Editor’s note: This article is an extract of a paper originally published at the MIT web site. All references and additional support for the ideas tested can be found in the original.
Abstract
Stop-loss rules—predetermined policies that reduce a portfolio’s exposure after reaching a certain threshold of cumulative losses—are commonly used by retail and institutional investors to manage the risks of their investments, but have also been viewed with some skepticism by critics who question their efficacy. In this paper, we develop a simple framework for measuring the impact of stop-loss rules on the expected return and volatility of an arbitrary portfolio strategy, and derive conditions under which stop-loss rules add or subtract value to that portfolio strategy. We show that under the Random Walk Hypothesis, simple 0/1 stop-loss rules always decrease a strategy’s expected return, but in the presence of momentum, stop-loss rules can add value. To illustrate the practical relevance of our framework, we provide an empirical analysis of a stop-loss policy applied to a buy-and-hold strategy in U.S. equities, where the stop-loss asset is U.S. long-term government bonds. Using monthly returns data from January 1950 to December 2004, we find that certain stop-loss rules add 50 to 100 basis points per month to the buy-and-hold portfolio during stop-out periods. By computing performance measures for several price processes, including a new regime switching model that implies periodic “flights-to-quality”, we provide a possible explanation for our empirical results and connections to the behavioral finance literature.
Introduction
Thanks to the overwhelming dominance of the mean-variance portfolio optimization framework pioneered by Markowitz (1952), Tobin (1958), Sharpe (1964), and Lintner (1965), much of the investments literature—both in academia and in industry—has been focused on constructing well-diversified static portfolios using low-cost index funds. With little use for active trading or frequent rebalancing, this passive perspective comes from the recognition that individual equity returns are difficult to forecast and trading is not costless. The questionable benefits of day-trading are unlikely to outweigh the very real costs of changing one’s portfolio weights. It is, therefore, no surprise that a “buy-andhold” philosophy has permeated the mutual-fund industry and the financial planning profession.[1]
However, this passive approach to investing is often contradicted by human behavior, especially during periods of market turmoil.[2] These behavioral biases sometimes lead investors astray, causing them to shift their portfolio weights in response to significant swings in market indexes, often “selling at the low” and “buying at the high”. On the other hand, some of the most seasoned investment professionals routinely make use of systematic rules for exiting and reentering portfolio strategies based on cumulative losses, gains, and other “technical” indicators.
In this paper, we investigate the efficacy of such behavior in the narrow context of stop-loss rules, i.e., rules for exiting an investment after some threshold of loss is reached and re-entered after some level of gains is achieved. We wish to identify the economic motivation for stop-loss policies so as to distinguish between rational and behavioral explanations for these rules. While certain market conditions may encourage irrational investor behavior—for example, large rapid market declines—stop-loss policies are sufficiently ubiquitous that their use cannot always be irrational.
This raises the question we seek to answer in this paper: When do stop-loss rules stop losses? In particular, because a stop-loss rule can be viewed as an overlay strategy for a specific portfolio, we can derive the impact of that rule on the return characteristics of the portfolio. The question of whether or not a stop-loss rule stops losses can then be answered by comparing the expected return of the portfolio with and without the stop-loss rule. If the expected return of the portfolio is higher with the stop-loss rule than without it, we conclude that the stop-loss rule does, indeed, stop losses.
Using simple properties of conditional expectations, we are able to characterize the marginal impact of stop-loss rules on any given portfolio’s expected return, which we define as the “stopping premium”. We show that the stopping premium is inextricably linked to the stochastic process driving the underlying portfolio’s return. If the portfolio follows a random walk, i.e., independently and identically distributed returns, the stopping premium is always negative. This may explain why the academic and industry literature has looked askance at stop-loss policies to date. If returns are unforecastable, stop-loss rules simply force the portfolio out of higher-yielding assets on occasion, thereby lowering the overall expected return without adding any benefits. In such cases, stop-loss rules never stop losses.
