# How we calculate average rate of return

#### If assumptions use the arithmetic mean:

'Arithmetic mean' values - in the absence of any volatility - will overstate the return received by the client(s), year-on-year. The degree of overstatement is a function of the assumed volatility, i.e. the SD value of the asset class/portfolio in question.

The formula for approximating the Geometric Mean is equal to: Arithmetic - Variance / 2(1+Arithmetic)

The application here is that all of the means listed are assumed to be arithmetic growth, division of the total growth by the number of sample years, and transforming it into an approximate Geometric Mean so that we can "accurately" estimate how compound growth impacts the asset class over the years

Example:

Asset Class A: Growth: 12% Standard Deviation: 36.11% Dividend Yield: 0% Interest Yield: 0%

100% Allocation in Asset Class A, Expected Average Return: 6.18%

The formula, in this case, would be 0.12 - (0.3611^2 / (2*1.12)) = 0.0617887, or the 6.18% you would see as the expected average return!  The software is 'correcting' (more-or-less) for the inherent overstatement within the assumed 'arithmetic mean value'. See page 8 of the linked document for a deeper dive. Formulate for calculating the average rate of return

The difference between an 'arithmetic' means return and the corresponding 'geometric' mean return 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) - the obvious difference being that the 'geometric' measure takes account of 'compounding', whereas the 'arithmetic' measure does not, and growing 100,000 by 5% pa for 10 years, will give you significantly more than 150,000 (so that's the 'overstatement'). It should be noted, therefore, that the arithmetic mean will, in practice, *always* be higher than the equivalent geometric mean.

With reference to the above - unless you're using the Monte Carlo - 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 (approximately 163k, in fact).

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 one 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 the market assumptions you are using will *already* include some assumptions about asset class volatility (the 'standard deviation' values).

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 (on Line 4 of the spreadsheet), that there is an option labeled 'Approximate Geometric Mean'. 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.

So, a difference 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 difference 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) the inherent overstatement that this entails. Having told the software to do so, you will then see a difference in the assumed rates of return being applied in your client's plan.

Finally, note that, if you *are* using the above functionality, just outlined, the software will automatically turn off this 'correction mechanism', *when* you run the Monte Carlo simulation, for the reasons mentioned previously, specifically that the result 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 correcting for, in the first place. Consequently, when we have the volatility, we don't need or want to use the (equivalent) geometric values, because then we would, essentially, be 'correcting twice' for the same thing.