How to find tangency portfolio (maximize sharpe ratio) using quadprog

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Hi,
We are performing a mean-var portfolio optimazation including serveral constraints. Using Mean-Variance Stochastic goal programming we have found the sub -optimal frontier accountin g for desierable weights in a subset of sustainable assets. Using the code below we were able to find the sub-optimal frontier changing the desired return level (r) . However, we now want to obtain the tangency portfolio as our selected optimal portfolio strategy, but we can´t figure out how. Given a risk free rate of 0.000412 does anyone know how we can find the tangency portfolio (i.e maximizing the sharp ratio) ?
Best regards ,
Agnethe and Guro
C = readtable('V strong.xlsx')
Covariance = table2array(C)
M = readtable('m.xlsx')
Mean_return = table2array(M)
M1 = readtable('h.xlsx')
Mean_return_h = table2array(M1)
nAsset = numel(Mean_return); r = 0.03
e_0 = 0.027900578
%Optimization Problem
portprob = optimproblem;
x = optimvar('x',nAsset,'LowerBound',0,'UpperBound',1);
objective = x'*Covariance*x;
portprob.Objective = objective;
sumcons = sum(x) == 1;
portprob.Constraints.sumcons = sumcons;
averagereturn = dot(Mean_return, x) >= r;
portprob.Constraints.averagereturn = averagereturn;
averagereturn2 = dot(Mean_return_h, x) >= e_0;
portprob.Constraints.averagereturn2 = averagereturn2;
%Run Optimization Problem
options = optimoptions('quadprog','Display','iter','TolFun',1e-10);
tic
[x1,fval1] = solve(portprob,'Options',options);
toc
C2 = readtable('V1.xlsx')
Covariance2 = table2array(C2) %Covariance Matrix All Assets
ReturnStrong = value'*Mean_return
VarStrong = value'*Covariance2*value
RiskStrong = sqrt(VarStrong)
  2 Comments
John D'Errico
John D'Errico on 15 Mar 2019
Edited: John D'Errico on 15 Mar 2019
Of course, now you just removed that line completely from your code. But now I note that the code you show never even assigns value, but you use it afterwards. If you just change your code randomly, nobody will ever be able to help you.

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Accepted Answer

Alejandra Pena-Ordieres
Alejandra Pena-Ordieres on 3 Dec 2021
Hi Guro,
I understand that you are trying to find the maximum Sharpe ratio portfolio using quadprog. Theoretically, the max Sharpe ratio portfolio is a nonlinear nonconvex problem that cannot be solved using quadprog (quadprog only solves quadratic problems). The natural formulation of the max Sharpe ratio problem looks as follows:
That being said, there is way to rewrite the max Sharpe ratio problem as a quadratic convex optimization problem given some mild assumtpions. The quadratic version of the max Sharpe ratio problem looks like:
and the weights will be given by .
Thus, the solution to the max Sharpe ratio problem can be obtained with the following code:
% Excess returns
mu = muGross - riskFreeRate;
% Portfolio problem
prob = optimproblem('ObjectiveSense','minimize');
% Variables
% Surrogate portfolio weights
y = optimvar('y',nAssets,1);
% Auxiliary variable
tau = optimvar('tau',1,1,'LowerBound',0); % tau >= 0
% Objective
% min y'*Sigma*y
prob.Objective = y'*Sigma*y;
% Constraints
% Sum of means equal to 1
prob.Constraints.sigmaSumToOne = mu'*y == 1;
% Linear equality constraints
prob.Constraints.equalities = A*y == b*tau;
% Linear inequality constraints
prob.Constraints.inequalities = D*y <= d*tau;
% Solve problem
options = optimoptions('quadprog','Display','iter'); % Any other options
x = solve(prob,'options',options);
w = x.y/x.tau;
Notice that to add the risk-free rate, the mean μ should represent the vector of expected excess returns, i.e., the difference between the gross return and the risk-free rate.
An easier way to solve this problem is to use the Portfolio object in MATLAB:
% Create Portfolio object
p = Portfolio('AssetCovar',Sigma,'AssetMean',muGross);
p = setDefaultConstraints(p); % Long-only fully invested
% Set equalities -- Ax = b
p = setEquality(p,A,b);
% Set inequalities -- Dx <= d
p = setInequality(p,D,d);
% Set risk free rate
p.RiskFreeRate = riskFreeRate;
% Find max Sharpe Ratio portfolio
w2 = estimateMaxSharpeRatio(p);
Hope this helps.
Alejandra

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