<p>Weights can't exceed 100%, so you know that <span class="math-container">$0 \le \alpha \le 1$</span>. If <span class="math-container">$\alpha =1$</span>, it means that 100% of the assets in the portfolio are risk-free assets. If <span class="math-container">$\alpha \neq 1$</span>, it mean that a share <span class="math-container">$\alpha$</span> of your portfolio is composed of risk-free assets, and a share <span class="math-container">$(1-\alpha)$</span> of your portfolio is composed of risky assets.</p>
<p>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 !</p>

CC BY-SA 4.0
2024-02-09T10:03:22.150Z
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.
Richard Hardy
CC BY-SA 4.0
2024-02-09T11:19:11.143Z
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 !
justaneconomist
CC BY-SA 4.0
2024-02-09T11:30:59.690Z
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?*
Richard Hardy
CC BY-SA 4.0
2024-02-09T11:56:11.240Z
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 !
justaneconomist
CC BY-SA 4.0
2024-02-09T12:05:03.557Z
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.
Richard Hardy
CC BY-SA 4.0
2024-02-10T03:22:48.747Z
Forcing the weight to be between $ -1.0$ and $+1$ would allow for shorting of the risk free rate which would be sort of be equivalent to raising cash during the optimization. This would only be relevant if the OP needed actual dollars in his-her optimization. This, in turn, would only be required if cash was needed for purchases. So, $\alpha$ possibly being negative is a valid possibility but it's probably not necessary if cash is already available.
mark leeds
CC BY-SA 4.0
2024-02-11T21:43:50.193Z
@krauuuus, that is what I thought at first, but then Luenberger later presented a different model where shorting was restricted.
wwjwjwjwj
CC BY-SA 4.0
2024-02-12T10:24:03.877Z
Does the latter model includes more than 2 assets ? Because my intuition is that, for $N=2$ assets, the restriction $\alpha \le 1$ is a sufficient condition for restriction of short-selling. However, for $N>2$ assets, it may not be sufficient thus another restriction  $\alpha \ge 0 $ is necessary ? If it's not, I don't think I have a clear answer unfortunately
justaneconomist
CC BY-SA 4.0
2024-02-12T12:02:00.173Z
It sounds like $\alpha$ is the weight given to the risk free asset. Therefore, $(1-\alpha)$  is the weight given to the non risk free asset(s) ( whether it's 1 other asset or more than one ). So, if $\alpha$ is restricted to be positive and less than 1.0, then the non risk free asset(s) cannot be shorted.
mark leeds
CC BY-SA 4.0
2024-02-09T09:48:30.400Z
2024-02-09T09:48:30.400Z
justaneconomist
<p>I am reading <em>Investment Science</em> 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?</p>
<p>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?</p>
<p><a href="https://i.sstatic.net/0dczl.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/0dczl.png" alt="enter image description here" /></a></p>

<finance><stock-market><bonds><asset-pricing><portfolio-theory>
Why is the portfolio weight of the risk-free asset capped at 1?
wwjwjwjwj
# Why is the portfolio weight of the risk-free asset capped at 1?

(Asked by: username_0 on 2024-02-08)

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?

[![enter image description here](https://i.sstatic.net/0dczl.png)](https://i.sstatic.net/0dczl.png)

**Tags:** finance, stock-market, bonds, asset-pricing, portfolio-theory

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## 1 answers

### Answer by username_1 on 2024-02-09. Score: 0

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 !

> **Comments:**
> **username_2**: 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.  
> **username_1**: 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 !  
> **username_2**: 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?*  
> **username_1**: 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 !  
> **username_2**: 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.  
> **username_3**: Forcing the weight to be between $ -1.0$ and $+1$ would allow for shorting of the risk free rate which would be sort of be equivalent to raising cash during the optimization. This would only be relevant if the OP needed actual dollars in his-her optimization. This, in turn, would only be required if cash was needed for purchases. So, $\alpha$ possibly being negative is a valid possibility but it's probably not necessary if cash is already available.  
> **username_0**: @krauuuus, that is what I thought at first, but then Luenberger later presented a different model where shorting was restricted.  
> **username_1**: Does the latter model includes more than 2 assets ? Because my intuition is that, for $N=2$ assets, the restriction $\alpha \le 1$ is a sufficient condition for restriction of short-selling. However, for $N>2$ assets, it may not be sufficient thus another restriction  $\alpha \ge 0 $ is necessary ? If it's not, I don't think I have a clear answer unfortunately  
> **username_3**: It sounds like $\alpha$ is the weight given to the risk free asset. Therefore, $(1-\alpha)$  is the weight given to the non risk free asset(s) ( whether it's 1 other asset or more than one ). So, if $\alpha$ is restricted to be positive and less than 1.0, then the non risk free asset(s) cannot be shorted.  

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