Portfolio Theory

Portfolio Theory - Many Risky Assets The purpose of this note is to luff you how to calculate the optimal investment portfolio and the e?cient frontier in the case of many risky assets and one risk faint asset. The examples in this note ar demonstrated in the buy the farm ?le portfolio theory.xls posted on Blackboard. I. Basic De?nitions We would like to mannikin an optimal portfolio out of many risky assets (possibly stocks). Suppose we fork out n risky assets (n?2). Using historical data we tooshie calculate the judge come backs and the variance-covariance intercellular substance of these n assets. The anticipate returns are given by a column transmitter of holding n × 1: ? ? ? R=? ? ? µ1 µ2 . . µn ? ? ? ?. ? ? The variance-covariance matrix is given by an n×n matrix: ? ? ? 11 ? 12 ... ? 1n ? ? 21 ? 22 ... ? 2n ? ? ? . ?. V =? . ? ? ? . . ? ? n1 ? n2 ... ? nn A portfolio is mediocre an array of proportions - the percentage of capital we apportion to each asset. Thus, a portfolio is a vector: ? ? ? x=? ? ? such that n x1 x2 . . xn ? ? ? ?, ? ? xi = 1. i=1 (*) 1 typically we use a column vector for a portfolio, just we can also sometimes use a path vector. This does not matter. Notice that xi can be negative. why? II. A.

Expectation, Variance and Covariance of Portfolio Returns Expected Return of a Portfolio The expected return of a portfolio x is µx = x1 µ1 + x2 µ2 + ... + xn µn . Using matrix notation we have µx = xT R. Example: Suppose that the vector of expected returns is ? ? 0.1 R = ? 0.12 ? . 0.08 prefigure the portfolio: ? 0.2 x = ? 0.5 ? . 0.3 The expected return of the por tfolio is ? 0.1 µx = (0.2 0.5 0.3) ? 0.12 ?! = 0.104 = 10.4%. 0.08 ? Consider the portfolio ? 0.2 y = ? ?0.3 ? . 1.1 The expected return on this portfolio is ? 0.1 µy = (0.2 ? 0.3 1.1) ? 0.12 ? = 0.072 = 7.2%. 0.08 In stand out: use TRANSPOSE( ) and MMULT( ). ? ? ? 2 B. Variance of a Portfolio The variance of portfolio x is given by ? 2 = xT V x. x Example: Consider the...If you want to get a all-inclusive essay, order it on our website: OrderEssay.net

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