The Optimizing 401(k)s project involves the analysis of 5 years of closing price data of six securities (VFINX, VEURX, VEIEX, VBLTX, VBISX and VPACX) from July 2011 to June 2016. The securities analyzed are a collection of mutual funds, which track a range of diversified financial indexes (both debt and equity) from geographically diverse markets around the world.

The price analysis performed on this data suggests the presence of trends that are perpetuated across all of the funds, in addition to fund-specific trends. This observation is expanded upon to identify the presence of two distinct types of risk: diversifiable and non-diversifiable.

Continuously compounded returns are calculated on a monthly basis for each asset, and are treated as samples of larger distributions. Under this assumption, descriptive univariate statistics (Constant Expected Return model parameters) are calculated for each mutual fund, and are analyzed to infer characteristics about the distribution of its prices. Additionally, by assuming the Gaussian form for the distributions of the prices, estimation errors are computed and analyzed to better understand the distribution of return of each ETF.

A measure of risk-adjusted return is determined (in the form of a Sharpe ratio), and is calculated to better gauge the performance of the mutual funds. Intensive computer-assisted resampling methods (i.e. Bootstrap) are used to calculate standard errors of measurement and 95% confidence intervals for this measure, and the mutual funds are ranked using this data.

Pairwise scatterplot of monthly returns of each of the mutual funds from July 2011 to June 2016

Pairwise scatterplots, covariance and correlation matrices are used to understand interactions between each of the mutual funds over the time horizon of the project. This analysis is used to reinforce the importance of diversification, and the ultimate goal of creating portfolios with these assets.

Cross-correlation diagram of each of the mutual funds

Values at Risk (VaR) is computed at the 1% and 5% risk levels for each asset at both the monthly and annual time horizons. This data is used to add a more tangible measure of asset volatility, by relating the mean and standard deviation of returns to real dollar values. Standard estimation errors for this metric are also determined using intensive computer-assisted resampling. The VaR for each mutual fund is analyzed within the scope of the precision of each of the estimators used to calculate them.

The project uses R as the primary programming language, with a range of packages being used for data analysis and visualization. R Markdown is used to generate the final document, combining text, code and output into a cohesive report. This allows for the document to be easily reproducible, with all of the code used for analysis and visualization being included in the report. The use of R and R Markdown allows for efficient and effective data analysis, and the ability to easily share and communicate results.

The key assumption of the Constant Expected Return (CER) model is that the processes it is modeling are covariance stationary. This hypothesis is tested by calculating and analyzing 24 month rolling estimates of the mean and standard deviation of returns for each of the mutual funds. Additionally, the rolling correlation between VFINX and VBLTX is analyzed to reinforce the importance of correlation measurement in the diversification of risk.

Efficient frontier built with monthly returns of each of the mutual funds from July 2011 to June 2016

After a brief introduction to portfolio theory, global minimum variance, tangent, and target return-equivalent portfolios are computed for scenarios where short sales are both allowed and disallowed. Brute-force computation is used to calculate the efficient frontiers for each of the scenarios, and these frontiers are analyzed to illustrate and emphasize the cost of not being able to short-sell assets. In addition to significant consideration and analysis of portfolio theory, descriptive statistics computed for each of the portfolios are used to reinforce the comparisons and conclusions drawn from the behavior of the different portfolios.

In conclusion, the project provides a detailed analysis of the performance of the selected mutual funds and their suitability for inclusion in 401(k) portfolios. The importance of good underlying assets and the impact of the restriction of short sales on portfolio construction are emphasized as key factors in the construction of effective 401(k) portfolios.