# Quant

Two questions: 1) Beta = COV(stock a, market) / VAR market which is different from regular correlation = COV(A,B) / std A , std B I’m thinking the reason is that you want to determine if the stocks movement is significantly different from the markets. 2) A different but related question, one of the requirements of linear regression is that your predicted value is unbiased. It then goes on to explain that a unbiased regression estimate is where b0 = 0 & b1 = 1. This makes sense if you are traking a stocks movement in relation to the market. If the stock you are trackings performance isn’t significantly different from the return of the market, you cant reject the null hypothesis that b1 = 1 But suppose you are regressing stock returns and the inflation rate. Beta should be negative. Obviously in this case your null cannot be that beta = 1, so what is it? Max a million Kerry.

1. It’s when you are doing one security against the market, or two securities to each other.
1. Beta has nothing to do with testing significance of anything. It’s just a multiplier, a measure of sensitivity to market moves You get it from doing a linear regression of the stock on the market, so you end up with the regression equation: E[r_stock_a] = alpha + beta * r_market (+error) So if alpha is 0, and the market moves 2%, the stock will be expected to move 2*beta % Correlation is a measure of how closely two variables move together, and is very different. For a start, it is bounded [-1,+1], whereas beta is not. 2) I’m not sure what b0 and b1 are. An unbiased estimator is one where its expected value is the same as the parameter is trying to estimate. So for example, taking the average of a sample is an unbiased estimator of the mean of a population: E[X_bar] = mu. I’m not sure where your variables come in.

Mabe I didn’t make myself clear enough, 1) E(R_stock A) = b0 + b1 (market risk premium) So in that formula, b1 = COV (stock A, Market) / VAR market which is different from regular COV which is = COV (stock A, Stock B) / std A*std B So my question is why isn’t the equation for b1 = COV (stock A, Market) / std A , std market I think it has to do with the fact that you want to determine if the stocks return is significantly different from the return of the market. so Ho = 1, if it isn’t significantly different you’d use the market risk premium added to the risk free rate to determine expected return of stock A. 2) My question for two is really when calcualting a t-test for significance of b1, when will I test for b1 = 0 and when for b1 = 1, because the CFA text specifically says that a unbiased estimator is where B1 = 1. A question in the Schweser text, asked for the value of the t-test and the solution has b1 = 0. Kerry.

Many mistakes here: Kerry1 Wrote: ------------------------------------------------------- > Mabe I didn’t make myself clear enough, > > 1) E(R_stock A) = b0 + b1 (market risk premium) > Ok, so b0 is the risk free rate, and b1 is beta. Standard CAPM. > So in that formula, b1 = COV (stock A, Market) / > VAR market > > which is different from regular COV which is = COV > (stock A, Stock B) / std A*std B > No, that’s not Covariance, that’s correlation. > So my question is why isn’t the equation for b1 = > COV (stock A, Market) / std A , std market Because you’re not using correlation, you are looking for the slope of the line of best fit for stock A and the market - usually referred to as “beta”. “Why” involves calculating it from first principles. You can either do that yourself or look it up. > > I think it has to do with the fact that you want > to determine if the stocks return is significantly > different from the return of the market. so Ho = > 1, if it isn’t significantly different you’d use > the market risk premium added to the risk free > rate to determine expected return of stock A. You really need to do some hard yards in Quant, there are so many misconceptions here. Start with hypothesis testing, and build your way back up to lvl 2 stuff. > > 2) My question for two is really when calcualting > a t-test for significance of b1, when will I test > for b1 = 0 and when for b1 = 1, because the CFA > text specifically says that a unbiased estimator > is where B1 = 1. > unbiased estimators and t-tests for significance of the variable are two separate topics. The t-stat is (observed value - hypothesised value)/(st error of estimate) I think, with n-2 df (again, I think). I described what an unbiased estimator is above. > A question in the Schweser text, asked for the > value of the t-test and the solution has b1 = 0. There’s no way I can possibly know what you are talking about here. If you want help with a specific question, post it up here. Snippets help noone. > > Kerry.

Whoa - No problem with what Chris wrote, but I seem to be doing this a lot this year (which means there’s something messed in the curriculum). >> 1) E(R_stock A) = b0 + b1 (market risk premium) > >Ok, so b0 is the risk free rate, and b1 is beta. Standard CAPM. So then we can do a whole bunch of stuff with CAPM and draw all kinds of interesting conclusions. But CAPM is a mathematical model not a statistical model. As soo as you start doing all this variance/covariance stuff you are talking about estimating the model. There is a pretty big literature on estimating beta but CAPM certainly doesn’t say anything like beta should be estimated using least squares.

