# Modern Portfolio Theory

January 27, 2018

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 control