ]; Holding = [ 42938; 24449; 42612; 15991 ]; UnitCost = [ 0.001; 0.001; 0.001; 0.004 ];
To analyze this portfolio, you can set up a blotter in a dataset object to help track prices, holdings, weights, and so forth. In particular, you can compute the initial portfolio weights and maintain them in a new blotter field called InitPort.
Blotter = dataset({Price, 'Price'}, {Holding, 'InitHolding'},'obsnames',Asset); Wealth = sum(Blotter.Price .* Blotter.InitHolding); Blotter.InitPort = (1/Wealth)*(Blotter.Price .* Blotter.InitHolding); Blotter.UnitCost = UnitCost; disp(Blotter);
Price InitHolding InitPort UnitCost Bonds 52.4 42938 0.3 0.001 Large-Cap Equities 122.7 24449 0.4 0.001 Small-Cap Equities 35.2 42612 0.2 0.001 Emerging Equities 46.9 15991 0.1 0.004
Since this is a hypothetical example, to simulate asset prices from a given mean and covariance of annual asset total returns for the asset classes, portsim is used to create asset returns with the desired mean and covariance. Specifically, portsim is used to simulate 5 years of monthly total returns. The mean and covariance of annual asset total returns are maintained in the variables AssetMean and AssetCovar. The simulated asset total return prices (which are compounded total returns) are maintained in the variable Y. All initial asset total return prices are normalized to 1 in this example.
AssetMean = [ 0.05; 0.1; 0.12; 0.18 ]; AssetCovar = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; X = portsim(AssetMean'/12, AssetCovar/12, 60); % monthly total returns for 5 years (60 months) [Y, T] = ret2tick(X, [], 1/12); % form total return prices
This plot shows the log of the simulated total return prices:
plot(T, log(Y)); title('\bfSimulated Asset Class Total Return Prices'); xlabel('Year'); ylabel('Log Total Return Price'); legend(Asset,'Location','best');
If working with actual historical asset prices, income, and corporate actions data, you would compute total returns for your assets by other means.
To explore portfolios on the efficient frontier, set up a Portfolio object using these specifications:
Portfolio weights are nonnegative and sum to 1.
Equity allocation is no more than 85% of the portfolio.
Emerging equity is no more than 35% of the portfolio.
These specifications are incorporated into the Portfolio object p in the following sequence of methods that starts with the portfolio constructor:
p = Portfolio('Name', 'Asset Allocation Portfolio', ... 'AssetList', Asset, 'InitPort', Blotter.InitPort);
The specification of the initial portfolio from Blotter gives the number of assets in your universe so you do not need to specify the NumAssets property directly. Next, set up default constraints (long-only with a budget constraint). In addition, set up the group constraint that imposes an upper bound on equities in the portfolio (equities are identified in the group matrix with 1s) and the upper bound constraint on emerging equities.
p = setDefaultConstraints(p); p = setGroups(p, [ 0, 1, 1, 1 ], [], 0.85); p = addGroups(p, [ 0, 0, 0, 1 ], [], 0.35);
Although you could have set the upper bound on emerging equities using the setBounds method, notice how you used the addGroups method to set up this constraint.
Finally, to have a fully specified mean-variance portfolio optimization problem, you must specify the mean and covariance of asset returns. Since starting with these moments in the variables AssetMean and AssetCovar, you could use the method setAssetMoments to enter these variables into your Portfolio object in the following way (remember that you are assuming that your raw data are monthly returns which is why you divide your annual input moments by 12 to get monthly returns).
p = setAssetMoments(p, AssetMean/12, AssetCovar/12);
To make things more interesting, however, you can use the total return prices and use the method estimateAssetMoments with a specification that your data in Y are prices, and not returns, to estimate asset return moments for your Portfolio object.
p = estimateAssetMoments(p, Y, 'DataFormat', 'Prices');
Although the returns in your Portfolio object are in units of monthly returns, and since subsequent costs are annualized, it is convenient to specify them as annualized total returns with this direct transformation of the AssetMean and AssetCovar properties of your object:
p.AssetMean = 12*p.AssetMean; p.AssetCovar = 12*p.AssetCovar;
Now, the Portfolio object is ready:
display(p);
p = Portfolio Properties: BuyCost: [] SellCost: [] RiskFreeRate: [] AssetMean: [4x1 double] AssetCovar: [4x4 double] Turnover: [] Name: 'Asset Allocation Portfolio' NumAssets: 4 AssetList: {'Bonds' 'Large-Cap Equities' 'Small-Cap Equities' 'Emerging Equities'} InitPort: [4x1 double] AInequality: [] bInequality: [] AEquality: [] bEquality: [] LowerBound: [4x1 double] UpperBound: [] LowerBudget: 1 UpperBudget: 1 GroupMatrix: [2x4 double] LowerGroup: [] UpperGroup: [2x1 double] GroupA: [] GroupB: [] LowerRatio: [] UpperRatio: []
An important step in portfolio optimization is to validate that the portfolio problem is feasible and the main test is to ensure that the set of portfolios is nonempty and bounded. Use the estimateBounds method to determine the bounds for the portfolio set:
[lb, ub] = estimateBounds(p); display([lb, ub]);
ans = 0.1500 1.0000 0.0000 0.8500 0.0000 0.8500 0.0000 0.3500
Since both lb and ub are finite, the set is bounded.
