Financial Analysis The Price of Risk. Skema Business School. Portfolio Management 1.

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1 Financial Analysis The Price of Risk Skema Business School Portfolio Management Course Outline Introduction (lecture ) Presentation of portfolio management Chap.2,3,5 Introduction to Bloomberg Modern Portfolio Theory (lectures 2-4) The risk return framework Chap. Efficient capital markets Chap.6 The price of risk Chap.7,8 Asset pricing models Chap.9 Fundamental Analysis (lectures 5-8) Analysis of financial statement Chap.0 Industry analysis Chap.2,3 Absolute and relative valuation analysis Chap., 4 Stock market valuation analysis Chap.2 Technical analysis (lecture 9) Chap.5 The asset management industry (lecture 0) Portfolio management strategies Chap.6 The different types of investment companies Chap.24 Evaluation of portfolio performance Chap.25 NB: chapters refer to Reilly & Brown 8th and 9th ed. Portfolio Management 2

2 Underlying assumption Stock returns follow a log-normal distribution Consequently they can be fully characterized by only two parameters The mean: expected return The variance of the return Portfolio Management 3 Portfolio Markowitz [952] Find the portfolios that maximize the return for a given level of risk (or minimize the risk for a given level of return) Notation E(r i ) = the expected rate of return for asset i s i = standard-deviation of asset i x i the weight of the asset i in the portfolio p Portfolio Management 4 2

3 Portfolio Expected Return for a Portfolio E(r p ) = S i x i E(r i ) Portfolio risk s p ² = s²(s i x i E(r i )) = S i S j x i x j cov ij where cov ij = E[ (r i -E(r i )) (r j -E(r j )) ] cov ij is the covariance between returns of asset i and asset j It s a measure of the degree to which two variables move together relative to their individual mean values over time Portfolio Management 5 Covariance and Correlation The correlation coefficient is obtained by standardizing (dividing) the covariance by the product of the individual standard deviations Correlation coefficient varies from - to + Covij ij s s where: the correlation coefficient of returns ij s i j i j the standard deviation of returns for asset i s the standard deviation of returns for asset j Portfolio Management 6 3

4 Correlation Coefficient It can vary only in the range + to -. A value of + would indicate perfect positive correlation. This means that returns for the two assets move together in a completely linear manner. A value of would indicate perfect correlation. This means that the returns for two assets have the same percentage movement, but in opposite directions Portfolio Management 7 Portfolio Portfolio risk (next) s p ² = S i S j x i x j s ij = S i x i ²s i ² + S i S j,ji x i x j cov ij The portfolio risk is composed by the weighted sum of Every individual asset return variances individual asset return covariance The portfolio risk doesn t only depend on asset risk by which it is composed, but also relations between these assets. Portfolio Management 8 4

5 Portfolio Portfolio risk (next) Example of a two asset portfolio r p = x r + x 2 r 2 s p ² = x ²s ² + x 2 ²s 2 ² + x x 2 cov 2 + x 2 x cov 2 = x ²s ² + x 2 ²s 2 ² + 2 x x 2 cov 2 = x ²s ² + x 2 ²s 2 ² + 2 x x 2 2 s s 2 Portfolio Management 9 Portfolio Risk-Return Plots for Different Weights E(R) With two perfectly correlated assets, it is only possible to create a two asset portfolio with riskreturn along a line between either single asset 2 = Standard Deviation of Return Portfolio Management 0 2 5

6 Portfolio Risk-Return Plots for Different Weights E(R) 0.20 With uncorrelated assets it is possible 0.5 to create a two asset portfolio with 0.0 lower risk than either single asset = 0 h i j k g f 2 2 = Standard Deviation of Return Portfolio Management Portfolio Risk-Return Plots for Different Weights E(R) With correlated assets it is possible to create a two asset portfolio between the first two curves 2 = 0 h i j k g f 2 2 = = Standard Deviation of Return Portfolio Management 2 6

7 Portfolio Risk-Return Plots for Different Weights E(R) With 2 = f 0.20 negatively g 2 correlated h assets it is = 0 i j possible to 2 = +.00 create a two k 0.0 asset portfolio with much 2 = lower risk than either single asset Standard Deviation of Return Portfolio Management 3 Portfolio Risk-Return Plots for Different Weights E(R) = = h 2 = 0 i j k 2 = With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk Standard Deviation of Return Portfolio Management 4 g f 2 2 =

8 Risk (standard-deviation) Portfolio Portfolio risk (next) When n, the asset portfolio number increases, the portfolio risk decreases and tends to the average covariance of the assets When n, Lim s p ² = S i S j,ji x i x j s ij The portfolio risk depend on the covariance more than on the assets variance Portfolio Management 5 Portfolio Portfolio risk (next) Number of assets and risk A limited number of assets is enough to obtain a very good diversification 20 assets => 90% of the possible maximum diversification 20 Number of assets Portfolio Management 6 8

9 Portfolio The diversification improve the couple risk/return For a same level of return, the risk decreases For a same level of risk, the return increases Diversification statement Chose uncorrelated assets Increase the number of asset in the portfolio Portfolio Management 7 Portfolio Portfolio Management 8 9

10 Portfolio Portfolio Management 9 The Efficient Frontier The Efficient Frontier : The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return Frontier will be portfolios of investments rather than individual securities Exceptions being the asset with the highest return and the asset with the lowest risk Portfolio Management 20 0

11 The Efficient Frontier When the number of assets increases, calculation become very cumbersome Use of the matrix calculation (n assets) R Expected return matrix V Variance-covariance matrix X p Weighting of the portfolio assets p Constraint X p 'U= (U vector column composed of ) r R r n V s s n sn s nn x p X p xnp U Portfolio Management 2 The Efficient Frontier Let p be a portfolio Portfolio return r p = X p 'R Portfolio variance s p ² = X p 'VX p Calculation of the efficient frontier Minimize s p ², for a given level of r p with the constraint X p 'U= Portfolio Management 22

12 The Efficient Frontier The efficient frontier is given by s² = (a - 2br + cr²) / (ac - b²) with a=r'v - R b=r'v - U c=u'v - U The minimum variance of the portfolio is X 0 = /c V - U r 0 = b/c s 0 ² = /c Portfolio Management 23 The Efficient Frontier Efficient Frontier maximum return for given risk E(r) The grey zone give all the possible risk/return combinations risk/return The red curve represent the Efficient frontier Risk = s² Portfolio Management 24 2

13 Asset Allocation Optimizer See XAAO.xls on Blomberg Excel Template XLTP Portfolio Management 25 Choice of the optimal portfolio The optimal portfolio choice with n risked assets E(r) U 3 U 2 U U 3 U 2 U B G Risk = s Each investor chose a portfolio located on the efficient frontier Mr Blue choses portfolio B and Mrs. Green choses portfolio G Portfolio Management 26 3

14 Assignment To post on Knowledge before the next session Create a correlation matrix on Bloomberg with your stock plus 4 other stocks to be chosen among the CAC mid 60 index. Look at the correlation and make a brief comment. Open a the XAAO.xls on Blomberg Excel Template XLTP, and create an efficient frontier with your stock and the 4 other ones that you selected above Portfolio Management 27 4