Chapter 5

1. Chapter 5 Finding the Efficient Set

2. Efficient set Attainable Portfolios • In the last chapter we identified the risk return relationships between different portfolios. This chapter is designed to determine which portfolio is best • Attainable portfolios (all possible combinations) • Fig 5.1 pg 93 entire shaded area & line • There are no portfolios that can be created with risky assets that have a level of risk and return outside the bullet • (IE. You cannot create a portfolio that has a E(r) of 20% and a standard deviation of 5%) • Minimum variance set - bullet shaped curve (only the line) • Given a particular level of return, has lowest standard deviation possible • There are two important components • 1) MVP • global minimum variance portfolio • 2) efficient set • given level of standard deviation, portfolios with highest return • Top half of the bullet These are the most desirable portfolios Efficient portfolio (on the efficient set) A) minimize risk for given return B) maximize return for given risk

3. Finding Efficient Set (with Short Selling) • Attainable set w 3 stocks • Start with • covariance matrix • Expected returns • Standard deviation (get from covariance matrix) • If you look at Fig 5.2 pg 95, shows the 3 stocks in the attainable set • The line between points are long positions in each security • The rest of the shaded area represents combinations of all three

4. Portfolio Weights • Computer plugs in different weights for each security • Fig 5.3 pg 96 (Weights of portfolios) • Pt R = 100% in B (Brown) • Pt T = 100% in A (Acme) • Pt S = 100% in C (Consolidated) • Inside triangle = + amounts of each stock PT L • On perimeter = + amounts in 2 & zero in third • PT Q + amount s in B & C nothing invested in A • Outside perimeter = short selling is taking place • Above line XY’ (northeast) -C • West of vertical -A • South of horizontal -B

5. Expected return Plane • B) Expected return plane • Solve the portfolio return line for the weight of 1 of the assets • Given that we know the returns of the individual assets we can further simplify this equation • Assume that we want a certain return; we place that number in the numerator (Rp) • Assume then that we choose to invest 90% in Asset A (XA) • The formula will then solve for the amount we must invest in Asset B • Since we know the weights for A & B we can solve for C • If we repeat this for another weight of XA; we will have 2 portfolios with equal returns

6. Returns Variance Standard deviationAmerican express .04% .0071 .0852 Anheuser Busch .54% .0035 .0593Apple computer .26% .0167 .1293 • We want a portfolio that Rp = .70 • Assume we invest 90 % of our funds in Asset A (Xa = .90) • Xa = .90 Xb = 2.277 Xc = 1 - .90 - 2.277 = (-2.177) • Assume want to invest 150% of our money in Asset A • Xa = 1.50 Xb = 2.748 Xc = 1 - 1.50 - 2.748 = (-3.248) • We would now repeat this but for a different return level and we would get another Iso- return line

7. Iso-Return - Variance

8. Iso-Return Lines • We would repeat this process for many different returns (thank goodness for computers) • We would then graph the lines and place it over the portfolio weight graph • Shaded areas – long positions in all 3 assets • all points on a line equal returns

9. Iso-Variance Elipses • Lets assume we want a portfolio with a variance of .02 • First invest 90% of our assets in Asset A • Multiply and rearrange terms • This is a quadratic equation, with some arranging of terms we can solve for the value of XB • quadratic formula has the following form: • In a quadratic formula X has two possible values use the following formula to find them

10. When we solve for quadratic formula it will provide 2 points with equal risk on an ellipse. • If we continue to change the weight of A and solve we would create the points of an ISO-variance ellipse • Now we would continue the process but this time change the desired variance • This set of ellipses would then be placed over the portfolio weight graph • Pts are all concentric about MVP (Fig 5.5 pg 100) • MVP is the bottom of a "valley“ • Each point on ellipse = risk

11. Portfolio Management / Iso return Variance

12. Critical line • Critical line - this line shows the portfolios of the Min variance set (equivalent to the bullet shaped curve) • MVP to Northwest = efficient portfolios • Superimpose the iso-return lines and the iso variance ellipses • Find pt on iso-variance ellipse tangent to iso-return • (Highest possible return given the risk ellipse) • Fig 5.7 pg 102 • pt Q on border of triangle • Invest positive amounts in brown and consolidated, 0 in acme • from pts Q to MVP , positive amounts in all three • to the south east of MVP not efficient • (Can find portfolio with higher return and the same risk) • To west of Q short Acme and positive amounts in other

13. Finding Efficient Set (without Short Selling) • for Fig 5.8 pg 107 must be on or inside the triangle • Get a line (SQZ) Minimum variance set • Fig 5.11 pg 111 • Note the two bullet curves • One superior to the other • Superior meaning more efficient

14. Property 1 of minimum variance set • If combine two or more portfolios from minimum variance set get another portfolio on the Minimum variance set • When discuss CAPM important because we assume that all investors hold efficient portfolios • If we combine all of them, we must then also get an efficient portfolio

15. Property 2 of the Minimum Variance Set • given the population of sec., there is a linear relationship between beta factors and their expected returns, if and only if we use a minimum variance portfolio as the index portfolio • Market index used for calculation of beta is minimum variance portfolio • Beta measures responsiveness of sec returns to a market portfolio ( all risky assets) • Must get a relationship as in right side of Fig 5.13 pg 113

16. Allocation to Risky Assets Investors will avoid risk unless there is a reward. The utility model gives the optimal allocation between a risky portfolio and a risk-free asset.

17. Risk and Risk Aversion • Speculation • Taking considerable risk for a commensurate gain • Parties have heterogeneous expectations • Gamble • Bet or wager on an uncertain outcome for enjoyment • Parties assign the same probabilities to the possible outcomes

18. Risk Aversion and Utility Values • Investors are willing to consider: • risk-free assets • speculative positions with positive risk premiums • Portfolio attractiveness increases with expected return and decreases with risk. • What happens when return increases with risk?

19. Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%) Each portfolio receives a utility score to assess the investor’s risk/return trade off

20. Utility Function U = utility – measures benefit Investors would like to maximize utility. Utility incorporates risk and return as well as individual sensitivity to risk Certainty Equivalent rate – rate willing to accept from a RF rate to buy it instead of the risky asset E ( r ) = expected return A = coefficient of risk aversion s2 = variance of returns ½ = a scaling factor

21. Utility

22. Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion

23. What is Risk Aversion? • Risk aversion measures how sensitive a person is to changing risk characteristics of an asset. They use this sensitivity to establish a difference in preference for an asset. In the utility formula it is the variable (could be negative) that determines the change in value necessary to compensate for the changes. • Risk Averse investors require higher levels of return as risk increases. • (A > 0) • Risk neutral investors pick securities solely by their expected utility • (A = 0) • Risk lovers are willing to engage in gambling • (A < 0) • Research has shown that most investors are between A = 2 & 4

24. Portfolio Dominance • What does dominance mean? • Mean Variance Criterion • Portfolio A dominates portfolio B if: • And