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Compute Maximum Reward-to-Risk Ratio for CVaR Portfolio

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Create a PortfolioCVaR object and incorporate a list of assets from CAPMUniverse.mat. Use simulateNormalScenariosByData to simulate the scenarios for each of the assets. These portfolio constraints require fully invested long-only portfolios (nonnegative weights that must sum to 1).

rng(1) % Set the seed for reproducibility.
load CAPMuniverse

p = PortfolioCVaR('AssetList',Assets(1:12));
p = simulateNormalScenariosByData(p, Data(:,1:12), 20000 ,'missingdata',true);
p = setProbabilityLevel(p, 0.95);
p = setDefaultConstraints(p);
disp(p)
  PortfolioCVaR with properties:

             BuyCost: []
            SellCost: []
        RiskFreeRate: []
    ProbabilityLevel: 0.9500
            Turnover: []
         BuyTurnover: []
        SellTurnover: []
        NumScenarios: 20000
                Name: []
           NumAssets: 12
           AssetList: {'AAPL'  'AMZN'  'CSCO'  'DELL'  'EBAY'  'GOOG'  'HPQ'  'IBM'  'INTC'  'MSFT'  'ORCL'  'YHOO'}
            InitPort: []
         AInequality: []
         bInequality: []
           AEquality: []
           bEquality: []
          LowerBound: [12x1 double]
          UpperBound: []
         LowerBudget: 1
         UpperBudget: 1
         GroupMatrix: []
          LowerGroup: []
          UpperGroup: []
              GroupA: []
              GroupB: []
          LowerRatio: []
          UpperRatio: []
        MinNumAssets: []
        MaxNumAssets: []
           BoundType: [12x1 categorical]

To obtain the portfolio that maximizes the reward-to-risk ratio (which is equivalent to the Sharpe ratio for mean-variance portfolios), search on the efficient frontier iteratively for the portfolio that minimizes the negative of the reward-to-risk ratio:

- portfolio return-risk free rateportfolio CVaR.

To do so, use the sratio function, defined in the Local Functions section, to return the negative reward-to-risk ratio for a target return. Then, pass this function to fminbnd. fminbnd iterates through the possible return values and evaluates their associated reward-to-risk ratio. fminbnd returns the optimal return for which the maximum reward-to-risk ratio is achieved (or that minimizes the negative of the reward-to-risk ratio).

% Obtain the minimum and maximum returns of the portfolio.
pwgtLimits = estimateFrontierLimits(p);
retLimits = estimatePortReturn(p,pwgtLimits);
minret = retLimits(1);
maxret = retLimits(2);

% Search on the frontier iteratively. Find the return that minimizes the
% negative of the reward-to-risk ratio.
fhandle = @(ret) iterative_local_obj(ret,p);
options = optimset('Display', 'off', 'TolX', 1.0e-8);
optret = fminbnd(fhandle, minret, maxret, options);

% Obtain the portfolio weights associated with the return that achieves
% the maximum reward-to-risk ratio.
pwgt = estimateFrontierByReturn(p,optret)
pwgt = 12×1

    0.0885
         0
         0
         0
         0
    0.9115
         0
         0
         0
         0
      ⋮

Use plotFrontier to plot the efficient frontier and estimatePortRisk to estimate the maximum reward-to-risk ratio portfolio.

plotFrontier(p);
hold on
% Compute the risk level for the maximum reward-to-risk ratio portfolio.
optrsk = estimatePortRisk(p,pwgt);
scatter(optrsk,optret,50,'red','filled')
hold off

Local Functions

This local function that computes the negative of the reward-to-risk ratio for a target return level.

function sratio = iterative_local_obj(ret, obj)
% Set the objective function to the negative of the reward-to-risk ratio.

risk = estimatePortRisk(obj,estimateFrontierByReturn(obj,ret));

if ~isempty(obj.RiskFreeRate)
    sratio = -(ret - obj.RiskFreeRate)/risk;
else
    sratio = -ret/risk;
end

end

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