CAPM's forumlum is E(R_i) = R_f + (E(R_m) - R_f) * beta, where R_i is the return of stock i R_m is the return of index R_f is the risk free rate In calcuation (least square regression) of beta, the expected values E(R_i) and E(R_m) can be calculated if we take longer period, say 6 years or longer of monthly returns. On the other hand, "risk free" requires short period, although we can take the average of interests of 1-year savings and loan as risk free rate. However, in CAPM, risk free rate is a constant, but I think it is not in a longer period of time. Therefore, the long period expectated returns and short period constant risk free rate is a natural contradiction. I can't find any mentioning yet, your comments are highly appreciated. Thanks in advance.

QuoteOriginally posted by: dino1019...However, in CAPM, risk free rate is a constant, but I think it is not in a longer period of time. Therefore, the long period expectated returns and short period constant risk free rate is a natural contradiction.If you borrow money for, let's say, T years by issuing a T-year maturity bond to finance an investment portfolio that you expect to liquidate right after the bond matures, your risk-free rate will be the interest rate on the bond. It will, by definition, be "risk free" because it remains constant until the bond matures, as well as throughout the time horizon of the investment.

I think you probably didn't answer my question, and your point seems not fully correct, and the doubt of it is related to my question.Yes, for a specific "risk asset" such as the bond we "shorted" (sold), there is a fixed rate to pay throughout the period.However, there are 2 misunderstanding here.Firstly, in CAPM, risk free rate is the baseline of all risk assets, and the risk is measured by "risk premium", the extra return for compensating the extra risk taken, i.e., the risk level above the risk free one; but the bond you sold is not necessarily a risk free asset, if the coupon rate is higher than the risk free rate.Secondly, even if your bond is sold at the current risk free rate, it is still not risk free throughout the lifetime, if the maturity is long. Bond is not risk free in the sense that interest rate might go down, and you pay higher than it later, that's not risk free. Practically speaking, in shorter period say 1 year or 3 months, it's ok to say the risk level is a constant.Let's go back to my question now. My purpose is to calculate the beta, a measure of risk of a specific risk asset, thus I can have the required return of the capital, or cost of capital for that risk asset. Beta is defined as the covariance of return of the risk asset and that of the total of all risk assets as whole. However, it's difficult to calculate the total of all risk assets, we usually take the index instead. When I am doing it, I found the problem that risk free rate or interest rate change substantially when the time frame is long, there is no reason to treat it as constant. But it'd better to calculate longer period to get beta, a contradiction.

dino, here's how the problem is treated in corporate finance:1. The risk-free asset is taken to be one that has no default risk and no reinvestment risk. Hence, the closest approximation would be a government zero coupon bond. The CAPM does not provide exact answers to what a risk-free asset is or what the correct figure for the risk-premium is. The convention is to match the duration of cash flows of the investment project to that of the zero. If you are pedantic, in principle, you could calculate the cost of equity capital for every single year in which your project yields cash flows (since the CAPM is originally a one-period model). However, for well-behaved term structures, this would be somewhat of an overkill.If you do company valuation, you take the bond of the longest maturity you can find, since firms are assumed to have infinite lives (unless you know the company will go out of business).2. Usually you calculate the beta by regressing the company's stock returns on a broad index of stocks, such as S&P500 or Morgan Stanley's World Index. Usually you want to take the period that is representative of the company's current operations, that is you take the last 5 years of monthly returns (the shorter the return period - weekly, daily - the more noise you have). For the stable growth period it is often assumed that the beta will converge to the that of the market, that is 1. However, I don't see why it should always be the case...Hope it helps.

First of all the risk free rate is completely irrelevant in finding the beta ratio. No matter what the risk free rate is and wheather it changes or not your beta ratio will stay the same. Forget about interest rates and do a regression of your stock against a relevant index (the right regression would of course be against the global index of stocks, commodities, properties, human capital etc....). If the alpha is significantly different (do t-student test) than any risk free rate that you feel is relevant it means that you may have used wrong interest rates (that as I said before has no connection with the beta). Financial theory says that the rate of a zero coupon bond with the same maturity (how long are you willing to keep the shares) as your investment is the most relevant rate. Say your investment is 5 years. An alternative (competitive) investment is to invest in a zero coupon bond with maturity 5 years and a yield say 4.5%. This is risk free investment (exclusively if you keep it until the end of the five years) because if you keep it until the end of the for sure will have a return of 4.5%. The yield of this specific bond may change over the 5 years but for you that will keep it until the maturity will be 4.5% FOR SURE (mean reverting process)! That of course doesn't mean that there is always a bond with the same maturity as your investment. By however using prices of different zero coupon bond and doing an extrapolation you can approximate risk free yields for any investment (that is called yield curve).

DimitrisLancs, olbi, rcohen, thanks so much for replies.I realized my mistake is to try to find the risk-free rates for all point in time in the past, and that is not necessary. I can decide the beta of a stock by doing regression on past data of the stock and the index. After that I have to decide the cost of equity capital at this moment, which depends on the period (from now into the future), say 5 years, in that case, a 5 year zero coupon bond is very suitable for the risk-free rate. I was told to use the average of 1-year savings and loan rate, which hense has two problems:firstly, 1-year period doesn't match my planned investment periodsecondly, even if I take 5-year savings and loan rate, probably by extrapolating, these rates are floating rather than fixed, in general, which are not desired.BTW, I am kinda "pedantic" in nature, because I am experienced at VLDB, I tend to take as longer period of data as possible.

If I am not wrong you are asked to find the cost of capital for the equities for each of the next 5 years. First of all bear in mind that the historic beta is not the best estimator of the future beta (adjusted beta, vasicek beta etc. are better - look in Elton, Gruber, Brown book). To find the right risk free rates for each of the next 5 years you need to calculate the future rates F(1,2), F(2,3), F(3,4), F(4,5). You can do so by using rates of different maturities. The future rates are risk free. You can be sure to obtain a F(4,5) yield (that will be used to find the cost of capital for the 5th year) by being short in a 4 year bond and long in a 5 year bond (or borrowing money for 4 years and lending for 5). Even if the interest rates change tommorow you will be guaranted to have a F(4,5) yield.

Dimitri,I think we need to discuss about the forward risk free rates a little more. When applied to CAPM, the forward risk free rates do not make sense,I think. The state of economy would change which would force the central bankers alter the theory of interest rates by force. WEll, can you post a paper where CAPM has been applied to market and have been verified.BUT, on the face of it what you wrote is absolutely correct. From an interest rate point of view where there is no equity exposure to be worried about, the process is correct and we alll know that. may be you can explain a bit more.

Six years of monthly returns is nowhere near long enough for precise estimates of E[R_m], much less E[R_i] for typical equities. With all historical data, it's hard to show convincingly that E[R_m] is even positive.You can subtract the 30-day T-bill rate (or equivalent for non-dollar currencies) from each monthly returns. Then you can force your regression through zero. The CAPM is a one-period model, so there are inconsistencies when applying it to periods longer than a year, or to considering intraperiod risk (like interest rate risk of a term bond). There are multi-period versions if that is important in your application.