A perfectly hedged position is meant to be totally insulated to market movements. Regardless of the market moves the, portfolio is supposed have no impact.
So the question is, under what circumstance do you initiate such a hedge? Well, you do that when you expect the market correct by a great degree.
Do remember, hedging is a way of deliberately insulating your position/portfolio. Hence any movement in the market does not have an impact on your position/portfolio. Therefore there is no question of making a profit or a loss when you are perfectly hedged. Your position/portfolio stays intact.
I think the hedged portfolio becomes perfectly hedged portfolio only in the case of scenario where the market starts behaving opposite to your portfolio. The capital required to hedge portfolio would be less than the capital required to have that portfolio. I think, hedging is done that way.
Is it like one already has certain number of shares or positions, but the person fears market would go down. So instead of squaring off his positions he would like to hedge and insulate it, so at a later time whenever he gains confidence on positive movement in his portfolio, will he release the hedge???
If the person fears market would move down and hedges his portfolio, instead if market moves up (his portfolio is gaining but his hedge is losing), can he come out of the hedge without any loss? Or should he wait until the market comes down back to his initial hedged state?
Absolutely, that is how a perfect hedge works.
When he hedges he is of the opinion market will correct. However if it moves up, the portfolio losses out on an opportunity make gains. The trader can release the hedge at any point he wishes.