In ten words or less for each what is the optimal portfolio(s)/position: 1) Sign Constrained MV Optimization with no leverage 2) Allowing for Borrowing/Lending

Note exactly sure what you mean but: 1) Along the frontier with return >= GMV portfolio return 2) On Capital Asset Line - Mix of Rf Asset & Tangency Portfolio

That’s basically what I’m asking… just looking for a quick and dirty way to remember what i need to do. I get the corner portfolio theorem… And I think at this point I get how we figure out what to do when we’re allowed to borrow/lend. But I’m just looking for some confirmation. The way i see it when borrowing/lending is allowed: You find the portfolio along the efficient frontier that has the highest sharpe ratio that still satisfies your investers return/risk requirements. Then set up an equation to solve for this and the risk free asset’s weights: Required Return = w(Return on Highest SR Portfolio) + (1-w)(risk free rate) And if “w” turns out to be negative for the risk free rate, we borrow. If its positive we lend. Is this correct? Or have i screwed it up

If borrowing is allowed you pick the portfolio with the highest sharpe ratio - which might be lower than your required return. But then you leverage up the return by borrowing. ie - the return requirement is 8%, the highest sharpe ratio portfolio has a return of 7. You do the same formula: 8 = w1(7) + (1-w1)*risk free rate, and you’ll find some number over 100% for the w1. You figure you’re getting that 7% and pay the risk free, in an amount so that you get that extra 1% return you need. Main idea is that if borrowing is allowed you can achieve the return with lower risk, than would have been the case if you picked another portfolio with a lower sharpe ratio instead of borrowing. On the other side if the portfolio with the highest sharpe ratio has a return higher than what you need, you can lend out and earn the risk free rate. I think it’s usually presented the other way in questions though.

you have it right…pretty much, except 1) it’s possible that that the highest sharpe ratio portfolio won’t meet the return requirement (but it will after you borrow) so don’t dismiss the highest sharpe portfolio just because it’s return is less than you need. 2) I think you meant to say if “1-w” turns out to be negative for risk free rate…

Thats how i see it, but if only lending is allowed then you have to forget all about the tangency portfolio and go back to corner portfolio weighting

what does GMV mean? also how do corner portfolios come up?

GMV = global minimum variance. Corner portfolios come up when there is restriction on short sale.

in less than 10 words, why dont we have a smooth line of efficient portfoilio when there is restriction on short sales?