You are here: Home - Investing -

Annualized returns

0
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.

This article is adapted from a chapter in the book No Fear Finance by Guy Fraser-Sampson, published by Kogan Page, RRP: £24.99. Special offer for Your Money Readers. Get 20% off No Fear Finance plus free p&p from now until 1st November. Phone: 01206 25 5678 and quote NOFF20 when you place your order. Or buy online www.amazon.co.uk

Tag Box

Debt

Pension

Spending

Financial fitness

There are 0 Comment(s)

If you wish to comment without signing in, click your cursor in the top box and tick the 'Sign in as a guest' box at the bottom.

Are you a first-time buyer looking for a mortgage?

Look no further, get the help you need by searching for your perfect mortgage

Which ISA is right for you? A round up of the six products available in 2017

From cash to innovative finance to lifetime, here's our guide to the ISA products available to savers this yea...

Guide to buy-to-let tax changes

In late 2015, former Chancellor George Osborne announced a range of  tax measures aimed at landlords, which t...

A guide to switching energy provider

All you need to know about switching from one energy supplier to another.

What will happen if rates change

How your finances will be impacted by a rise in interest rates.

Regular Savings Calculator

Small regular contributions can build up nicely over time.

Online Savings Calculator

Work out how your online savings can build over time.

Five fund tips for a 0.25% interest rate environment

With interest rates stuck at a record low 0.25% and expectations rates could fall to close to zero, here are ...

Protecting family wealth: 10 tips for cutting inheritance tax

Inheritance tax - sometimes known as 'death tax' - can cause even more heartache for bereaved families. But th...

Travel insurance: Five tips to ensure a successful claim

Ahead of your summer holiday, it’s important to make sure you have the right level of travel cover or you co...

Investing your money

Alliance Trust Plc gives you smart insight into how to invest your money

Money Tips of the Week

Read previous post:
The truth about bonds

The basics every investor should know about bonds - adapted from No Fear Finance

Close