However, for non-random-walk portfolios, we find that stop-loss rules can stop losses. For example, if portfolio returns are characterized by “momentum” or positive serial correlation, we show that the stopping premium can be positive and is directly proportional to the magnitude of return persistence. Not surprisingly, if conditioning on past cumulative returns changes the conditional distribution of a portfolio’s return, it should be possible to find a stop-loss policy that yields a positive stopping premium. We provide specific guidelines for finding such policies under several return specifications: mean reversion, momentum, and Markov regime-switching processes. In each case, we are able to derive explicit conditions for stop-loss rules to stop losses.
Of course, focusing on expected returns does not account for risk in any way. It may be the case that a stop-loss rule increases the expected return but also increases the risk of the underlying portfolio, yielding ambiguous implications for the risk-adjusted return of a portfolio with a stop-loss rule. To address this issue, we compare the variance of the portfolio with and without the stop-loss rule and find that, in cases where the stop-loss rule involves switching to a lower-volatility asset when the stop-loss threshold is reached, the unconditional variance of the portfolio return is reduced by the stop-loss rule. A decrease in the variance coupled with the possibility of a positive stopping premium implies that, within the traditional mean-variance framework, stop-loss rules may play an important role under certain market conditions.
To illustrate the empirical relevance of our analysis, we apply a simple stop-loss rule to the standard asset-allocation problem of stocks vs. bonds using monthly U.S. equity and bond returns from 1950 to 2004. We find that stop-loss rules exhibit significant positive stopping premiums and substantial reductions in variance over large ranges of threshold values—a remarkable feat for a buy-high/sell-low strategy. For example, in one calibration, the stopping premium is approximately 1.0% per annum, with a corresponding reduction 2 in overall volatility of 0.8% per annum, and an average duration of the stopping period of less than 1 month per year. Moreover, we observe conditional-momentum effects following periods of sustained losses in equities that seem to produce scenarios where long-term bonds strongly dominate equities for months at a time. These results suggest that the random walk model is a particularly poor approximation to monthly U.S. equity returns, as Lo and MacKinlay (1999) and others have concluded using other methods.
Motivated by Agnew’s (2003) “flight to safety” for household investors, which is similar to the well-documented “flight to quality” phenomenon involving stocks and bonds, we propose a regime-switching model of equity returns in which the Markov regime-switching process is a function of cumulative returns. We show that such a model fits U.S. aggregate stock index data better than other time-series models such as the random walk and AR(1), and can explain a portion of the stopping premium and variance reduction in the historical data.
Literature Review
Before presenting our framework for examining the performance impact of stop-loss rules, we provide a brief review of the relevant portfolio-choice literature, and illustrate some of its limitations to underscore the need for a different approach.
The standard approach to portfolio choice is to solve an optimization problem in a multiperiod setting, for which the solution is contingent on two important assumptions: the choice of objective function and the specification of the underlying stochastic process for asset returns. The problem was first posed by Samuelson (1969) in discrete time and Merton (1969) in continuous time, and solved in both cases by stochastic dynamic programming. As the asset-pricing literature has grown, this paradigm has been extended in a number of important directions.[3]
However, in practice, household investment behavior seems to be at odds with finance theory. In particular, Ameriks and Zeldes (2004) observe that
. . . a great deal of observed variation in portfolio behavior may be explained by the outcome of a few significant decisions that individuals make infrequently, rather than by marginal adjustments continuously.
