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.

In our last post, we ran simulations on our 1,000 randomly generated return scenarios to compare the average and risk-adjusted return for satisfactory, naive, and mean-variance optimized (MVO) maximum return and maximum Sharpe ratio portfolios.1 We found that you can shoot for high returns or high risk-adjusted returns, but rarely both. Assuming no major change in the underlying average returns and risk, choosing the efficient high return or high risk-adjusted return portfolio generally leads to similar performance a majority of the time in out-of-sample simulations.

In our last post, we explored mean-variance optimization (MVO) and finally reached the efficient frontier. In the process, we found that different return estimates yielded different frontiers both retrospectively and prospectively. We also introduced the concept of satsificing, originally developed by Herbert Simon. Simply put, satisficing is choosing the best available solution afforded by a messy reality, when an optimal solution requires hard to know (or unknowable) information. Alternatively, an optimal solution might be easy to find, but requires conditions so abstracted from normal phenomena as to be devoid of reality.

In our last post, we ran through a bunch of weighting scenarios using our returns simulation. This resulted in three million portfolios comprised in part, or total, of four assets: stocks, bonds, gold, and real estate. These simulations relaxed the allocation constraints to allow us to exclude assets, yielding a wider range of return and risk results, while lowering the likelihood of achieving our risk and return targets. We bucketed the portfolios to simplify the analysis around the risk-return trade off.

Our last few posts on portfolio construction have simulated various weighting schemes to create a range of possible portfolios. We’ve then chosen portfolios whose average weights yield the type of risk and return we’d like to achieve. However, we’ve noted there is more to portfolio construction than simulating portfolio weights. We also need to simulate return outcomes given that our use of historical averages to set return expectations is likely to be biased.

In our last post, we analyzed the performance of our portfolio, built using the historical average method to set return expectations. We calculated return and risk contributions and examined changes in allocation weights due to asset performance. We briefly considered whether such changes warranted rebalancing and what impact rebalancing might have on longer term portfolio returns given the drag from taxes. At the end, we asked what performance expectations we should have had to analyze results in the first place.