Jensen Alpha

Jensen's Alpha

motivated by e-mail from Ron Mc.

There's this thing called Jensen's alpha, designed to measure how well a mutual fund manager does ... compared to "the market".

>The market?
Yeah, maybe the Wilshire 5000, maybe the DOW, maybe the S&P500, maybe ...
>And does Jensen's thing work?
Define "work".

Anyway, let's take a look at Jensen's alpha (invented my Michael Jensen in the 1970s):

We collect the returns for some stock (or mutual fund) over the past n months (weeks? years?)
We do the same for some market index, say the S&P500.
Then, using these returns, we generate a scatter plot and identify the slope of the "best line" fit: that's called beta.
Then we calculate the CAPM according to the magic formula:
CAPM Return = R_f + beta (R_mkt - R_f)
where R_f = risk-free return and R_mkt = (average or "expected") market return

CAPM (presumably) gives the "expected" stock return ... unless that stock didn't behave like the market as a whole.

>Huh?
If the stock behaved "like the market", then we'd expect beta = 1, meaning a 1% change in the market return would be reflected in a 1% change in the stock return.
After all, the slope of that "best line fit" is (change in GE return) / (change in mkt return).
In Figure 1, since beta is significantly larger than 1, it seems that market changes are amplified in GE returns.
Indeed, for the situation in Fig. 1 we have:
beta = 1.470, R_mkt = 0.117% and (let's assume) R_f = 0.01% so CAPM return = 0.01+ 1.470 (0.117 - 0.01) = 0.167%

Figure 1 (using daily returns over one year)
>And if the stock did NOT behave like the market?
Aah, that's where Jensen comes in.
Now we take a look at the (average or "expected") stock return to see if'n it did much better (or worse) than would be the case if it just followed market machinations.
To do that, we ...

>You calculate how much better the stock did, right?
Yes.
Jensen's alpha = R_stock - [ R_f + beta (R_mkt - R_f)]
where R_stock = (average or "expected") stock return

CAPM = 0.167% says that we'd expect the (daily) returns of GE stock to average 0.167% if it just followed the market.
However, if we calculate the actual (average) GE return over the past year, we get 0.066%.
>So GE underperformed the market, right?
That's how it'd be interpreted. In fact, we'd get J-alpha = R_stock - CAPM = 0.066 - 0.167 = - 0.101% and that ain't good.
>Uh ... but I can see that from the chart in Fig. 2 so why that ritual to calculate J-alpha?
You asking me? I just write about these things. I don't have to understand 'em.
Our conclusion about how well (or poorly) a stock or fund did (compared to "the market"), will depend upon beta.
In our GE vs S&P example, I've used daily returns over the past year.
One normally uses monthly returns ... maybe over several years.
However, beta depends upon whether you take end-of-month returns or mid-month returns ... or whatever.
For example, check out Beta here, Beta there.

Figure 2 : performance for year 2009
It's worthwhikle noting that beta is also a measure of correlation and volatilities of the two stocks involved.
If GE, for example, had a high correlation with the market (which is usually does) and its volatility is significantly larger than the market volatility, then beta will be large.
That's because:
Beta[x,y] = r SD[y] / SD[x]
where r is the (Pearson) correlation and SD[y]/SD[x] is the ratio of Standard Deviations.

Another thing: we look at historical returns. That tells us what happened in the past, but does it tell us anything about what'll happen in the future?
Further, the derivation of the magic formula for the Capital Asset Pricing Model (CAPM) involves the standard deviation of returns as the measure of risk.
I really dislike that association. Besides ...

>Do you have a spreadsheet?
Of course. It looks like this:

Click the picture to download

Note the following:

Three years worth of data is downloaded from Yahoo and monthly returns are calculated.
The values of beta and CAPM and the mean stock return R_stock are calculated.
In the above example (GE vs S&P): beta = 1.688, CAPM = -1.07%, annual risk-free = 4% (so monthly rate = 0.33%) and mean GE return = -1.66%
Then Jensen's alpha is calculated. (In the above example: J-alpha = - 0.59%)

Note: If you'd like to consider only deviations from the R_f (the risk-free return), stick a y into cell O5.
Then, the risk-free return is subtracted from all returns and beta is calculated as the slope of the modified regression line.
Note that, if J-alpha = 0%, then (R_stock - R_f) / (R_mkt - R_f) = beta.
Note, too, that the spreadsheet uses the S&P500 as the "market".
If you calculate the S&P500 vs the S&P500, you get (surprise!) J-alpha = 0%.
If you calculate the DOW vs the S&P500, you get J-alpha = 0.07%.

I might point out the Jensen's "alpha" ain't the same as the intercept of the regression line which is also called alpha ?!#$@!*?

>So what do you conclude?
GE didn't do good over the past three years ... compared to the S&P500.
However, note something interesting: monthly GE stock returns amplified changes in market returns, 'cause beta = 1.688 was pretty large.
In other words, if an S&P return changed by X% then we'd expect GE return (for the same period) to change by 1.688X%.
That's the interpretation of the regression line: that "best line fit" in the right graph.

>So you'd expect that to hold in the future?
Me? Of course not.
In fact, if you change the End Date for the data download (in cell B3) to a few years earlier your conclusion may be quite different.
For example, GE did worser than the S&P over the period 2004-2006, yet J-alpha was positive.
So don't think that positive J-alpha means the stock done good, compared to the market.

>So what do you think?
Me? Think?

I should mention that, at this particular time and this particular "definition" of J-alpha (i.e. using monthly returns over three years),
our friend GE has the worsest J-alpha on the DOW.

Did I mention that (somewhere on the spreadsheet) you can calculate J-alpha for a gaggle of stocks?