Optimization problem for assigning portfolio weight for each stock price

optmiz_pb.md

Portfolio Optimization (Stock Market)

Basically, we want to assign weight to history stock price, higher weight means good stock. So the question is: I need to maximize cost D, where we define as,

D = w' * std / sqrt (w' * Sigma * w)

where

• w is portfolio weight

• w' is the transpose of w

• w is N-by-1 vector (weight vector corresponded to each stocks)

• w' is 1-by-N vector

• std is N-by-1 vector (standard deviation of history stock prices)

• sigma is N-by-N matrix (covariance matrix of historical stock data)

• (w' * std) is inner product of weight and standard deviation of stocks, which is scalar

• sqrt(w' * Sigma * w) is also a scalar (called risk... I think)

The maximization of D needs to keep std and Sigma constant, and it only needs to vary "w". After max D is obtained I need to be able to see the final w and use it for future calculations.

The constraints are:

• sum of the w in the vector any time equals = 1 (sum(w) = 1)
• bound of each individual w in the vector: -1 =< w <= 1

Note Minimization Function from Matlab

• http://www.mathworks.com/help/optim/ug/fmincon.html

Solution

Here I tested on random matrix. Note that we use cost = -D instead since fmincon solve minimization problem so it's equivalent to solve maximization problem for cost = D

% initial condition
N = 10;
Sigma = randn(N,N)'*randn(N,N);
w0 = randn(N,1);
std = abs(randn(N,1));
Aeq = ones(N,1)';
beq = 1;
lb = -ones(N,1);
ub = +ones(N,1);
calculate_cost = @(w) -w'*std/ sqrt(w'*Sigma*w); % inline function

% using fmincon to solve
w_final = fmincon(calculate_cost, w0, [], [], Aeq, beq, lb, ub);

Here is the solution with actual problem

[N, M] = size(std);
Aeq = ones(N,1)';
beq = 1;
lb = -ones(N,1);
ub = +ones(N,1);
w0 = randn(N,1);
calculate_cost = @(w) -w'*std/abs(sqrt(w'*Sigma*w)); % assuming we know Sigma and std

w_final = fmincon(calculate_cost, w0, [], [], Aeq, beq, lb, ub);

So w_final is the final weight solution, fmincon can stuck at local minima you can either run multiple time or seed with different initial condition w0

To test, with eyeball, corresponded w should has low value if std is high. And also cost of final weight w_final should be lower than initial w0

calculate_cost(w0)
calculate_cost(w_final) % should have lower cost than line above

Another easy way to assign weight to each stock

w_i = (1/sigma_i)/sum(1/sigma_i)

where sigma_i is standard deviation of history of stock price. Suppose stock is matrix where each column contains stock price history

std_vec = std(stock, 1); % standard deviation over column
w = (1./std_vec)/sum(1./std_vec); % final weight

Reference

• I like this article http://www.nag.com/doc/TechRep/Pdf/tr2_00.pdf