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


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 ...
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?
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?
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 ...
Uh ... did I mention there's a money-back guarantee on the spreadsheet?