1. Portfolio Theory & Related Topics

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

3. 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

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

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

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

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

9. Finding the Optimal Complete Portfolio Using Indifference Curves

10. 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 = P = ω1 r1 + ω2 r2
Our portfolio

11. Thus, we have
E(rP) = ω1 E(r1) + ω2 E(r2)
And
σP = SQRT ( ω12 σ12 + ω22 σ 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.

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

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

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

16. proof here

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

18. 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

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