# Beta

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Slide 1

﻿Beta Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES

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Measuring the danger of an individual resource The measure of danger of an individual resource in a portfolio needs to consolidate the effect of expansion . The standard deviation is not a right measure for the danger of an individual security in a portfolio. The danger of an individual is its methodical hazard or market chance, the hazard that can not be dispensed with through expansion. Keep in mind: the ideal portfolio is the market portfolio. The danger of an individual resource is measured by beta . The meaning of beta is: A.Farber Vietnam 2004

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Beta Several understandings of beta are conceivable: (1) Beta is the responsiveness coefficient of R i to the market (2) Beta is the relative commitment of stock i to the change of the market portfolio (3) Beta demonstrates whether the danger of the portfolio will increment or reduction if the heaviness of i in the portfolio is somewhat altered A.Farber Vietnam 2004

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Beta as an incline A.Farber Vietnam 2004

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A m easure of efficient hazard : beta Consider the accompanying direct model R t Realized profit for a security amid period t  A steady : an arrival that the stock will acknowledge in any period R Mt Realized profit for the market all in all amid period t  A measure of the reaction of the arrival on the security to the arrival available u t An arrival particular to the security for period t (idosyncratic return or unsystematic give back)- an irregular variable with mean 0 Partition of yearly return into: Market related part ß R Mt Company particular section a + u t A.Farber Vietnam 2004

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Beta - outline Suppose R t = 2% + 1.2 R Mt + u t If R Mt = 10% The normal profit for the security given the arrival available E[ R t | R Mt ] = 2% + 1.2 x 10% = 14% If R t = 17%, u t = 17%-14% = 3% A.Farber Vietnam 2004

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Measuring Beta Data: past returns for the security and for the market Do straight relapse : slant of relapse = assessed beta A.Farber Vietnam 2004

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Decomposing of the difference of a portfolio How much does every benefit add to the danger of a portfolio? The difference of the portfolio with 2 unsafe resources can be composed as The fluctuation of the portfolio is the weighted normal of the covariances of every individual resource with the portfolio. A.Farber Vietnam 2004

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Example A.Farber Vietnam 2004

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Beta and the deterioration of the fluctuation The change of the market portfolio can be communicated as: To figure the commitment of every security to the general hazard, isolate every term by the difference of the portfolio A.Farber Vietnam 2004

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Marginal commitment to hazard: some math Consider portfolio M . What happens if the division put resources into stock I changes? Consider a division X put resources into stock i Take first subordinate as for X for X = 0 Risk of portfolio increment if and just if: The peripheral commitment of stock i to the hazard is A.Farber Vietnam 2004

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Marginal commitment to hazard: delineation A.Farber Vietnam 2004

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Beta and minimal commitment to hazard Increase (sightly) the heaviness of i : The danger of the portfolio increments if: The danger of the portfolio is unaltered if: The danger of the portfolio diminishes if: A.Farber Vietnam 2004

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Inside beta Remember the relationship between the connection coefficient and the covariance: Beta can be composed as: Two determinants of beta the connection of the security come back with the market the instability of the security in respect to the unpredictability of the market A.Farber Vietnam 2004

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Properties of beta Two importants properties of beta to recollect (1) The weighted normal beta over all securities is 1 (2) The beta of a portfolio is the weighted normal beta of the securities A.Farber Vietnam 2004