Omega ... and its use in portfolio allocation

motivated by email from Roberto F.
In a recent paper
(in PDF format), Keating and Chadwick introduce an interesting measure
of stock return distributions called Omega.
It compares the average value of returns above some threshold return r
(such as a Riskfree return) with the returns less than r.
It goes like this:
 Look at the return distributions for a stock, as in Figure 1.
It shows the distributions(s) of a thousand GE daily returns ... both the probability
density and the cumulative distribution.
 This density chart shows the percentage of returns which lie in each return interval of size 0.2%
 100% of returns lie between 10% and +10%
... so the TOTAL area under the density graph is "1", or 100%.
 Each point on the cumulative distribution (at return r) gives the area beneath the density distribution
to the left of r.
Example: Figure 1 says that 0.73, or 73% of returns are less than r = 1%.
That 0.73 is the area shown in light green, under the density distribution.
>So what about Omega?
Patience!
 Figure 1

 We pick a return, say r = 1%, and calculate two areas:
I_{1} and I_{2}
associated with the cumulative distribution, as shown in Figure 2, below.
 Then we calculate the ratio: Omega = I_{2} / I_{1}
 A plot of Omega versus r would look like Figure 3
plotted just from from 2% to 3%
Figure 3
 Figure 2

>A plot from 2% to 3%? Why not 10% to +10% ... the whole thing?
The whole thing? Okay, that's Figure 3A.
>So F(x) is the area under f(x), and now you calculate the area under F(x)?
Yeah. Strange, eh? However, I_{2} is actually an area above F(x).
But consider this:
 I_{2} is equal to:
(The area under the line y = 1) minus (The area under y = F(x))
from x = r to x = U
... U = 10%, the Upper limit of our GE returns.
 However the area under the line y = 1 is just (U  r) or, in our example, it's (0.10  0.01)
... for 1%, we use r = 0.01 as in Figure 4.
 Hence I_{2} = (U  r)  (Area under F(x) ... from x = r to x = U.
 Let's call A[a,b] the area under y = F(x), from x = a to x = b.
 Then I_{2} = (U  r)  A[r,U]
and I_{1} = A[L,r]
... where L is the Lower limit of the returns.
 Note that A[L,U] is the entire area under y = F(x)
... and doesn't depend upon r
 Then A[L,r] + A[r,U] =
A[L,U]
... the two areas add up to the entire area, eh?
so we can write: A[r,U] = A[L,U] 
A[L,r]
 Figure 3A
Figure 4

>zzzZZZ
Wait!
Figure 5 gives a typical picture of I_{1} and I_{2}
As we move to from r = L to r = U,
I_{1} increases from 0 to A[L,U]
and I_{2} decreases from U  L  A[L,U] to 0.
Okay, now we can write Omega = I_{2} / I_{1} like so:
Omega =
[(U  r)  A[r,U]] /
A[L,r] =
[(U  r)  A[L,U] +
A[L,r]] /
A[L,r] =
1 + [U  A[L,U]  r
] /
A[L,r]
 Figure 5

Now U  A[L,U] is a constant (for our stock, over the selected time interval).
Let's call it C. We then can write:
Omega = 1 +
[C  r ] / A[L,r]
where r is some return ... between the Minimum and Maximum return
L and U are the Lower and Upper limits on returns
... namely the Minimum and Maximum return
A[L,U] is the area under the cumulative distribution of returns
... from the Minimum return to the Maximum return
C = U  A[L,U]
...which, for a given stock and time period, is a constant.

The shape of the Omegacurve, as shown in Figure 3, is explained by the fact that the
expression above has a 1/A[L,r] in it !
>Yeah, it looks simple enough, but what good is it, and what on earth does it mean
... and what about financial stuff?
Check out Omega Math
where we show that:
for Part II
