Beta here Beta there
motivated by a paper noted by mathguy

On a recent Morningstar discussion there was a reference to an interesting paper which noted that financial rituals based upon a calculation of beta from historical data may lead to misleading (if not totally inaccurate) conclusions.
If we plot the daily (weekly? monthly?) returns of some stock versus the returns for "the Market", the slope of the best-fit straight line is beta.

One normally takes as "the Market" something like the Wilshire 5000 or maybe the S&P 500.
Figure 1A shows the monthly returns for GE vs those for the S&P 500 for the period Jan 1, 2000 to Dec 31, 2003.

>The slope of the red line is 0.988, so that's beta?
Yes, for this stock and time period and that market ... using monthly returns where we look at the end-of-month prices only.
We look at the end-of-month price for Jan, 2000 and Feb, 2000 etc. ending with the end-of-month price on Dec, 2003 and ...
>And calculate the monthly returns with these prices.

Yes. Now look at Figure 1B. We still use the period Jan, 2000 to Dec, 2003 and the S&P as "the Market". Guess what stock it is? I'll give you a hint. It's GE again, but now we look at the mid-month prices (rather than the end-of-month prices).
>Like Jan 15, Feb 15, etc.?
Well, yes, if the market is open on the 15th, else we use the day before. In any case, you'll notice that beta is different, eh?

>And that makes a difference?
Well, yes, if you use that CAPM thing.
In fact, if you compare the end-of-month beta with end-of-month minus 2 days you'd get (for GE and S&P): beta = 1.252

Figure 1A

Figure 1B

>What's "end-of-month minus 2 days"?
For example, we consider the closing price on Jan 29, 2000 instead of Jan 31, 2000
and the price on Feb 27, 2000 instead of Feb 29, 2000
and the price on ...

>Yeah, I get it. Minus 2 days ...

Right. The CAPM prescription is:

[1]       E[R] = Rf + beta (E[Rm] - Rf)

where, Rf is a risk-free return (say 4%)
and E[Rm] is the Expected Market return (say 9% for the S&P 500)
and E[R] is the Expected return of your stock (like GE, for example) and beta is ... uh, some beta.

If you use end-of-month beta = 0.988 you'd get an expected GE return of E[R] = 4 + 0.988(9 - 4) = 8.94%
If you use end-of-month minus 2 days beta = 1.252 you'd get an expected GE return of E[R] = 4 + 1.252(9 - 4) = 10.26%

>I''ll take minus 2 days! Uh ... how 'bout minus 3 days or even ...?

You can play yourself with a spreadsheet (comparing your favourite stock with the S&P 500 for Jan, 2000 - Dec,2003)
and varying the day of the month when prices are considered and see what it does to beta. Quite intriguing.

This is for GE:

The green ones are the ones we've considered above: minus 0, 2 and 15 days.

>I'll take minus 10, okay? CAPM says my expected GE return is ... uh ...
That'd be 11.1%.

Anyway, click here to download a spreadsheet to play with.

>But what if I don't want to use the S&P 500 as "the Market" or I don't want Jan, 2000 - Dec, 2003 or ... ?
The spreadsheet let's you change them.

>But what if ... ?
Just try it!

I should point out that, according to CAPM, a bigger beta implies a bigger expected stock return.
Yet, during the last of the 1990s (when the market really took off), a bigger beta (obtained by changing the day of the month when the returns were calculated) ... these bigger betas were associated with smaller returns
Look at the chart
There we used the four years of monthly returns from 1996 to 2000, changing the day of the month when the prices were considered.
It shows the Compound Annual Growth Rate and the beta (as we change the day-of-the-month).
You'll notice that a bigger beta meant ...

>A smaller return!
You got it.

One other thing:
Sometimes beta is calculated using, say, monthly returns, reduced by some risk-free rate (called "excess" returns) ... like short term treasuries. If the risk-free rate is constant (say 4% month after month), then it has no affect on
      beta = COVAR[stock,market] / SD2[market] = r SD[stock] / SD[market]
where r is the correlation between the stock and market returns.

If we subtract a constant risk-free rate from both stock and market returns, none of r, SD[stock] or SD[market] will change. (See stat stuff.)

But, if the risk-free rate varies wildly (month to month) then it's necessary to use the "excess" returns.
Note that the CAPM equation given in [1] can be written:

[2]       (E[R] - Rf) / (E[Rm] - Rf) = beta

so the "excess" returns   E[R] - Rf   and   E[Rm] - Rf   play a prominent role.

Indeed, the fact that beta is a "slope" is clear ... as illustrated here          
where we plot "excess" returns - the stock versus the market.

Note that the yellow dot is determined from the "excess" of the Expected Market Return:   E[Rm] - Rf.
The "excess" of the Expected Stock Return (that's E[R] - Rf) is then determined so that the slope is beta.

