Why losses should be kept small - A mathematical view

"Cut your losses short and let your profits run."

I keep repeating this cliche ad nauseum. Seasoned traders swear by it. Amateurs flout it (to their own detriment). I believe it is key to investment success. Intuitively we might agree that it is important, but can we really prove it's worth?

Investors seek returns from their investments and trading decisions. Amateur investors like to make money on every trade/investment. While we might want our investments to be winners, there are too many imponderables that could put a spanner in their wheels. Ergo, we can land up with losses as well (whether or not we book them). In the markets all outcomes are uncertain. Profits and losses, both, are part of the trading landscape. (In fact, successful speculators lose more often than they win)

Since every outcome is uncertain, winning merely has probability, as has losing. So over time, we would have some winning investments and some losing investments. Lets consider the following:

W = Probability of Winning (0<=W<=1)

L = Probability of Losing (0<=L<=1, L=1-W assuming no breakeven trade)

AP = Average size of profits (AP>0)

AL = Average size of losses (AL<0)

This brings us the what is termed as the Mathematical Expectation (E) of our investing decisions. E is the Expected Profit per trade.

E = W*AP + L*AL

(Note that the product L*AL is negative since AL has a negative value.)

E denotes what profit we will make on an average per trade. All other things being equal, the larger the value of E, the better it is for our returns. Traders should strive to achieve a high value for E from their trades.

Look at the right hand side of the above equation. For E to be high, the first part (W*AP) of the sum needs to be high while the second part (L*AL) needs to be kept to a small (negative) value.

For W*AP to be high, either W needs to be high or AP needs to be high or both.

But W has an upper limit and cannot get higher than 1. (1 implies all winning investments)

AP in contrast does not have any upper bound. Though profits don't go to the moon, large profits are not uncommon.

Just to illustrate, say,

AP = 4% and W = 0.8 (80% chance of success), W*AP = 3.2%

AP = 8% and W = 0.6 (60% chance of success), W*AP = 4.8%

AP = 16% and W = 0.4 (40% chance of success), W*AP = 6.4%

AP = 50% and W = 0.2 (20% chance of success), W*AP = 25%

Your profits go up even as your winning percentage goes down. It is easier to find investments with a lower success rate. Most people want to maximise the chance of winning. Professionals focus on maximising profit instead, not the chance of winning. If risk is properly managed (which it should be), it is much easier to find trades that have a lower chance of making money than ones that have 80%, 90% or higher chance of success.

In the product, W*AP, it is AP that plays a much dominant role compared to W in determining the product. Average Profit is more important than the Winning Percentage. Try seeking a high value for average profits - LET YOUR PROFITS RUN

Using the same logic, on the other part of the equation, L*AL, average losses are more important than chance of loss. For example,

L= 2% and L= 0.6 (40% chance of success), L*AL= 1.2%

L= 5% and L= 0.5 (50% chance of success), L*AL= 2.5%

L= 10% and L= 0.4 (60% chance of success), L*AL= 4%

L= 20% and L= 0.3 (70% chance of success), L*AL= 6%

The chance of success is increasing, but your losses are also going up.

An 5 fold increase in average loss needs to be compensated by a 80% decrease in your losing percentage. If average loss increases from 2% to 10%, your winning percentage (initial value = 50% = chance of heads on a coin flip) needs to increase to 90%, not an easy task. It is much easier to keep a loss down to 2% and get a 50% success rate than to let losses increase to 10% in an attempt to get a 90% success rate.

The bottomline is: Keep average losses small - CUT LOSSES SHORT

Let profits run and cut your losses short! When you have a profit, don't sell under the fear that the markets would take away your profits. Let profits become big. When you have a loss, don't hope that it would turn around. Don't let losses get big.

It does not matter whether you win or lose. What matters is how much money you make when you win and how much money you lose when you don't win.