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“MORE RISK, MORE EXPECTED REWARD.”

If you naively applied this central principle of financial theory to investing in equities over a 40-year career, you’d probably want to hold a portfolio consisting mostly of high-risk stocks. Given your long time horizon, the extra volatility most likely wouldn’t be a problem. You might start by buying a diversified collection of the most volatile stocks within an investable universe such as the Standard & Poor’s 500 Index. Once in a while, you’d look at your portfolio and find that some of your stocks had low volatility or had dropped outside the investable universe, so you’d replace them with newly available high-volatility stocks to stay true to your investment style. (Stock volatility here is measured on an absolute basis, not relative to a benchmark.)

To your horror, you would have significantly underperformed a cap-weighted index portfolio even within a decade. Adding insult to injury, holding the low-volatility stocks would have been better still. What’s going on here?

The historical performance of different asset classes over the long term is indeed consistent with the rule of thumb that greater risk leads to greater reward. However, it appears that this principle does not extend to individual equities. Many academic studies, as well as historical backtests of low-volatility indexes, indicate that portfolios composed purely of low-volatility stocks tend to outperform portfolios of high-volatility stocks over the long term. This phenomenon is usually interpreted to imply that low-volatility stocks have higher returns, on average. As this flies in the face of standard ideas about asset pricing, it is usually referred to as the “low-volatility anomaly.”

In fact, this interpretation is premature: The anomaly can be better understood by considering portfolio-level effects. This may seem a little counterintuitive. After all, if portfolios consisting of low-volatility stocks perform so well over the long term, doesn’t this mean that the low-volatility stocks must themselves generally perform well? Not necessarily. Compounding and rebalancing effects complicate the picture. To continue to hold low-volatility stocks within a given investable universe without style drift, an investor must periodically sell stocks that have increased in volatility or fallen out of the universe. Applied diligently, this requirement becomes a rebalancing rule for portfolio construction. The difficulty in understanding what drives the performance of the resulting portfolio is that the rebalancing itself contributes to the portfolio return substantially.

How can we tell if the low-volatility anomaly is the result of individual stock behavior or portfolio rebalancing effects — or both — and why should we care? In a nutshell, the rebalancing explanation would provide more confidence that past observations will persist in the future. This insight has more than merely academic interest. Billions of dollars have flowed into low-volatility equity portfolios over the past few years. For instance, by 2011 pension funds and other institutions had invested almost $10 billion in low-volatility U.S. equity strategies, up from virtually nothing in 2005. The story has been similar outside the U.S., with more than $6 billion invested in low-volatility equity strategies in continental Europe alone. Index providers such as MSCI, Russell Investments and S&P offer low-volatility indexes. On the retail side the most popular new exchange-traded fund in the U.S. for 2011 was a low-volatility strategy designed to replicate the S&P 500 Low Volatility Index.

It is easy and tempting to dismiss low-volatility equity strategies as a fad. Only time will tell, but in our opinion, the category merits considerable attention, as these strategies may indeed offer superior risk-adjusted returns than capitalization-weighted indexes. The low-volatility equity products offered today — and there are frequently new ones — generally fall into one of two categories: portfolios composed of low-volatility stocks and equity portfolios with low volatility.

Portfolios of the first type are generally constructed by selecting the least volatile stocks from an eligible universe, then combining them in some manner. There may be fundamental or quantitative overlays, and possibly some risk control in the choice of stock weighting. Because correlations are not always taken into account during the stock selection process, the overall portfolio volatility may be higher than one might expect, but it should still be lower than the volatility of a cap-weighted index. The S&P 500 Low Volatility Index is an example of a benchmark that tracks the hypothetical value of a portfolio of this type.

The second category includes minimum-variance portfolios. They are constructed by a methodology that seeks to lower or minimize the overall portfolio variance. MSCI’s Minimum Volatility indexes are an example of such an approach, which generally addresses portfolio risk in a more effective manner than portfolios of low-­volatility stocks because lowering that risk is specifically targeted. Many proponents of low-volatility portfolios attribute their historical and simulated outperformance versus cap-weighted benchmarks to the low-volatility anomaly. However, these portfolios typically include some high-volatility stocks that have low correlations to the other stocks in the portfolios. This means that the low-volatility anomaly, as described above, is unlikely to fully account for the performance of such portfolios. Instead, optimizing to minimize the volatility may well be expected to provide a more efficient portfolio than the cap-weighted index.

Interest in low-volatility-type equity strategies, which comprise both of the above classes of portfolios, has never been higher. Regardless of the specifics, these strategies are widely perceived to exploit the low-volatility anomaly. Indeed, they are often advertised in exactly this way, prompting reactions ranging from enthusiastic acceptance to outright skepticism.

It’s hard to argue that a portfolio-level phenomenon isn’t real: Papers noting the existence of the low-volatility anomaly appeared as early as the 1970s. It is easy to confirm that portfolios of high-volatility stocks have underperformed portfolios of low-volatility stocks over the long term. These findings have been replicated in studies of markets in many different countries.

Finance academics and practitioners have advanced many explanations for these findings. They tend to fall into two categories: stock mispricing and systematic risk factors. Mispricing — meaning that high-volatility stocks are overvalued relative to low-volatility stocks — is sometimes attributed to behavioral effects. For example, overconfidence and irrational preference for lotteries may lead to demand for high-volatility stocks greater than that indicated by the stocks’ fundamentals. The fact that the resulting mispricing isn’t arbitraged away is attributed to lower liquidity in low-volatility names or to the benchmark-centricity of institutional investors, creating a limit to the amount of capital that can flow into lower-beta stocks. These effects are hard to quantitatively verify, however, and fail to explain the presence of the anomaly in the historical data for time periods before benchmarks were in common use or when investors had higher risk aversion. Systematic risk factors may be based on actual volatility or idiosyncratic (beta-adjusted) volatility; exposure to these factors would contribute to future lower returns. The rationale for the existence of these risk factors is less clear. In none of these cases are any portfolio-level effects, such as return contribution as a result of rebalancing, taken into account.

