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Unformatted text preview: MS&E111 Lecture 9 Introduction to Optimization May 13, 2006 Prof. Amin Saberi Scribed by: Yusuf Ozuysal 1 Applications of Linear Programming in Finance Linear programming is one of the most widely used tools in finance. Using LPs for solving optimization problems encountered in finance has been a common practice for many years and a great percentage of the courses thought on optimization in finance are devoted to LPs. While dealing with LPs in finance one mostly needs a well established background on probability theory and statistics also. However in this lecture, we will be visiting a simplified version of a general problem linear problem in finance and using this simple instance to get an idea of the general properties of such problems in finance. 1.1 Portfolio Optimization A portfolio can be defined as a collection of financial assets where the cost for the assets are known at the beginning of a period, but there are only statistical data available about the change in these costs over the period. In portfolio optimization we are trying to buy from a certain set of stocks in such a way that the value of the portfolio at the end of this period will be maximized. Example: Let us look at the matrix of relative percentage gains given below. In the matrix each column corresponds to a different asset and each row corresponds to a different event or a scenario for the period we are interested in. So the entry P ij is the percentage gain we will receive from asset j , if scenario i happens. We are trying to find the optimal distribution of assets that maximizes the estimated overall gain. P = . 2 1 1 . 2 8 1 . 05 1 . 1 1 . 2 1 . 1 1 1 . 05 1 1 1 . 1 0 . 6 1 . 05 1 . 3 0 . 9 0 . 7 0 . 2 1 . 05 1 . 2 1 . 5 0 . 8 0 . 2 1 . 05 As can be seen from the matrix, the assets offer a variety of percentage gains for different scenarios. For example, the asset in the last column can be thought as a bank investment 2 MS&E 111: Introduction to Optimization  Spring 06 where we are gaining 5 percent of our investment no matter what happens in the market.where we are gaining 5 percent of our investment no matter what happens in the market....
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