Data Science

Risk-constrained optimization

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.

Parsing portfolio optimization

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.

Round about the kernel

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.

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.

Sequential satisficing

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.

I like to MVO it!

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.

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.

Discounted expectations

After our little detour into GARCHery, we’re back to discuss capital market expectations. In Mean expectations, we examined using the historical average return to set return expectations when constructing a portfolio. We noted hurdles to this approach due to factors like non-normal distributions, serial correlation, and ultra-wide confidence intervals. While we highlighted these obstacles and offered a few suggestions to counteract such drawbacks, on first blush it didn’t seem like historical averages were all that satisfactory.