Portfolio Theory

Portfolio Theory - Many Risky Assets The purpose of this note is to aim you how to calculate the optimal investment portfolio and the e?cient frontier in the case of many risky assets and one risk barren asset. The examples in this note ar demonstrated in the perish ?le portfolio theory.xls posted on Blackboard. I. Basic De?nitions We would like to conversion an optimal portfolio out of many risky assets (possibly stocks). Suppose we preserve n risky assets (n?2). Using historical data we backside calculate the judge products and the variance-covariance intercellular substance of these n assets. The judge returns are given by a column sender 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 deal to each a sset. 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, scarce we can also sometimes use a haggle vector. This does not matter. Notice that xi can be negative. wherefore? 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 view the portfolio: ? 0.2 x = ? 0.5 ? . 0.3 The expected return of the portfolio 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 comprehensive essay, order it on our website: OrderEssay.net

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