Moreover, other documented empirical characteristics of investor behavior include nonparticipation (Calvet, Campbell, and Sodini 2006); under-diversification (Calvet, Campbell, and Sodini 2006); limited monitoring frequency and trading (Ameriks and Zeldes 2004); survival-based selling decisions or a “flight to safety” (Agnew 2003); an absence of hedging strategies (Massa and Simonov, 2004); and concentration in simple strategies through mutual-fund investments (Calvet, Campbell and Sodini 2006). Variations in investment policies due to characteristics such as age, wealth, and profession have been examined as well.[4]
In fact, in contrast to the over-trading phenomenon documented by Odean (1999) and Barber and Odean (2000), Agnew (2003) asserts that individual investors actually trade infrequently. By examining asset-class flows, she finds that investors often shift out of equities after extremely negative asset returns into fixed-income products, and concludes that in retirement accounts, investors are more prone to exhibit a “flight to safety” instead of explicit return chasing. Given that 1 in 3 of the workers in the United States participate in 401(k) programs, it is clear that this “flight to safety” could have a significant impact on market prices as well as demand. Consistent with Agnew’s “flight-to-safety” in the empirical application of stop-loss, we find momentum in long-term bonds as a result of sustained periods of loss in equities. This suggests conditional relationships between stocks and bonds, an implication which is also confirmed by our empirical results.[5]
Although stop-loss rules are widely used, the corresponding academic literature is rather limited. The market microstructure literature contains a number of studies about limit orders and optimal order selection algorithms (Easley and O’Hara, 1991; Biais, Hillion, and Spatt, 1995; Chakravarty and Holden, 1995; Handa and Schwartz, 1996; Harris and Hasbrouck, 1996; Seppi, 1997; and Lo, MacKinlay, and Zhang, 2002). Carr and Jarrow (1990) investigate the properties of a particular trading strategy that employs stop-loss orders, and Tschoegl (1988) and Shefrin and Statman (1985) consider behavioral patterns that may explain the popularity of stop-loss rules. However, to date, there has been no systematic analysis of the impact of a stop-loss rule on an existing investment policy, an oversight that we remedy in this paper.
Editor’s note: the paper continues by developing a framework for analyzing stop-loss rules and presents the results of various rules.
Empirical Analysis
To illustrate the potential relevance of our framework for analyzing stop-loss rules, we consider the performance of (rules) when applied to the standard household asset-allocation problem involving just two asset classes: stocks and long-term bonds. Using monthly stock- and bond index data from 1950 to 2004, we find that stop-loss policies produce surprising conditional properties in portfolio returns, stopping losses over a wide range of parameter specifications. Our simple stop-loss rule seems to be able to pick out periods in which long-term bonds substantially out-perform equities, which is especially counterintuitive when we consider the unconditional properties of these two asset classes historically.
For our empirical analysis, we use the monthly CRSP value-weighted returns index to proxy for equities and monthly long-term government bond returns from Ibbotson and Associate to proxy for bonds. We also consider Ibbotson’s shortterm government bond returns for purposes of comparison. Our sample runs from January 1950 to December 2004, the same time span used by Ang and Berkart’s (2004) study of regime-switching models and asset allocation. In Section 5.4, we consider the longer time span from January 1926 to December 2004 to reduce estimation error for our behavioral regime-switching model estimates.
We conclude that stop-loss rules apparently exploit momentum effects in equities and long-term bonds following periods of sustained losses in equities.
Basic Results
The empirical analysis we perform is straightforward: consider investing 100% in equities in January 1950, and apply the simple stop-loss policy (rules) to this portfolio on a monthly basis, switching to a 100% investment in long-term bonds when stopped out, and switching back into equities 100% when the re-entry threshold is reached. We run this simulation until December 2004, which yields a time series of 660 monthly returns {rst} with which we compute the performance statistics.
Specifically, we compute performance measures for the simple stop-loss strategy (rules) for cumulative-return windows J =3, 6, 12, and 18 months over stop-loss thresholds γ =4–14% and re-entry thresholds δ = 0% and 2%. The performance measures ∆µ, ∆σ, ∆µ po , and po are graphed in Figure 1. Two robust features are immediately apparent: the first is that stopping premiums ∆µ are positive, and the second is that the volatility differences ∆σ are also negative. This suggests that stop-loss rules unambiguously add value to mean-variance portfolio optimizers. Moreover, the robustness of the results over a large range of parameter values indicates some significant time-series structure within these two asset classes.
