Hi guys, A really quick and possibly pretty amateur question. I have estimated the Beta value of a particular stock (NWG.L) to the FTSE 250 index over a 2 year time horizon. My regression gives me the following equation : y= 0.53 + 0.29 (X) Where 0.53 is the alpha value and 0.29 is the Beta sensitivity to the index.Now, am I completely wrong in thinking that if was to sub into my equation the index return (X) over the same two year period of my regression, that it should equal the actual stock return over that period??? I have tried it a few times using different time periods and horizons but to no avail. (FYI - I have conducted my simple linear regression in Bloomberg under the command BETA GO)I just need to know if I am totally wrong in thinking this should work? I am not trying to predict stock returns, just using the results of my regression to decompose previous stock returns. I am quite new at this so please forgive me if this is a stupid question! Please help. Nik.

Three issues prevent the beta equation from giving the exact numbers that one might expect:1) the actual regression equation is Y_i= 0.53 + 0.29 (X_i) + eps_i, where eps_i is epsilon or residual error in the data points. The regression for beta is never exact so any use of the equation will be off by some epsilon. The amount of error when you plug in the index return to estimate the stock return will be a function of the correlation coefficient which tell you how much of the variation in the stock's returns were explained by variations in the index's returns.2) Plugging in a return for a long time period will only come close to working (within epsilon) if the regression and the returns are defined in a logarithmically (log returns). In a arithmetic model of returns (i.e., Return_i = ((Price_i-1) - (Price_i))/(Price_i-1) ) the returns don't add. A 50% return plus a 50% return is not a 100% return. If you measure returns logarithmically (e.g., Return_i = ln( (Price_i)/(Price_i-1) ) ), then they will add up to within epsilon. I'm not sure which model Bloomberg uses.3) The beta value isn't stationary - it changes over time so that if one plugs-in returns for a period different from the one used to estimate beta, then the computed return won't equal the observed return. Only if you compute beta with log returns over an X year period and then plug in the exact same X-year period returns into the equation will you get the right answer because the sum of the epsilon values across the period will be zero.

Hi Traden4Alpha, Thanks a lot for your help!! I have implemented all of the points you mentioned and they have improved my results a little bit but not by much. I think my problem lies somewhere in the fact that the R squared in my regressions is consistenly small (less than 5%) hence my regression results may not be very accurate. Any ideas? Thanks again for your help so far. Nik.

If your goal is to predict stock returns, then using just market return as a factor will hardly suffice. Try to use additional factors (google search for factor models might help).

QuoteOriginally posted by: nikkI think my problem lies somewhere in the fact that the R squared in my regressions is consistenly small (less than 5%) hence my regression results may not be very accurate. Any ideas? Yes: a very small R squared means very large values of epsilon and thus large discrepancies between actual returns versus those predicted using the regression equation. That stock appears to not track the index very well and thus the estimates of alpha and beta will not be very good.If you want to decompose the returns for NWG.L, then you will need to find other data streams that correlate better with NWG.L.

Hi, Thanks again for the advice. But I am only trying to estimate beta to try and hedge out FTSE 250 market risk by taking a short position in the FTSE 250 index. So my main goal is not to predict returns just to get an accurate picture of the market risk of this and quite a few other stocks. Any other ideas?Nik.

QuoteOriginally posted by: nikkBut I am only trying to estimate beta to try and hedge out FTSE 250 market risk by taking a short position in the FTSE 250 index. So my main goal is not to predict returns just to get an accurate picture of the market risk of this and quite a few other stocks. The low R squared means there's very little (historical) market risk in the position. Moreover, if the value of beta is inaccurate or unstable, then your hedging could be adding risk. For example, if you assume that your current historical estimate of beta is correct, you would short the index by 29 pence for every Pound invested in NWG.L. But if the "true" future value of beta less than 0.29/2, then you have just added market risk, increased transaction costs, and lost market gains. The fact that R square is so low means that hedging the market risk will do little to reduce the volatility of the position.You might have more success hedging market risks in a diversified portfolio of stocks, rather than looking at market risks in individual stocks.

Ok thanks very much. I will produce a price series for my entire stock portfolio and then estimate an overall level of market risk versus the index. Hopefully that will increase my R squared.Cheers. Nik.