When Do Stop-Loss Rules Stop Losses?

Motivation(s)

The stopping premium characterizes the marginal impact of stop-loss rules on any given portfolio’s expected return. Under a random walk model, returns are unforecastable, so stop-loss rules force the portfolio out of higher-yielding assets on occasion. Hence stop-loss rules never stop losses in this market, which is why financial planners recommend buy-and-hold lifecycle funds.

However, passive investors often behave irrationally during market swings resulting in selling at the low and buying at the high. These behavioral patterns have been studied previously in the context of stop-loss rules, albeit the overall literature is rather limited despite the popularity of stop-loss orders.

Proposed Solution(s)

The authors propose a simple stop-loss heuristic that is not derived from any optimization problem. The goal is a systematic analysis of the impact of a stop-loss rule on an existing investment policy. This idea of overlaying one set of heuristics on top of an existing portfolio strategy have recently grabbed the interest of institutional investors (e.g. portable alpha strategies).

The heuristic assumes the returns \(\{ r_t \}\) for the portfolio strategy \(P\) are stationary with finite mean \(\mu\) and variance \(\sigma^2\). The expected return \(\mu\) of \(P\) is greater than the risk-free rate \(r_f\); \(\mu - r_f \equiv \pi\) denotes the risk premium of \(P\). See the paper for the intuitive derivations of the stop-loss policy, stopping premium, stopping ratio, and variance difference of a stopping strategy.

Evaluation(s)

The authors analyze the proposed heuristic under different return-generating processes using an autoregressive model of order one. Note that the proposed policy focuses solely on binary asset allocation.

Under the Random Walk Hypothesis, the heuristic will always reduce the portfolio’s returns. The derivations show that the only effect of a stop-loss policy in this market is to replace the portfolio strategy \(P\) with the risk-free asset when the strategy is stopped out, thereby reducing the expected return by the risk premium of \(P\).

When the underlying process exhibits mean-reversion, the stop-loss policy hurts expected returns to a first-order approximation. This is consistent with the intuition that mean-reversion strategies benefit from reversals, and thus a stop-loss policy that switches out of the portfolio after certain cumulative losses will miss the reversal and lower the expected return of the portfolio.

Markets with momentum may yield positive stopping premium. This is also consistent with the intuition that in the presence of momentum, losses are likely to persist; therefore, switching to the risk-free asset after certain cumulative losses can be more profitable than staying fully invested.

When the portfolio return follows a two-state Markov regime-switching, the only source of potential value-added for the stop-loss policy is if the equity investment in the low-mean regime has a lower expected return than the risk-free rate.

The authors also tested their heuristic on daily futures prices between 1993 and 2011. The IMM S&P futures contract is used for a position in U.S. Equities and the 10-year CBT Treasury note futures contract. The results indicate the use of longer term stop-loss strategies might have improved performance consistent with how investors use the strategy in practice. Shorter term, lower frequency stop-loss policies have negative stopping premiums over large ranges of parameters. Longer term stop-loss at frequencies above one month perform better and can achieve positive stopping premiums. The decision to exit and the stopping threshold seems to have a larger impact on the variation in results than the re-entry threshold.

Future Direction(s)

  • What kind of heuristics would deep reinforcement learning learn in the context of stop-loss rules?

  • How to learn the market condition in an unsupervised manner?

  • How does applying this heuristic over contrarian or momentum strategies affect profits?

Question(s)

  • How often do models with little theoretical insights succeed at backtesting and yield profits? Note that backtesting includes different market conditions (e.g. momentum, reversal, trending, ranging).

Analysis

The effectiveness of stop-loss rules depends on the market conditions. When there is market momentum, stop-loss heuristics may yield positive gains and reduce volatility.

The authors would have a stronger argument if the experiments included stocks in addition to futures. Futures are slowly varying compared to stocks, so perhaps the latter will demonstrate stop-loss rules are better than buy-and-hold strategies.

One interesting point is the claim that there is now substantial evidence from the cognitive sciences literature that losses and gains are processed by different components of the brain. This newfound explanation for market irrationality is more amenable to analysis than the adaptive market hypothesis. Overall, the derivations of the heuristic are crystal clear, very intuitive, and practical.

References

KL07

Kathryn Kaminski and Andrew W Lo. When do stop-loss rules stop losses? In EFA 2007 Ljubljana meetings paper. 2007.