What is the difference between arithmetic and geometric returns?

Firstly, the difference between 'arithmetic' and 'geometric mean' values can be easily illustrated, as follows: if the value of my house has gone from 100,000 to 150,000, over 10 years, it has therefore increased by 50%, over 10 years. We can evaluate this increase by saying either (1) that the house grew at 5% per annum (arithmetic), or (2) that it grew by 4.14% per annum (geometric). It should be noted, therefore, that the arithmetic mean will *always* be higher than the equivalent geometric mean (or, at least, will *not* be less than).

When one is creating a client plan, and using 'asset allocations', in the process, the underlying assumptions, about long-term 'mean' returns, can be either arithmetic values, or geometric values and - with reference to the above, it will be the case, if one is using 'arithmetic' mean values, that one will be consistently 'overestimating' the client's actual return, for the same reason that, if I begin with a property valued at 100k, and grow this at 5% per annum, year-in, year-out, my house will be worth significantly more than 150k (more in the region of 163k). In reality, of course, no-one gets the same return, year-in, year-out, so what makes the difference here is 'volatility', or the lack of it. For normal purposes, therefore, i.e. when you're assuming the client gets the same return year-in, year-out, one would ideally want to use 'geometric' mean values. For the purposes of the Monte Carlo simulation, on the other hand, where returns are 'randomized', and it is therefore *not* assumed that the client is getting the same return, year-in, year-out, one would ideally want to use 'arithmetic' mean values, i.e. when there is 'volatility', the 'gap' between these two values starts to close, because of 'volatility drag', and one's market assumptions will *already* include some assumption about volatility (i.e. standard deviation).

Because of the above 'dichotomy', we would suggest to users (if they have any interest in using the Monte Carlo simulation, at least), that they use 'arithmetic' mean values for their assumptions and, having done so, note (in Preferences > Market Assumptions), that there is a small 'tick box' labelled 'Approximate Geometric Mean for default calculation'. This is an instruction (to the software) to take the assumed arithmetic mean values, and 'convert' them to geometric mean values, and there's a standard formula for doing so, which takes account of the assumed standard deviation values, for each of the asset classes being employed.

This formula is: G = (approximately) R - V/2(1+R)

NB: G = Geometric Mean; R = Arithmetic Mean, and V = Variance. Variance is equivalent to SD^2. (SD = Standard Deviation)

So, a difference (a visible impact of some description) in the software does *not* arise just from telling the software that the values are either arithmetic, or geometric, because this does nothing, in and of itself. A visible impact only arises when you tell the software that you would like it to take your arithmetic mean values and apply the above formula, to correct (so far as possible) for the likely overstatement that this built-in to the arithmetic values. Having told the software to do so, one will then see a difference in the assumed rates of return being applied in your client's plan.

The difference that this corrective mechanism makes would be visible within the Let's See screen, if one looks into Detailed Info > Investments, or Detailed Info > Pensions, to see what rates of return are, in fact, being applied to one's accounts. One can also see the impact, at the level of an individual investment, or pension account, by opening Advanced Settings > Asset Allocation, where you will see the assumed 'mean' return for the account, based on the selected asset allocation.

Finally, note that, if you *are* using the 'corrective' function, outlined above, the software will automatically turn-off this 'correction mechanism', *when* you run the Monte Carlo simulation, for the reasons already alluded to, i.e. that the impact of introducing volatility, via the Monte Carlo, is to introduce 'volatility drag', which automatically reduces one's overall return (depending on the extent of the volatility), and it's the absence of this volatility that one is trying to correct for, in the first place.

Consequently, when we have the volatility, one doesn't need, or want to use the (equivalent) geometric values, because to do so would, essentially, be 'correcting twice' for the same thing.

To conclude, therefore, it's the precisely the *absence* of volatility (in the normal course of the plan) that necessitates a degree of correction, to reduce the overstatement that is typically inherent in an arithmetic mean value. Once that volatility is introduced (via the Monte Carlo simulation, for example), we no longer need the software to apply that artificial, corrective function (for the overstatement), and so it turns-off that function, automatically (and uses the arithmetic mean values).