Excess Returns as Zero-Cost Spreads and CAPM Risk Premia
Summary
The document distinguishes common uses of excess return: an asset's return minus the risk-free rate, and one asset's return minus another benchmark's return, such as the market. Both are differences between returns and can be interpreted as the payoff on a portfolio that is long one position and short an equal-dollar amount of another, giving the portfolio zero initial cost under the stated setup. The term risk premium is commonly used for return above the risk-free rate, while market excess return often refers to the market's return above the risk-free rate.
For CAPM, the response gives alpha as the asset's average risk-free excess return less beta times the market's average risk-free excess return. It also notes that excess returns form a vector space: adding them or scaling them preserves the property. The discussion is conceptual and does not resolve which measure best suits every statistical factor; the choice depends on the model and benchmark. The zero-cost interpretation assumes matching dollar exposures and abstracts from financing, transaction costs, and implementation constraints.
Key ideas
- The difference between any two returns can be treated as an excess return.
- An equal-dollar long-short portfolio has zero initial cost and pays the difference in returns.
- Asset return minus the risk-free rate is a common measure of risk premium.
- CAPM alpha compares the asset's average risk-free excess return with its beta-adjusted market risk premium.
- Excess returns remain excess returns when added or scaled, under the stated definitions.
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# Interpretation of Excess Return
# Interpretation of Excess Return
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 $ER^{(2)} = R_i - R_M$
## Answer by fni (score 2)
https://quant.stackexchange.com/a/28420
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 $R^i_{t+1}-R^j_{t+1}$.
## Answer by Matthew Gunn (score 1)
https://quant.stackexchange.com/a/40592
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.