Figure 1 also shows that ∆µ seems to decrease with larger cumulative-return windows, especially for J =6 and 12 months. This finding is consistent with ∆µ increasing in po when the risk-free rate rf is higher than the conditional expected return of equities, conditioned on being stopped out.
For the shorter window lengths, smaller thresholds provide less value-added but the value remains positive. However, for the 18-month window, larger thresholds perform better. This connection between stop-loss threshold and cumulative-return window size suggests that there is some fundamental relation—either theoretical or behavioral—between the duration of losses and their magnitude.
Our findings seem to imply momentum-like effects for large negative equity returns, except in the case of large losses over short periods of time which seems to imply reversals. However, since the main focus of our attention is on means and variances, a natural concern is the undue influence of outliers. Even during stop-out periods, we find that the kurtosis of stock and bond returns to be in the range of 2 to 3. We also find that the stop-out periods are relatively uniformly distributed over time, refuting the obvious conjecture that a small number of major market crashes are driving the results. Surprisingly, when we exclude the “Tech Bubble” by limiting our sample to December 1999, we find increased performance for our stop-loss policy in most cases. One explanation is that during significant market declines, our stop-loss policy may get back in too quickly, thereby hurting overall performance.
Figure 1 also includes a plot of ∆σ, which shows that volatility is always reduced by 20 the stop-loss policy, but the reduction is smaller for larger stopping thresholds γ. This is to be expected because larger thresholds imply that the stop-loss policy is activated less often. Nevertheless, the reduction in variance is remarkably pronounced for a strategy which so rarely switches out of equities. This reduction seems to be coming from two sources: switching to a lowervolatility asset, and avoiding subsequently higher-volatility periods in equities.
The stop-loss policy has a significant impact on the portfolio’s Sharpe ratio even in this simple two-asset case. The relation between ∆SR and window size underscores the potential connection between the amount of time losses are realized and appropriate stop-loss thresholds. Based on our empirical analysis, we conclude that stop-loss policies could indeed have added value to the typical investor when applied to equities and long-term bonds from 1950 to 2004.
Editor’s note: the paper then provides a more detailed analysis of these results by conducting a performance attribution for the two assets, and by examining several models for asset returns to gauge how substantial these effects are.
Conclusion
In this paper, we provide an answer to the question when do stop-loss rules stop losses? The answer depends, of course, on the return-generating process of the underlying investment for which the stop-loss policy is implemented, as well as the particular dynamics of the stop-loss policy itself. If “stopping losses” is interpreted as having a higher expected return with the stop-loss policy than without it, then for a specific binary stop-loss policy, we derive various conditions under which the expected-return difference—which we call the stopping premium—is positive. We show that under the most common return-generating process– 31 the Random Walk Hypothesis—the stopping premium is always negative. The widespread cultural affinity for the Random Walk Hypothesis, despite empirical evidence to the contrary, may explain the general indifference to stop-loss policies in the academic finance literature.
However, under more empirically plausible return-generating processes such as momentum or regime-switching models, we show that stop-loss policies can generate positive stopping premia. And when applied to the standard household asset-allocation decision between U.S. equities and long-term bonds from January 1950 to December 2004, we find a substantially positive stopping premium with a correspondingly large reduction in variance. These empirical results suggest important nonlinearities in aggregate stock and bond returns that have not been fully explored in the empirical finance literature. For example, our analysis suggests elevated levels of momentum associated with large negative returns, and asymmetries in asset returns following periods of cumulative losses.
