Omega ... and its use in portfolio allocation
motivated by e-mail 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 Risk-free 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.
    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?

Figure 1

  • We pick a return, say r = 1%, and calculate two areas: I1 and I2 associated with the cumulative distribution, as shown in Figure 2, below.
  • Then we calculate the ratio: Omega = I2 / I1
  • 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, I2 is actually an area above F(x).
But consider this:

  • I2 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 I2 = (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 I2 = (U - r) - A[r,U]
    and I1 = 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

Figure 5 gives a typical picture of I1 and I2  
As we move to from r = L to r = U,
I1 increases from 0 to A[L,U] and
I2 decreases from U - L - A[L,U] to 0.

Okay, now we can write Omega = I2 / I1 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 Omega-curve, 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