In PM we know we can borrown at free risk rate so we can invest more than 100% in the market portfolio. But my questions is can we do opposite? go short the market portfolio and invest more than 100% in the risk free rate??
Sorry for the misspelling
Yes.
Why would you want to?
In that case your portfolio have risk = 0 right? becasuse youre investing all in free risk asset.
Not if you’re short the market.
pls can you explain it better.
because in that case you would have negative standar deviation
Um . . . no.
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If you are short the market you’re exposed to market risk (if market goes up your short position loses you money). You’re investing that money into a risk-free asset. You are basically performing a service that governments provide, taking risk-free bonds on your balance sheet. You’re short a higher risk/higher return asset and long a low return risk-free asset… not the strategy you’d want to be in honestly. If you’re bearish on the market it’s better to short the stocks you are bearish on and find some other investment (alternative investments, or counter-cyclical stocks, etc) to seek a return exceeding the risk you’re taking by shorting some equity index or however you plan to short the stock market. Or use options to monetize your bearish view through spreads, strangles or straddles etc.
Cheers 
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