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Unformatted text preview: Markowitz MeanVariance Portfolio Theory 1. Portfolio Return Rates An investment instrument that can be bought and sold is often called an asset . Suppose we purchase an asset for x dollars on one date and then later sell it for x 1 dollars. We call the ratio R = x 1 x the return on the asset. The rate of return on the asset is given by r = x 1 x x = R 1 . Therefore, x 1 = Rx and x 1 = (1 + r ) x . Sometimes it is possible to sell and asset that we do not own. This is called short selling . It works somewhat as follows. Suppose you wish to short (or short sell) a particular stock XXX . You begin by asking your stock broker if their firm is holding XXX in the total pool of stocks owned by all of their customers. If the brokerage does hold (or manage) some of stock XXX , you can ask them sell any number of stock XXX up to the number that they hold. This sale is credited against your account as a debt equal to the number of stock XXX they sell on your behalf. That is, your debt is not denominated in dollars, but rather in the number of stock XXX that you are shorting (i.e. your account is short by the given number of stock XXX ). On your account asset sheet, this short sale appears as a negative number associated with the shorted asset. Remember, this negative number is not denominated in dollars, but rather in the number of stocks , or assets, shorted. Due to the sale of stock XXX you have received x dollars. Eventually, you must ask the brokerage to buy the same number of stock XXX back as you originally asked them to sell and return this stock to the pool of assets that they are holding for their customers. On the date at which you return stock XXX you ask your broker to repurchase it at its current going value of x 1 dollars and return it to the brokerages asset pool. If x 1 < x , then you have made a profit on this transaction; otherwise, you have a loss. The return and rate of return on this transaction are given by R = x 1 x = x 1 x and r = ( x 1 ) ( x ) x = x 1 x x , respectively. Short selling can be very risky, and many brokerage firms do not allow it. Nonetheless, it can be profitable. 1 2 Let us now consider constructing a portfolio consisting of n assets. We have an initial budget of x dollars that we wish to assign to these assets. The amount that we assign to asset i is x i = w i x for i = 1 , 2 ,...,n , where w i is a weighting factor for asset i . We al low the weights to take negative values, and when negative it means that the asset is being shorted in our portfolio. To preserve the budget constraint we require that the weights sum to 1: n i =1 w i = 1. That is, the sum of the investments = n X i =1 w i x = x n X i =1 w i = x ....
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 Fall '09
 STAFF
 Variance

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