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I have calculated my optimal portfolio weights following the mean-variance framework where I go $w_1$ in the risky asset and $1-w_1$ in the risk free rate.

I get the following result: $w_1$ = 1.5, 1-$w_1$ = -0.5

My question is, how can I interpret this? Obviously I need to short the risk-free rate and go long in the risky asset. However, if I have 100 dollars to use for my portfolio, how do I allocate this amount in practice?

Moreover, it should follow that I go long with 150 dollars and short 50, but the question is, where does this money come from? How is it possible that I get money when I go short?

One option would be to standardize the sum of the weights. Hence this is not possible as then you never will invest in the risk-free rate.

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You are aware of what a "margin account" is, right? –  chrisaycock Feb 1 '13 at 18:16
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Can't questions directly be closed that are asked by someone who is obviously not working in this industry nor has sufficient exposure to quant finance? I thought that was the whole idea of this site to provide a platform for exchange between quants. –  Matt Wolf Feb 2 '13 at 1:48
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closed as off topic by Bob Jansen, Matt Wolf, John, Alexey Kalmykov, phi Feb 2 '13 at 12:47

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2 Answers

You borrow the money against the risk free rate.

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In your set up where you have just two assets, risky asset and risk-free asset, where weight of risky asset is w1, and consequently weight of risk-free asset is 1 - w1:

w1 = 1, => You invest all your money in the risky asset.

w1 = 0, => You invest all your money in the risk-free asset

0 < w1 < 1, => You invest some of your money in the risky asset and some of it in the risk-free asset

w1 > 1, => Your case : you invest all you money (100) in the risky asset, plus you borrow (50) at risk free-rate and invest it in the risky asset.

w1 < 0, => implies that you short the risky asset to invest in the risky free rate, this behavior is irrational, because we can find combinations of risky asset and risk-free asset that yield higher mean return at lower variance, thus w1 < 0, should never happen.

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