Estimating Beta through Statistical Regression
One of the most straightforward ways to estimate beta takes a lot of computing power.
The __market model__,
$$\displaystyle R_i = \alpha_i + \beta_iR_M + e_i $$
can be estimated using __statistical regression__. The best linear connection between these two variables of the market return and the asset return will be estimated using a method known as __least squares regression__, where the sum of the squared error terms is minimized to find the best fit line.
Suppose the sum of squared errors is very small. How would you interpret this result?
Right!
No, all errors are squared, and so are positive. The squared terms couldn't cancel each other out in summation at all.
Since this method sums the squared errors, all terms are positive. So to end up with a sum of squared errors that is relatively small has to mean that there are very small errors.
And in the real world, there certainly are errors.
Any asset has variance, and won't match up perfectly with market movements.
Here, security returns are plotted against market returns, and the best fit line is shown. The slope of this regression line is beta. A security with returns more sensitive to the market, for example, would have more market risk. This would show on the plot as more vertical dispersion in the scatter plot points, and the best fit line would then have a higher slope, indicating a higher beta.
Suppose you started with the best fit line as shown, and then the very next observation of the return pairing was given as observation X. What do you think would happen to the beta estimate?
Yes!
This observation would clearly "pull" the line into a flatter position than it currently is, meaning a lower slope, and a lower beta.
How else could you say this?
No, consider that the new best fit line would be flatter, with a lower slope.
No, the beta measure would be different with this observation. The best fit line would have to adjust to this new information.
No, the market return was relatively high, not low. It was the asset's return which was low. But also, the level of the market return by itself will never cause beta to change. It's the _relative returns_ between the market and the asset which affect beta.
Yes, good work!
This raises another interesting question: how long should the observation period be for measuring beta? There is no perfect answer here, since it just depends on too many things. One standard timeframe is five years, but this is a very long time if the company has changed recently, and will not provide the best idea of future price sensitivity.
On the other hand, a 12-month window gathers the most recent events, but will also ignore sizable past events which may repeat themselves. This is not a small issue either: beta can change a lot depending on the time period chosen.
If a firm made large structural changes about four years ago, and has been relatively stable since then, what would you think is the best time period for estimating beta?
To summarize this discussion:
[[summary]]
No, this is pretty short. It would probably be best to look at a longer time frame, since things have been stable for the past four years or so.
No, this would include the period before the restructuring, which is likely to skew the beta measure going forward.
Correct! Three years will cover a fairly long recent history following the restructuring process.
Another thing that will really change the beta estimate is the return frequency. Daily returns, weekly returns, and monthly returns will all provide different estimates. Computing power is cheap, so it's best to consider multiple estimates of a firm's systematic risk when possible.
There must be very small errors
The squared errors largely cancel each other out when summed to end at a small number
Beta would decrease
Beta would increase
Beta would be unchanged
The market return was very low, reducing beta
A high market return was met with a relatively low asset return, indicating less sensitivity to the market
One year
Three years
Five years
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