# Combination of factors

Let's say I have 10 factors and I want to find a combination (basically sum of exposures) of factors (of any length) from this set which has max sharpe. Is there an easy way to find this out rather than running simulations of all 2^10 combinations?

This seems like a portfolio optimization problem, but with factor portfolios. What do you mean by best performing? Do you refer to:

• Minimum variance
• Maximum Sharpe
• Maximum returns

From what I recall, only minimum variance has a closed-form solution to find the weights and therefore there is no need to run simulations. There might be a variation of the maximum Sharpe where you have a closed form solution but I think it does not involve short-selling - honestly not sure for the maximum Sharpe part.

You can go into some portfolio optimization books or literature to find out, it should not be that hard. Good luck!

• By best performing, I mean maximum Sharpe. How will portfolio optimization help me choose a combination of m factors with max sharpe from set of n factors. Isn't portfolio optimization use to find optimal weights with given a given alpha and covariance matrx? Commented Mar 9 at 6:16
• Yes, the optimal weights is your vector of exposure to your factors. You would need historical returns to generate your expected returns (not alpha) and covariance matrix. Depending on your objective function, you can obtain a vector of exposures that produces the max Sharpe. Commented Mar 9 at 9:36
• Hi @Anonymous, if you feel my answer has helped you, maybe you can give me an upvote or accept my answer as the solution (: Commented Mar 13 at 13:48