It is often claimed that one reason to cut losers quickly is that after a -20% fall you need a rise of 25% to recover and after a -33% fall you would need a +50% rise to recover - with the implication that it is therefore harder to recover.

These number pairs are simply reciprocals expressed as "percentages minus 100". For example:

• -20% is equivalent to 0.8 and 1/0.8 = 1.25 (+25%)
• -33% is equivalent to 0.66 and 1/0.66 = 1.5 (+50%)

So "geometrically" the pairs are equivalent. But in real life, do the stocks move in a way which suggests they are fighting an arithmetical imbalance or a geometrical equality?

By chance I was looking at the deciles of the "RS 1y" metric and noticed something interesting which prompted me to write this short post.

 Decile RS 1y log(1+rs/100) 10% -33.3 -0.405 20% -20.6 -0.231 30% -12.1 -0.129 40% -5.5 -0.056 50% +0.6 0.006 60% +7.2 0.069 70% +15.4 0.143 80% +26.8 0.237 90% +48.3 0.394

What struck me instantly was how closely the quantile points above 50% (the median) match the reciprocals of their counterparts below 50%. For example the 80% percentile is near 25% which matches the -20.6% of the 20% percentile.

If you are not used to working out reciprocals and converting back into percentages in your head (who is?) I have included the "log" of each value in the second column. If the matching reciprocal was perfect its log would be exactly the negative of the value it matches.

Needless to say the correspondence is almost perfect. By using the relative strength rather than absolute yearly change we have removed the underlying market movement and it reveals that stock price movements are most definitely following a geometric rule not an arithmetic one. They really do treat +25% and -20% as a matched pair.

Obviously this does not mean that every stock which falls by -20% is going to recover. It just means that the asymmetry in the numerical value is not in itself a good justification for using very tight stop-losses.

(In fact -20% falls are really not that uncommon. About 40% of all stocks will haveĀ  a fall of -20% or more from their highest value at some point during any given year.)

There is a nice little exploitable consequence to this demonstration. Although the median (50% percentile) of RS 1y is very close to zero (as you would expect) the (arithmetic) mean value is +9%. That difference comes from averaging pairs like (-20,+25) and (-33,+50) across all the stocks.

This…