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Beta. Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES. Measuring the risk of an individual asset. The measure of risk of an individual asset in a portfolio has to incorporate the impact of diversification .

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## Beta

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**Beta**Prof. André FarberSOLVAY BUSINESS SCHOOLUNIVERSITÉ LIBRE DE BRUXELLES**Measuring the risk of an individual asset**• The measure of risk of an individual asset in a portfolio has to incorporate the impact of diversification. • The standard deviation is not an correct measure for the risk of an individual security in a portfolio. • The risk of an individual is its systematic risk or market risk, the risk that can not be eliminated through diversification. • Remember: the optimal portfolio is the market portfolio. • The risk of an individual asset is measured by beta. • The definition of beta is: A.Farber Vietnam 2004**Beta**• Several interpretations of beta are possible: • (1) Beta is the responsiveness coefficient of Rito the market • (2) Beta is the relative contribution of stock i to the variance of the market portfolio • (3) Beta indicates whether the risk of the portfolio will increase or decrease if the weight of i in the portfolio is slightly modified A.Farber Vietnam 2004**Beta as a slope**A.Farber Vietnam 2004**A measure of systematic risk : beta**• Consider the following linear model • RtRealized return on asecurity during period t • Aconstant :areturn that the stock will realize in any period • RMtRealized return on the market as awhole during period t • Ameasure of the response of the return on the security to thereturn on the market • utAreturn specific to the security for period t(idosyncratic returnor unsystematic return)- arandom variable with mean 0 • Partition of yearly return into: • Market related part ßRMt • Company specific part a+ut A.Farber Vietnam 2004**Beta - illustration**• Suppose Rt = 2% + 1.2 RMt+ ut • If RMt= 10% • The expected return on the security given the return on the market • E[Rt|RMt] = 2% + 1.2 x 10% = 14% • If Rt= 17%, ut = 17%-14% = 3% A.Farber Vietnam 2004**Measuring Beta**• Data: past returns for the security and for the market • Do linear regression : slope of regression = estimated beta A.Farber Vietnam 2004**Decomposing of the variance of a portfolio**• How much does each asset contribute to the risk of a portfolio? • The variance of the portfolio with 2 risky assets • can be written as • The variance of the portfolio is the weighted average of the covariances of each individual asset with the portfolio. A.Farber Vietnam 2004**Example**A.Farber Vietnam 2004**Beta and the decomposition of the variance**• The variance of the market portfolio can be expressed as: • To calculate the contribution of each security to the overall risk, divide each term by the variance of the portfolio A.Farber Vietnam 2004**Marginal contribution to risk: some math**• Consider portfolio M. What happens if the fraction invested in stock Ichanges? • Consider a fraction Xinvested in stock i • Take first derivative with respect to X for X = 0 • Risk of portfolio increase if and only if: • The marginal contribution of stock i to the risk is A.Farber Vietnam 2004**Marginal contribution to risk: illustration**A.Farber Vietnam 2004**Beta and marginal contribution to risk**• Increase (sightly) the weight of i: • The risk of the portfolio increases if: • The risk of the portfolio is unchanged if: • The risk of the portfolio decreases if: A.Farber Vietnam 2004**Inside beta**• Remember the relationship between the correlation coefficient and the covariance: • Beta can be written as: • Two determinants of beta • the correlation of the security return with the market • the volatility of the security relative to the volatility of the market A.Farber Vietnam 2004**Properties of beta**• Two importants properties of beta to remember • (1) The weighted average beta across all securities is 1 • (2) The beta of a portfolio is the weighted average beta of the securities A.Farber Vietnam 2004

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