# Nuke Geometric Returns

When I was a kid 'Chinese Torture' was known to be extremely subtle.

For example, a victim would be tied up and a drop of water would be allowed to slowly drip on someone's forehead for days until they went insane.

Terrifying.

Of course, actual torture in Imperial China was subtley different.

Death by a thousand cuts entailed literally slicing a man apart little by little while he was alive.

When you have a big idea, it's a good strategy to keep the details sketchy for as long as possible, don't allow it to be picked apart.

Case in point: the Iran deal. Light on details, in the hope that historical momentum will float it out of the reach of hardliners.

Usually, I am not a details person. I'd rather let people explore their ideas. But there are some basics which can't be ignored.

Financial time series analysis only makes sense with logarithmic returns.

Why?

The quick answer is that any statistical analysis assumes that we are comparing apples with apples.

The long answer requires and example to show some complications.

A price which jumps from

`100, 80, 100`

results in the following geometric returns

`-20%, 25%`

and these log returns

`-22%, 22%`

The average geometric return is given by

`-1 + [ (1-20%) * (1+25%) ] ^ 0.5`

The log average is

`(-22% + 22%) / 2`

Now try to calculate something simple like variance.

Usually it's defined as

`( SUM (xi - mu) ^ 2 ) / (n-1)`

We already have the average geometric return. We 'summed' up by multiplying the returns together.

A little shakey about how you subtract one geometric return from another?

How about squaring?

Don't know? Good. If you use geometric returns, and calculate anything at all, you've just proved my point.

Oh, you just use Excel's variance function? Well that expects common garden numbers also.

Log returns are nice because they are work as we would expect. Geometric returns are a God awful mess.

That doesn't stop most people from reporting them unfortunately.