## Tuesday, June 19, 2012

### The Coefficient of Variation

Coefficient of Variation: A relative measure of variation measuring the standard deviation relative to the mean.

Recall the standard deviation is the square root of variance.

CV = (standard deviation / mean) * 100

This metric is useful for comparing variables that have different standard deviations and different means.

For instance, let’s assume we have two corn hybrids that average 150 bushels an acre, and hybrid A’s yield has a standard deviation of 10 bushels per acre, while hybrid B’s yield has a standard deviation of 50 bushels an acre.  If we are about to plant one of these hybrids, we know that hybrid A is going to provide us more certainty about our expected yields in the fall.  The empirical rule tells us that about 95% of the time if we plant hybrid A, our yields will be between 130 and 170 bu/acre. If we plant hybrid B, 95% of the time yields will be between 50 and 250 bu/acre.

So, with two hybrids with the same average yield, we know that the one with the lower standard deviation will perform more consistently. But what if they both have different yields and different standard deviations?

Hybrid A:  mean 185 bu;/acre  std dev: 5 bu/acre
Hybrid B: mean: 200 bu/acre std dev: 30 bu/acre

Choosing the most reliably yielding hybrid in this case is not quite so easy. But the CV helps us make the comparison:

Hybrid A: CV = (5/185) *100 = 2.7%
Hybrid B: CV =  (30/200)*100 = 15%

We can see by the CV that hybrid A, even though it yields on average less than hybrid B,  will deliver more consistent results.

In the context of finance, we can think of the CV as a measure of relative dispersion that can be used to compare the risks of assets  that have different mean (expected )returns.
Reference: Principles of Managerial Finance.  11th Edition. Lawrence J. Gitman.