Markowitz's investment theory , Mean square model , It's a theory of portfolio selection , Its basic content is ： Under the assumption that there is no risk-free lending , Find out the effective frontier of portfolio based on the mean and variance of individual stock return , The optimal portfolio under certain conditions when investors allocate portfolio on the effective frontier .
Effective frontier μ, σ A hyperbola on a plane of coordinates , Opening right , The above points must be fully decentralized and eliminate the non systematic risk of the portfolio .
The basic theoretical formula is as follows ：
objective function ：min σ^2=∑∑ (Vij*Wi*Wj)—— Minimum variance
①μ=E(μ)—— Under certain expectation
②ΣWj=1—— The sum of the weights is 1
therefore , The Markowitz mean square model can be regarded as ①,② The linear programming problem of realizing objective function under two constraints .
It can also be regarded as the dual problem of the linear programming problem , soon ① The maximization of , Take the current objective function as the constraint condition .
The following shows the effective frontier in the way of simulation ：
function Port1=PortSimu(M,N,mu,sigma) % Yes N Medium assets simulation M Group weight M=100;N=3; X=zeros(M,N) for
i=1:M X(i,:)=rand(1,N) X(i,:)=X(i,:)/sum(X(i,:)); end
% Simulated yield , Based on this, the expected return and covariance matrix of three assets are calculated mu=10; sigma=0.6; R=normrnd(mu,sigma,M,N);
% Calculated and simulated M Portfolio risk and return under group weight for i=1:M [PortRisk(i), PortReturn(i)] =
portstats(ExpReturn, ExpCov, X(i,:)); end % Draw a simulation plot(PortRisk, PortReturn,'r.')
The picture is as follows , We can see clearly the effective frontier curve