# Portfolio Theory

### One-N against the world!

We’re taking a short break from neural networks to return to portfolio optimization. Our last posts in the portfolio series discussed risk-constrained optimization. Before that we examined satisificing vs. mean-variance optimization (MVO). In our last post on that topic, we simulated 1,000 60-month (5-year) return series using the 1987-1991 period for our four assets: stocks, bonds, commodities (gold), and real estate. We then iterated through the samples using weights derived from the naive portfolio, the satisficing algorithm1, and the maximum Sharpe ratio portfolio on the previous sample to create portfolios on the next sample.

### 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.

### More factors, more variance...explained

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.

### Macro variance

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.

### Explaining variance

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?

### 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.

### Satisficing and optimizing

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.

### 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.