motivated by e-mail from Roberto F.
Omega ... and its use in portfolio allocation
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?
- 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%
>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] =
... the two areas add up to the entire area, eh?
so we can write: A[r,U] = A[L,U] -
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]
I2 decreases from U - L - A[L,U] to 0.
Okay, now we can write Omega = I2 / I1 like so:
[(U - r) - A[r,U]] /
[(U - r) - A[L,U] +
1 + [U - A[L,U] - r
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