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I don't understand what is the mathematical and financial role of risk free rate in the CAPM formula . Why do we need to add 10 years treasury yield to the formula then substract it again from the market return and multiply it by beta . What's the point of adding a different financial product to the formula to calculate an expected return of a stock that is related onlt to the stock market?

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The CAPM simply states that, in equilibrium, $$\mathbb{E}[R_i]-R_f=\beta_i(\mathbb{E}[R_m]-R_f),$$ that is, expected excess returns of assets are proportional to the expected market excess returns. It's much more convenient to think in terms of excess returns.

You can interpret the risk-free rate as funding cost. You somehow gotta buy the asset first. This requires you to borrow some money (and pay the risk-free rate to the lender). Equivalently, you may want to interpret the infamous $$\mathbb{E}[R_i]=R_f+\beta_i(\mathbb{E}[R_m]-R_f)$$ as $$\text{Expected Return} = \text{Time Premium} + \text{Risk Premium}.$$ So, what do you expect Apple to pay over the next period? You want to be compensated for not investing your money somewhere else (opportunity cost). Secondly, you want to be compensated for holding a risky asset. The CAPM states that you receive more returns depending on the systematic risk of the asset (measured by market beta).

Treasury yields are indeed a common proxy for a risk-free rate. Ken French's website provides the one-month Treasury bill rate (from Ibbotson Associates). Bob Merton once said the risk-free rate is only asset that's clearly defined. All other assets (so called risky assets) are simply defined by not being risk-free.

The existence of risk-free rate is crucial for the CAPM derivation. However, you can write down a CAPM version without a risk-free rate. You essentially only need a zero-beta asset. This was found by Fischer Black (1972).

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When it comes to return on investment, there are two dimensions one needs to consider - time and risk. The kind of questions we want an answer for is: for an investment such that I expect payout in this amount of time with these kinds of risks, what kind of return should I expect, or what kind of return does the market demand? If we don't consider risk, say you lend to US government through purchasing some treasury securities (which is generally regarded as having no chance of default), you will get some rate of return to compensate for your lending them your money, and this is the baseline rate of return you would expect on a risk-free investment. CAPM adds in risk - it tells you for an additional unit of risk that you bear through some investment (here the risk specifically is co-variability with the general market, CAPM shows that you only get compensated for bearing systematic risk), how much additional expected return you get for bearing that risk. This is in addition to the baseline risk-free return that you would expect from a risk-free investment, and CAPM says that this expected excess return is some $\beta$ multiplied by the market expected excess return.

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Think of the risk-free asset as a benchmark used by the CAPM. Think of the CAPM as a tool for measuring excess expected returns (NOT expected returns). What are excess expected returns in excess of, according to the CAPM? The risk-free rate. Why care about the risk-free asset? Because it is what we are measuring risky asset returns in excess of, as just said, but also because the risk-free asset has 0 volatility unlike the risky assets. That's why the risk-free asset is called risk-free because it has 0 volatility (zero risk). Risky assets on the other hand have a volatility greater than 0 and therefore are risky.

With this said, what better benchmark is there for CAPM to use, for measuring excess expected returns of risky assets, than their polar opposite: the risk-free asset?

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