Fmincon - Value-at-Risk Minimization - Incorrect values returned

Hi all,

I'm having troubles using fmincon. A little help would be reaally appreciated.

So i'm trying to minimize the following function (Semi-Parametric Conditional Value-at-Risk) over w, the weight between two assets return series :

function adjcvar=adjcvar(rets,alpha,w) 

%Portfolio return
rp=w*rets(:,1)+(1-w)*rets(:,2);
%Mean return and StDev
mu=mean(rp);
sig=std(rp);
%Skewness and Kurtosis
S=skewness(rp);
K=kurtosis(rp);

%Standard Normal inverse cumulative distribution function 
z=norminv(alpha,0,1);

%Estimated Quantile adjusted for Skewness and Kurtosis
Q=z+(z^2 - 1)*S/6+(z^3 -3*z)*(K-3)/24 - (2*z^3 - 5*z)*(S*diag(S))/36;

%Adjusted Value-at-Risk
adjvar=min(mu+sig*diag(Q),0);

%Adjusted Conditional VaR       
adjcvar=mean(rp(rp<=adjvar));

Fmincon is used as follows:

[w]=fmincon(@(w)adjcvar([ret.mscimin,static.vsfret],alpha,w),w0w,[],[],[],[],lbw,ubw,[],options);

with

options = optimset('Display','notify','Algorithm','active-set');
lbw=0; %Upper-bound
ubw=1; %Lower-bound
w0w=0.5; %Starting Value

What happens is that if I set w0w, the starting value, between 0 and 0.61, fmincon returns w=0. If i set w0w at 0.62 or above, fmincon returns w=1.

I know empirically that the right weight is around 0.61-0.62.

Fmincon returns the following message:

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the default value of the function tolerance,
and constraints were satisfied to within the default value of the constraint tolerance.

<stopping criteria details>

Active inequalities (to within options.TolCon = 1e-006):
  lower      upper     ineqlin   ineqnonlin
    1

I'm quite new to fmincon and don't really understand why itjust goes to the upper or lower boundary and stops.

Thanks for reading and any advice you could give.

Have a nice day.

Edouard

Answer 1

[Amit]

This is a single variable minimization. You should try fminbnd instead of fmincon.

Also, try plotting adjcvar([ret.mscimin,static.vsfret],alpha,w) for w values between 0 and 1 and see is the difference in sizes.

[Matt J]

Moreover, the objective function is not smooth, due to this operation

 adjvar=min(mu+sig*diag(Q),0);

so FMINCON does not apply.

In addition, I wonder, why this

 function adjcvar=adjcvar(rets,alpha,w)

isn't creating problems. The same name 'adjcvar' is being used for both a variable and function, i.e., both the name of the objective function and its output.

[Amit]

I didn't even notice that. Strange.

[Edouard]

Thank you so much, fminbnd works way better!

Actually, in the sample I work on mu+sig*diag(Q) is never greater than zero, such that the min(x,0) is always at x within the boudaries set for w.

The function is a nice, smooth bell.

Finally, and you might find this ridiculous, I'm mnimizing CVaR whereas CVaR is a negative value.

Hence I changed adjcvar to abs (mean(rp(rp<=adjvar))) and it works fine.

Note that fmincon returns a close but different value than fminbnd (0.6144 against 0.6183).

Anyway, thanks to you both.

[Matt J]

Actually, in the sample I work on mu+sig*diag(Q) is never greater than zero, such that the min(x,0) is always at x within the boudaries set for w.

That doesn't eliminate the hazards of non-smoothness. FMINCON doesn't confine itself to the bounds during its search. Not by default anyway. Moreover, min(x,0) is only smooth where x is strictly less than zero, and it's not clear whether your bounds would guarantee that, even if obeyed.

Note that fmincon returns a close but different value than fminbnd (0.6144 against 0.6183).

That could be a matter of the stopping tolerances you've chosen in each.

Answer 2

[Moad]

Hello Sir can u provide a code of minimazing Value at Risk with the fmincon

i actualy want to minimize portfolio Parametric VaR for many assest returns

thank you very much