Investment Analysis & Portfolio Management: Risk, Variance, Covariance

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 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

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

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

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 xjwith probability pj • 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[rEsso] = .2(2) + .3(6) + .3(9) + .2(12) = 7.3

21. Population Return and Variance • The expectation can be applied to functions of X • For the dice example applied to X2 • And to X3

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 rj • Probability of state j occurring is pj • 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

26. Portfolio Expected Return • N assets • M states of the world • Return on asset i in state j is rij • Probability of state j occurring is pj • Xi 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:

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 rij • Probability of state j is pj • 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

33. Portfolio Variance • Two assets A and B • Proportions XA and XB • Return on the portfolio rP • Mean return • Portfolio variance

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

35. Portfolio Variance • The portfolio return is • So • Collecting terms

36. Portfolio Variance • Squaring • Separate the expectations • Hence

37. Portfolio Variance • 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 rB rA

39. Correlation Coefficient • perfect negative correlation • Variance of the return of a portfolio rB rA

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 Xi • Portfolio variance is • But so

42. Effect of Diversification • Diversification: a means of reducing risk • Consider holding N assets • Proportions Xi = 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