Hi all,Let's say we have N assets, for each asset we have M strategies, how do we form an "optimal" portfolio?I am thinking of the following:1. Treating each of the N*M assets as a seperate and individual strategy, and run optimization for each of them, (need to be careful not to do over-fitting, but I don't know what's the best way to do optimization without overfitting; let's discuss about this also). 2. For each of the optimized strategies (there are N*M in total), (optimized in the sense of highest possible Sharpe ratio for each one), obtain the returns time series.3. Run the correlation on the returns time series and obtain the covariance matrix of size N*M x N*M.And do a mean-variance analysis with a target expected return and an arbitrary number of risk aversion parameter. Solve for the optimal weights. The data are daily data; but somehow I do monthly rebalancing. So the optimal weights are sought for each month. The above procedure is again optimized in a robust way (but not sure how to do it robustly without getting into overfitting), to get the highest possible robust Sharpe ratio.-------------------\Any thoughts on the above procedure? Thanks a lot!Would you rather spend time optimizing individual asset+strategy w.r.t sharpe ratio, or you would spend time constructing a better portfolio, w.r.t to sharpe ratio?Any practical way of constructing a good portfolio striking a good balance of "optimality" and "practicality"?Any pointers? I just don't know a realistic (non-simplistic) portfolio is constructed...Thanks! For example, do you do daily return covariance calculation with monthly rebalance, etc.?

Hi Mizhael,I personally don't know anything about this subject; looks like a nice mathematical problem, but your question is valid: is it practical to try to solve it? Anyway, I just read in the book "Inside the Black Box" (Rishi Narang) that many quants simply assign equal weights. Here's what the book says: "The basic premise behind an equal-weighting model is that any attempt to differentiate one position from another has two potentially adverse consequences, which ultimately outweigh any potential benefit [...] The first potential problem with unequal weighting is that it assumes implicitly that there is sufficient statistical strength and power to predict not only the direction of a position in the future but also the magnitude and/or probability of its move relative to the other forecasts in the portfolio.[...] The second potential problem is that it generally leads to a willingness to take a few large bets on the "best" forecasts and many smaller bets on the less dramatic forecasts. " However, the book goes on to describe some optimization techniques that people use, so it would be a good source to look at if you are interested. Best,V.

Folks, What's a good way to obtain the expected return in the mean variance framework? That's to say, in the mean-variance framework, a covariance matrix and a vector of expected return is needed as the inputs to the optimization problem. My friend says "using historical mean of return" as a substitute for the expected return here is not a good idea...Any other thoughts? Similar to the covariance matrix?Thanks!

I think you just opened a big can of worms here because how to actually apply the modern portfolio theory to find mean-variance efficient portfolios has been a research topic for a long time. Of course, the most simple way to do it is to use the historic mean vector and the sample covariance matrix of all assets. That's the way I had done it for a finance course homework assignment. Obviously, this is simplistic. There are many problems. The expected returns may be time varyings. More than two decades of GARCH research had shown that the conditional variances are most likely time-varying too. Finally, the sample covariance matrix of historic returns may be a very noisy estimator even for the unconditional covariance matrix. There is a lot of papers on this topic. One of the more recent is by Oliver Ledoit and Michael Wolf:"Improved Estimation of the Covariance Matrix of Stock Returns With an Application to Portfolio Selection" (2003)They propose a covariance estimator based on shrinkage estimation, and then compare its performance to a few other covariance matrix estimators. This may be a good start if you're interested in applying the theory. I am sure there may be other relevant papers out there..

QuoteOriginally posted by: APSI think you just opened a big can of worms here because how to actually apply the modern portfolio theory to find mean-variance efficient portfolios has been a research topic for a long time. Of course, the most simple way to do it is to use the historic mean vector and the sample covariance matrix of all assets. That's the way I had done it for a finance course homework assignment. Obviously, this is simplistic. There are many problems. The expected returns may be time varyings. More than two decades of GARCH research had shown that the conditional variances are most likely time-varying too. Finally, the sample covariance matrix of historic returns may be a very noisy estimator even for the unconditional covariance matrix. There is a lot of papers on this topic. One of the more recent is by Oliver Ledoit and Michael Wolf:"Improved Estimation of the Covariance Matrix of Stock Returns With an Application to Portfolio Selection" (2003)They propose a covariance estimator based on shrinkage estimation, and then compare its performance to a few other covariance matrix estimators. This may be a good start if you're interested in applying the theory. I am sure there may be other relevant papers out there..Yeah, lots of papers, but how do they perform in real-world? Any good methods/models there?

Your are right, there are many papers related with this topic.Regarding the estimation of the vector of means, the estimation error is much larger than that of the covariance matrix. So, less to do with the expected returns.Regarding the covariance matrix, the best methods are based on the shrinkage techniques by Ledoit and Wolf:Ledoit, O., and M. Wolf, 2003, "Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection," Journal of Empirical Finance, 10,603-621.Ledoit, O., and M. Wolf, 2008, "Robust Performance Hypothesis Testing with the Sharpe Ratio," Journal of Empirical Finance, 15, 850-859.Regarding the estimation of the portfolio weights directly, the best paper I know is:DeMiguel, V., L. Garlappi, F. Nogales, and R. Uppal, 2009, "A Generalized Approach to Portfolio Optimization: Improving Performance By Constraining Portfolio Norms,"Management Science, 55, 798-812.