Sanity check on regressions

[TeenageDiplomat]

Hi - just need some input on this one. Simple practice q from Kaplan:

Coefficient Standard Error Intercept 2.1 2.01 Index 1.9 0.31

Q: If the return on the industry index is 4%, the stock’s expected return would be

1. 11.2%
2. 7.6%
3. 9.7%

A: 9.7% © because

Y = b₀ + b₁X Y = 2.1 + 1.9(4) = 9.7%

I get that if you follow the formula (Y = mX + b) and plug and chug, you’ll get that answer. My confusion comes from the following: what does that coefficient of 1.9 represent? It is the slope of the regression (aka this stock’s beta). It represents a 1.9% expected change in the stock for any 1% change in the index.

Therefore if the index changes 4%, why isn’t our expected change simply 4 x 1.9 = 7.6% ? Why does the y-intercept (the value if our index changed by 0%, which we know if didn’t) required in this calculation? Do you get what I’m saying from a step back and away from the simple formula? Thanks

[S2000magician]

@TeenageDiplomat

My confusion comes from the following: what does that coefficient of 1.9 represent? It is the slope of the regression (aka this stock’s beta). It represents a 1.9% expected change in the stock_ 's return _ for any 1% change in the index_ 's return _.

That’s better.

@TeenageDiplomat

Therefore if the index_ 's return _ changes 4%, why isn’t our expected change _ of return _ simply 4_ % _ x 1.9 = 7.6% ?

It is.

@TeenageDiplomat

Why does the y-intercept (the value if our index changed by 0%, which we know if didn’t) required in this calculation? Do you get what I’m saying from a step back and away from the simple formula? Thanks

So that we know the _ value _ of our stock’s return, not merely the change in its return. When the index’s return changes by 4% from, say, 5% to 9%, does the return on our stock (which changes by 7.6%) change from 0% to 7.6%? Or from 35% to 42.6%? Or from −18.4% to −10.8%?