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Beta

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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  1. Beta Prof. André FarberSOLVAY BUSINESS SCHOOLUNIVERSITÉ LIBRE DE BRUXELLES

  2. 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

  3. 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

  4. Beta as a slope A.Farber Vietnam 2004

  5. 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

  6. 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

  7. Measuring Beta • Data: past returns for the security and for the market • Do linear regression : slope of regression = estimated beta A.Farber Vietnam 2004

  8. 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

  9. Example A.Farber Vietnam 2004

  10. 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

  11. 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

  12. Marginal contribution to risk: illustration A.Farber Vietnam 2004

  13. 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

  14. 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

  15. 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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