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# Annualized returns

Written by:
28/09/2011
How to work out the returns of a particular share over a given period of time - adapted from No Fear Finance.

Periodic returns, of whatever nature, record the return of one single period. So, what do we do when we want to look at the returns of more than one period? One answer of course is simply to use a longer period. If we want to look at lots of daily returns we might look instead at the return of the month, or quarter, or even year that they represent.

There are many reasons, however, why this approach will not normally be appropriate (for example, we may wish to calculate the average and Standard Deviation). Frequently we will be looking at any number of annual returns, for example – perhaps the returns of a particular share in every year over 10 or 20 years.

How do we approach this? Annualized returns are the traditional tool used. What are these? Average annual returns. We simply add up whichever periodic return figures we are looking at, and then divide the result by the number of individual observations or return figures. It is exactly the same exercise that we carried out when calculating the average (or mean) of the height of our schoolchildren.

Incidentally, this sort of average that we have just calculated is called the arithmetic mean. There is actually a different way of calculating an average, which is called the geometric mean, and mathematicians quite correctly argue that it is this, and not the arithmetic mean, which we should use when looking at investment returns.

If we want to find the average height of a schoolchild then it is perfectly valid to take a group of schoolchildren, add up their heights, and then divide the total by the number of observations, the number of children in the group.

However, if we think about the way in which investment returns operate, and are usually stated, they are a percentage rate by which the value of an asset is increased (or decreased) during a given period. In other words, we are multiplying the starting value of our asset by the rate of percentage return. It is as if, instead of measuring the heights of all the children in a group, we measure them all once a year and want to know not the average height at a particular time, but the average rate of growth.

The geometric mean

When we wish to find the average of a list of values, such as the average height of a group of schoolchildren, it is quite correct to add up the observations to find their total value and then divide by the number of observations to find their average or mean. If you think about the way in which investment returns work, however, there is an argument that this approach does not reflect reality.

Suppose that we have an investment that yields a return of 14% in Year 1, 8% in Year 2, and 6% in Year 3. What is actually happening here in arithmetic terms? The answer surely is multiplication, not addition.

In Year 1 we need to multiply the starting value of our investment by 14% (or 0.14) to find the return or 114% (or 1.14) to find its closing value. Would it not then make more sense to use a multiplication-based approach for our calculations rather than one that works by means of addition?

The geometric mean is that tool. Unlike the arithmetic mean, the geometric mean multiplies all the numbers together and then calculates what is called ‘the nth root’ or ‘root n’ of that number. Remember that, just like powers, a root can have any value.

Negative returns

Astute readers will at this stage doubtless be asking ‘Ah yes, but what happens if one of your returns is negative? This could give some very strange results’. Indeed it could. In fact, as you rightly suspected, it is not possible to calculate a geometric mean for any sequence of numbers where one of them is negative.

There is a way around this, though. Suppose in Year 2 you suffer a negative return of 5% rather than a positive one of 8%. Think about what is actually happening. If we suffer a negative return of 5% we have reduced the value of our asset by 5%. It is now worth 95% of what it was worth before and, as we have already seen, this is the same as multiplying by 0.95.

So, we can simply restate our three annual returns as 1.14, 0.95, and 1.06 respectively and proceed as before. This time we calculate: (1.14 × 0.95 × 1.06) and you will find that once you calculate the bits in the brackets first this becomes: 1.14798 which equals 1.047. In other words, the geometric mean of the annual returns is now 4.7%, since our asset value is increasing on average by that amount every year.

So, you can see that it is indeed possible to handle negative periodic returns when calculating a geometric mean, provided that we re-state the percentage rates first.

In practice, however, most investors and their advisers typically use the arithmetic mean, and only a cynic would suggest that this might be because it will always tend to produce a higher figure, thus flattering apparent investment Return  performance.

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