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I have question of “complete hedge strategy” as described in FRM books. Optimal number of contracts to short sell is calculated as follows: if you have 1,000 USD portfolio, Beta is 1.3 and S&P index price is assume 50$ and multiplier is 2 and If you would like to make “complete hedge” (i.e. aim for beta of zero) you should short 13 contracts (calculated as = 1,000*1.3/30*2) but I don’t see how this works in practice. Let’s assume risk free rate (RF) is 5% and Market return (KM) is 11%, then without applying any hedge, my return applying CAPM model would be: 1,000*(5%+(11%-5%)*1.3) = 128 USD. Now, assuming I have shorted 13 contracts i.e. 1,300 USD (calculated as 13*50*2) and assuming my market return is 11%, then my loss will be 143 USD (1,300 * 11%) which I will have to pay because I shorted 13 contracts (I placed bet that share price would be constant and in reality market returned 11% growth). Bottom line is that I have gained 128 USD on my portfolio and lost 143 USD on my futures hedge. My net loss is 15 USD. My question is - how is the loss of USD 15 a “complete hedge”? Should not it be the case that compete hedge leads to 5% return - which is the risk free rate of return? 

Thanks in advance.

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First, you’ve shorted 13 futures contracts. There’s no indication of futures prices in your question - so 11%*1300 isn’t correct.
Secondly, the hedge is for prices falling which is why you’ve shorted the contracts and not for prices rising.
Third, if you short less than 13 contracts, you’re underhedged according to the formula.

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