Arbitrage in a Single Index Model

Question

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 1

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.