One possible solution is seen here on page 45 http://www.kellogg.northwestern.edu/faculty/papanikolaou/htm/finc460/ln/lecture1.pdf

But the question of the benefit of Machine Learning (as other alternative methods) for this type of questions remains...

As I wrote the question, I got to the solution my self. So - I'll rephrase the question and be more specific.

I have a risk score of only 100 of the companies out of 1.000. So, I need to generalize the 900 out of the training set of 100.

The behaviour of all the 1000 companies is available.

Which Machine Learning method would be more appropriate? Would you build 100 categories? or try to group the 100 training set in a few group categories, like A, B, C, D, E, F ?

Clearly, after having a risk for each of the 1000 companies, the Sharpe ratio could be used to solve the matrix equation.

@DS A important and relevant question...

* I think you mean 'where' not 'were' in your question...

Anyway, I don't see 'time' in your equation...given we are talking about credit, you may want to look at the half-life of a loan [Discounted Payback Period]...

Also, you may like the IC [information coefficient] better than the Sharpe ratio [it substitutes benchmark return for risk-free return].

Hope this stimulates some thoughts. Cheers!

Thanks!. Sorry for the English typo. I'll check further on information coefficient...

Dear Daniel, if you wish to optimize risk/return, then Sharpe ratio is the classical way to go. Unfortunately, the estimation of the numerator of the ratio (excess returns) is very complicated. Moreover, the closed-form solution will always give very leveraged portfolios which will in turn yield considerable turnover.

I would rather recommend a Minimum Variance approach. This is justified not only because the related portfolio are less risky, but also because they generate competitive P&L (you can type "Low vol anomaly" in Google and many references will pop up).

However, this does not solve the turnover problem. Nor does it generate Long-Only portfolios. You can try either norm constraints (see DeMiguel et. al in Management Science), or numerical optimization using lower bounds for the weights.

With regards to Machine Learning, here is a nice link : http://www-stat.wharton.upenn.edu/~steele/Courses/9xx/Resources/MLFinancialApplications/MLFinance.html

I am not sure that you should be headed in this direction at this stage.

I did not fully understand how you compute your risk score. Usual indicators are volatility, , Value-at-Risk, Expected Shortfall... Do you want (for instance) to replace the vol of the Sharpe Ratio by your risk indicator ?

Guillaume, First thank you very much for taking the time and writing me back. I'll read it carefully later on and post my comments here.

At the moment, I have the probability of default as parameter. The assets are not publicly traded, so volatility itself is not available. thanks again.

Daniel, in this case, if probability of default is the only information you can easily obtain, I suggest you try the classical weighting scheme which assigns to each asset a magnitude inversely proportional to its risk. If p_i is the probability of default of asset i, then its weight should first be set to 1/p_i and then all the weights should be normalized so that they add up to one. Consequently, the asset with the highest probability of default has the smallest weight, etc. It's pretty standard and works well if the risk indicator is the volatility (this approach is equivalent to the Equal Risk Contribution concept introduced by Maillard et. al 2010 when all the correlations are equal).

Guillaume,

Do you mean the following article or something else?

http://www.thierry-roncalli.com/download/lwp-rbi.pdf

In fact, I meant

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1271972

But all papers of this page are worth looking at:

http://www.thierry-roncalli.com/riskparity.html

Thank you so much. And this social network is so cool, because other people with similar research would not have to ask again :-)