1. And portfolio variance
Covariance
And portfolio variance

2. Review question
Define the internal rate of return.

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

4. 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.

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

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

8. Variance and standard deviation
Form deviations
Take their expectation.

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

12. 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.

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

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

15. 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.

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

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

20. Variance of portfolio return
Diversification effects

21. Portfolio risk and return,

22. Portfolio deviation
Deviation squared
Remember

24. Portfolio variance

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

26. Correlation coefficient

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