Covariance

1. Covariance And portfolio variance

2. The states of nature model • Time zero is now. • Time one is the future. • At time one the possible states of the world are s = 1,2,…,S. • Mutually exclusive, collectively exhaustive states. • This IS the population. No sampling.

3. The states of nature model • States s = 1,2,…,S. • Probabilities ps • Asset j • Payoffs Rj,s • Expected rate of return

4. = rate of return on j in state s = probability of state s = expectation of rate on j

6. Variance and standard deviation • Form deviations • Take their expectation.

8. Covariance • Form the product of the deviations • (positive if they both go in the same direction) • and take the expectation of that.

10. Covariance • It measures the tendency of two assets to move together. • Variance is a special case -- the two assets are the same. • Variance = expectation of the square of the deviation of one asset. • Covariance = expectation of the product of the deviations of two assets.

11. Correlation coefficient • Like covariance, it measures the tendency of two assets to move together. • It is scaled between -1 and +1.

12. Correlation coefficient • = covariance divided by the product of the standard deviations. • Size of deviations is lost.

13. Intuition from correlation coefficients • = 1, always move the same way and in proportion. • = -1, always move in opposite directions and in proportion. • = 0, no tendency either way.

14. Portfolio Risk and Return • Portfolio weights x and 1-x on assets A and B.

16. An amazing fact • Mixing a risky asset with a safe asset • is often safer than the safe asset.

18. Variance of portfolio return • Diversification effects

19. Portfolio risk and return,

20. Portfolio deviation Deviation squared Remember

22. Portfolio variance

23. Portfolio variance depends on covariance of the assets. Positive covariance raises the variance of the portfolio.

24. Correlation coefficient

25. Skipped the following in 2006

26. Historical data • Holding period return • Equivalent annual return • Not the same

27. Holding period return1926-1929 • Rhp is the holding period return • 1+Rhp = (1+r26)(1+r27)(1+r28)(1+r29) • = 1.1162*1.3749*1.4362*.9158 • = 2.0183592 • Rhp = 101.83592%.

28. Question: • Is Rhp the return from holding 4 years at the sample average rate? • No. • 4 years at 21.075% would yield (1.21075)^4-1 =1.1489084 • i.e. 114.89 %, instead of Rhp = 101.83592%

29. Review question • Define the internal rate of return.

30. Answer: • The internal rate of return of a project is r such that, given the cash flows CFt of the project,

31. Review item • In the first year a portfolio has a rate of return of -30%. • In the second year it has a rate of return of +30%. • What is the holding period return?

32. Answer: • Solve 1+Rhp=(.7)(1.3). • Then Rhp = .91 - 1 = -.09.

34. Equivalent annual rate is the geometric average • Solve for x in (1+x)^4 =2.0183592 • Solution 19.19269%. • approximately. • It answers the question: what is the equivalent rate over 4 years? • Population mean answers the question: • What is the average for next year?