Averages are useful, but often we can do better by using more information than just the average. This applies to financial advice as much as any other discipline.

This article follows on from the previous two articles: *Financial Projections, getting them right: Simple v Compound – What you need to know! PARTS 1 & 2:*

We have already established in those articles that in the simple case of a lump sum invested for a fix period, projecting an assumed rate of return based on a simple average of historical returns will overstate, possibly substantially, the projected value at the end of the period. However, by definition, using a compound average of historical returns will result in a projected value at the end of the period that aligns perfectly with the historical returns.

As an example, the Chart 1 below shows the value at the end of 20 years of $100 invested in the ASX 200, using the following return assumptions:

- Historical yearly returns from the last 20 years
- The simple average of those historical returns
- The compound average of those historical returns

**Chart 1: Simple Average vs Compound Average returns**

Chart 1 shows how the compound average produces a result exactly the same as the historical returns from which it was calculated, whereas using the simple average produces a much higher result (almost 20% higher).

However, in more realistic financial modelling scenarios, whether using a simple average or compound average return overstates the projected value at the end of the period (relative to the historical returns) will depend on the pattern of historical returns themselves and the particular set of cash flows (positive and negative) being modelled. We can’t be sure one way or the other (see previous article, *Financial Projections, getting them right: Simple v Compound – What you need to know! PART 2).*

In addition, apart from any bias that might be introduced by using an average return assumption in our financial projections, there is the issue of variability underlying the results. Even if we were modelling a lump sum investment and we knew with certainty what the future long-term compound average return would be, our projection over any selected period – say the next two years – would still be subject to significant variability and therefore significant uncertainty.

For example, the compound average return of the ASX 200 over the last 20 years is 7.38% p.a. (the simple average return is 8.26% p.a.). Suppose we project the value of a $100 investment over 2 years and we assume the average return of 7.38% p.a. in our projections. And suppose we also project the value based on returns for each of the consecutive 2-year periods that occurred in the last 20 years. Well, Chart 2 below shows the results – the thick black line is the projection based on the compound average of 7.38% p.a. and the other lines represent projections based on all of the 2-year periods in the last 20 years (there are 19 of these).

**CHART 2: Range of Outcomes based on Historical Returns**

As you can see in Chart 2, there is quite wide dispersion in projected value after just 2 years – the projected values range from $70 to $160 and approximately half the results lie above the compound average value ($115) and approximately half lie below.

This illustrates that when an adviser shows a client some projections based on an assumed average rate of return, the results they are showing are approximately 50/50. That is, even if the assumed average rate of return is correct there is, approximately, a 50% chance the actual outcome will be lower than shown in the projections. Put another way, whether or not the client will achieve the goals shown is roughly equivalent to the toss of a coin. Most people are not comfortable with that, particularly when the goals relate to their retirement. A one in two chance of running out of money early is just too high!

So, what can we do? How can we ensure that our projections are consistent with our assumptions and how they were determined? And how can we deal with the uncertainty underlying projections – something which has not been discussed much to date?

Well, we could simulate a large number of sets of future returns based on historical data and with an average over all sets equal to the average of the historical data or equal to our assumed average return, if that is different to the historical average. It is called Stochastic Modelling and is based on the following premise.

- We don’t know what future returns will be, but we do know for sure that they will not be constant over time – rather, they will vary year by year.
- We know that those variations in returns can have a significant impact on our projections.
- We know that if we run our projections based on assumed averages of those variable returns, even using the compound average, there is a 50% chance we are overstating the results.
- So instead of projecting average returns let’s model the varying returns.

For example, suppose we simulate 2,000 sets of 2 year returns based on historical data, and apply those simulated returns to the cash flows we wish to model. We would end up with 2,000 different results reflecting 2,000 plausible different futures. Now, if each of those different futures is equally likely, then we can draw some conclusions about the likelihood of any particular result. For example, if 1,000 of those results showed a value of less than $115 at the end of the 2 years then we could estimate the likelihood of a result lower than $115 as being 1,000/2,000 or 50%. If the client’s goal is to have at least $115 at the end of 2 years then there is a 50% risk they will not achieve their goal. Or, put another way, they can be only 50% confident they will achieve their goal. Confidence and risk are two sides of the same coin.