>The green dot?

Beta and CAPM and Equity Risk Premium
Okay, suppose that there's a stock that has consistently given spectacular annual returns ... like 20%.
Suppose, too, that this stock is completely uncorrelated to "the Market", so beta = 0.

>Is that possible?
Well, that'd be the case if the stock price increased at a constant annual rate, like 20%. Then its Volatility would be 0 and its Covariance with "the Market" (call it COVAR[stock, market] ) would be 0 and since beta = COVAR[stock, market] / Variance[market] then beta = 0.

Anyway, from that CAPM model, namely   E[R] = Rf + beta (E[Rm] - Rf),   we'd have an Expected stock return of E[R] = Rf.

>You call that spectacular?
No, I don't ... but it points up a problem with taking the CAPM model as an "Expectation" for the stock return.

>Then what should I take it as? I mean, what good is the CAPM model?
Good question. Let's think about it:

  • Suppose the "excess" stock returns, plotted against the "excess" market returns, gives a regression line similar to Figures 1A, 1B.
  • Suppose, too, that the points are close to this line. Then you'd expect the stock to behave much like the market.
  • If that's the case, and the market returns are expected to be, say, 3% above the risk-free rate Rf, then it's reaonable to expect that the stock returns will be above Rf as well.
  • In fact, since beta is the slope of the regression line, it gives an estimate of the ratio of "excess" returns.
  • That means that if beta = 1.23 (for example), then you'd expect the "excess" stock return to be 1.23 x 3% or 3.69% above Rf.

>Yeah, so?
So, if the "excess" stock returns are NOT close to the regression line, then how can you come to that conclusion? How can you have ANY expectation that the stock returns behave in a manner similar to the market ... with beta as some sort of scale factor? How can ...?

>Yeah, so?
So beta alone is practically useless in attempting to estimate "excess" stock returns ... in my humble opinion
In addition to the problems noted above, about "when" one considers the returns.

I suggest that one first look at how closely these stock returns are to the regression line ... and that depends upon some measure of the distance between the points and the regression line. One measure of the error, namely the sum of the squares of the (vertical) distances of the points to the regression line, is:

Error2 = SD2[stock] - COVAR2[stock,market] / SD2[market] = SD2[stock] (1 - r2)

where SD is the Standard Deviation and r is the (Pearson) Correlation between the stock and the market.   (See Best Line Fit )

>So if that Error is small, or zero, then one should have some faith in the Expected "excess" stock return?
As per CAPM ? I don't think so. For our example (where the stock has a constant 20% return) the regression line is horizontal (since the stock returns are constant!) and every stock return point lies precisely on that line so the Error is 0 yet ...

>Yet the expected "excess" stock return still isn't the Risk-free return, eh?
Yes. I mean NO, it isn't.

>So when would you have some faith in the CAPM model?
Me? I have little faith in any predicted values. However, if there's a high correlation between stock and market returns (either positive or negative ... so r is close to +1 or -1) and beta isn't close to 0 (implying a respectable Covariance between stock and market returns) and the Error is small ... then I guess you'd expect stock returns to be similar to market returns.

>So is the market is 3% above Risk-free you'd expect the stock to be 3% above Risk-free, right?
No, you'd expect the stock to be beta x 3%. Remember that beta is an estimate of the RATIO:   (stock excess) / (market excess).

Did I mention that the "excess" we've been talking about has a name? It called:

Equity Risk Premium = "The extra return that the stock provides over the Risk-free rate to compensate for market risk."

If the "excess" stock return (or Equity Risk Premium) is, say, 4% (that's above the Risk-free rate) then it's the extra return you'd want because of the extra risk you take by investing in the stock ... rather in a risk-free asset.

>To compensate for market risk? What's that about?
If the stock behaves much like the market, and market returns goes up and down, then you'd expect your stock to go up and down as well ... with changes in stock returns being beta times the changes in market returns. That's the risk associated with being in "the market".

>And the "market" might go up or down depending upon economic factors or some war somewhere or some terrorist activity or ...?
Yes, and your stock will probably participate to some extent in this up and down movement of the market.

>Assuming your stock is intimately connected to the market.
Well, yes, and it's things like beta and r and Error which measure this intimacy.

>Can you ... uh, summarize, like when to trust CAPM and ...?
Well, one often reads things like:

"An R-squared measure of 35, for example, means that only 35% of a fund's movements can be explained by movements in its benchmark index."

Yes, that's r2. You'll note that our Error depends upon R-squared = r2.

>How come that google link has the exact same wording ... about "35% of a fund's movements"?
That's typical. Some respected financial guru says something (having presumably justified her statement) and everybuddy picks up the statement and repeats it (again and again and again) ... but avoiding the justification that the guru (presumably) provided.

>And should I swallow it?
That depends upon your appetite.

See also Covariance.