# Linear programming and factor models vs M-V optimization?

I have been recently researching about portfolio optimization problems and it is unclear to me what is currently the state of art modeling choices when it comes to this topic.

On one hand, I've researched linear factor models and linear programming. These approaches generally revolved around the idea of modeling the mean $$\mu=E[R_{p}]$$ of the portfolio as a factor model, i.e., explaining it either through some financial factors (Fama-French portfolios), or some statistical procedure PCA or Factor Analysis, etc. The idea seems to revolve around the fact that if we could model individual returns as $$E[R_{i}]=\beta X+e_{i}$$, and $$E[R_{p}]=wE[R_i]$$, then $$E[R_p]=w\beta X$$. This is a bit abuse of notation, but this article describes it better. In this case, the variance is not accounted for, therefore, we do not need to model it (which is a big advantage), since we can just maximize $$\mu$$. We then try to control the variance by specifying constraints, either by constraining factors, or industries, or stock themselves.

On the other hand, we have the mean-variance optimization problems. I will deviate from Markowitz's term mean-variance and state that any objective that involves the mean and the variance is a mean-variance problem (so, optimizing Sharpe or Kelly is also a mean-variance). This will require us to model both the moments, which is significantly more difficult. However, it seems that the covariance/variances are not as important, so misestimation there might not be such a large issue (correct me if I am wrong). There are, obviously, many modifications to this (e.g., Fabozzi has an entire textbook on robust portfolio optimization). This can come in many formst, from robust estimators (James-Stein for mean, Ledoit-Wolf for covariance) to robust optimization (including confidence intervals for the estimates as a min-max problem). VaR optimization, for me, would also be a mean-variance optimization, since you would inevitably need to model a mean and a variance (e.g., in a case where VaR is a simulation of a GBM, parameters for mu and sigma still need to be fitted).

Stemming from all I've said above, I have the following questions:

(1) Both methods require estimation of $$E[R_i]$$, which I believe would be better notated as $$E[R_{i,t+1}]$$, where $$t+1$$ is the next period when you are going to reoptimize and rebalance your weights. If so, what is best practice when it comes to the choice of horizon? Do we model daily returns and scale them to weekly, monthly? Or do we model monthly returns? If so, do we consider monthly as overlapping or not?

(2) Are we particularly interested in the forecast $$E[R_{i,t+1}]$$ or are we interested in simply explaining (think of factor models) $$E[R_{i,t}]$$? It seems that we do the former in M-V optimization and the latter in linear programming?

(3) Is there a consensus of what is a better approach currently in the industry? It seems that linear programming and linear factor models are way simpler way of modeling, so perhaps less model risk? On the other hand, M-V optimization has evolved quite a lot since Markowitz yielding stable weights if done properly?

Here are just some musings so someone can correct me if anything I've said is incorrect:

1. You are always estimating future returns of your assets. This can be done via a factor model, fundamental analysis, or other methods. The wider your horizons, the more Gaussian your returns will look. Monthly returns will therefore tend to look more normal than daily or even hourly returns. If you are assuming Gaussian returns (for instance in modern portfolio theory) you want your asset returns to line up with assumptions.
2. Factor models can be used in explaining risk and returns (attribution) but can also be used in forecasting. I've commonly seen factor models used to construct a Variance-Covariance (VCV) matrix as it tends to be more stable period-to-period than a VCV matrix derived from the asset's returns. This is important because small changes in your VCV matrix can have a big effect when plugged into an optimizer.
3. Generally you will want factor constraints in active management as you are managing to benchmark. These could be anything from beta, to sectors, to firm size etc. Therefore you will usually use factors in your risk model even if you don't in your return model. In industry, you will have quant PMs that will largely use factor models to forecast alpha. Then you will have fundamental PMs that use fundamental analysis to forecast alpha. These are different approaches trying to obtain the best result possible. I think you are getting confused between your objective function (such as mean-variance, shape ratio, global minimum variance etc.) and the inputs to your model which are how to estimate asset returns and VCV.
• Can you please expand on (2). I have recently experimented with singular value decomposition. Applying SVD on the data and constructing the VCV is the equivalent of MLE estimator of VCV under Gaussian. Aug 30 at 7:49
• If you have factor exposures for your assets and a factor VCV matrix, you can use these to estimate the asset VCV. I remember reading that this tends to be more stable period-to-period as exposures don't change frequently and factors themselves are generally chosen because they are orthogonal. Here is the math describing the transformation: web.stanford.edu/~wfsharpe/mia/fac/mia_fac3.htm Aug 30 at 12:28
• Also as a subsequent example, it is easier for PMs to have a forecast on 50ish factor returns than a forecast of 1000+ asset returns. This helps just based on number of forecasts required but also because you are estimating systemic components (factors) instead of systemic and idiosyncratic components (stock returns for example). What this means is you need fewer forecasts and the forecasts are easier to get 'right' (i.e. it is easier to predict how the auto industry as a whole will perform than to predict Tesla's returns). Aug 30 at 12:57

I have an academic background, but I'm also familiar with practical applications in the field. Large hedge funds like Citadel or AQR often employ academics in think-tank environments to develop ideas that are later tested by practitioners. These ideas go through multiple phases before becoming a part of the traders' toolkit. I will use momentum strategies as a reference point since they provide good illustration of the topic and conflicting ideas.

1. Return Periods
• When doing research, it's generally advisable to collect as much data over as long a time horizon as possible. Because stock markets are volatile and noisy, daily returns might not provide an accurate representation of actual performance. Weekly or monthly data should be sufficient.

• For Long-only Portfolios: If your research focuses on long-only portfolios based on long-term firm characteristics, using monthly returns over a 20-year period for in-sample parameterization could be effective. Methods like Fama-MacBeth can further refine your model for out-of-sample period.

• For Momentum Strategies: You'll need to consider the length of the period that restricts your data collection. Finding expected stock returns for the last 3-12 months is not enough. You also need to quantify the momentum's strength and stability. Diversification across industries is recommended; think about finding good quality momentum stocks just to have them to be sensitive to the same exact systematic factor.

• Sidenote: Individual stocks usually exhibit positively skewed returns (long right-tail), whereas market returns are generally negatively skewed.

1. Covariance Matrix
• I'm not sure what exactly you're asking, but @LattePrincess seems to have a good idea. Properly estimating the covariance matrix is crucial. If you only use past actual returns and your portfolio has 15-30 stocks, you're better off equally weighting them or using the inverses of individual standard deviations as portfolio weights.
• Suggested Reading: Paper by Demiguel et al. (2007) http://faculty.london.edu/avmiguel/DeMiguel-Garlappi-Uppal-RFS.pdf
1. Strategy-Specific Factors The strategy you're employing will dictate which factors you need to consider.
• For Systematic Risk: You'll likely want to derive some estimates. However, remember that momentum assets can produce absolute returns, provided extreme events are excluded.

• Diversification Concerns: Momentum strategies/assets tend to perform poorly overall in adverse conditions. Factor models can only offer limited help.

• Real-World Restrictions: Consider constraints like maximum weights per asset, long/short ratios, and broker requirements.