Monte Carlo Weight Sampling for Mean–Variance Portfolios
Summary
The document considers an interview prompt asking for a Monte Carlo method to construct an efficient frontier, minimum-variance portfolio, and maximum-Sharpe portfolio under long-only, fully invested constraints. Expected returns, variances, and covariances are already supplied, raising the question of what should be simulated. One answer is to randomly sample feasible portfolio weights, calculate each portfolio’s expected return and volatility, and inspect the resulting risk–return cloud for candidate optima.
The responses also question whether Monte Carlo adds value for classical mean–variance optimization, since analytic optimization with Lagrange multipliers can produce the efficient frontier directly. Random weight sampling may illustrate feasible portfolios, but can miss optimal portfolios and does not by itself establish that the frontier is complete. The discussion mentions resampling as a possible extension but gives no implementation details, convergence analysis, or comparison of sampling schemes. Thus, it clarifies one interpretation of the prompt while highlighting that its intended method and success criteria are underspecified.
Key ideas
- When returns and covariances are given, Monte Carlo can sample feasible portfolio weights rather than asset returns.
- Each sampled weight vector can be evaluated by its expected return and volatility.
- The long-only, fully invested constraints require nonnegative weights that sum to the full portfolio value.
- For classical mean–variance optimization, analytic methods can make random sampling redundant for finding the frontier.
- A finite random sample may fail to capture the true efficient frontier or its optimal portfolios.
Tags
Full text
# Monte Carlo based mean variance optimization # 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: - Produce a set of efficient portfolios with increasing volatility / return. - Find the minimum variance portfolio - Find the highest sharpe ratio portfolio. Portfolios should be subject to the following constraints: - No shorting (all weights >= 0) - 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? ## Answer by Bob Jansen (score 4) https://quant.stackexchange.com/a/24468 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. ## Answer by Alon Benach (score 2) https://quant.stackexchange.com/a/78159 It seems the MC method is only used to namedrop theory in this case. Yes, you can simulate 10,000 sets of weights to form a cloud of ER to risk plots, but since you're going to solve it with Lagrangian multipliers to get the efficient frontier, which is the only item of interest there, the simulation is redundant and adds nothing to your model. ## Answer by Sivaji (score 0) https://quant.stackexchange.com/a/41712 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
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.