optimal allocation of two assets by minimizing shortfall probability using fmincon

I want to determine the optimal allocation of two assets (here "x") such that shortfall probability is minimized.

Shofrtfall probability is derived by applying condition that (if B<b) then sum all incidences and divide by total trials for each "t" and then finally sum the probability of all "t". The value of shortfall probability is 50.55.

When I run fmincon, I get fval=50.55 and the same initial guess of the two weights of two assets.

Following is my code.

%Defined constants inlcude m_e,v_e, m_b,v_b,rho, L,trials,b       
x1 = 0.5; %weights of assets 
x2 =0.5;
t =1:35;                % columns
T = 35;                 % Total time
x = [eq_share, bd_share]; % total 

%simulations 
RNx1=normrnd(); % n x m normalized random numbers for first asset
RNx2=normrnd();  % n x m normalized random numbers for second asset

Returnx1= exp(m_e + v_e*(RNx1)); % return on first asset
Returnx2 = exp(m_b+(rho*v_b*RNx1+((1-rho^2)^0.5)*v_b*RNx2)); % return on second asset

Returnportfolio = x1*Returnx1+(x2)*Returnx2; % portfolio returns

W = max(L*Returnportfolio,0); % Matrix of wealth
B = min(b,L*Returnportfolio); % Realized Benefit
p= B<b
S=(cumsum(B<b, 2) == 1) .* (B<b)
shortfallprobability =sum(S)./trials
 
%% Optimal allocation 

options = optimoptions('fmincon','Display','iter','Algorithm','sqp');
opts.Algorithm = 'sqp';
f = @(x) shortfallprobability
[n,fval] = fmincon(f,[0.5 0.5],[],[],[],[],[],[],[],options)

When I run the program, I get

  function_handle with value:
  
    @(x)  sum((sum((cumsum(B<b,2)==1).*(B<b))./sim))
 Iter  Func-count            Fval   Feasibility   Step Length       Norm of   First-order  
                                                                       step    optimality
    0           3    5.055392e-01     0.000e+00     1.000e+00     0.000e+00     0.000e+00  
    

Initial point is a local minimum that satisfies the constraints.

Optimization completed because at the initial point, the objective function is non-decreasing 
in feasible directions to within the value of the optimality tolerance, and 
constraints are satisfied to within the value of the constraint tolerance.

<stopping criteria details>

n =

                       0.5                       0.5
fval =

         0.505539191954337

Can you please check the code and my mistake. I am sorry to bother as I have already spent much time on it but could not figure out.

Is my fmincon setup ok?

can I use fmincon for this problem?

Am I definining objective function wrongly?

Answer 1

[Alan Weiss]

The problem is the you have not defined your objective function as a separate file that the optimization calls. You need to write a file something like this:

function f = shortfallprobability(x,RNx1,RNx2,m_e,m_b)
Returnx1= exp(m_e + v_e*(RNx1)); % return on first asset
Returnx2 = exp(m_b+(rho*v_b*RNx1+((1-rho^2)^0.5)*v_b*RNx2)); % return on second asset
Returnportfolio = x1*Returnx1+(x2)*Returnx2; % portfolio returns
W = max(L*Returnportfolio,0); % Matrix of wealth
B = min(b,L*Returnportfolio); % Realized Benefit
p = B<b
S = (cumsum(B<b, 2) == 1) .* (B<b)
f = sum(S)./trials
end

Take the corresponding lines out of your code. Then call the solver like this:

f = @(x)shortfallprobability(x,RNx1,RNx2,m_e,m_b);
[n,fval] = fmincon(f,[0.5 0.5],[],[],[],[],[],[],[],options)

Alan Weiss

MATLAB mathematical toolbox documentation

[susman]

Thank you for the suggestion. I am trying to do that but still getting the error that my initial function or varaible is unkown. I try to firgure it out and will tell you when the code will work.