JoeyDVivre Wrote: ------------------------------------------------------- > Whoa - > > No problem with what Chris wrote, but I seem to be > doing this a lot this year (which means there’s > something messed in the curriculum). > > >> 1) E(R_stock A) = b0 + b1 (market risk premium) > > > > > >Ok, so b0 is the risk free rate, and b1 is beta. > Standard CAPM. > > So then we can do a whole bunch of stuff with CAPM > and draw all kinds of interesting conclusions. > But CAPM is a mathematical model not a statistical > model. As soo as you start doing all this > variance/covariance stuff you are talking about > estimating the model. There is a pretty big > literature on estimating beta but CAPM certainly > doesn’t say anything like beta should be estimated > using least squares. Fair enough. I think CAPM’s a load of obsolete nonsense anyway, but I suppose you have to know where you’ve been to know where you’re going. Out of interest, how do you estimate beta if not by regressing it?

Well thanks for not helping out there guys, pretty sure I’m not talking absolute bull. What I was looking for, if anyone else is interested is: Correlation (A,M) = COV (A,M) / std A*std M beta apparently (haven’t seen this in the text) = corr (A,M)*std A / std M which then becomes = COV (A,M) / VAR M (given in the text)

he he, easy tiger

Shame they deleted it. Perhaps you ought to go elsewhere if you don’t like the help you get here.

I also like the fact that the US swear filter misses most of my more fruity vocabulary!

chrismaths Wrote: ------------------------------------------------------- > JoeyDVivre Wrote: > -------------------------------------------------- > ----- > > Whoa - > > > > No problem with what Chris wrote, but I seem to > be > > doing this a lot this year (which means there’s > > something messed in the curriculum). > > > > >> 1) E(R_stock A) = b0 + b1 (market risk > premium) > > > > > > > > > >Ok, so b0 is the risk free rate, and b1 is > beta. > > Standard CAPM. > > > > So then we can do a whole bunch of stuff with > CAPM > > and draw all kinds of interesting conclusions. > > But CAPM is a mathematical model not a > statistical > > model. As soo as you start doing all this > > variance/covariance stuff you are talking about > > estimating the model. There is a pretty big > > literature on estimating beta but CAPM > certainly > > doesn’t say anything like beta should be > estimated > > using least squares. > > Fair enough. I think CAPM’s a load of obsolete > nonsense anyway, but I suppose you have to know > where you’ve been to know where you’re going. > > Out of interest, how do you estimate beta if not > by regressing it? Well, most people aren’t really interested in estimating beta in the usual sense; they are interested in predicting beta going forward over the next time period. At the very least people use shrinkage estimators, but there is a whole literature on how to do even better. It’s really dull.

Kerry1 Wrote: ------------------------------------------------------- > Well thanks for not helping out there guys, pretty > sure I’m not talking absolute bull. > > What I was looking for, if anyone else is > interested is: > > Correlation (A,M) = COV (A,M) / std A*std M > > beta apparently (haven’t seen this in the text) = > corr (A,M)*std A / std M > > which then becomes = COV (A,M) / VAR M (given in > the text) Read more carefully - beta isn’t that at all. An estimate of beta might be that but beta is a parameter and mixing your estimators and your parameters is bad medicine.

JoeyDVivre Wrote: ------------------------------------------------------- > Kerry1 Wrote: > -------------------------------------------------- > ----- > > Well thanks for not helping out there guys, > pretty > > sure I’m not talking absolute bull. > > > > What I was looking for, if anyone else is > > interested is: > > > > Correlation (A,M) = COV (A,M) / std A*std M > > > > beta apparently (haven’t seen this in the text) > = > > corr (A,M)*std A / std M > > > > which then becomes = COV (A,M) / VAR M (given > in > > the text) > > > Read more carefully - beta isn’t that at all. An > estimate of beta might be that but beta is a > parameter and mixing your estimators and your > parameters is bad medicine. Is a shrinkage estimator similar to using mean reversion for betas?

Yes - same (you “shrink” toward 0).

Thanks for that Joey, I used it this morning on a concept question and it worked… So without getting too involved in Quant I’ll delude myself and use it as fact.