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

2. A.Farber Vietnam 2004 |2|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:

3. A.Farber Vietnam 2004 |3|3 Beta Several interpretations of beta are possible: (1) Beta is the responsiveness coefficient of R i to 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

4. A.Farber Vietnam 2004 |4|4 Beta as a slope

5. A.Farber Vietnam 2004 |5|5 A measure of systematic risk : beta Consider the following linear model R t Realized return on a security during period t  A constant : a return that the stock will realize in any period R Mt Realized return on the market as a whole during period t  A measure of the response of the return on the security to the return on the market u t A return specific to the security for period t (idosyncratic return or unsystematic return)- a random variable with mean 0 Partition of yearly return into: –Market related part ß R Mt –Company specific part  + u t

6. A.Farber Vietnam 2004 |6|6 Beta - illustration Suppose R t = 2% + 1.2 R Mt + u t If R Mt = 10% The expected return on the security given the return on the market E[R t |R Mt ] = 2% + 1.2 x 10% = 14% If R t = 17%, u t = 17%-14% = 3%

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

8. A.Farber Vietnam 2004 |8|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.

9. A.Farber Vietnam 2004 |9|9 Example

10. 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

11. A.Farber Vietnam 2004 | 11 Marginal contribution to risk: some math Consider portfolio M. What happens if the fraction invested in stock I changes? Consider a fraction X invested 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

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

13. 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:

14. 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

15. 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