Stochastic Dominance
motivated by e-mail 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 r-values, 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 F-value is less than the average G-value, over each r-range: Min ≤ r ≤ R.

To get an average, we'd integrate F(r) and G(r) and divide each integral by the length of the r-range: 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 r-value (for the portfolios) and R is some r-value between the Min and Max r-values.

Okay, so second order dominance is defined like so:
 If F and G are two cumulative distributions, and where F1(r) ≤ G1(r) then we say that: F has second order stochastic dominance over G.
>What about Figure 2? Does F have that second order stuff? It sure don't have first order because at r = -2.1% I can see ...
If we plot both F1(r) and G1(r) we'd get Figure 3.

>Okay, so what about third order dominance?
We could integrate again and get third order dominance.
 If F and G are two cumulative distributions, and and where F2(r) ≤ G2(r) then we say that: F has third order stochastic dominance over G.

Figure 3

>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 ...