# Monte Carlo based mean variance optimization

I was asked this question in an interview some years ago. It struck me as a poorly formed question. I thought I would put it out there to the community to see if I just simply missed something.

Problem Statement For n assets, you are given expected returns (ER), variances (V) and covariances. Your task is to write Monte Carlo based mean variance optimization that will:

1. Produce a set of efficient portfolios with increasing volatility / return.
2. Find the minimum variance portfolio
3. Find the highest sharpe ratio portfolio.

Portfolios should be subject to the following constraints:

1. No shorting (all weights >= 0)
2. No leverage (sum of all weights = 100%).

Why I think this is poorly stated problem I understand MVO and MC. The only context I have seen MC in a MVO concept is where MC is utilized to make random draws from a chosen distribution to arrive at an ER and Covariance Matrix. Those however are given here.

If I am wrong here then what is the MC random draws in this case - the asset weights?

• The way you've put it, I agree that it is poorly phrased. Even if you sampled the returns with Monte Carlo, you could still pass the expected returns and covariances of that result to any portfolio optimizer. They could have meant Michaud resampling, but just saying MC-based MVO is too vague. – John Feb 22 '16 at 18:31
• Thanks John. I thought perhaps the intent was to generate a traditional MVO with the provided inputs and then generate resampled MVO using f permutations on the starting inputs like this. corporate.morningstar.com/ib/documents/MethodologyDocuments/… I did ask the interviewer but was given no additional input. I've certainly done Google searches over time for some guidance but found nothing definitive. Thus I put it out here. – Eric Bruce Feb 22 '16 at 18:38
• The resampling procedure in that paper is the same that I was thinking of. I wouldn't stress about it too much: would you really want to work for someone who can't make their interview questions clear? – John Feb 22 '16 at 22:05

I believe the question to be too vague to be a good interview question. If you want to do Mean Variance Optimization (MVO) it's hard to see the point of Monte Carlo simulation. One of the good thing of MVO is its analytic tractability. Clearly, the topic is not widely discussed as this Google Search has this question as the first result (I was in incognito mode). The first linked paper by Xu would not be appropriate for any interview. Wikipedia mentions the usage of Monte Carlo for extensions of MVO but not for classical MVO itself.

To conclude: I don't believe you're wrong. They could have meant other things but then there is much to choose from, e.g. Michaud resampling as suggested by John and the above.

• Wrt to your point about Monte Carlo extensions to MVO. Monte carlo simulations are also used for mean-CVar/ES optimization. However, that's not mean-variance. If someone gives you the mean and covariance, then they are clearly implying you don't need to deal with vine copulas (to pick on one thing in the Wikipedia section). Even if they did give you vine copulas, then you could still generate the distribution from Monte Carlo, calculate the mean and covariance of the simulations and pass that to the optimizer. – John Feb 22 '16 at 22:52

what is the MC random draws in this case - the asset weights?-Yes How will solve a optimization problem 1.by traditional methods like Lagrangian multipliers 2. Monte Carlo Simulation: You are given expected returns and co-variance matrix, Now run a Monte Carlo Simulation (say 10000 times) by randomly selecting the weights of the portfolio. For every randomly chosen set of weights calculate portfolio return and volatility i.e you will have 10000 portfolio returns and volatility pairs. Identify the optimal portfolio based on the given conditions