Choosing Contemporaneous Risk-Free Rates for Beta Estimation
Summary
The document asks how to select a risk-free rate when estimating a company’s beta against the S&P 500 from daily returns over a multiyear period. It considers using a five-year Treasury yield observed at the start date, averaging yields over the sample, or updating the rate through time.
One answer recommends using the rate contemporaneous with each purchase or return date, so the benchmark changes over the sample. It notes that the appropriate maturity depends on what risk the analyst intends the risk-free asset to exclude: a longer Treasury rate can carry inflation risk, while a short bill may be more suitable under that definition. Another answer favors a long-term rate and a sample average, showing that practitioners may make different choices. The discussion gives no empirical comparison of these choices, and its final critique of CAPM is an opinion rather than evidence. The practical implication is to match rate timing and maturity to the return frequency and risk-free assumption.
Key ideas
- A risk-free rate used in excess returns can be updated to match each return date.
- The selected maturity depends on the analyst’s definition of risk-free investing.
- Long-maturity government yields may expose investors to unexpected inflation risk.
- The document presents differing recommendations and does not compare their empirical effects on beta.
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Full text
# How to determine the risk free rate for the calculation of Beta # How to determine the risk free rate for the calculation of Beta I want to calculate the beta of a computer vendor using return data from 31st Jan 2008 to 31 Jan 2013 (period of five years) against the return of S&P500. I will be regressing the excess return of company against excess return of S&P500 (where the company is the dependent variable, and S&P is the independent variable). Using the daily treasury yield curve rates as the source of my risk free rates. I am a bit dumbfounded as to exactly which rates to use. First I have determined that I will use the five maturity constant maturity treasury rates, which is because the duration of the five year maturity treasury rates is similar to the length of the project. but now, I don't know which date I should choose my rate from. Should I choose the 5 year maturity on 31st Jan 2008, which is 2.82% (and divide it by 360 to get the daily rate) and use this rate for the next five years. Or should I get the average of 5 year maturity yield from 31st Jan 2008 to 31st Jan 2013 (which is 1.8053% using excel) and use it as my risk free rate? Or is there a better method for the estimation of the risk-free rate. ## Answer by Dave Harris (score 1) https://quant.stackexchange.com/a/34503 Use the risk-free rate contemporaneous to the purchase date, so it would keep moving. Just a reminder, the "risk-free" rate is an asset that pays out the same amount of money in all states of nature. If the Earth is struck by a meteor and the recipient is the sole survivor, a computer will still be functioning to print the check. There is an argument that the five-year rate is not free of risk because there could be a sudden shock to inflation that could not be compensated for and therefore consumption would not be held constant. In that case, the 90-day bill is appropriate. An alternative would be the overnight LIBOR, but that number is monkeyed with and so a good number for it may not exist. Nonetheless, except in hyperinflation, the LIBOR bears almost no risk. As a footnote, I advise everyone to ignore the CAPM. The short-form preaching of it is that it is not supported empirically. ## Answer by Naveen Potnuru (score 0) https://quant.stackexchange.com/a/34011 IMO, investor wants to invest in a company based on the expectations that this stock provids returns above long-term interest rate offered by debt investments. So beta shall be calculated using long-term interest rates (may be 10/20 years)..and also I would suggest to chose average of long-term interest rates from the 31st Jan 2008 to 31st Jan 2013 to use it as risk free rate.
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