Given the constructed Portfolio object, use the method plotFrontier to view the efficient frontier. Instead of using the default of 10 portfolios along the frontier, you can display the frontier with 40 portfolios:
plotFrontier(p, 40);
Notice gross efficient portfolio returns fall between approximately 6% and 16% per years.
The Portfolio object p does not include transaction costs so that the portfolio optimization problem specified in p uses gross portfolio return as the return proxy. To handle net returns, create a second Portfolio object q that includes transaction costs:
q = setCosts(p, UnitCost, UnitCost); display(q);
q = Portfolio Properties: BuyCost: [4x1 double] SellCost: [4x1 double] RiskFreeRate: [] AssetMean: [4x1 double] AssetCovar: [4x4 double] Turnover: [] Name: 'Asset Allocation Portfolio' NumAssets: 4 AssetList: {'Bonds' 'Large-Cap Equities' 'Small-Cap Equities' 'Emerging Equities'} InitPort: [4x1 double] AInequality: [] bInequality: [] AEquality: [] bEquality: [] LowerBound: [4x1 double] UpperBound: [] LowerBudget: 1 UpperBudget: 1 GroupMatrix: [2x4 double] LowerGroup: [] UpperGroup: [2x1 double] GroupA: [] GroupB: [] LowerRatio: [] UpperRatio: []
To be more concrete about the ranges of efficient portfolio returns and risks, use the method estimateFrontierLimits to obtain portfolios at the endpoints of the efficient frontier. Given these portfolios, compute their moments using estimatePortMoments. The following code generates a table that lists the risk and return of the initial portfolio as well as the gross and net moments of portfolio returns for the portfolios at the endpoints of the efficient frontier:
[prsk0, pret0] = estimatePortMoments(p, p.InitPort); pret = estimatePortReturn(p, p.estimateFrontierLimits); qret = estimatePortReturn(q, q.estimateFrontierLimits); fprintf('Annualized Portfolio Returns ...\n'); fprintf(' %6s %6s\n','Gross','Net'); fprintf('Initial Portfolio Return %6.2f %% %6.2f %%\n',100*pret0,100*pret0); fprintf('Minimum Efficient Portfolio Return %6.2f %% %6.2f %%\n',100*pret(1),100*qret(1)); fprintf('Maximum Efficient Portfolio Return %6.2f %% %6.2f %%\n',100*pret(2),100*qret(2));
Annualized Portfolio Returns ... Gross Net Initial Portfolio Return 9.70 % 9.70 % Minimum Efficient Portfolio Return 5.90 % 5.77 % Maximum Efficient Portfolio Return 13.05 % 12.86 %
This result shows that the cost to trade ranges from 14 to 19 basis points to get from the current portfolio to the efficient portfolios at the endpoints of the efficient frontier (these costs are the difference between gross and net portfolio returns.) In addition, notice that the maximum efficient portfolio return (13%) is less than the maximum asset return (18%) due to the constraints on equity allocations.
A common approach to select efficient portfolios is to pick a portfolio that has a desired fraction of the range of expected portfolio returns. To obtain the portfolio that is 30% of the range from the minimum to maximum return on the efficient frontier, obtain the range of net returns in qret using the Portfolio object q and interpolate to obtain a 30% level with interp1 to obtain a portfolio qwgt:
Level = 0.3; qret = estimatePortReturn(q, q.estimateFrontierLimits); qwgt = estimateFrontierByReturn(q, interp1([0, 1], qret, Level)); [qrsk, qret] = estimatePortMoments(q, qwgt); fprintf('Portfolio at %g%% return level on efficient frontier ...\n',100*Level); fprintf('%10s %10s\n','Return','Risk'); fprintf('%10.2f %10.2f\n',100*qret,100*qrsk); display(qwgt);
Portfolio at 30% return level on efficient frontier ... Return Risk 7.90 9.09 qwgt = 0.6252 0.1856 0.0695 0.1198
The target portfolio that is 30% of the range from minimum to maximum net returns has a return of 7.9% and a risk of 9.1%.
