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 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.
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.
In our last post, we took a quick look at building a portfolio based on the historical averages method for setting return expectations. Beginning in 1987, we used the first five years of monthly return data to simulate a thousand possible portfolio weights, found the average weights that met our risk-return criteria, and then tested that weighting scheme on two five-year cycles in the future. At the end of the post, we outlined the next steps for analysis among which performance attribution and different rebalancing schemes were but a few.
In our last post, we compared the three most common methods used to set return expectations prior to building a portfolio. Of the three—historical averages, discounted cash flow models, and risk premia models—no single method dominated the others on average annual returns over one, three, and five-year periods. Accuracy improved as the time frame increased. Additionally, aggregating all three methods either by averaging predictions, or creating a multivariate regression from the individual explanatory variables, performed better than two out of the three individual methods.
Over the past few weeks, we’ve examined the three major methods used to set return expectations as part of the portfolio allocation process. Those methods were historical averages, discounted cash flow models, and risk premia models. Today, we’ll bring all these models together to compare and contrast their accuracy.
Before we make these comparisons, we want to remind readers that we’re now including a python version of the code we use to produce our analyses and graphs.
In our last post, we applied machine learning to the Capital Aset Pricing Model (CAPM) to try to predict future returns for the S&P 500. This analysis was part of our overall project to analyze the various methods to set return expectations when seeking to build a satisfactory portfolio. Others include historical averages and discounted cash flow models we have discussed in prior posts. Our provisional analysis suggested that the CAPM wasn’t a great forecasting model.