Constructing a Zero-Cost Position Against Positive Single-Index Alpha
Summary
This exchange explains how a portfolio with positive alpha in a single-index model can be paired with market exposure to isolate its modeled excess return. For each unit invested in the portfolio, the construction shorts 1.4 units of the market portfolio and invests 0.4 units at the risk-free rate. The short market position offsets the portfolio’s 1.4 beta, while the risk-free investment balances the initial cash flows, leaving no net upfront investment under the stated setup.
The resulting expected profit is the model’s 0.04 alpha. The answer emphasizes that this amount is earned only in expectation, so the position is not a guaranteed arbitrage in the strict sense of a risk-free payoff. The argument depends on the single-index relation being valid, the portfolio and market positions being available at the assumed terms, and the risk-free borrowing or lending rate being applicable. It is an algebraic illustration, not evidence that such a mispricing can be traded profitably after costs or estimation error.
Key ideas
- A portfolio’s beta exposure can be offset by taking an opposing position in the market portfolio.
- The market short is scaled to match the portfolio’s stated beta.
- A risk-free position can balance the initial cash flows in the illustrated construction.
- The residual expected return equals the stated alpha under the model.
- A positive expected payoff does not guarantee a risk-free realized profit.
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Full text
# Arbitrage in a Single Index Model
# Arbitrage in a Single Index Model
Simple question really, but I'm very confused by the starting point. Let's assume that we have a portfolio whose excess returns can be described by the following equation from the single index model:
E(R) = .04 + 1.4*(Risk Premium of the Market)
Obviously, alpha is .04, beta is 1.4. This portfolio is underpriced, as it lies outside the Security Market Line and has a positive alpha.
Now, if I wanted to exploit this and earn that .04 alpha, I understand I'd create some sort of tracking portfolio to mimic the 1.4 beta. This is where I'm confused, I see these questions and they state that we'd borrow .4 at the risk free rate and buy a portfolio with 1.4 beta. This is where all my questions start.
How does this make any sense?? Firstly, what's the base assumption of how much money we have? 1 unit? This means 1 unit lets us buy a portfolio of beta = 1? If we borrow .4 at the risk-free rate, how does that allow us to buy a 1.4 beta portfolio (doesn't this assume price is strictly proportional to beta and NOTHING else?)
Any insight would be very much appreciated.
## Answer by Xiaohuolong (score 2)
https://quant.stackexchange.com/a/60538
Suppose the risk-free rate of return is $R^f$, the rate of return of the portfolio here is $R^p$, and the market return is $R^m$. We know $$\mathbb{E}[R^p-R^f]=0.04+1.4\mathbb{E}[R^m-R^f]$$
If we put \$1 into the portfolio, short \$1.4 the market portfolio, and invest the \$0.4 at the risk-free rate, then the expected wealth will be $$\mathbb{E}[X]=R^f+0.04+1.4(\mathbb{E}[R^m]-R^f)-1.4\mathbb{E}[R^m]+0.4R^f=0.04$$ Note that no initial investment is required upfront. Arbitrage by definition means making something out of nothing, although here that profit happens only in expectation.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.