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Unformatted text preview: IEOR 4500 Introduction to Portfolio Optimization References: The classical reference is Portfolio Selection: Efficient Diversification of Investments, by Harry Markowitz. A more modern reference is: Modern Portfolio Theory and Investment Anal- ysis, by Elton, Gruber, Brown and Goetzmann, ISBN 0471238546. The general ob jective in static portfolio optimization is to make investments in assets (e.g. stocks) so that the overall portfolio provides good return, that is to say, it has a high growth rate per dollar invested while providing good protection against ”risk”. Typically, the total amount of money to be invested will be large, and, at the end of our selection process, the quantity of each asset that we actually invest in will also be large (or, rather, will not be tiny). In order to guide us in this process, all we have is a time series of data for each asset: its price history for, say, the past three years (for experimental purposes, we might instead simulate past data, e.g. using the geometric Brownian motion model). Thus, at first blush what we want is to pick a portfolio whose expected return is large. Experience says that we should also consider the standard deviation of the return, and in the end we will use an ob jective that trades-off expected return and standard deviation (the risk component). Consider any given asset, i . Let its (closing) price at day t be p ( t ) i . If we own k i units of this asset at the end of day t , then by the end of day t +1 our investment will have become p ( t +1) i k i . In other words, the return on asset i , on day t , (i.e. how much we gain, per dollar invested at day t ) is: μ ( t ) i = p ( t +1) i k i- p ( t ) i k i p ( t ) i k i = p ( t +1) i- p ( t ) i p ( t ) i (1) Suppose we were to observe the behavior of this asset (and others) over T days, say. Then we can compute its average return, ¯ μ i = 1 T T X t =1 μ ( t ) i !...
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- Fall '09
- Optimization, WI, optimization problem, portfolio optimization