Wednesday, February 16, 2011

When A Stop-Loss Makes Sense

Tuesday I posted on the stop loss myth, but it's actually a more complex issue. The post criticized the thought that given an expected return on an asset, a stop loss rule can change an asset's expected return. It isn't often stated this boldly, but implied, and this is just wrong. A stop-loss merely adjusts the expected return by the expected time horizon you are in the trade. An asset does not have a different expected return because of what the holder does with it by definition in the paper referenced.

Yet, stop loss rules can be very useful when you do not know the expected return of the asset, or strategy, you are applying. In those cases, almost everyone prudently stops doing something if it loses some arbitrary amount. I hear Millenium had a rule that once a trader had a 3% loss of his initial capital allocation, he was gone; harsh, but with hundreds of traders of uncertain ability, a reasonable rule of thumb.

The idea comes from the optimal stopping problem, which has a large literature, and has been applied to picking secretaries. The idea is, you sample things, and can move on, and you want to choose the secretary/asset that will generate the greatest annuity. When you sample from a strategy/asset, it is like pulling a ball labeled x from an urn that are distributed with mean μ and standard deviation σ. Each sample costs c>0, which can be considered an opportunity cost. An any stage you can stop and keep your asset/secretary for the remainder of the 'game', generating x each period. In this case, the optimal rule is to sample until x>f(c,σ), where f'(c)<0, so the criteria is a decreasing function of the cost of sampling. Further, f'(σ)>0, so as the volatility goes up, you would do better searching still further, hoping to get a really high valued secretary/strategy, etc.

This has been applied to optimal dating strategies, where you date X women, then after that, proposing to the next one who is 'better' than any of those X (this assumes each of X would accept your proposal). X is derived by taking the number of chances you think you will get and dividing by e (2.72). So, if you date 10 girls a year, and figure you'll date for 10 years (100 total), take the first 37 to find your max, and of the next 63 take the first greater than the best of the max (you aren't allowed to 'go back' at any time). That maximizes your chances of finding your soulmate.

A related problem with similar intuition is if you think returns can have a mean of -μ or +μ, then a rule that exits a strategy when the cumulative return is X standard deviations below zero over any horizon will get you out of the low mean return strategies with a greater probability than the high mean return strategies, surely a good thing over the long run. The exit rule is a function of volatility and the costs of sampling.

This is a great strategy towards life in general. You should expect to pay to find what you are specially suited for, whether it's a career or mate. This is why people don't mind paying to take risks as they frequently do, because such plays are not made in a one-time mean-variance setting, but as part of a strategy of finding your alpha or comparative advantage. To never take risk but merely work really hard is to never find your best, but rather, do best in what may be a very suboptimal match (eg, I would be a bad piano player no matter how hard I practiced). When young, failing is OK, you just have to know when to stop (not too soon or too late) and try something else. Life is finite, and so at some point you have to settle down at what you are best at, or what works best for you, even if you aren't great at it, and stick with it. Optimizing does not mean you will be very successful, it just means you are maximizing your potential, which is all anyone can do.

John said...

I thought this was a very good post. Honestly spent over an hour just thinking about optimal stopping problems.

I was wondering how to actually implement something like this in practice. For instance, assume you have some prior distribution and some forecast distribution over a specific time horizon that together you use to create a portfolio. I assume you blend the prior and the forecast distribution into a final distribution with some confidence. Stopping in this sense could be setting the confidence in the forecast distribution equal to 0%. However, the prior distribution and the forecast distribution may change often and this change may occur before the time horizon has been reached. This means that, in practice, it would not make sense to think of it like flipping a coin over and over to determine whether it is fair. The relevant distribution is changing as you throw it and you may not even see the final result once, let alone watch it a bunch of times.

Perhaps a better way to think about the optimal stopping strategy would be to form a portfolio based on your forecast distribution (hard to say whether this should be 100% confidence in the forecast distribution or the optimal confidence with just these two distributions, you may have n others) and regress the returns of it against the returns of the portfolio formed from your prior to get the alpha over some (possibly short) history. If there is greater than a 95%/99% probability that the alpha of the forecast portfolio is less than 0% (or some minimum bound), then you would set the confidence in that distribution equal to 0%. Since there are only two distributions here, then you would set the prior to 100%, but if you are looking at more than one potential forecast distribution, then you may need to recalculate the confidences in the other distributions with these ones fixed at 0%. Then when you optimize you can incorporate transaction costs. If costs are high, then you still might retain some exposure to the above portfolio.

J said...

The optimal dating strategy you mention assumes that all 37 girls you dated during the last three years are still waiting for your decision. May be I didnt understand the idea. Anyway, since the girls are playing the same game, the dating strategy is not optimal. Your "optimal" girl has many chances better than you and may not be available for long.

People think that reality is following cold probability rules when it is malignantly playing against you.

Anonymous said...