Optimization problem - Sequential Approximation - f(x(i)) = x(i) * E[r(i)|rp(x)<=VaR] - find the x such that all f(x) are equal

Dear all, I am trying to solve the following optimization/sequential approximation problem concerning an ERC portfolio.

I have n assets in portfolio. r(i) is a random variable which identify the asset's i return and rp( x ) is the portfolio's return, function of assets' returns ( x '* r ). x is the vector of portfolio weights.

Now, I have a Risk Contribution function which is:

 RiskContribution(i) = x(i)* MarginalContribution(i) 
                     = x(i) * E[r(i)|rp(x)<=VaR(rp(x))]

What I am looking for is the x * portfolio allocation such that the vector of Risk contributions RC = [RC(1) RC(2) … RC(n)] have all the same elements (RC(1)=RC(2)=…=RC(n)).

Does anybody have an idea of how to solve this problem?

I think that is impossible to solve it as a minimization problem because once I change the portfolio's weights, also portfolio returns rp(x) and VaR Value-at-Risk change.

PS: naturally, I need x(i)>0 for every i since short selling is not allowed in my problem [..and at least a minimum weight should be allocated to every asset]. An additional constraint could be the sum(x(i))= 1 but it is not compulsory (I can normalize weights in a second moment).

Thanks to everybody.

[Matt J]

An additional constraint could be the sum(x(i))= 1 but it is compulsory (I can normalize weights in a second moment).

Did you mean to write that it is not compulsory? If so, then x(i)=0 for all i is a trivial solution.

I also don't see how you are free to post-scale the x(i). Does VaR(x) satisfy the following homogeneity property

     VaR(c*x)= c*VaR(x)

for c>0. Otherwise, I don't see it.

[Andrea]

Dear Matt,

thanks for your reply. I have changed the post [just added NOT COMPULSORY..I did a mistake, sorry]. I have also corrected the part in which I define the weights constraint [x(i)>0 for every asset].

a) I supposed to post scale the weights as y(i)=x(i)/sum(x(i)) with y(i) the new weights (which will sum up to 1).

b) Value-at-Risk satisfies the homogeneity property.

Do you have any idea of how to proceed?

[Matt J]

Do you already have a means of evaluating

 MarginalContribution(i,x) =  E[r(i)|rp(x)<=VaR(rp(x))]

Is this a function you already have code for?

[Andrea]

Yes, sure.

I have coded it in this way. Please, if the code looks not professional, I apologize (I am not an experienced MATLAB end user. I have learn it in less than 1 month).

the idea is to find the expected value of all the ith asset's returns in corrispondence of the portfolio returns lower than VaR.

So, easily I structured it in the following way:

%1) compute portfolio returns

      returnsPortfolio= x*vertcat(returnsAsset1, returnsAsset2,..., returnsAssetN)

%2) build a matrix of all the returns as:

returnsAll = [returnsAsset1, returnsAsset2,..., returnsAssetN, returnsPortfolio]

%3) order the returns matrix, sorting for the portfolio returns in an increasing way

[Y,I] = sort(returnsAll(:,nAssets+1))
returnsAllSorted = returnsAll(I,:)

%4) compute the PortfolioVaR which will be used to select only portfolio returns <= PortfolioVaR

PortfolioVaR = quantile(returnsPortfolio, alpha)

%5) compute the marginal risk contribution of each asset i [marginal contribution will be a row vector of nAssets elements]. Below, consider that column nAssets+1 indicates the column of portfolio returns.

for i=1:nAssets
for j=1:FinalTime
if returnsAllSorted(j,nAssets+1) <= portfolioVaR
sum(i)=sum(i)+returnsAllSorted(j,i)
end
end   
marginalContribution(i)=sum(i)./(nTrials*alpha)
end

% in vector "sum", each element is the sum of returns of asset i which are in corrispondence with portfolio returns <= VaR. Then, dividing by the number of elements summed, we obtain the expected value/mean.

Hope this could help you (and me).

Regards.

PS: this code works correctly. I have already checked.

[Andrea]

Dear Matt,

I have just solved the problem applying the fmincon procedure and following an SQP algorithm.

Since this problem concerns Equal Risk Contribution, this means that RC(i)=RC(j) for every j and i. Moreover, we know that sum(RC(i))=ES [Expected Shortfall Portfolio].

So, I have minimized error=sum((RC(i)/ES - 1/n)^2) using fmincon and setting up 0 as lower bound and 1 as upper bound for weights.

Answer 1

[Matt J]

Glad you got it working, Andrea. I'll just add as an Answer that LSQNONLIN might be better for an objective of the form

 error=sum((RC(i)/ES - 1/n)^2)

and for constraints consisting only of upper and lower bounds. LSQNONLIN is more customized for least squares minimization than FMINCON.