# Why is the portfolio weight of the risk-free asset capped at 1?

I am reading Investment Science by David Luenberger, and in it he creates a portfolio with a risk-free asset and a risky asset. α is the weight of the risk-free asset, and he sets α ≤ 1. Why is that?

I know α ≥ 0 corresponds to lending at the risk-free rate, but could you not theoretically short the risky asset and use those funds to lend at the risk-free rate?

Weights can't exceed 100%, so you know that $$0 \le \alpha \le 1$$. If $$\alpha =1$$, it means that 100% of the assets in the portfolio are risk-free assets. If $$\alpha \neq 1$$, it mean that a share $$\alpha$$ of your portfolio is composed of risk-free assets, and a share $$(1-\alpha)$$ of your portfolio is composed of risky assets.
• Under short-selling, the weights of an asset can be negative, meaning that $\alpha$ can be higher than one (for N=2 assets). The conditions $\alpha \le 1$ implies that short-selling is prohibited. It is often a (strong) hypothesis in the portfolio models ! Feb 9 at 11:19