But why do we need to simulate sets of returns rather than just use historic returns (as shown in Chart 2)? Well, there are several reasons for this:

- For longer projections, as required in financial advice, we simply don’t have enough historical periods to produce statistically meaningful estimates of the probability of achieving goals. For very long projection periods, we don’t have enough history to produce even one projection into the future, let alone many projections from which we can estimate probabilities.
- We may believe that future average returns, or variability in returns, will differ from the historic returns, perhaps significantly. So we need to be able to use assumptions that may differ from those implied by the historic returns.

Ok, but how do we generate those simulated returns? Well, that requires building models of how returns vary from period to period. This is normally done using statistical models fitted to historical data. Those models may include some of the following:

- A measure of the average return
- A measure of the variability in returns from period to period (often the standard deviation or related statistic)
- An auto-correlation factor – this measures any tendency for the return in one period to be related in some way to the return in previous periods. There is in the historical data an indication that we get “runs” of good returns and poor returns.
- Correlation of returns between asset classes, products or securities – whatever investment types we are modelling. For example, there is a strong correlation between Australian equity returns and Global equity returns.

Once we have fitted our models (there are plenty of model-fitting applications around) we can then produce simulated returns using those models together with random number generators. These simulated returns are then fed into a cash flow projection engine and thousands of simulated futures are produced for the client, based on their financial inputs and goals (derived from the fact find). From those simulated futures we can, for any given investment strategy, find the probability the client will achieve their goals at their chosen confidence level.

The chart below shows the results for 19 simulations of a 2 year period based on ASX 200 returns over the last 20 years and an initial investment of $100. This can be compared to Chart 2 above, which is the same but based on projections using historical 2 year periods (of which there are 19).

**Chart 3: Simple Average vs Compound Average returns**

As you can see in Chart 3, the range and spread of outcomes are broadly similar between the projections using historical 2 year periods and the projections using simulated 2 year periods. The advantage of the simulation approach is that, based on the 20 years of historical data we can produce as many projections as we like, whereas using the historical data in this example we can only produce 19 projections (there are 19 consecutive 2 year periods in the 20 years of historical data). So, simulation provides much better estimates of the probabilities of achieving goals and also enables us to use assumptions about the future that are different to the past.

The final chart below shows projections over 20 years atselected confidence levels and also projections using the simple average and compound average returns. Again, these are based on the 20 year historical returns for the ASX 200 and a lump sum investment of $100.

**Chart 4: Simple Average vs Compound Average returns**

In the above chart, for example, the 90% confidence line (purple) shows the value of the investment at each period at the 90% confidence level. That is, based on the modelling there is a 90% chance (9 out of 10 times) that the value of the investment will be at the level shown or higher. Or, put another way, there is a 10% risk that the value of the investment will fall below this line. Some points to note:

- The 50% confidence projection and the projection using the compound average return are almost identical (hard to distinguish in this chart as they lie one on top of the other).
- The projection using the simple average return is above the 50% confidence level i.e. there is LESS than a 50% chance that the investment would be as high as that shown by using the simple average return.
- The difference between the 50% confidence level (approximately what is shown in most financial advice projection tools) and the 90% confidence level (what many people want when planning for retirement) is very large. After 20 years the values are $415 for the 50% confidence level and $234 for the 90% confidence level – just over half the value shown for the 50% confidence level.

Stochastic Modelling is incredibly powerful because it enables a more informed decision by the client as to how to invest their funds. It allows people to make a decision based on modelling that reflects reality (returns varying from period to period) and that properly accounts for risk in the context of achieving goals. It starts to move the question away from “how fearful are you of short-term fluctuations in investment performance” – i.e. the current approach – to the question “what level of confidence do you want to plan with when setting your goals and choosing an investment strategy to support those goals?”

For the advisers, this makes for a better conversation when there is a market downturn. When clients are scared and ask “why did you put me in that portfolio, it has lost 20% in value over the last few months” the adviser can answer “yes, as we discussed the value of your portfolio will vary over time – but we have modelled that as best we can and the modelling indicates that you can be X% confident of meeting your goals despite the ups and downs of short-term investment performance”.

Of course, this then begs the question – “what investment strategy will optimise my client’s financial goals for their chosen level of confidence?” To answer that properly, we then need to overlay an optimisation process on top of the stochastic modelling. Not easy to do, but well worth it for both client and adviser. We will explore optimisation in the next article.