R is the rate of return on the portfolio measured in each period t, F is the risk free rate of return (return on a 90 day U.S. Treasury Bill), and M is the rate of return on the market portfolio, which is often approximated by the S&P 500 total return index or the MSCI World Index etc. Usually monthly data is used to compute the series of returns - typically 36 to 60 months of data. Alpha and beta are then the coefficients of the ordinary least squares regression of R-F on M-F. You can do this in Excel using the function LINEST.
Beta shows the sensitivity of the portfolio to the market rate of the return. Alpha shows the average of the returns that are not explained by the market rate of return. The last term epsilon is a series of error terms that represents the variations in return not explained by the market rate of return - the noise in the regression.
But what does alpha actually look like? The following chart shows a scatter of my rates of return for the last 36 months against the returns of the MSCI All Country Gross World Index:
The trendline computed by Excel shows the part of returns explained by the MSCI index - its slope is beta. Alpha is the height of the trendline above the zero, zero point on the chart (I haven't deducted the risk free return here by the way). This is a monthly alpha - the annual alpha here is something like 12%, which compared to most mutual funds is rather high. But I am mostly struck by how fragile alpha looks when surrounded by the wild monthly fluctuations in the portfolio returns.
2 comments:
Is there a statistical significance test to determine is the alpha is significant compared to the monthly "noise" in monthly portfolio returns?
Also, to be a measure of "true" alpha doesn't it have to be based on risk-adjusted returns? ie. although MSCI All Country Gross World Index is a convenient benchmark to compare your returns to what you could achieve with a passive, index investment, you should really use a composite index with the same asset allocation and risk profile as your actual investments in order to measure alpha?
This is the problem often found with Mutual Fund managers who have "outperformed" the benchmark - you often find they have been gaining the extra "alpha" by taking on more risk than the benchmark contains.
Regards
http://enoughwealth.com
1. Yes you can do a t-test on the alpha coefficient. The t-statistic I come up with is 1.44. This means that there is a 7.9% probability that alpha is actually zero or less. The Sharpe ratio gives a similar test against the risk free rate. My Sharpe Ratio for the same period is 1.37. Most mutual funds who are getting that kind of a Sharpe Ratio in the last three years have lower alpha and so the alpha they do have is likely less statistically significant still. The TFS Market Neutral Fund has a Sharpe Ratio of 1.47 since inception and claims a beta of 0.26 and an alpha of 14.8% to their benchmark. Their alpha is likely more statistically significant than mine. That's why I added the fund to my portfolio :)
2. I debated the choice of the benchmark recently here with Rich Gates and some other commenters. I feel that the MSCI gives me a very steep benchmark to overcome and likely would be my default purely passive investment - globally diversified stocks. It is the most closely correlated index to my actual returns. I don't think I am taking on more risk that isn't measured by beta as half my portfolio is in bonds. I am heavily concentrated in the Australian market - so that could be a source of risk. Constructing a benchmark based on my asset allocation would make sense if it was fixed. But three years ago I had no bonds in the portfolio and a year from now probably won't either. I want to be able to measure whether market timing and trading adds return over a buy and hold stock portfolio.
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