Standard Deviation

While the monthly and annual compound returns accurately calculate the rate at which an investment has profited (or lost) over a given period of time, they don't in any way reflect the degree to which that investment may have fluctuated in value in doing so. The reason for this is that the CROR assumes a steady growth in the value of the investment over the time period, in that it increases in value by a set amount each month (in the case of the previous example, by 6.57%.)

As you probably well know, share trading is anything but stable in either its monthly or annual returns. As an example, look at the results in the calculation box. Once again, they are real results from one of the services we track. The largest monthly gain was 60.7% (month 15) while the largest monthly loss was -35.6% (two months later in month 17). That is a difference of 96.3%, while between months 10 and 11 it was 39.8%.

That demonstrates perfectly the volatility of trading. And while it is a reality of trading, it is not necessarily a detriment. The service we just quoted, for example, has a monthly CROR of 11.2% (more than twice the average) and an annual CROR of 258.3%. Good trading by anyone's standards.

Yet that volatility, given that it is such an inherent part of trading, needs to be taken into account. Otherwise, the results are misleading and not truly reflective of the industry. That is where the standard deviation is useful.

The standard deviation is a measure by which the results could have been expected to vary from either the monthly or annual CROR (depending on which is being measured). In that way, it is a measure of volatility (or risk) and therefore useful for analysing trading results.

The formulae for calculating the standard deviation is more complex than the formulas we have used previously and beyond our present scope (click here for more detail). It is, however, easily accessed through the Excel STDEVP function - where the range of monthly returns is A1:A19, =STDEVP(A1:A19) generates the standard deviation. Using the data from the calculation box, it would be 21.3%.

Knowing the standard deviation of a service's returns allows us to "adjust" them to take into account their volatility. This, in effect, becomes a risk-adjusted rate of return. The formulae for this is:

These, then, are rates of return for the service that take into account not just its performance but also the risk (or volatility) involved. Both figures, the CROR and the risk-adjusted CROR, should always be noted, in particular the degree to which the latter varies from the former.