After asset allocation and rebalancing, choosing weights for each asset in the portfolio is next in line among the critical decisions that determine investment results. Once you make reasonable choices on the asset classes to hold, and how and when to rebalance those assets, your focus should turn to asset weights. But choosing weights is tricky, considerably more so compared with asset allocation or rebalancing. As a practical matter, you can use some intelligent rules of thumb for designing a portfolio that’s probably in the neighborhood of long-term optimal (the highest expected return for a given level of risk) with regards to asset allocation and rebalancing. Deciding how to weight the assets, by contrast, carries a much higher analytical burden for engineering a satisfying outcome.

The stakes are certainly high. You can make great choices with asset allocation and rebalancing, but poor decisions on weighting the assets could override the other design advantages and leave you with mediocre results or worse.

Thinking about portfolio weights starts by considering the fundamental insight that Harry Markowitz initially outlined back in the 1950s. Every investor is looking for a portfolio that sits on the efficient frontier — i.e., a portfolio that’s designed to capture the highest return for the lowest risk for the set of assets in question. This is true whether you’re running a hedge fund or a plain vanilla balanced fund of stocks and bonds. There’s a lot of light and heat about the details in the world of money management, but all of investing boils down to the framework that Markowitz forged all those years ago. The problem (or the opportunity, depending on your perspective) is that there are many portfolios along the efficient frontier. The main question that confronts every investor (or institution): Which optimal portfolio do I want?

Quite a lot of the answer is bound up with your asset allocation plan. Are you going limit your portfolio to US stocks and US bonds? Or will you broaden your horizons by diversifying across the major asset classes? There are lots of possibilities. Whatever your preference, there are multiple ways to optimize the mix of the assets that you’re targeting. Indeed, every portfolio, no matter how extreme or exotic, has its own set of optimal combinations and it’s all about choosing weights.

How to choose? The default is to go with Mr. Market’s strategy of letting the relative market values determine weights. This tends to be a competitive strategy, by the way, particularly if you own a wide selection of the major asset classes. In conventional finance theory, holding Mr. Market’s asset allocation (all the major asset classes weighted by market value) is the super-optimal portfolio that will dominate all others over the long term. But this is a controversial idea, although I think you can do quite well with this default strategy. In any case, most investors choose to customize the mix, which is partly a function of choosing weights.

Theoretically speaking, if you had no insight about expected return and risk, the standard choice for weighting assets would lead you to hold equal amounts of your targeted list of assets. If we have no confidence in our ability to add value by estimating future returns and risk, equal weighting is a reasonable plan. In fact, equal weighting has an encouraging history as a real world strategy, but that’s another story.

Let’s say that you’ve rejected market-value and equal-weighting schemes. Now what? The traditional approach is to look for portfolios on the efficient frontier with the highest expected return. This basic formulation describes the lion’s share of how investors and institutions manage money, even if they don’t think in such Markowitzian terms. But there’s a growing body of research that tells us to look for portfolios that emphasize another goal: minimize risk (volatility). These portfolios are still efficient, but they’re on the other side of the efficient frontier. In fact, it turns out that the minimum-volatility mix for a given set of assets tends to deliver superior results vs. trying to maximize return.

A picture’s worth a thousand words on this point. The black line in the chart below shows the idealized efficient frontier. A variety of asset mixes along the black line deliver optimal results. The problem is that many investors end up with portfolios along the red line—suboptimal results. Why? Because forecasting returns is difficult… really difficult. Projecting risk, by contrast, is a bit easier. The bottom line: looking for a mix of assets that will deliver the lowest level of volatility has shown itself to be a practical strategy that will outperform most if not all of the other weighting schemes for the same set of assets.

This is hardly a revelation at this point. Indeed, you can find a number of ETFs based on minimum vol (MV) strategies (here’s a list of funds). The related research is certainly persuasive for thinking that MV is a practical alternative to the usual suspects for designing and managing a portfolio. (For a quick overview of MV and related research, see Edhec’s summary.)

Estimating MV mixes for a set of assets is relatively straightforward. It’s really a simple optimization process. The basic path is looking for a quantitative solution by one of several methodologies. The possibilities include quadratic-programming and regression-based solutions that will tell us how to weight the assets to produce a MV portfolio. Crunching the numbers is easy enough. You could do so in Excel, although I prefer to R via the quadprog package. But there’s a glitch: in order to come up with reasonable results, you’ll need a decent set of estimates about how the asset returns in the portfolio are likely to covary with one another in the future. In other words, you’ll need to generate a robust covariance matrix (CM).

The details aren’t complicated, although the garbage-in-garbage-out rule applies. Fortunately, there are some practical methods for helping us steer clear of trouble in this critical process. Indeed, choosing a covariance estimator is the key challenge when it comes to designing a minimum-vol strategy. We could use the historical data set as is, although there are several reasons why we should “shrink” the associated CM via any one of several techniques, such as the Ledoit-Wolf estimator. (For an overview of the rationale behind this procedure, see this paper: “Honey Honey,I Shrunk the Sample Covariance Matrix.”)

I’ll dive into the numbers in a future post. Meanwhile, the main takeaway here is that a minimum-vol strategy is a strong alternative for weighting assets when you’re looking for something other than market-value or equal weights. One useful line of research is to compare the mix in your current portfolio with a minimum-vol-implied set of weights. Do they differ radically? Are you comfortable with the difference? Why?

It’s also valuable to compare how minimum-vol weights stack up against market-value weights. The idea of holding all the major asset classes, or something comparable, and weighting this portfolio in order to minimize vol provides quite a lot of valuable perspective.

The bottom line: There’s a lot we can learn by studying how a given strategy compares with its minimum-vol counterpart. The relative simplicity and encouraging track record with MV puts in on the short list when looking for an alternative weighting strategy. Why? As one answer, consider a recent study that offers an intriguing explanation for the persistence of the “low volatility anomaly”:

Over the past 41 years, high volatility and high beta stocks have substantially underperformed low volatility and low beta stocks in U.S. markets. We propose an explanation that combines the average investor’s preference for risk and the typical institutional investor’s mandate to maximize the ratio of excess returns and tracking error relative to a fixed benchmark (the information ratio) without resorting to leverage. Models of delegated asset management show that such mandates discourage arbitrage activity in both high alpha, low beta stocks and low alpha, high beta stocks. This explanation is consistent with several aspects of the low volatility anomaly including why it has strengthened in recent years even as institutional investors have become more dominant.