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

We created this feature after some discussions with some advisers.  Check out the resources below for an explanation of the difference between an arithmetic average (the standard average we all know, add all the numbers, divide by the number of numbers), and the geometric mean.

These sites do a good job explaining:

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 the fact that the we believe that most of the data provided to us by various advisers is likely arithmetic assumptions, we decided that providing an option to approximate the geoemetric mean given arithmetic market assumptions would allows us to come up with better projections in our standard calculation, while still allowing our monte carlo simulation to work as before.

The formula we use to approximate a geometric average from an arithmetic average comes from this paper

DIVERSIFICATION, REBALANCING, AND THE GEOMETRIC MEAN FRONTIER

The formula for our geometric mean approximation(G) is:  (from page 8 of the paper)
where
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 most likely your assumptions are arithmetic averages.