## Monday, August 6, 2012

### Why Don't We Annualize Returns of Less Than One Year?

Most performance analysts have heard it said that we should never annualize returns of less than one year.  Some may also be aware that GIPS provision I.5.A.4 (2010 edition of GIPS) forbids compliant firms from annualizing returns for periods of less than a year?

But why shouldn't this be done?  Certainly, we could do it mathematically.  The formula for annualized return (from the CIPM Expert Level curriculum) is:

where each r(i) is an annual return and N is the number of annual periods.  Or, more generically:

where R(ann) is the annualized return, R(cum) is the cumulative return and d is the number of days in the evaluation period.

There are two reasons why we don't annualize periods of less than a year.  The first reason is ethical.  The second reason is conceptual.

The ethical reason is that it is not considered best practice to take a return that is for a period of less than a year, and project such a return to an annual time frame.  For example, let's assume a manager has a return of 11% for the month of January.  We can certainly use the above formula to extend (i.e., project) the return to an annual timeframe:

First of all, projecting returns is not good practice when presenting actual performance, and that is exactly what annualizing is in such a scenario.  Secondly, in a case such as this, while it is very possible to earn a return of 11% in a single month, most would argue that sustaining this kind of performance through a full year is highly unlikely.  Thus, annualizing in this kind of situation is ethically inappropriate.

The conceptual reason we don't annualize returns of less than a year gets to the heart of what an annualized return is:  a geometric mean return; i.e., a geometric average.  In general, the formula for geometric mean is:

In our case, the X(i) are the wealth relatives (aka unit values) of the sub-period returns.  By calculating a geometric mean return, we are calculating an (annual) average return that reflects the impact of return compounding.  But, it is still an average return.  As such, it does not make sense to calculate an annual average when we don't even have a full (annual) period under consideration.  Looking at the above example, it doesn't make sense to calculate an annual average return (geometrically) when we only have an evaluation period of .08 years (i.e., 31 days), does it?

I think while most people are aware of the ethical reason, thinking about the conceptual reason may help candidates understand this important concept.