Quick Joke.

Q: What is the difference between a banker and a trader?
A: A banker will try and tell you a 10% loss followed by a 10% gain is breaking even.

This is a bit less arcane than some of the issues we usually discuss on the Win-Vector Blog, but it is a fun one. And it does take some effort to disabuse yourself of the banker’s fallacy.

It turns out that a lot of our instincts about something as simple as ratios is not quite right. Likely this is because the innate skills of counting leads to a deep understanding of addition and not of multiplication. Take for example the opening joke: a 10% loss followed by a 10% gain sounds like it should be exactly breaking even. But in fact it is exactly a 1% loss.

To compute the loss and gain on $100 we would say after the 10% loss we have $100*(1-0.1) = $90. A 10% gain on this remaining portion would be written as $90*(1 + 0.1) = $99 which, as predicted, is missing a dollar. An incorrect explanation would be something along the lines “well the loss was first, so it applied to a larger number than the gain.” But relative losses and gains work by multiplying and therefore is insensitive to order. It is a fact that a 10% loss followed by a 10% gain is exactly the same as a 10% gain followed by a 10% loss (which eliminates the attempted explanation). The correct explanation is the flaw was far earlier than you would think: you should not believe that the opposite of 10% loss is a 10% gain. To undo the effect of a 10% loss you need just over an 11% gain (a 11.1111111% gain).

For a more dramatic example consider the Dow Jones Industrial Average. It was at $12827 on January 7th 2008, by March 5th of 2009 it had fallen to $6594 or a 48% loss. By January 4th 2010 it had experienced a 60% gain relative to March 5th 2009- but that only got us to $10583, still a 17% loss relative to January 7th 2008. The opposite of 48% loss is in fact 192% gain (which obviously has not happened).

Bankers typically quote interest rates as if they were additive. Things like points and fees are all added. This is almost correct for small interest rates. This nearly right (but actually wrong) language is why we have a bestiary of confusing terms describing interest: simple interest, compound interest and yield. The bankers need some way to signal which numbers will actually be used for computing your mortgage payments versus which numbers will be used for advertising (and in the US they tended not to tell you many of the more important numbers until they were required to by law).

Traders, on the other hand, are very comfortable with multiplying relative losses and relative gains. The main trick of achieving such mastery is to convert multiplication into addition. The way to do this is the log() function (or the logarithm).

The log() function is simple function that has the property that log(a*b) = log(a) + log(b). For connivence lets pick our notation so that log(10) = 1. From this we can deduce that it must be the case that:

log() can not be used on zero or negative numbers (at least not if you expect a real number as a result). For other values we use our calculator.

A trader uses logarithms to think additively about relative changes (also called “returns”). Breaking even is represented as 0 (our friend log(1)), relative increases are represented as positive numbers and relative decreases are represented as negative numbers. For example a 10% loss is represented additively using logarithms as log(1- 0.1) = -0.0458. Now in this logarithm notation the additive opposite of a -0.0458 is in fact (as you would hope) +0.0458. You can even double check: log(1 + 0.1111111) = 0.0458. In this notation the mathematics and the language work together- the opposite of a loss is a gain with the same magnitude (and positive sign).

Returning to our initial example: a 10% loss is represented as -0.0458 and a 10% gain is represented as log(1 + 0.1) = 0.0414, so if we add them (how we combing operations in the logarithmic notation) we get -0.0044. Notice this is not zero, and is in fact equal to log(0 – 0.01) or a net-loss of 1%.

The point is that even trivial math becomes difficult if you are forced, by language or convention, to work from false premises.

Categories: Opinion

Tagged as: Finance logarithm paradox percent change relative returns

jmount

Data Scientist and trainer at Win Vector LLC. One of the authors of Practical Data Science with R.