Although you could accept this result, suppose you want to target values for portfolio risk. Specifically, suppose you have a conservative target risk of 10%, a moderate target risk of 15%, and an aggressive target risk of 20% and you want to obtain portfolios that satisfy each risk target. Use the estimateFrontierByRisk method to obtain targeted risks specified in the variable TargetRisk. The resultant three efficient portfolios are obtained in qwgt:
TargetRisk = [ 0.10; 0.15; 0.20 ]; qwgt = estimateFrontierByRisk(q, TargetRisk); display(qwgt);
qwgt = 0.5407 0.2020 0.1500 0.2332 0.4000 0.0318 0.0788 0.1280 0.4682 0.1474 0.2700 0.3500
Use estimatePortRisk to compute the portfolio risks for the three portfolios to confirm that the target risks have been attained:
display(estimatePortRisk(q, qwgt));
ans = 0.1000 0.1500 0.2000
Suppose you want to shift from the current portfolio to the moderate portfolio. You can estimate the purchases and sales to get to this portfolio:
[qwgt, qbuy, qsell] = estimateFrontierByRisk(q, 0.15);
If you average the purchases and sales for this portfolio, you can see that the average turnover is 17%, which is greater than the target of 15%:
disp(sum(qbuy + qsell)/2)
0.1700
Since you also want to ensure that average turnover is no more than 15%, you can add the average turnover constraint to the Portfolio object:
q = setTurnover(q, 0.15); [qwgt, qbuy, qsell] = estimateFrontierByRisk(q, 0.15);
You can enter the estimated efficient portfolio with purchases and sales into the Blotter:
qbuy(abs(qbuy) < 1.0e-5) = 0;
qsell(abs(qsell) < 1.0e-5) = 0; % zero out near 0 trade weights
Blotter.Port = qwgt;
Blotter.Buy = qbuy;
Blotter.Sell = qsell;
display(Blotter);
Blotter = Price InitHolding InitPort UnitCost Port Buy Sell Bonds 52.4 42938 0.3 0.001 0.18787 0 0.11213 Large-Cap Equities 122.7 24449 0.4 0.001 0.4 0 0 Small-Cap Equities 35.2 42612 0.2 0.001 0.16213 0 0.037871 Emerging Equities 46.9 15991 0.1 0.004 0.25 0.15 0
The Buy and Sell elements of the Blotter are changes in portfolio weights that must be converted into changes in portfolio holdings to determine the trades. Since you are working with net portfolio returns, you must first compute the cost to trade from your initial portfolio to the new portfolio. This can be accomplished as follows:
TotalCost = Wealth * sum(Blotter.UnitCost .* (Blotter.Buy + Blotter.Sell))
TotalCost = 5.6248e+003
The cost to trade is $5,625, so that, in general, you would have to adjust your initial wealth accordingly before setting up your new portfolio weights. However, to keep the analysis simple, note that you have sufficient cash ($60,0000) set aside to pay the trading costs and that you will not touch the cash position to build up any positions in your portfolio. Thus, you can populate your blotter with the new portfolio holdings and the trades to get to the new portfolio without making any changes in your total invested wealth.
First, compute portfolio holding:
Blotter.Holding = Wealth * (Blotter.Port ./ Blotter.Price);
Next, compute number of shares to Buy and Sell in your Blotter:
Blotter.BuyShare = Wealth * (Blotter.Buy ./ Blotter.Price); Blotter.SellShare = Wealth * (Blotter.Sell ./ Blotter.Price);
Notice how you used an add-hoc truncation rule to obtain unit numbers of shares to buy and sell.
Finally, clean up the blotter by removing the unit costs and the buy and sell portfolio weights:
Blotter.Buy = []; Blotter.Sell = []; Blotter.UnitCost = [];
The final result is a blotter that contains proposed trades to get from your current portfolio to a moderate-risk portfolio. To make the trade, you would need to sell 16,049 shares of your bond asset and 8,069 shares of your small-cap equity asset and would need to purchase 23,986 shares of your emerging equities asset.
display(Blotter);
Blotter = Price InitHolding InitPort Port Holding BuyShare SellShare Bonds 52.4 42938 0.3 0.18787 26889 0 16049 Large-Cap Equities 122.7 24449 0.4 0.4 24449 0 0 Small-Cap Equities 35.2 42612 0.2 0.16213 34543 0 8068.8 Emerging Equities 46.9 15991 0.1 0.25 39977 23986 0
The final plot uses plotFrontier to display the efficient frontier and the initial portfolio for the fully specified portfolio optimization problem. It also adds the location of the moderate-risk or final portfolio on the efficient frontier.
plotFrontier(q, 40); hold on scatter(estimatePortRisk(q, qwgt), estimatePortReturn(q, qwgt), 'filled', 'r'); h = legend('Initial Portfolio', 'Efficient Frontier', 'Final Portfolio', 'location', 'best'); set(h, 'Fontsize', 8); hold off
addGroups | estimateAssetMoments | estimateBounds | estimateFrontierByRisk | estimateFrontierLimits | estimatePortRisk | plotFrontier | Portfolio | setAssetMoments | setBounds