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Markowitz's Modern Portfolio Theory (MPT), or mean-variance portfolio, involves finding the best balance between the profit and risk of portfolio returns over a future period. This strategy looks for the lowest risk given a targeted return. For simplicity, we apply it to a long-only portfolio to test for alpha. To calculate expected return, weights are assigned to each stock in the portfolio. Current backtested performance shows that Modern Portfolio Theory does not outperform a naive equal weight strategy. In this research, we improve on MPT by adding an additional function to reduce error in risk and expected return calculations. Through regularization, backtracking, and cross-validation, our results show this strategy to outperform both MPT and, in some cases, a naive equal weighted one.

First, let us walk through the calculations for MPT. Assume a portfolio of n assets, with expected return µ. wi is the weight of an asset i in the portfolio. The sum of these weights should be equal to one. To prevent shorting, set the weights to be greater than or equal to zero. Negative weights mean the asset is a liability, which indicates shorting.

Portfolio return has a mean of wTµ and a variance of wTΣw, where Σ is the covariance matrix of asset returns. Covariance helps determine what assets to include in the portfolio. For a given target value R, we can characterize the wanted portfolio by its weight vector, wn:

Now, we use machine-learning techniques to optimize it further.

First, we do regularization on the above equation and conditions. Regularization corrects overfitting and renders the model less likely to fit any noise in the training data, making it more accurate and efficient. We use performance-based regularization, or PBR, to restrict the variances of sample data to be less than or equal to a constant that is set by the algorithm. This constraint will maximize the out-of-sample Sharpe ratio in a testing dataset.

We calculate variance as our measure of risk. To do so, we take the sum of all of the returns of a stock that deviate from the mean return. Assuming many variances, we take the sum of all deviations from the mean and we have sample variance. The sample variance of the sample variance is then calculated to remove any outliers.

Next, we have to constrain that to a constant, which we will call U that controls the degree of regularization and returns the maximum Sharpe ratio. To find this U, we use backtracking line search.

Here, we start at the maximum, U1, and backtrack by descending by step size U, where is the step function. You find U1 by choosing a random number, or constant. Eventually, U will start to decrease and produce a Sharpe ratio that is smaller than the previous U’s. That indicates that the Sharpe ratio prior to this smaller one is the greatest Sharpe ratio in the set of data, which in this case belongs to Un-1. If this Sharpe ratio is not good enough, we try another U1 and backtrack from there until we are satisfied with the Sharpe ratio.

We use these U’s in out-of-sample performance-based k-cross validation. We split our training data set into k equal-sized bins and calibrate U on each of them. The goal is to get the best U for the entire training dataset. We train on every combination of k-1 bins and test on the remaining bin. Below is an example of training and testing on k bins:

We then find the optimal U overall by averaging the best U found for each combination of bins for the training data. This optimal U maximizes the Sharpe ratio on the remaining bin.

At the end, we get the equation SVar(wTΣw) U, which we add on to other modern portfolio equations to get:

Results

We test subsets of five randomly selected large capitalization NASDAQ listed companies. Equal weight, Ensemble, Markowitz Modern Portfolio Theory, and MPT with Regulation are each tested on the portfolio of companies.

The objective function is first to maximize risk-adjusted returns, calculated by the Sharpe Ratio. This is a popular metrics used in portfolio management. A Sharpe consistently above 1 represents good results, while a ratio of 2 shows exceptional performance. The second goal is to maximize absolute portfolio returns.

Monthly historical data from January 1, 2004 - January 31, 2017 is used. The returns for each company are normalized to give a common scale

The training period is 36 observations and the test period is 120 for testing. Figure 2 below shows the cumulative return of the testing period for each of the portfolio strategies. The results show that an equal weighted portfolio with 20% invested in each security significantly outperforms all other methods. Other iterations of the test show differing, but unstable returns between the different portfolio techniques.

Figure 3 below shows the weights applied to each individual security across the time horizon. Companies with greater volatility are penalized in the both Regularization and Markowitz Modern Portfolio Theory.

The results of this iteration are not consistent with tests on other baskets of securities. Equal weight portfolio performs best in this test, while Regularization method is best in others. We are currently researching in more depth to better understand why performance is not stable.

Reference: http://www.optimization-online.org/DB_FILE/2014/11/4625.pdf