Our analytical and empirical results contain several points of intersection with the behavioral finance literature. First, the flight-to-safety phenomena—best illustrated by events surrounding the default of Russian government debt in August 1998—may create momentum in equity returns and increase demand for long-term bonds, creating positive stopping premia as a result. Second, systematic stop-loss policies may profit from the disposition effect and loss aversion, the tendency to sell winners too soon and hold on to losers too long. Third, if investors are ambiguity-averse, large negative returns may cause them to view equities as more ambiguous which, in relative terms, will make long-term bonds seem less ambiguous. This may cause investors to switch to bonds to avoid uncertainty about asset returns.
More generally, there is now substantial evidence from the cognitive sciences literature that losses and gains are processed by different components of the brain. These different components provide a partial explanation for some of the asymmetries observed in experimental and actual markets. In particular, in the event of a significant drop in aggregate stock prices, investors who are generally passive will become motivated to trade because mounting losses will cause them to pay attention when they ordinarily would not. This influx of uninformed traders, who have less market experience and are more likely to make irrational trading decisions, can have a significant impact on equilibrium prices and their dynamics. Therefore, even if markets are usually efficient, on occasions where a significant number of investors experience losses simultaneously, markets may be dominated temporarily by irrational forces. The mechanism for this coordinated irrationality is cumulative loss.
Of course, our findings shed little light on the controversy between market efficiency and behavioral finance. The success of our simple stop-loss policy may be due to certain nonlinear aspects of stock and bond returns from which our strategy happens to benefit, e.g., avoiding momentum on the downside and exploiting asymmetries in asset returns following periods 32 of negative cumulative returns. And from the behavioral perspective, our stop-loss policy is just one mechanism for avoiding or anticipating the usual pitfalls of human judgment, e.g., the disposition effect, loss aversion, ambiguity aversion, and flight-to-safety.
In summary, both behavioral finance and rational asset-pricing models may be used to motivate the efficacy of stop-loss policies, in addition to the widespread use of such policies in practice. This underscores the importance of learning how to deal with loss as an investor, of which a stop-loss rule is only one dimension. As difficult as it may be to accept, for the millions of investors who lamented after the bursting of the Technology Bubble in 2000 that “if I only got out earlier, I wouldn’t have lost so much”, they may have been correct.
[1] This philosophy has changed slightly with the recent innovation of a slowly varying asset allocation that changes according to one’s age, e.g., a “lifecycle” fund.
[2] For example, psychologists and behavioral economists have documented the following systematic biases in the human decision making process: overconfidence (Fischoff and Slovic, 1980; Barber and Odean, 2001; Gervais and Odean, 2001), overreaction (DeBondt and Thaler, 1986), loss aversion (Kahneman and Tversky, 1979; Shefrin and Statman, 1985; Odean, 1998), herding (Huberman and Regev, 2001), psychological accounting (Tversky and Kahneman, 1981), miscalibration of probabilities (Lichtenstein et al., 1982), hyperbolic discounting (Laibson, 1997), and regret (Bell, 1982a,b; Clarke et al., 1994).
[3] For a comprehensive summary of portfolio choice see Brandt (2004). Recent extensions include predictability and autocorrelation in asset returns (Brennan and Xia, 2001; Xia, 2001; Kim and Omberg, 1996; Wachter, 2002; Liu, 1999; and Campbell and Viceria, 1999), model uncertainty (Barberis, 2000), transaction costs (Balduzzi and Lynch, 1999), stochastic opportunity sets (Brennan, Schwartz, and Lagnado, 1997; Brandt, Goyal, Santa-Clara, and Stroud, 2005; and Campbell, Chan, and Viceria, 2003), and behavioral finance (see the references in footnote 2).
[4] For example, lack of age-dependence in allocation, lower wealth and lower education with greater nonparticipation and under-diversification, and greater sophistication in higher wealth investors have all been considered (see Ameriks and Zeldes, 2004).
[5] Although excess performance in long-term bonds may seem puzzling, from a historical perspective, the deregulation of long-term government fixed-income products in the 1950’s could provide motivation for the existence of these effects.
