1. Portfolio Theory & Related Topics

2. Some Basic Ideas
Return
Risk
Risk Free Rate
Utility Function
(continuation)

3. SITUATION A -- do nothing
Consider these two situations…
SITUATION A -- do nothing
Payoff = $ 100
You have $ 100
P = Probability = 100%
E (payoff) = 100
St Deviation (payoff) = (It is s sure thing !!!!)
In essence, you just keep your money, don’t do anything with it

4. SITUATION B – take a gamble
Payoff = $ 150
P = 50%
Payoff = $ 50
Initial “investment”, $ 100
P = 50%
E (payoff) = 0.5 x x 50 = 100
St Deviation (payoff) = SQRT ( 0.5 x x 2500) = 50
Would you go with B? … after all, the Expected value of the payoff is the same in both cases, and you can earn something “extra” with B… What Do You Think? Or Feel?

5. Most people would probably go with Situation A (the sure thing), it is less risky (less “variation”)
What is RISK?
Uncertainty
The possibility that things will go wrong
The St Deviation of the outcome, somehow, gives as an idea about the risk involved

6. Very risky
less risky
This is a fair game (expected return = 0) but with different risk levels

7. Let us introduce the concept of “Utility”
Utility, refers, loosely speaking, to the “amount of pleasure or satisfaction” that one obtains from a given amount of wealth (w)
A utility function, U = F (w), must satisfy the condition that
“more wealth” is better than “less wealth.”
In other words, F must be a monotonically increasing function.

8. U = F (w)
Risk neutral person
Risk lover
U” is positive
Risk averse
U” is negative w

9. Suppose, in the previous situation, we have a utility function (U), for a risk averse person, defined as
U = sqrt (w)
Situation A
E (U) = sqrt ( 100) = 10
Situation B
E (U) = 0.5 x sqrt (150) x sqrt (50) = 9.65
Thus, he refuses to play the game… (he has decided based on the Expected Utility rather than the Expected Return)
A risk averse player will refuse to engage in a game in which the E [Return] is 0, that is, a fair game.

10. We call certainty equivalent (c) an amount such that, if received with certainty, will make the player (investor) indifferent between playing or not
In our example, Scenario B has an expected utility E (U) = 9.65
Solving for c in
9.65 = sqrt (c), we get c = 93.1
This means that if our player had to choose between receiving 93.1 dollars with 100% assurance OR playing the game he would be indifferent
More precisely U (c) = E ( U (w) ) is the equation to solve, in general

11. More broadly, in the context of a portfolio of investments, (a portfolio characterized by…
[1] its E(return); and
[2] its St. Deviation…
an appropriate utility function should involve these two parameters
For instance…..

12. Utility Function Where U = utility
E ( r ) = expected return on the asset or portfolio
A = coefficient of risk aversion
s2 = variance of returns
How can you tell between Risk Lover and Risk Averse ?

13. The Indifference Curve
Same utility 13

14. Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4

15. Suppose now we have n assets (possible investment choices) and that each asset return is a random variable characterized by its mean and st. deviation. Also, we assume that we know the covariance matrix
Covariance matrix

16. To make life easier we will assume that all the weights are the same, 1/n; we just want to see the “diversification effect”
In this case, the portfolio variance can be expressed as…
average variance
average “co-variance”

17. average variance
average “co-variance”
IF n becomes very large, it goes to zero
IF n becomes very large, it “converges” to “average co-variance”, NOT ZERO

18. Risk of a typical asset
Portfolio St Dev
St Dev of Portfolio returns
n, number of assets

19. Risk eliminated thru diversification
Diversifiable Risk
Portfolio St Dev
St Dev of Portfolio returns
Systematic Risk (we cannot eliminate it), non-diversifiable risk
n, number of assets
The power of DIVERSIFICATION !!!!