Monday, March 19, 2007

Time-Varying Alpha and Beta

When estimating alpha and beta, a big question is how much data to use? 36 months? 60 months? Different periods will result in different estimates and those estimates will change as new data comes in. Advanced time series econometrics methods can use all the data available to estimate alpha and beta coefficients that vary over time. To produce the chart above I used the Diffuse Kalman Filter to estimate alpha and beta. I assumed that alpha and beta could each be modeled by what is called a local linear trend model.

Alpha increases smoothly over time. The model couldn't distinguish any changes in the slope of this learning curve. With more data over time it might be able to. It could distinguish a lot of variation in beta. I increased my beta from when I started investing from around 0.35 to a maximum of 1.35 in September 2001. Increasing my beta in this period after March 2000 was not a good idea! In recent years beta has fallen to its current level of 0.42. Alpha is currently 10.3%. The expected rate of return is, therefore, 17.6%.

We can also use this model and the actual monthly returns to the MSCI Index and the risk free rate to compare actual returns to those that are predicted by the model and the MSCI index:

Deviations from the predictions are the "noise" that is not explained by the MSCI and the risk-free rate or the smoothly trending estimate of alpha.


Anonymous said...

Moom, are you using Matlab to get this data? just curious.

Btw, you do sound like a finance professor, now that I am getting a feel of your writing style. :)

mOOm said...

No I use an econometrics package called RATS ( to estimate both this and my NDX trading model. The chart was produced by Excel.