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We propose a simple analytical framework to measure the value added or subtracted by stop-loss rules—predetermined policies that reduce a portfolio’s exposure after reaching a certain threshold of cumulative losses—on the expected return and volatility of an arbitrary portfolio strategy. Using daily futures price data, we provide an empirical analysis of stop-loss policies applied to a buy-and-hold strategy using index futures contracts. At longer sampling frequencies, certain stop-loss policies can increase expected return while substantially reducing volatility, consistent with their objectives in practical applications.

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 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-and-hold” 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 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 re-entering 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 (e.g., 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 a standard asset-allocation problem of stocks versus bonds using daily futures data from January 1993 to November 2011. We find that stop-loss rules exhibit positive stopping premiums over longer sampling frequencies over a larger range of threshold values. These policies also provide substantial reduction in volatility creating larger Sharpe ratios as a result. This is a remarkable feat for a buy-high/sell-low strategy. For example in one calibration, using stop loss over monthly intervals in daily data can increase the return by 1.5% and decrease the volatility by 5%, causing an increase in the Sharpe ratio by as much as 20%. These results suggest that stop-loss rules may exploit conditional momentum effects following periods of losses in equities. These results suggest that the random walk model is a particularly poor approximation to U.S. stock returns and may improperly value the use of non-linear policies such as stop-loss rules. This is consistent with Lo and MacKinlay (1999) and others using various methods to examine limitations of the random walk.

In this paper, we provide a concrete answer to the question of 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, 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. When applied to a standard buy-and-hold strategy using daily U.S. futures contracts from January 1993 to November 2011 where the stop-loss asset is U.S. long-term bonds futures, we find that at longer sampling frequencies, certain stop-loss policies add value over a buy-and-hold portfolio while substantially reducing risk by reducing strategy volatility, consistent with their objectives in practical applications. These empirical results suggest important nonlinearities in aggregate stock and bond returns that have not been fully explored in the empirical finance literature. Our analytical and empirical results contain several points of intersection with the behavioral finance literature. First, the flight-to-safety phenomena, which is 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 non-linear 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 of negative cumulative returns). 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 apparent effectiveness 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 many investors who lamented, after the subprime mortgage meltdown of 2007–2008, that “if only I had gotten out sooner, I wouldn't have lost so much,” they may have been correct.