We start a new series on neural networks and deep learning. Neural networks and their use in finance are not new. But are still only a fraction of the research output. A recent Google scholar search found only 6% of the articles on stock price price forecasting discussed neural networks.1
Artificial neural networks, as they were first called, have been around since the 1940s. But development was slow until at least the 1990s when computing power rapidly increased.

Our last post parsed portfolio optimization outputs and examined some of the nuances around the efficient frontier. We noted that when you start building portfolios with a large number of assets, brute force simulation can miss the optimal weighting scheme for a given return or risk profile. While optimization finds those weights (it should!), the output can lead to infinitesimal contributions from many assets, which is impractical or silly. Placing a minimum on the weights helps a bit.

Our last few posts on risk factor models haven’t discussed how we might use such a model in the portfolio optimization process. Indeed, although we’ve touched on mean-variance optimization, efficient frontiers, and maximum Sharpe ratios in this portfolio series, we haven’t discussed portfolio optimization and its outputs in great detail. If we mean to discuss ways to limit our exposure to certain risks (presumably identified in the risk factor model) while still shooting for a satisfactory (or optimal) risk-adjusted return, we’ll need to investigate optimization in more detail.

Risk factor models are at the core of quantitative investing. We’ve been exploring their application within our portfolio series to see if we could create such a model to quantify risk better than using a simplistic volatility measure. That is, given our four portfolios (Satisfactory, Naive, Max Sharpe, and Max Return) can we identify a set of factors that explain each portfolio’s variance relatively well?
In our first investigation, we used the classic Fama-French (F-F) three factor model plus momentum.

In our last post, we looked at using a risk factor model to identify potential sources of variance for our 30,000 portfolio simulations. We introduced the process with a view ultimately to construct a model that could help to quantify, and thus mitigate, sources of risk beyond a simplistic volatility measure. In this post, we’ll look at building a factor model based on macroeconomic variables to see if such a model does what it says on the tin.

We’re returning to our portfolio discussion after detours into topics on the put-write index and non-linear correlations. We’ll be investigating alternative methods to analyze, quantify, and mitigate risk, including risk-constrained optimization, a topic that figures large in factor research.
The main idea is that there are certain risks one wants to bear and others one doesn’t. Do you want to be compensated for exposure to common risk factors or do you want to find and exploit unknown factors?

In our last post, we took our analysis of rolling average pairwise correlations on the constituents of the XLI ETF one step further by applying kernel regressions to the data and comparing those results with linear regressions. Using a cross-validation approach to analyze prediction error and overfitting potential, we found that kernel regressions saw average error increase between training and validation sets, while the linear models saw it decrease. We reasoned that the decrease was due to the idiosyncrasies of the time series data: models trained on volatile markets, validating on less choppy ones.

In our last post, we looked at a rolling average of pairwise correlations for the constituents of XLI, an ETF that tracks the industrials sector of the S&P 500. We found that spikes in the three-month average coincided with declines in the underlying index. There was some graphical evidence of a correlation between the three-month average and forward three-month returns. However, a linear model didn’t do a great job of explaining the relationship given its relatively high error rate and unstable variability.

We recently read two blog posts from Robot Wealth and FOSS Trading on calculating rolling pairwise correlations for the constituents of an S&P 500 sector index. Both posts were very interesting and offered informative ways to solve the problem using different packages in R: tidyverse or xts. We’ll use those posts as a launchpad to explore the rolling correlation concept with respect to forecasting returns. But we’ll be using Python to do a lot of the heavy lifting.

We’re taking a break from our portfolio series and million sample simulations to return to a subject that we haven’t discussed of late despite its featured spot in this blog’s name—options. In this post, we’ll look at the buy-write (BXM) and put-write (PUT) indices on the S&P 500, as conceived, calculated, and published by the CBOE. Note: we’ve discussed the buy-write strategy in the past here and here. In those posts, we analyzed the performance of the buy-write relative to its underlying index, the S&P 500.