motivated by email from Hank S.
We talked a lot about measures of Risk and ...
>It's standard deviation, right?
That's one (lousy) way to measure risk, but there's also
Value at Risk and Drawdown
and Coherent Risk and other Assorted Variations and
>So?
So there's this neat paper where the author defines a new measure of risk like so:
 You look at a bunch of monthly returns. (Example: The last 10 years worth.)
 You see what fraction p, are nonnegative (Example: p = 52.9%)
 You calculate the Mean of those nonnegative returns: G% (Example: G = 3.13%)
 You calculate the "Expected" nonnegative return: p G (Example: p G = (52.9%)(3.13%)= 1.67%
 Then you repeat for the other, negative returns:
q% negative returns with a Mean of L% hence an Expected negative return of (q%)(L%)
(Example:
q = 47.1% negative returns with a Mean of L = 3.33% giving an Expected negative return of (q%)(L%) = (47.1%)(3.33%%) = 1.57%
>Don't tell me! You add them together, right?
Wrong. You want the difference between the expected Gain and the expected Loss and that's their difference: GLS = pG  qL
In our example, that'd be: GSL = (1.67%)  (1.57%) = 3.24%.
See how it works?
>What's GLS?
The GainLoss Spread.
>Where's the spreadsheet?
Some observations:
 Consider a given return distribution f(x), like so
where the Means of positive and negative returns are identified (G and L)
and the two areas provide the percentages p and q.
>Two areas? You mean the green and red areas?
Yes. That's the density distribution and the entire area beneath the curve is 1.
In other words, p + q = 1.
The area to the left of 0% is the percentage of returns which are less than 0%.
[A] That'd be: q = ∫ f(x) dx where the integration is from ∞ to 0.
>Wake me when you're finished ... zzzZZZ.
 
Anyway, we now have: GLS = pG  qL = I_{2} + I_{1}.
The interesting thing is that these two areas, I_{1} and I_{2}, occur in the definition of Omega.
Indeed, Omega = I_{2} / I_{1}. See? Omega uses the ratio rather than the sum. Isn't that interesting?
>Uh ... yes ... fascinating.
Okay, one thing that's really nasty about using Standard Deviation as a measure of "RISK" ... it doesn't change when you add 10% to all the returns.
At least, with both GLS and Omega, adding a constant return changes them both.
>That's what you'd want for something called "RISK", eh?
I would. In fact, adding a contstant return will just shift the distribution to the right and that tends to increase both GLS and Omega
... but not always. Check out this spreadsheet
Okay, let's stare intently at the moving distributions, both density f(x) and cumulative F(x):
>And all them areas, eh?
Exactly. We're increasing and decreasing returns, thereby shifting the distributions.
After some cogitation, let's define something we'll call RISK.
We'll want it to decrease as the distribution shifts to the right (since any selfrespecting RISK would decrease).
>Because all returns are being increased, right?
Yes.
 Figure 1 
>Okay, I pick q / p. It decreases as we shift right and ...
But it could be infinite if p is 0. See the allred situation, in Figure 1? p = 0 there.
How about ...?
>Wait! I pick q, all by itself. It goes from q = 1 to q = 0 as the returns go from allnegative to allpositive. Good, eh?
Uh ... look at these two distributions.
They have the same qvalue, namely 0.20 or 20% and that means ...
>That means that 20% of the returns are negative ... for both distributions.
Yes, but would you assign the same RISK to each?
>Uh ... one of them, the bottom one ... it has a lot more of them really negative returns.
So our definition should assign a greater risk to that one, eh? It's got fat tails.
>So we gotta include the "spread" of returns ... somehow.
You won't believe this, but Standard Deviation measures the "spread", so maybe we should ...
>I got it! RISK = q * (Standard Deviation). How 'bout that?
Well, if you get a constant positive return the Standard Deviation would be 0 so your RISK would be 0 ... and that's good.
But if you get a constant negative return your RISK would also be 0 ... and that's NOT good.
>Okay, how about ...?
Before you get too excited, let me explain what we're after.
 
