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: [12×1 double]
                    UpperBound: []
                   LowerBudget: 1
                   UpperBudget: 1
                   GroupMatrix: []
                    LowerGroup: []
                    UpperGroup: []
                        GroupA: []
                        GroupB: []
                    LowerRatio: []
                    UpperRatio: []
                  MinNumAssets: []
                  MaxNumAssets: []
    ConditionalBudgetThreshold: []
        ConditionalUpperBudget: []
                     BoundType: [12×1 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
         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

Figure contains an axes object. The axes object with title Efficient Frontier, xlabel Conditional Value-at-Risk of Portfolio, ylabel Mean of Portfolio Returns contains 2 objects of type line, scatter. This object represents Efficient Frontier.

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

See Also

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