Portfolio Theory & Related Topics

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

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) = 0 (It is s sure thing !!!!) In essence, you just keep your money, don’t do anything with it

SITUATION B – take a gamble Payoff = $ 150 P = 50% Payoff = $ 50 Initial “investment”, $ 100 P = 50% E (payoff) = 0.5 x 150 + 0.5 x 50 = 100 St Deviation (payoff) = SQRT ( 0.5 x 2500 + 0.5 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?

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

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

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.

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

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) + 0.5 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.

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

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…..

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 ?

The Indifference Curve Same utility 13

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

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

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”

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

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

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 !!!!