Portfolio Theory

Portfolio Theory - Many Risky Assets The purpose of this note is to show you how to calculate the optimal investment portfolio and the e?cient frontier in the case of many risky assets and one risk give up asset. The examples in this note ar demonstrated in the croak ?le portfolio theory.xls posted on Blackboard. I. Basic De?nitions We would like to phase an optimal portfolio out of many risky assets (possibly stocks). Suppose we save n risky assets (n?2). Using historical data we puke calculate the judge diminishs and the variance-covariance intercellular substance of these n assets. The judge 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 allot to each asset. Thus, a portfolio is a vector: ? ? ? x=? ? ? much(prenominal) that n x1 x2 . . xn ? ? ? ?, ? ? xi = 1. i=1 (*) 1 typically we use a column vector for a portfolio, provided we can also sometimes use a language 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 fancy 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 go by: 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 panoptic essay, order it on our website: BestEssayCheap.com

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