# The maze of volatility

Volatilityis taken to be standard deviation (SD), a mathematical term describing the amount of variance around a mean of a set of data. It is widely used for many purposes in statistical analysis but in financial services we commonly use it to describe the extent to which an asset’s price has varied from its average return over a period.

Many advisers use volatility data as one input when building portfolios for clients. In these portfolios a lower overall volatility figure is often taken to predict reduced variance – or movement – in returns in the future.

If a large series of data points (monthly price changes, for example) is plotted on a chart according to how frequently a particular value occurs, the results will typically plot as a bell curve, as in Graph 1. While a few values will be extremely high or extremely low, most will be clustered somewhere in the middle. By adding all the values together and dividing the result by the number of values, we find the mean, which is the average return in each period, although it is entirely possible the average return figure was not actually achieved in any of the periods.

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As we can see in Graph 1, The SD of the data series measures the extent to which the actual values are clustered around the mean value – a low SD means that they are all close together while a high SD will arise if they are widely dispersed and the returns vary a lot from one month to the next.

Assuming a normally distributed data set, where the curve is symmetrical, two thirds of the time, the actual return for a period will fall within one standard deviation either side of the mean. 95 per cent of the time, the return will be within two standard deviations of the mean and 99 per cent of the time within three. This is best illustrated with a few examples:

Table 1 shows the average returns and SDs of the FTSE All Share Index between February 1955 and January 2015, sampled at various frequencies.

Let’s look at the monthly data (top row in Table 1). The average return is 1.07 per cent for each month. However, during that period, these monthly returns had a standard deviation of 5.38 per cent. That meant that two thirds of the time, it varied between +/- 5.38 per cent either side of 1.07 per cent, so between -4.31% and 6.45%.

As we look at the other frequencies, we must remember we are sampling the same data series less frequently, so the period returns and the period standard deviation will be different. Same data; different frequency. Money Management publishes average monthly volatility data for 36 months (top row in the Table). Other providers (for example, Morningstar’s website) provide average annual data over three years (bottom row in the Table).

One frequency is not right or more accurate and the others wrong or less accurate, as they all use the same data; they just express it differently. While the underlying data is the same, different data sets are not interchangeable.

The third row (labelled ‘Annualised return’) is highlighted in the table. In this instance it has been estimated from the monthly data which is why the annualised return and the actual annual return are slightly different.