A factor regression and what a beta meansফ্যাক্টর রিগ্রেশন আর বিটা মানে কী

Excess return, and why it is not the raw return

A regression of one return on another is easy to write and slightly wrong. Holding cash already earns something: a treasury bill yield, a savings rate, whatever the safest asset on the market pays. What a stock has to earn to be worth the bother is its return above that floor. So both sides of the regression are excess returns: the stock's return minus the risk-free rate for the same month, and the market's return minus the same risk-free rate. Subtracting a number that moves with time keeps the intercept honest across periods when the risk-free rate itself was high or low, rather than folding a changing floor into a coefficient that is supposed to describe only the stock.

The market model

The market model is one regression with one right-hand variable: the stock's excess return equals alpha plus beta times the market's excess return, plus noise. It is ordinary least squares exactly as in the farms example elsewhere in this room, with rain replaced by the market's excess return and yield replaced by the stock's. Beta is the slope: covariance of the two excess returns divided by the variance of the market's. Alpha is the intercept: the mean stock excess return minus beta times the mean market excess return.

Worked example: twelve months

Beta is the sum of the products divided by the sum of the squared market deviations: 99 divided by 66, which is 1.5. Alpha is the mean stock excess return minus beta times the mean market excess return: 2.0 minus 1.5 times 1.0, which is 2.0 minus 1.5, which is 0.5. The line is stock excess return equals 0.5 plus 1.5 times market excess return.

What R squared says

Put each month's market excess return into that line: 5.0, -2.5, 6.5, 0.5, -1.0, 8.0, 3.5, -4.0, 2.0, 0.5, 5.0 and 0.5. The gaps between what happened and what the line predicts are the residuals: 0, 0.5, 0.5, 0.5, 0, 0, -0.5, 1.0, 3.0, -0.5, 0 and -4.5. Square them and add: the total is 31.5. The sum of squared deviations of the stock's own excess return, from the deviation column above, is 180. R squared is one minus the ratio of the two: 1 minus 31.5 over 180, which is 1 minus 0.175, which is 0.825. Eighty two and a half per cent of this stock's month to month swing moved with the market. The rest, months nine and twelve especially, is this firm's own news landing on a day the market itself barely moved.

R squared here comes from twelve months, and twelve months is not enough to call it stable. Treat it as a description of this particular year, not a forecast for the next one.

Beta, said as a sentence about risk

A beta of 1.5, the number this workbook found, says the stock's excess return tends to move one and a half times as far as the market's, in the same direction. A beta of 1.3, to take the round number analysts actually quote most often, says the same thing at a gentler slope: if the market's excess return rises ten per cent, expect this stock's excess return to rise about thirteen per cent over the same stretch, and to fall by about the same multiple when the market falls. A beta above one is not a promise of more return. It is a promise of more swing, on the way down as much as on the way up. A beta below one moves less than the market in both directions; a beta near zero barely moves with it at all; a negative beta is rare enough that it is worth checking the arithmetic before trusting it.

The three-factor extension, briefly

One right-hand variable is a start, not the whole story. Small companies have tended to outrun large ones by more than their market beta explains, and cheap companies, measured by book value against price, have tended to outrun expensive ones the same way. Fama and French's three-factor model adds both: a size factor, the return on small stocks minus large ones, and a value factor, the return on cheap stocks minus expensive ones, alongside the market factor already here. The regression is the same idea with three slopes instead of one, run exactly as before, and alpha is now the return the three together cannot explain, which is a stricter test than the single-factor alpha above.

Dhaka Stock Exchange cautions

Many counters on this market do not trade every session. A day with no trade is not a day with no news: the exchange simply carries the last traded price forward, so that day's return is recorded as zero whether the true price moved or not. A stock with several such days a month looks, on paper, less connected to the market than it is, because half its real reaction to a market move is sitting in the next session's price rather than that day's. The effect is not random noise, it is a one-directional pull: a beta estimated this way is usually biased towards zero, understating both how defensive a thinly traded stock looks and how risky it actually is once it does move. One way through it is to widen the window: sum returns over a week instead of a day, so a stale price has more time to catch up, or add a lead and a lag of the market's excess return to the same regression and add the three slopes together. Neither is free: a wider window throws away timing, and the lead-lag fix needs more months of data than the simple regression above to hold its own standard errors steady.

Where it goes wrong

• Regressing raw returns instead of excess returns, and calling the intercept alpha anyway.
• Reading R squared as a verdict on the stock rather than on how much of its movement is systematic.
• Quoting a beta with no mention of the window it was estimated over: a beta from a falling market and a beta from a calm one can differ for the same stock.
• Trusting a beta from a stock that barely trades, with no correction and no warning.
• Reporting alpha as skill without saying over what stretch, and without a standard error next to it.

Checklist

• Both columns are excess returns, the risk-free rate already taken out.
• Beta is stated with the window it was estimated over.
• R squared is reported as how much of the movement is explained, not as a grade.
• Thin trading has been checked before the beta is trusted.
• If a factor beyond the market is added, it is named and its own slope is reported.

The lab runs a factor regression, including the single-factor CAPM case, from the method list and writes alpha, beta and R squared into the APA table on its own; its market button fetches a daily series for a ticker so twelve months of prices become excess returns without being typed in by hand, and the workshop's Returns tool turns a run of prices into the daily or monthly returns a factor regression needs before it can run at all.