# Market Assumptions - Approximate Arithmetic vs. Geometric mean for default calculation

We created this feature after discussions with some our subscribers. Please see the resources below for an explanation of the difference between an arithmetic average, on the one hand (i.e. the simple mean value we all know, whereby the inputs are summed and then divided by the number of inputs), and the geometric mean, on the other hand (which takes account of compounding in a way that the simple, arithmetic 'mean' value does not).

The following sites do a good job explaining the difference:

allfinancialmatters.com/2006/06/06/average-vs-geometric-aver age/<allfinancialmatters.com/2006/06/06/average-vs-geometric-aver age/>

After discussions with some other advisers, we came to a couple of conclusions:

1) For our standard projection, market assumptions using geometric averages would give a better result, as arithmetic averages skew the projection to the positive.

2) For our monte carlo simulation, arithmetic based averages are most appropriate

Given that we believe most of the data provided to us by our subscribers are likely to be arithmetic assumptions, we decided that providing an option to approximate the geometric mean, given arithmetic market assumptions, would allow us to provide better projections in our standard 'cash flow' calculation, while still allowing our monte carlo simulation to work as before.

The formula we use to estimate the 'true' geometric mean figure from an arithmetic mean figure is pretty standard, but an explanation is provided in the following paper written by William Bernstein:

'DIVERSIFICATION, REBALANCING, AND THE GEOMETRIC MEAN FRONTIER'

Bernstein, William J. and Wilkinson, David J., Diversification, Rebalancing, and the Geometric Mean Frontier (November 24, 1997). Available at SSRN: https://ssrn.com/abstract=53503 or http://dx.doi.org/10.2139/ssrn.53503

The formula for calculating the approximate geometric mean (G) is:  (from page 8 of the paper):

V = stdDev squared
R = arithmetic mean

V
G ~ R -  ______
2(1+R)

or

stdDev*stdDev
R - _____________
2 + 2 *R

When you provide updates for your market assumptions, we generally recommend you enable this feature as (in the absence of reasons to think otherwise) your assumptions are most likely to be arithmetic mean values.