1. Investment Analysis and Portfolio Management Lecture 3 Gareth Myles

2. FT 100 Index

3. £ and $

4. Risk Variance The standard measure of risk is the variance of return or Its square root: the standard deviation Sample variance The value obtained from past data Population variance The value from the true model of the data

5. Sample Variance General Motors Stock Price 1962-2008

6. Sample Variance Year93-9494-9595-9696-9797-98 Return %36.0-9.217.67.234.1 Year98-9999-0000-0101-0202-03 Return %-1.225.3-16.612.7-40.9 Return on General Motors Stock 1993-2003

7. Sample Variance Graph of return

8. Sample Variance With T observations sample variance is The standard deviation is Both these are biased estimators The unbiased estimators are

9. Sample Variance For the returns on the General Motors stock, the mean return is 6.5 Using this value, the deviations from the mean and their squares are given by Year93-9494-9595-9696-9797-98 29.5-15.711.10.727.6 870.25246.49123.210.49761.76 Year98-9999-0000-0101-0202-03 -7.718.8-23.16.2-47.4 59.29353.44533.6138.442246.76

10. Sample Variance After summing and averaging, the variance is The standard deviation is This information can be used to compare different securities A security has a mean return and a variance of the return

11. Sample Covariance The covariance measures the way the returns on two assets vary relative to each other Positive: the returns on the assets tend to rise and fall together Negative: the returns tend to change in opposite directions Covariance has important consequences for portfolios AssetReturn in 2001Return in 2002 A 102 B 2

12. Sample Covariance Mean return on each stock = 6 Variances of the returns are Portfolio: 1/2 of asset A and 1/2 of asset B Return in 2001: Return in 2002: Variance of return on portfolio is 0

13. Sample Covariance The covariance of the return is It is always true that i. ii.

14. Sample Covariance Example. The table provides the returns on three assets over three years Mean returns Year 1Year 2Year 3 A 101211 B 101412 C 69

15. Sample Covariance Covariance between A and B is Covariance between A and C is

16. Variance-Covariance Matrix Covariance between B and C is The matrix is symmetric

17. Variance-Covariance Matrix For the example the variance-covariance matrix is

18. Population Return and Variance Expectations: assign probabilities to outcomes Rolling a dice: any integer between 1 and 6 with probability 1/6 Outcomes and probabilities are: {1,1/6}, {2,1/6}, {3,1/6}, {4,1/6}, {5,1/6}, {6,1/6} Expected value: average outcome if experiment repeated

19. Population Return and Variance Formally: M possible outcomes Outcome j is a value x j with probability  j Expected value of the random variable X is The sample mean is the best estimate of the expected value

20. Population Return and Variance After market analysis of Esso an analyst determines possible returns in 2010 The expected return on Esso stock using this data is E[r Esso ] =.2(2) +.3(6) +.3(9) +.2(12) = 7.3 Return26912 Probability0.20.3 0.2

21. Population Return and Variance The expectation can be applied to functions of X For the dice example applied to X 2 And to X 3

22. Population Return and Variance The expected value of the square of the deviation from the mean is This is the population variance

23. Modelling Returns States of the world Provide a summary of the information about future return on an asset A way of modelling the randomness in asset returns Not intended as a practical description

24. Modelling Returns Let there be M states of the world Return on an asset in state j is r j Probability of state j occurring is  j Expected return on asset i is

25. Modelling Returns Example: The temperature next year may be hot, warm or cold The return on stock in a food production company in each state If each states occurs with probability 1/3, the expected return on the stock is StateHotWarmCold Return101218

26. Portfolio Expected Return N assets M states of the world Return on asset i in state j is r ij Probability of state j occurring is  j X i proportion of the portfolio in asset i Return on the portfolio in state j

27. Portfolio Expected Return The expected return on the portfolio Using returns on individual assets Collecting terms this is So

28. Portfolio Expected Return Example: Portfolio of asset A (20%), asset B (80%) Returns in the 5 possible states and probabilities are: State12345 Probability0.10.20.40.10.2 Return on A 26912 Return on B 51043

29. Portfolio Expected Return For the two assets the expected returns are For the portfolio the expected return is

30. Population Variance and Covariance Population variance The sample variance is an estimate of this Population covariance The sample covariance is an estimate of this

31. Population Variance and Covariance M states of the world, return in state j is r ij Probability of state j is  j Population variance is Population standard deviation is

32. Population Variance and Covariance Example: The table details returns in five possible states and the probabilities The population variance is State12345 Return5263 Probability0.10.20.40.10.2

33. Portfolio Variance Two assets A and B Proportions X A and X B Return on the portfolio r P Mean return Portfolio variance

34. Population covariance between A and B is For M states with probabilities  j Portfolio Variance

35. The portfolio return is So Collecting terms

36. Squaring Separate the expectations Hence Portfolio Variance

37. Example: Portfolio consisting of 1/3 asset A 2/3 asset B The variances/covariance are The portfolio variance is

38. Correlation Coefficient The correlation coefficient is defined by Value satisfies perfect positive correlation rArA rBrB

39. Correlation Coefficient perfect negative correlation Variance of the return of a portfolio rBrB rArA

40. Correlation Coefficient Example: Portfolio consisting of 1/4 asset A 3/4 asset B The variances/correlation are The portfolio variance is

41. General Formula N assets, proportions X i Portfolio variance is But so

42. Effect of Diversification Diversification: a means of reducing risk Consider holding N assets Proportions X i = 1/N Variance of portfolio

43. Effect of Diversification N terms in the first summation, N[ N-1] in the second Gives Define Then

44. Effect of Diversification Let N tend to infinity (extreme diversification) Then Hence In a well-diversified portfolio only the covariance between assets counts for portfolio variance