If you are a financial adviser, you need to understand the difference between how average rates of return can be represented. One of the key differences is simple vs compound averages (also known as arithmetic v geometric averages).

In this article we will highlight the problem of using simple average returns in financial projections. In the next article we will look at compound average returns. And in the third article we will explore stochastic simulations.

The best way to understand the impact of using simple average returns is through some simple examples.

**Example 1**

Suppose I invest $100 over two years and in the first year I earn 6% p.a. and in the second year I earn 14% p.a. What is my “average” return?

Well, one approach would be to take an average the two rates of return i.e. calculate the simple average, as follows

- simple average return = (6% + 14%) / 2 = 10% p.a.

If I now applied that simple average rate of 10% p.a. to each year, then I would calculate the value of my investment after two years as:

- value calculated using simple average return = $100 x 1.10 x 1.10 = $121.00

But if I applied the actual rates of return to each year then I would calculate the actual value of my investment after two years, which is:

- actual value = $100 x 1.06 x 1.14 = $120.84

“So what” you say, “yes there is a difference but it is very small, not significant really”.

**Example 2**

Agreed – in the previous example the difference could be seen as a rounding error. But what if we extended the example to a 20 year period? Imagine (because it would never really happen) that the actual rates of return alternate between 6% p.a. and 14% p.a. over the next 20 years. The simple average is still 10% p.a, right? Well, at the end of the 20 years we would have (note, ^ means “to the power of”):

- value calculated using simple average return = $100 x 1.10^20 = $673
- actual value = $100 x 1.06^10 x 1.14^10 = $664

The difference is a bit more significant now, but still not dramatic – less than 2% difference between the two values. In terms of overall uncertainty in projecting a client’s future out to 20 years, we might not be concerned by this difference.

**Example 3**

But now let’s look at more extreme returns, say for asset classes with high short-term volatility such as equities. Take the example of a correction followed by a rebound. Let’s say that my returns are -20% p.a. and +40% p.a. The simple average return is still 10% p.a. over the two years. So , the value of my investment after two years would be:

- calculated using simple average return = $100 x 1.10 x 1.10 = $121 (as before)
- actual value = $100 x 0.80 x 1.40 = $112

Now we have a big difference in relative terms – 8% difference in value after just two years.

**Example 4**

Like before, if we now assume the -20% p.a. and +40% p.a. returns alternate over a 20 year period then the difference would be:

- calculated using simple average return = $100 x 1.10^20 = $673 (as before)
- actual value = $100 x 0.8^10 x 1.40^10 = $311

That is a big difference – the value calculated using the simple average return is over 200% of (i.e. more than 2 x) the actual value. The chart below shows the values over all 20 years… That sort of difference could be a problem in financial projections.

But of course, over a 20 year time frame equity returns don’t alternate between just two values. And we don’t get negative returns one out of every two years nor do we get huge positive returns one out of every two years. So the “error” caused by using simple average returns in financial projections lies somewhere between the above examples. Over a 20 year period it is likely to be more than 2% but less than 200%!

**Example 5**

Now let’s take a look at actual historic data. Over the 20 years to June 2020, the simple average annual return on Australian Equities (based on ASX 200 Accumulation Index) has been 8.26% p.a (excluding Franking Credits). Over the same period, the Accumulation Index has increased by approximately 415%. Assume we invested $100 twenty years ago, back on 1 July 2000, then we would have at 30 June 2020:

- value calculated using simple average return = $100 x 1.0826^20 = $489
- actual value = $100 x 415 / 100 = $415

The value calculated using the (constant) simple average return would be almost 20% higher than the actual value. The chart below shows the comparison over the full 20 years.

Well, some might say “We have to make assumptions about future returns anyway, and they may be different to historic averages and are also subject to significant uncertainty, so what does it matter? It all comes out in the wash, doesn’t it?” Well, yes and no. Yes, there is significant uncertainty surrounding assumptions regarding the future, and we can’t avoid that.

But what we can avoid is knowingly including bias in our calculations – and using simple average rates of return calculated from historic data will often produce higher results than using compound average rates of return calculated from the same data (unless the actual returns are themselves constant, but which investment provides that?).

What is the point of all of this? Well, for long-term projections if you are going to assume a constant rate of return for each period into the future it is better in principle not to use an assumption based on the simple average of historic returns. Doing so will, generally, over-estimate projected asset values for your clients. Better instead to use an assumption based on the compound average of historic returns, which we’ll discuss in the next article.