Prospect Theory: The Mathematical Reason You Cut Winners and Ride Losers

Answer both of these quickly, without arithmetic.

You are up 0.9R on a trade. Your target is 1.0R and it is nine tenths of the way there. Do you take the 0.9R now, or hold for the last tenth?

You are down 0.9R. Your stop is at 1.0R and price is nine tenths of the way to it. Do you close for 0.9R now, or hold and see?

Most traders take the profit and hold the loss. They then describe this to themselves as a failure of discipline, resolve to do better, and do exactly the same thing next week.

It is not a failure of discipline. Those two choices have the same expected value, and your brain is running a well-documented calculation that produces both answers correctly. The calculation is not broken. It is just optimising a different quantity from the one your account balance measures.

Once you can see the calculation, the behaviour stops being mysterious. It also stops being fixable by willpower, which is the more useful discovery.

The function you are actually maximising

In 1979 Daniel Kahneman and Amos Tversky published a description of how people actually choose under risk, as opposed to how economics had assumed they choose. It won a Nobel Prize and it describes your trading better than any book written about trading.

Three properties matter here.

Outcomes are felt as changes, not as states. You do not experience an account of $52,000. You experience being up $2,000, or down $8,000 from the high. The reference point does all the work, and you will move it without noticing.

Sensitivity diminishes as you move away from that point. The difference between 0R and 1R feels enormous. The difference between 4R and 5R barely registers. Both are one R.

Losses hurt more than equivalent gains please. Not slightly. The parameter estimated across the experimental literature is about 2.25, which means a loss carries roughly two and a quarter times the psychological weight of an identical gain.

Sit with that number, because it explains the shape of every retail equity curve you have ever seen. Your system may produce 2R winners against 1R losers, a perfectly good business. Emotionally, you are running at a deficit on every single round trip. The account is compounding and the trader is bleeding.

The second engine: you do not use the probabilities you were given

Loss aversion is the famous half. The half that does the damage at the trading desk is what happens to probability.

People do not weight outcomes by their probability. They weight them by a distorted version of it. Small probabilities are inflated, which is why lottery tickets and 20R lottery-ticket trades both sell. And large probabilities are deflated, which almost nobody talks about, and which is the entire mechanism of this article.

Look at the bottom row. A 90% probability of winning feels like 71%. A 90% probability of losing feels like 78%. Both are underweighted, and the near-certain loss is underweighted less than the near-certain win.

Which means: when a good outcome is nearly certain, you do not quite believe it. And when a bad outcome is nearly certain, you believe it a little more, but still not enough.

The reflection effect, at your desk

Now put the two engines together and run the two questions from the top of this page. Both choices have identical expected value. Both are a coin the market has already mostly flipped.

There it is. The same person, on the same day, with the same information, is risk-averse when he is winning and risk-seeking when he is losing. Kahneman and Tversky called it the reflection effect: flip the sign of the outcomes and preferences flip with them.

This is not a metaphor for trading. It is the arithmetic of trading. Every trade you take places you on one side of a reference point, and which side you are on silently rewrites what you want.

An honest caveat, because the theory does not say what people claim it says

You will read, in a hundred trading blogs, that prospect theory proves traders always cut winners short. It does not, and the overreach is worth correcting because the truth is more useful.

Run the same calculation on a position that is up 0.6R with a 2R target and a 40% chance of getting there. The model does not prefer taking the profit. It prefers holding, because at moderate probabilities the weighting distortion runs the other way and the 2R payoff is large enough to survive the concavity.

The reflection effect bites near certainty. It is at its most violent precisely when the outcome is nearly settled, when a trade is inches from its target or inches from its stop. Which is, unfortunately, exactly where every trade must eventually go.

So the correct statement is not “you will always cut winners.” It is: you will systematically mishandle the endgame of every trade, in a predictable direction, and the closer you are to resolution, the worse your decision-making becomes.

What it costs

Take a trader with a real edge. A 40% win rate, 2R winners, 1R losers. Expectancy is +0.20R a trade, and over a hundred trades he makes 20R. Good business.

Now let the reflection effect do its work. He starts closing at 1.2R because 90% of the way to target feels like enough. He starts giving losers a little room because the stop is nearly touched and it is right there, so his average loss creeps to 1.35R.

Note the second row on its own. Cutting winners from 2R to 1.2R, while keeping every stop perfectly, converts a 20R year into a 12R loss. The stops were flawless. The discipline everybody talks about was intact. He still lost.

And note the third row. Widening stops alone costs him almost nothing here, because his winners are still running. This is why traders are so certain that the stop is the thing to work on: the stop is the loss they can see. The money is being taken somewhere else.

Why willpower is the wrong tool

Everything above happens while the position is open. That is the whole problem, and it points at the whole solution.

Before you enter, you have no reference point. There is no gain, no loss, no side of the line to be standing on. Your judgement in that state is the best judgement you own. The moment the fill prints, a reference point exists, the value function switches on, and every subsequent decision is made by a different animal.

Resolving to be stronger next time asks the animal to overrule itself. It will not. It is not a defect in the animal.

The reference point is the only thing you truly choose

Here is the last turn of the screw, and it is the one worth carrying away.

Prospect theory says outcomes are evaluated against a reference point. It does not say where the reference point has to be. That is set by framing, and framing is negotiable.

A trader whose reference point is his entry price experiences every open trade as a running verdict on his judgement. A trader whose reference point is the expected value of his system over the next hundred trades experiences an individual loss as a scheduled payment. Same trade. Same tick. Entirely different function being maximised.

You cannot argue yourself out of loss aversion. Nobody has. What you can do is choose what the loss is being measured against, and the honest answer is that a single trade is not being measured against anything at all. It is one draw from a distribution you already agreed to.

Cut winners and ride losers, and you are not undisciplined. You are simply still measuring from the wrong place.