Portfolio Performance

Tool to test the out-of-sample performance of different portfolio optimization strategy

This project comprises of three part, a Python package, a GUI and a web server. The package is the core of the project, it contains the functions to optimize the portfolio and to test the out-of-sample performance of the portfolio. The GUI is a simple interface to interact with the package, and the web server is a locally run api which fetches the latest data from the internet and runs the optimization and performance test.

Based on the paper: Optimal Versus Naive Diversification . You can check out the PyPi page here.

The GUI is available at sidnand.github.io/Portfolio-Lab/.

The web server is available on GitHub.

This project was went to turn the original research paper into a general purpose application that could be used by finance hobbiests and researchers alike. It includes 11 pre-built models and for the tech savvy, the ability to create their own models. The models included are talked about below.

Models

We will let:

• \(n\) be the number of assets in the portfolio
• \(r\) be the expected returns of the assets
• \(C\) be the covariance matrix of the assets
• \(w\) be the weights of the assets in the portfolio
• \(r_p\) be the expected return of the portfolio
• \(r_f\) be the risk free rate
• \(\sigma_p\) be the standard deviation of the portfolio

Equal Weighted (Benchmark)

This is the simplest model, it assigns equal weight to all the assets in the portfolio.

\[w = \frac{1}{n}\]

Minimum Variance

This model minimizes the variance of the portfolio.

\[min \quad w^TCw\]

such that:

\[w^T1 = 1\]

Minimum-varaince with shortsale constraints

Minimum-varaince with generalized constraints (Jagannathan Ma)

Kan Zhou equal weighted model

Mean-varaince (Markowitz) model

This model maximizes the return of the portfolio for a given level of risk.

\[max \quad w^Tr - \lambda w^TCw\]

such that:

\[w^T1 = 1\]

Mean-variance with shortsale constraints

Kan Zhou (2007) “three-fund”

Bayes-Stein

Bayes-Stein with shortsale constraints

MacKinlay and Pastor’s (2000)

Statistics

To compare the performance of the different models, we use the Sharpe ratio, which is the risk adjusted return of the portfolio. Defined as:

\[\frac{r_p - r_f}{\sigma_p}\]

Shapre Ratio Statistical Significance

To test the statistical significance of the Sharpe ratio, we use the Jobson Korkie Z test . The null hypothesis is that the Sharpe ratio is zero, and the alternative hypothesis is that the Sharpe ratio is not zero. The test statistic is:

\[Z = \frac{\hat{SR}}{\sqrt{\frac{1}{T}}}\]

Where:

• \(\hat{SR}\) is the sample Sharpe ratio
• \(T\) is the number of observations