downside Beta
motivated by email from Peter U.
While we're talking about Beta I thought we should mention "downside beta".

>Huh? Never heard of it.
Yeah, I'd never heard of it before, either.

Anyway, we get monthly (or weekly or annual etc.) returns for the past umpteen years and plot the return of some asset or portfolio
against the return of some index (or benchmark) ... like the DOW, for example.
Remember Beta?

The slope of the regression line (or best fit line to these points) is called Beta.
If the slope is 1.36, as in Figure 1, we interpret this to mean that if the DOW increases by 1%, then IBM will increase by 1.36% so that ...

>I assume that means "roughly", not exactly, right?
Uh ... yes, that's the common interpretation of Beta. See the wee box? That's a piece of the regression line.
It has slope equal to Beta. It describes some sort of "average" relationship between historical returns and ...

>And if DOW drops 1% then we might expect IBM to drop "roughly" 1.36%, right?
Yes ... and that's exactly what we want to talk about!

Figure 1
Suppose we were interested in how the DOW's influence on IBM behaves (using historical data) when their monthly returns are, say less than 3%.

>So you ignore the returns greater than 3%, eh?
You got it! Then we'd get a subset of the points in Figure 1.
In fact, we'd get Figure 2. Do you see the points retained? We just tossed out the returns greater than 3% and did another regression.

>You tossed out the DOW returns, too!
Sure. Why not? We're interested in those returns in both our portfolio asset(s) ... and the Index against which we're measuring.
Maybe 3% is what I can get in a bank or some other risk-free investment.
Any portfolio or index returns less than that are lousy ... so maybe we'd like to know how our portfolio and the index compare when their returns are lousy.

>But the slope, Beta, is just 0.89 in Figure 2.
Yeah, nice eh? We can interpret this to mean that, for lousy returns, when the DOW goes down by 1% then IBM only goes down by 0.89%.

Figure 2
>And when the returns are good, say greater than 3%?
Yeah, that'd be interesting, too. See Figure 3? Beta for them "good" returns is 1.95 so that means ...

>IBM does MUCH better than the market when the market is good, right?
That's how one might interpret it, yes ... assuming that the DOW is our "market".

>But then we're not talking "downside" Beta ... we're talking "upside" Beta.?
Yeah ... I guess so.


>I assume there's a spreadsheet?
Yes. So far it's a modified version of that 4-stock-regression.xls spreadsheet we talked about before... except it's got a piece that looks like this:

>What's all that stuff about solving for x and y?
Just in case you think that downsideBeta + upsideBeta= Beta, it ain't true.
In fact, since Beta and Alpha are the slope and intercept for the regression line, we can solve for x and y such that:
x*downsideBeta + y*upsideBeta = Beta
x*downsideAlpha + y*upsideAlpha = Alpha
... and that's done in the spreadsheet.