## Sunday, 17 April 2011

### Volatility is not uncertainty.

Twelve months ago a hedge fund was created.  Eleven months ago it made a 1% first month return; for -2% for its second month, and so on.  By now, its 12 monthly percent returns look like this: $\left\{1,-2,2,\frac{1}{2},\frac{3}{4},-\frac{1}{2},0,0,0,1,-3,-4\right\}$.  It is useful to distinguish where the uncertainty lies.

The sequence of returns are certain enough.  They're a part of history.  You can calculate their sample variance as $\sigma^2 = \frac{1}{11} \sum_{m=1}^{12}(r_i - \bar{r})^2$ where $\bar{r}$ is the average, $-0.27$ in this case.  And the sample historical volatility $\sigma = 1.8$.  These are all certain.  The calculated volatility tells you with certainty just how variable those returns were.

If I got the measurements of the sizes of the planets in our solar system, I could likewise calculate the population variance and volatility.  With certainty.  I'm not saying anything whatsoever about their likelihood of change in the future.

In the world of finance and investing, we usually perform two extra operations which introduce uncertainty.  First, we decide we want to consider the unknown future and rope in history to help us.  Second, we construct a reference model, random in nature, which we hypothesise has been generating the returns we have seen so far and which will continue to generate the returns likewise into the unknown future.  That's a big second step.

Without wanting right now to go into issues about how valid this is, or even what form the model might take, I'll jump right in and suggest that next month's returns are expected to come in between  $-2$% and $1.4$%.  As soon as we decided to make a prediction about the unknown future, we added a whole bunch of uncertainty.  By picking our model (which, after all, might be an inappropriate choice), we've added model uncertainty.  By assuming that the future is going to be like the past, we've expressed a level of trust in reality which emboldens us to apply volatility to reduce all the uncertainties we just introduced.

A second way you could introduce uncertainty was to create a guessing game.   Write all 12 returns down on pieces of paper and put them in a hat.  Let a glamorous assistant pull a piece of paper out of the hat.  Then let people bet cash to profit or lose from the difference between the drawn number and the mean.  In those circumstances the volatility of the original returns would help you size your bet.