In the first article we looked at how taking a simple average return derived from historical data and using that for projections of the long-term future will yield a higher result than would be expected based on that historical data.

That is because simple averages only tell us what the average return is for one period. But when you are projecting multiple consecutive periods (i.e. compounding returns), as is done in financial advice, the simple average is not appropriate. An extreme example would be two years where the return is -50% p.a. in the first year and +50% p.a. in the second year. The simple average return is 0% p.a. If you used that 0% p.a. rate of return to do a projection of \$100 invested for two years, you would show:

• value in 2 years using simple average return = \$100 x (1 + 0/100) x (1 + 0/100) = \$100

But if you used the actual historical returns (-50% p.a. and +50% p.a.) you would show:

• value in 2 years using historical returns = \$100 x (1 – 50/100) x (1 + 50/100) = \$75

You can see that the value using the historical returns is much lower than the value using the simple average of those historical returns.

Whilst this is an extreme example, the principle always applies. The projection of a lump sum investment using the simple average return of historical data will ALWAYS produce a result higher than using the historical (actual) returns on which the average was based (or equal if the historical returns are constant).

So, what is the alternative? The answer is the compound average return. In the example shown above, the simple average return was calculated as

• simple average return = (-0.50 + 0.50) / 2 x 100 = 0.00% p.a.

whereas the compound average return would be calculated as:

• compound average return = ((0.50 x 1.50) ^(1/2) – 1) x 100 = -13.40% p.a.

[Note: the ^ sign denotes “to the power of”]

Now, if we apply this compound average rate of return to our \$100 investment we would project the value in two years time as:

• value in 2 years using compound average return = \$100 x 0.8660 x 0.8660 = \$75

which, of course, is what we get using the historical returns (see above).

So far so good. If we use compound average returns then that should remove the bias inherent in simple average returns, right? Yes, BUT this only applies, strictly, to the situation of projecting the value of a single upfront investment over a fixed period . What if there are a range of cash flows over time? Well, then it will depend on the timing of the cash flows relative to the pattern of returns as to whether the compound average return produces higher or lower values compared to the simple average return and the historical returns.

As an example, take the following scenario:

• \$200,000 invested upfront
• \$8,000 invested at end of each year for next 10 years
• \$75,000 withdrawn at end of each year for the 10 years following that

The chart below shows the balance at the end of each year based on:

1. historical returns (calculated from ASX 200 total return index excluding fees, tax and franking credits)
2. simple average of historical returns (8.26% p.a.) and
3. compound average of historical returns (7.38% p.a.)

In this particular scenario, the final balance (close to zero) using the compound average return is very similar to that using the historical returns. The final balance using the simple average return is much higher – over \$100,000 higher.

But now take a different scenario:

• \$200,000 invested upfront
• \$8,000 invested at end of each year for next 5 years
• \$37,000 withdrawn at end of each year for the 15 years following that

Now the chart looks like this:

So, in this scenario the final balance using the simple average return is very similar to that using the historical returns. This time it is the final balance using the compound average return that is quite different – around \$100,000 lower than the other two.

Clearly, then, we can’t be sure what will be the impact, relative to historic returns, of using the simple average return compared to the compound average return when there is a wide range of cash flows over time. So what do we do?

Well, there is another way that better takes into account the impact of variable returns, and that is Stochastic Modelling, which we will discuss in our next article.