I haven't posted much since the start of school. I'm still working on Portfolio Management but much of what I have learned aren't that worth blogging about since it's nothing new and different. The only piece of thing I have on my blog is in relations with my work on The Fund. I also haven't updated the Docs as well on MPT because I haven't read much of that book recently. I'm currently focusing on getting a better overview of the available Black-Litterman literature. There are also other methods I need to get to learning such as Portfolio Sorts by Almgren, Entropy Pooling by Meucci, Risk Parity portfolios, etc.
However, I did come up with a new weighting method which I thought may be sort of a novel thought in terms of portfolio management. It is nothing new and I am simply recycling existing ideas. The resulting portfolio does have some nice properties and is shown to outperform in my out-of-sample but that is in one case I have surveyed and many more tests need to be done before I can confirm its validity. The rest of this article will be split up in four sections: The Mean-Variance Setting, Net Neutral portfolios, Methodology of study and finally results.
In the problem of Asset Allocation, the investor will need to decide how much wealth to allocate to each asset. According to utility theory, the investor allocates these weights that maximizes his/her utility function where is an vector of asset weightings. The most common Utility function is the power function
Where is a vector of asset mean returns, is an covariance matrix and is the Investor's risk aversion coefficient. Under optimal weights , the Utility function becomes:
In finding the derivative, the first term is just . For the second term, recall that the covariance matrix is symmetrical thus and can be aggregated into one term for every . Thus, the derivative simplifies to a gradient:
Setting in the original power function, we arrive upon the above equation.
Where is the Portfolio Sharpe Ratio thus maximizing the utility function would be maximizing the numerator and thus the highest Sharpe Ratio. Later on in my empirical work, I will maximize the sharpe ratio rather than the utility function as the result is the same.
Net Neutral Portfolios
In a perfect world, this is easy, but we live in a random world and thus need to estimate our mean asset return and covariance matrix. These two inputs vary over time and are hard to forecast. In the covariance case, research by Elton and Gruber (2006) show that an improved estimation of covariance matrix can be done through the constant correlation model. It assumes that 1) sample estimates create large imbalances in actual covariance between assets and 2) since we cannot accurately forecast these imbalances, assuming the average correlation provides a better estimation than samples. In the asset return case, empirical returns are used although there are many proposed methods in forecasting returns (I'm still working on it folks).
The formulas used in mean-variance optimization assumes perfect information and 100% certainty in inputs. However, estimation risk persists and often mean-variance optimization will yield large long and short positions in certain stocks that has experienced significant price movements. In out-of-sample performance, these concentrated portfolios often under perform the benchmark and even the 1/N heuristic which has gained traction in recent years.
In designing an optimal portfolio for The Fund, I often run into these issues of largely imbalanced positions and little diversification (in a long only fund portfolio). I came up with easy rules to deal with this such as maximum weights constraints, combination of a GMVP Portfolio and Tangency Portfolio, etc. In searching for a solution, I thought of a novel idea that tries to incorporate an equally weighted portfolio as the starting optimal one. It is a simple solution to highly concentrated portfolios by incorporating the 1/N portfolio.
Two portfolios are constructed, one equal weighted portfolio and one that is Mean-Variance optimized. However, this mean-variance optimized portfolio (net neutral) is done such that net position is zero and short sale is restricted from extreme values. More specifically subject to constraints . is a short sale constraint and severely limits the degree that the optimizer can short sell an asset. This was originally proposed by Lintner (1965) where under the terms of a short sale, the proceeds received is kept by the broker/dealer and an additional margin is added (of same value) by the investor. Thus, the net position is equal to the short sale and all short sales have a position contribution. The final portfolio is constructed such that:
where is a scaling factor to scale up to the Net Neutral Portfolio. I set to 0.5 in my studies but I do not know of an efficient way to estimate this scaling factor. The Net Neutral Portfolio is said to "trade" assets based on given information on a mean-variance scale. For every position it longs, it must sell a position in the portfolio. Therefore, the optimizer is forced to pick out favourable securities through long positions and unfavourable securities through short positions.
In my study, I surveyed this method with an stock portfolio screened from the S&P 500 Index. The screener scanned for the lowest P/E percentile (5%) 2 years ago (as stated on 1/1/2013). I performed an out-of-sample test from Dec 2013 to Dec 2015 and sample my estimates from a non-overlapping five year window (Dec 2008 to Dec 2013). Constant correlation model is utilized and empirical returns and standard deviations are fed as inputs. In addition to the NNP, it's returns and Sharpe ratios are benchmarked against an equally weighted portfolio of same assets, S&P 500 Index and a MVO optimized portfolio with same inputs and assets. All returns are monthly. The NNP is calculated using Excel Solver with standard GRG Non-linear algorithm. The multi-start option is used with 10 trials to increase the chance that a global max is reached.
Over the out-of-sample testing period, the Net Neutral Portfolio generated the second highest return in comparison to the benchmark portfolios and second highest Sharpe. Similarly, MVO turned to be the worst performer with the worst Sharpe. Equally Weighted Portfolio generated an almost similar return to NNP but a lower Sharpe. I then took the monthly returns of each portfolio and regressed them against the Fama/French 3 factor model to arrive upon the risk adjusted Alpha. This is shown in the third column.
Results are listed below:
|Portfolio||Total Return||Sharpe Ratio||Alpha|
|S&P 500 Index||41.89%||17.142||0|
Here is a picture of the relative portfolio performance time-series:
In conclusion, the Equally-Weighted portfolio seems to outperform all other portfolios which adds strength that 1/N heuristics still tend to outperform all other portfolio weightings. However, NNP and EWP are able to show consistent gain over the benchmark and MVO portfolios. I hypothesize this may be because of the screening criteria for Value stocks (Top P/E) that tend to out-perform. A study on separate quantiles of P/E will be needed to determine the efficacy of a NNP.
There are still much tests left to conduct before NNP can be accepted as a superior adjustment method, I still don't have much faith in it as I believe this result is just pure luck. I will also be doing more screens and historical studies on a variety of portfolios with the NNP adjustment. A good friend of mine asked me to try this model on data in the late 90s as that is when value stocks underperformed growth stocks. If NNP is able to out-perform then it should show some validity in its derivation. Analyst forecasts can be included as well to see if it generates extra risk premium. More work should also be focused on picking the optimal tau and alpha values. Lastly, I also want to experiment with bootstrapping these statistics to see if I can conduct non-parametric testing on mean return difference. I hope you enjoyed this post and stay tuned for a potential part two! 🙂