Mathematical Formulation
Although the hump at -σ is difficult to model, behavior induced modifications manifest themselves in the shape of the return histogram around a small neighborhood of zero. It is approximated by a straightforward formula.
Let: = the closed interval from zero to +1 standard deviation of returns (including zero)
Let: [-σ, 0) = the half open interval from -1 standard deviation of returns to zero (including -σ and excluding zero)
Let:
- return in month i, 1 ≤ i ≤ n, and n = number of monthly returns
Then:
The Bias Ratio roughly approximates the ratio between the area under the return histogram near zero in the first quadrant and the similar area in the second quadrant. It holds the following properties:
- a.
- b. If then BR = 0
- c. If such that then BR = 0
- d. If the distribution is Normal with mean = 0, then BR approaches 1 as n goes to infinity.
The Bias Ratio defined by a 1σ interval around zero works well to discriminate amongst hedge funds. Other intervals provide metrics with varying resolutions, but these tend towards 0 as the interval shrinks.
Read more about this topic: Bias Ratio (finance)
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