We suppose we have a collection of returns and wish to define RISK.
It'll be computed from that collection of returns. What characteristics should it have?
 It should be a positive number between 0 and 1 ... where 0 means no risk and 1 means lots of risk.
If'n it were a number between 0 and 1, we could compare risks ... for this asset and that one.
 It should decrease if all returns are increased.
 It should be zero if the returns are all positive and 1 if all the returns are negative ... just like q.
 It should reflect the RISK characteristics of real, live returns, not normallydistributed proxies.
The pretty pictures above are all normal distributions because ... uh ... they're pretty.
 It should increase if we increase the number of really negative returns ... meaning fat tails for negative returns.
 It should decrease if we increase the number of really positive returns ... meaning fat tails for positive returns.
 ...
 Figure 2 
>What's an outlier?
See the wee dots on the xaxis, in Figure 2? They're 3 standard deviations from the Mean return. Beyond that, returns should be very rare.
For those GE returns, there are 10x the number of outliers that you'd expect from a Normal distribution.
>Okay, so what RISK definition do you suggest?
I'm thinking. Let's see ...
RISK = function(p, q, G, L, outliers, standard deviation, unknown variables, phases of the moon, ...)
If we just include negative outliers, left of the yaxis, then when they shift right ...
... maybe we should weight the negative returns by how far to the left of 0 they are
... uh ... we may have to give up having our RISK lying between 0 and 1 ...
>Yeah, but where's the cumulative distribution for those real, live GE returns, and p and q and ... ?
Good question. They look like this:
>Huh? You mean that ragged f(x) distribution for GE, in Figure 2, looks that smooth in Figure 3?
Nice, eh?
>Okay, so what RISK definition do you suggest?
I'm still thinking ...
 Figure 3 
Let's play with them negative returns. After all, since RISK should imply losses, the negative returns should play an important role.
>Wait! What if there are no negative returns. Then what's A?
It'd be 0, but don't worry about dividing by 0. The numerator is also 0 and is smaller than the denominator 'cause it's got squared returns.
In other words, as the negative returns approach 0, their average will approach 0 ... but so will their standard deviation.
>But isn't A the same as L, the "ordinary" average of negative returns?
Aha! So it is!
Then let's rename our newest variables like so:
>Where are you going with this?
Uh ... I'll know when I get there!
However, note the following (for general, continuous distributions):
[B1]:
The Mean of a variable x with density distribution f(x) is: M = ∫ x f(x) dx / ∫ f(x) dx.
[B2]:
The Standard Deviation is SD, where: SD^{2} = ∫ (x  M)^{2} f(x) dx / ∫ f(x) dx = ∫ x^{2} f(x) dx / ∫ f(x) dx  M^{2}.

G = the Average of positive returns
SD = the Standard Deviation of positive returns
p = fraction of returns which are positive
L = the Average of negative returns
SD = the Standard Deviation of negative returns
q = fraction of returns which are negative
Then define:
M =
{ SD^{2} + L^{2}} / L
... the weighted average of negative returns
and
M =
{ SD^{2} + G^{2}} / G
... the weighted average of positive returns


Note that f(x) refers to the whole distribution, from x = ∞ to x = +∞.
If we're considering just a particular xinterval (like x > 0), we divide everything in sight by ∫ f(x) dx integrated over that interval.
Normally, ∫ f(x) dx = 1 over = ∞ to +∞.
For the GE example in Figure 3, the mean of negative returns was L = 5.6%. The weighted mean was M = 8.2%.
The mean of positive returns was G = 5.5%, whereas the weighted mean was M = 9.7%.
>It's them outliers, eh?
Seems that way, don't it?
RISK Geometry

Here's something interesting. Suppose we want to determine the centroid of the area shown
Each wee vertical strip has area: (1  F(x)) dx.
The moment about the vertical axis is then: x (1  F(x)) dx.
The total moment of the entire area is then: Moment = ∫ x (1  F(x)) dx (the integration being from x = 0 to x = ∞).
Then, integrating by parts, Moment = [ (x^{2}/2) (1F(x))] + (1/2)∫x^{2} f(x) dx = (1/2)∫x^{2} f(x) dx
where the first part, evaluated between 0 and ∞, is 0.
 Figure 4 
Using [B2], we see that:
[B3]: Moment = (1/2) [ SD^{2} + M^{2} ] ∫ f(x) dx
But the centroid is: Moment / Area and the Area is: ∫ (1  F(x)) dx = [ x (1F(x))] + ∫x f(x) dx
where, again, the first part, evaluated between 0 and ∞, is 0.
Using [B1], we see that:
[B4]: Area = ∫x f(x) dx = M ∫ f(x) dx
Dividing [B3]/[B4] gives:
Centroid = (1/2) [ SD^{2} + M^{2} ] / M

 Figure 5 
>Hey! Ain't that them weighted average return things?
Yes ... and that's fascinating, eh what?
The magic formula [SD^{2}+M^{2}]/M looks funny, but it's just the centroid.
>It's twice the xcoordinate of the centroid.
Uh, yes ... I stand corrected.
Note, however, that [SD^{2}+M^{2}]/M is just: (Average Squared Return) / (Average Return).
Note, too, that: [SD^{2}+M^{2}]/M > M if M > 0
and [SD^{2}+M^{2}]/M < M if M < 0.
>So what's the ycoordinate of the centroid?
I have no idea.
However ... let's stare at all the nice numbers we have, so far:
Now, can you invent a nice definition for RISK?
>Uh ... no. Can you?
Not yet.
 Figure 6 
