Portfolio Theory & Related Topics

Risk & Return: a funny mix-and-match game The question is: What is the right mix of risk and return (for a given investor) ?

Recall that in general terms the return of a risky asset (r), that is, an asset whose return is subject to uncertainty, can be written as E(r) = RF + { E(r) – RF } Risk free rate Risk Premium

Consider a portfolio P made up of two assets a risk free asset (return = RF) and a risky asset (return = r) Portfolio

E(rP) slope 1 E(r) The CAPITAL ALLOCATION LINE RF

Utility Function Where U = utility E ( r ) = expected return A = coefficient of risk aversion s2 = variance of returns

Utility as a Function of Allocation to the Risky Asset, and A=4

Finding the Optimal Complete Portfolio Using Indifference Curves

ω1 + ω2 = 1 P = ω1 r1 + ω2 r2 1 r1 σ1 and 2 r2 σ2 and also Consider now a portfolio that is made up of two risky assets 1 r1 σ1 and 2 r2 σ2 and also assume we know the covariance σ12 = cov (1, 2) = ρ12 σ1σ2 ω1 + ω2 = 1 P = ω1 r1 + ω2 r2 Our portfolio

Thus, we have E(rP) = ω1 E(r1) + ω2 E(r2) And σP = SQRT ( ω12 σ12 + ω22 σ22 + 2 ω1 ω2 σ1 σ2 ρ ) Let us use ω1 as a parameter, we can call it ω for simplicity (note that ω2 = 1 – ω). For a given value of ω we can calculate E(rP) and σP and see what happens.

For a given value of ρ E(rP) σP 100% in (1) 100% in (2) Efficient frontier σP

For a given value of ρ E(rP) σP 100% is (1) 100% in (2) Interesting point σP

What is the portfolio (that is, the ω) that will minimize the variance ? Answer

proof here

For a given value of ρ E(rP) σP 100% is (1) 100% in (2) MINIMUM VARIANCE PORTFOLIO

Suppose now we have two risky assets (1 and 2) plus the risk free asset. Lets us look at possible combinations or portfolios. CAL 1 E(rP) (1) O CAL 2 (2) RF σP O = optimal risky portfolio

RF E(rP) σP Indifference curves (1) O (2) Lending Borrowing O = optimal risky portfolio