Indeed, in contrast to these stock-based explanations, portfolio-­based explanations allow quantification of the anomaly, suggest reasons for its long-term persistence and pave the way for more-efficient low-volatility portfolio construction. The key observation is that compounding has a significant impact on long-term returns. For example, during the past four years, many investors have learned the hard way that it takes a lot more than a 40 percent positive return to recover from a 40 percent downturn. This is because returns after the big downturn are compounding from a smaller base. There is no way to avoid this with an individual security in a buy-and-hold setting; the compound return really does reflect long-term performance.

Diversification into varied risky assets helps in this regard. When multiple stocks are combined in a long-only portfolio, their interaction actually causes the portfolio to have a higher compound growth rate than the weighted average compound growth rate of the stocks in that portfolio. The difference in growth rates is usually small for concentrated, infrequently traded portfolios. For diversified, regularly rebalanced portfolios, this difference can amount to hundreds of basis points per year. (The difference between portfolio and stock compound returns, or “excess growth rate,” was first explained in a 1982 paper by Robert Fernholz and Brian Shay, giving rise to the discipline of Stochastic Portfolio Theory. The phenomenon has subsequently been independently rediscovered several times and is sometimes known as “volatility pumping” or “diversification return.”)

The full impact of diversification and rebalancing on compound returns can be quantified using highly technical mathematics. Fortunately, the low-volatility anomaly can be analyzed using much simpler techniques. Let’s focus only on the impact on portfolio returns of the simple requirement that whenever we trade a portfolio, we are allowed to hold stocks only within the investable universe. Suppose that this universe consists of the 1,000 largest stocks (by capitalization) in the U.S. market. If we hold a stock that drops out of the universe, we will certainly be selling it at a relative loss, because the stock has failed to keep up with the other stocks. After all, it has lost its place in the top 1,000. In essence, the strategy requires that we sell low in this case, creating a drag on performance.

Of course, even a cap-weighted index of the same stocks is subject to this reconstitution drag. The question is, do portfolios of low-volatility stocks have less drag than the index? There is good reason to believe that they should and, furthermore, that portfolios of high-volatility stocks have more drag than the index. Everything follows from the basic observation that low-volatility stocks have less variability of outcomes, while high-volatility stocks have more variability. The effect on the reconstitution drag should be as follows:

• Low-volatility stocks tend to move less in price than the average stock, so there is less risk of them dropping out of the universe than the average stock. On the other hand, high-volatility stocks have a higher risk of dropping out than the average stock.

• When a low-volatility stock does drop out of the universe, it is likely to fall less than the average stock. The opposite is true for a high-volatility stock.

• Low-volatility stocks are probably larger in capitalization, therefore less likely to drop out of the universe than the average stock. Conversely, high-volatility stocks are typically smaller in capitalization, therefore more likely to drop out of the universe than the average stock.

To get some sense of the effects of the reconstitution drag on performance, let’s imagine investing in a very simple strategy in which we buy and hold a single stock, replacing it only when it is no longer in the investable universe at the time of reconstitution. To simplify the situation even further, we’ll define this universe as consisting of stocks with capitalizations greater than $10 billion, and we will assume a monthly reconstitution. We start with one such stock. At the beginning of every subsequent month, there are two possibilities:

• The market capitalization of the company remains above $10 billion: Continue to hold the stock.

• The company’s market capitalization has dropped below $10 billion: Sell the stock and use the freed capital to buy the stock of the company whose market capitalization has most recently risen above $10 billion.

Not knowing anything further about the stocks involved, it is impossible to say what the short-term performance of this investment strategy will be. However, over many decades most large companies eventually falter for one reason or another. Whenever that happens for the stock being held, the strategy is to sell the stock at a loss and buy another stock after it has risen in value.

How much will this sell-low-buy-high trading hurt long-term portfolio performance? One way to think about it is to decompose the past monthly return of the stock into two pieces: the part that drove the capitalization down to $10 billion and the part that drove the capitalization even lower (see figure, above). It’s not fair to include the first piece in the cost analysis. Had the stock gone down so that the market capitalization was exactly $10 billion, we would have held on to it for another month, during which time it may have recovered. Instead, it is the excess negative return — labeled as “sell cost” in the figure — that really hurts. Moreover, the replacement stock was likely bought at a premium above the $10 billion threshold, contributing to the overall cost.

As a very long-term investment, this strategy is dubious at best: It can only work if the stock held never drops below $10 billion and will suffer greatly from reconstitution drag otherwise. The short-term performance will be poorest when the identity of the stock changes every month, as we will be losing money on every trade. If the stock we are holding is of high volatility, this is more likely to happen.

Of course, a single stock could go years, even decades, without dropping below the $10 billion threshold. We can speed up the process by investing in many stocks, which at the same time will allow us to cast new light on the persistent long-term underperformance of the megacap stocks with respect to broader stock indexes. Suppose we hold the largest 100 stocks at cap weight. Keeping the same monthly rebalancing schedule, we therefore sell the stocks that have dropped out of the top 100 at the end of the month and buy the stocks that have come into the top 100. This diligent rebalancing keeps us true to our megacap style, but it is also the sort of sell-low-buy-high trading that hurts the performance of the single-stock strategy. How much is the reconstitution drag in this case? For each stock that drops out of the universe, we can estimate the negative impact beyond the reconstitution boundary as follows:

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