Wednesday, January 25, 2012
Is Arithmetic Return Bias Basis of Low Vol Anomaly?
I created an index of the highest beta stocks from 1962 to present. Every 6 months I took those 100 highest beta stocks, excluding the lowest 20% in market cap (to get rid of dumb stocks you can't trade). The results are in the chart above, and summary data are as follows:
The top line, "AnnReturn", is the arithmetic return, and here the monthly returns for the high beta stocks are about 0.14% higher than the S&P500, which when multiplied by 12 is a 1.7% difference. But looking at the chart which shows a total return chart, and the geometric annualized return, we see a very different picture, with the high beta stocks underperforming by 3.5% annually.
The basis for this is the difference between geometric and arithmetic returns, which is
Geometric Return =Arithmetic Return - Variance/2
Thus, the differential annualized variance (in this case, 12% vs. 2%), generates the differential annualized return. Interestingly, the return rankings for these data are different depending on the horizon!
Mutual funds and individual investor holding horizons average about 1 year, and I think that's a good assumption for an investment horizon. It seems that 1 year would be the obvious horizon to apply data against, but the problem is there is so little of it. There's like twelve times as much monthly data! A simple fix would be to use log returns, but this doesn't always happen, and I think those who still find the Security Market Line to have a positive slope in general are looking at monthly percent return data, and this is why they see what they do.