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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?

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1 Answer 1

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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.

Also, it is one of the hypothesis of many portfolio models, where you want to maximize return (or minimize variance) under the constraint that the sum of the weights in the portfolio are strictly equal to 1 !

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    $\begingroup$ To quote the OP, but could you not theoretically short the risky asset and use those funds to lend at the risk-free rate? The sum of the weights remain =1 even under shorting. $\endgroup$ Feb 9 at 10:03
  • $\begingroup$ 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 ! $\endgroup$
    – krauuuus
    Feb 9 at 11:19
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    $\begingroup$ You explain what the condition implies but the OP already understands that. Their question is, why is that?, followed by a reasonable could you not theoretically short the risky asset and use those funds to lend at the risk-free rate? $\endgroup$ Feb 9 at 11:30
  • $\begingroup$ I mean you can, theoretically, short the risky asset and use the funds to lend at the risk-free rate, but this is a different model. Here, apparently D. Luenberger presents the baseline model without short selling. Maybe other models explore further conditions with short selling ! $\endgroup$
    – krauuuus
    Feb 9 at 11:56
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    $\begingroup$ Your argument seems to boil down to, this is this way because it is this way, and a different way would be a different model. Such an argument sounds tautological to me, hence my comments. Anyway, thank you for elaborating. $\endgroup$ Feb 9 at 12:05

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