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How is excess return defined for a given asset?

There are altogether two different definitions for excess return used in the calculation of alpha and beta and I'm unable to understand which one should we apply to generate valid statistical factors.

Definition 1: Given by the difference between return rate of the stock and risk-asset (such as Treasury Bill) or $ER^{(1)} = R_i - R_f$

Definition 2: Given by the difference between return rate of the stock and return rate of the market index with same risk or $ER^{(2)} = R_i - R_M$

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They are both excess returns even though the standard convention is to talk about risk premium for $R_{t+1}-R^f$ and excess return on the market for $R_{t+1}-R_M$. If you believe in CAPM, then you need the former to compute: $$\alpha= (\bar{R}-R_f) - \beta(\bar{R}_M-R_f)$$

By definition, the excess return is the payoff of a portfolio with price zero today, i.e. if today you buy asset i you pay 1\$ to receive $R_{t+1}^i$ tomorrow, the same goes for asset j. Going long 1\$ in i and short 1\$ in j we obtain today 1\$-1\$=0 and tomorrow $R^i_{t+1}-R^j_{t+1}$.

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For any returns $R_i$ and $R_j$, we call the difference $R_i - R_j$ an excess return. The difference between any two returns is an excess return. An excess return is the return on a zero cost portfolio (because you are equally long and short).

In particular, using returns in excess of the risk free rate is quite common: $$R^e_i = R_i - R_f$$

Mathematical convenience of excess returns

Excess returns are often more mathematically convenient/elegant to work with than regular returns because the space of excess returns is a vector space, hence adding excess returns together or scaling them still gives you an excess return.

Let $R^e_1$ and $R^e_2$ be excess returns. Let $a$ and $b$ be scalars. Then:

  • $a R^e_1$ is an excess return (closure under scalar multiplication)
  • $R^e_1 + R^e_2$ is an excess return (closure under addition)

The difference $R_{\mathrm{Apple}}-R_f$ is an excess return. Multiply by the scalar $2$ and you get $2R_{\mathrm{Apple}}-2R_f$ which is also an excess return.

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