High-Frequency Trading and High Returns

This guest post is contributed by Ricardo Fernholz, a professor of economics at Claremont McKenna College. Some of his other work was profiled on this blog here. 

The rise of high-frequency trading (HFT) in the U.S. and around the world has been rapid and well-documented in the media. According to a report by the Bank of England, by 2010 HFT accounted for 70% of all trading volume in US equities and 30-40% of all trading volume in European equities. This rapid rise in volume has been accompanied by extraordinary performance among some prominent hedge funds that use these trading techniques. A 2010 report from Barron’s, for example, estimates that Renaissance Technology’s Medallion hedge fund – a quantitative HFT fund – achieved a 62.8% annual compound return in the three years prior to the report.

Despite the growing presence of HFT, little is known about how such trading strategies work and why some appear to consistently achieve high returns. The purpose of this post is to shed some light on these questions and discuss some of the possible implications of the rapid spread of HFT. Although much attention has been given to the potentially destabilizing effects of HFT, the focus here instead is on the basic theory behind such strategies and their implications for the efficiency of markets. How are some HFT funds such as Medallion apparently able to consistently achieve high returns? It is natural to suspect that such excellent performance is perhaps an anomaly or simply the result of taking significant risks that are somehow hidden or obscured. Indeed, this is surely the case sometimes. However, it turns out that there are good reasons to believe that many HFT strategies are in fact able to consistently earn these high returns without being exposed to major risks.

To understand how this works, let’s consider the S&P 500 U.S. stock index. Suppose that we wish to invest some money in S&P 500 stocks for one year. Currently, Apple has a total market capitalization of roughly $500 billion, making it the largest stock in the S&P 500 and equal to approximately 4% of the total capitalization of the entire index. Suppose that we believe it is very unlikely or impossible that either Apple or any other corporation’s capitalization will be equal to more than 99% of the total S&P 500 capitalization for this entire year during which we plan to invest. As long as this turns out to be true, then it is actually pretty simple to construct a portfolio containing S&P 500 stocks that is guaranteed to outperform the S&P 500 index over the course of the year and that has a limited downside relative to this index. In essence, we can construct a portfolio that will never fall below the value of the S&P 500 index by more than, say, 5% and that is guaranteed to achieve a higher value than the S&P 500 index by the end of the year.[1]

This is not a trivial proposition. If we combine a long position in this outperforming portfolio together with a short position in the S&P 500 index, then we have a trading strategy that requires no initial investment, has a limited downside, and is guaranteed to produce positive wealth by the end of the year. According to standard financial theory, this should not be possible.[2] Furthermore, the assumptions that guarantee that our portfolio will outperform the S&P 500 index appear entirely reasonable. After all, not for one day in the more than 50-year history of the S&P 500 has one corporation’s market capitalization come anywhere close to equaling even 50% of the total capitalization of the market. A 99% share of total market capitalization would essentially amount to there being only one corporation in the entire U. S. for an entire year. This seems like neither a likely outcome nor one that investors should take seriously when constructing their portfolios.

What does a portfolio made up of S&P 500 stocks that is guaranteed to outperform the S&P 500 index look like? There are many different ways in which such a portfolio can be constructed, but one feature common to all such portfolios is that relative to the S&P 500 index itself, they place more weight on those stocks with small total market capitalizations and less weight on those stocks with large total market capitalizations. The weight that an index such as the S&P 500 places on each individual stock is equal to the ratio of that stock’s total market capitalization relative to all stocks’ total market capitalizations taken together. In the case of Apple, then, the S&P 500 index would place a weight of roughly 4% in this individual stock while those portfolios that use HFT to outperform this index would instead place a weight of less than 4% in Apple stock.

The second key feature of these outperforming portfolios is that they must be constantly rebalanced to maintain the chosen weights for each stock. This implies that any time the price of one stock increases relative to the price of other stocks, some of this stock must immediately be sold to maintain that stock’s prescribed weight. This constant rebalancing is what makes these trading strategies part of HFT. Consider, for example, an equal-weighted portfolio that invests the same dollar amount in each of the 500 stocks that make up the S&P 500 index.[3] If the price of one of those stocks increases relative to the others, it is necessary to sell some of that stock and purchase a small amount of the other 499 stocks in order to rebalance the portfolio and maintain the equal weights for all stocks. Trading frequently in order to rebalance the portfolio in this way plays a crucial role in outperforming the market.

