Machines

Kernel of error

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.

Corr-correlation

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.

Weighting on a friend

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.

Testing expectations

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.

Performance anxiety

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.

Portfolio simulations

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.

Mad 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.

Implied risk premia

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.

Machined risk premia

Over the last few posts, we’ve discussed methods to set return expectations to construct a satisfactory portfolio. These methods are historical averages, discounted cash flow models, and risk premia. our last post, focused on the third method: risk premia. Using the Capital Asset Pricing Model (CAPM) one can derive the required return for a particular asset based on the market price of risk, the asset’s risk, and the asset’s correlation with the market.

Risk premia

Our last post discussed using the discounted cash flow model (DCF) as a method to set return expectations that one would ultimately employ in building a satisfactory portfolio. We noted that if one were able to have a reasonably good estimate of the cash flow growth rate of an asset, then it would be relatively straightforward to calculate the required return. The problem, of course, is figuring out what the cash flow growth rate should be.