motivated by email from Mike E.
Suppose there were two $1,000 portfolios, like Figure 1 where the growth is shown over the last few years. Which would you choose?
>Figure 1? I'd choose Portfolio 1 ... of course.
Me too. So now let's look at the cumulative distribution of returns for each portfolio, in Figure 2:
Figure 2 
 Figure 1 
Note that Portfolio 1 is generally smaller than Portfolio 2.
>Uh ... shouldn't it be larger?
Remember; the cumulative distribution indicates the fraction of returns which are less than something. For example, for Portfolio 1, there are fewer returns less than r = 1%.
>Aha! That's why I like Portfolio 1!
Pay attention.
If we stare at the two cumulative distributions, which we'll call F(r) and G(r), then you like F because, generally, F(r) ≤ G(r).
>Actually, I like Portfolio 1 because it grows more than Portfolio 2. I'm looking at Figure 1, not Figure 2.
Well, we're doing Stochastic Dominanace, so we're looking at Figure 2.
If two cumulative distributions F and G satisfy F(r) ≤ G(r) for all rvalues, then we say that:
F has first order stochastic dominance over G.

>But, in Figure 2, F isn't always less than G. Look at r = 2.1% and you'll see that ...
Yes, and that's where second order dominance comes in.
Suppose that the average Fvalue is less than the average Gvalue, over each rrange: Min ≤ r ≤ R.
To get an average, we'd integrate F(r) and G(r) and divide each integral by the length of the rrange: R  Min. Like so:
Of course, if we're just interested in which is smaller, we can ignore the division by (R  Min).
>Uh ... what's that Min and R stuff?
Min is the minimum rvalue (for the portfolios) and R is some rvalue between the Min and Max rvalues.
Okay, so second order dominance is defined like so:
If F and G are two cumulative distributions, and
where F_{1}(r) ≤ G_{1}(r)
then we say that:
F has second order stochastic dominance over G.

>Okay, so what about fourth order dominance?
That's left as an exercise ... for you!
>So where's the spreadsheet? You don't expect me to do all those cumulative distribution calculations and ...
The spreadsheet looks like this:
Click on the picture to download the spreadsheet
You pick four stocks, assign some percentage allocation to each (so they add to 100% !) and that generates Portfolio 1.
You assign some other allocations to each and that generates Portfolio 2.
You can then play with the allocations with them slide things to find a nice allocation and ...
>Did I mention that I prefer Portfolio 1 ... based upon the growth?
But isn't this stochastic dominance interesting?
>No!
But whereas the growth chart reflects the order in which the returns occur, the cumulative distribution doesn't ... so it's interesting to consider, no?
>zzzZZZ
After all, in the future, those returns may not occur in the same order, so we shouldn't pay that much attention to the order and ...
>ZZZZZZ
Uh ... did I mention there's a moneyback guarantee on the spreadsheet?