The two portfolio characteristics described above – more weight on smaller stocks and high-frequency rebalancing in order to maintain those weights – are not particularly complex and do not rely on information that is not readily available to the public. Of course, the true strategies behind most HFT are more sophisticated and must address real-world issues such as trading costs.[4] Despite its simplicity, however, this discussion describes a valid method of investing that achieves very high returns without major exposure to risk. This point is clearly demonstrated in Figure 1. The figure plots the log return of a portfolio that combines long positions in stocks that decrease in value with short positions in stocks that increase in value and that is rebalanced every minute and a half, much like in the previous discussion.[5]Based on a simulation that uses data from U.S. stock prices in 2005, this portfolio earns a compound return of more than 100% over the course of the year.Surely, the ability of HFT strategies to achieve high returns by exploiting the relative movement that is natural among stock prices in this way explains much of both the rapid spread of HFT and the consistent success of prominent HFT funds such as Medallion. Barring a significant change in financial regulation, then, there is little reason to think that the spread of HFT will reverse itself.

Figure 1: The  compound return of a portfolio that combines long positions in stocks that decrease in value with short positions in stocks that increase in value and that is rebalanced every minute and a half.

Nothing about our discussion of HFT strategies and high returns is inconsistent with a market that is efficient in the sense that stock prices reflect the public’s full knowledge about fundamentals. According to standard financial theory, anytime an individual stock price does not reflect that stock’s fundamentals, rational investors will buy or sell that stock and earn high returns until the stock price shifts to a value that does reflect fundamentals so that such returns are no longer possible. The HFT strategies that achieve high returns with limited risk, however, do not rely on deviations between prices and fundamentals. Indeed, as long as fundamentals are such that no one stock dominates the entire market for a full year, investors are able to consistently earn these returns without any knowledge of fundamentals and their deviations from prices. In this case, there is neither a contradiction between market efficiency and the ability of HFT to consistently outperform the market nor is there necessarily anything about HFT that makes markets more efficient.

The fact that HFT represents a highly effective investment strategy that has little to do with market fundamentals raises several challenging questions. How is it possible for HFT to consistently earn high returns with limited risk in an efficient market? What happens as more and more people pursue these high returns with HFT? How does HFT on a large scale like this affect stock prices? These are difficult “general equilibrium” questions that have yet to be answered satisfactorily by financial economists. There is little doubt, however, that a deeper understanding of these issues is likely to yield important theoretical and practical insights about the true workings of financial markets.

Footnotes

1. For these portfolios, there is a tradeoff between the extent to which the portfolio will outperform the index, the length of time before this outperformance is to occur, the maximum possible underperformance during this period, and the upper bound on the relative capitalization of the largest stock in the index. See Fernholz, Karatzas, and Kardaras (2005) for details.

2. A discussion of some of these “equilibrium’’ issues is provided by Karatzas and Kardaras (2007).

3.  In fact, this simple equal-weighted portfolio is, under slightly stronger assumptions, guaranteed to outperform the S&P 500 over a sufficiently long time period.

4. For example, high-frequency traders are often broker/dealers in order to reduce potentially significant trading costs.

5. In addition to combining long and short positions in this way, the portfolio also closes all of its outstanding positions at the end of each day. This helps to reduce the volatility of the portfolio’s return. For a more detailed description of how this portfolio works, see Fernholz and Maguire (2007).

References

Fernholz, R., I. Karatzas, and C. Kardaras (2005, January). Diversity and relative arbitrage in equity markets. Finance and Stochastics 9(1), 1-27.

Fernholz, R. and C. Maguire, Jr. (2007, September/October). The statistics of statistical arbitrage. Financial Analytics Journal 63(5), 46-52.

Karatzas, I. and C. Kardaras (2007, October). The numeraire portfolio in semimartingale financial models. Finance and Stochastics 11(4), 447-493