1. Finding the Efficient Set
Chapter 5
Finding the Efficient Set

2. Attainable Portfolios
Efficient set
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. Computer plugs in different weights for each security
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. We want a portfolio that Rp = .70
Returns Variance Standard deviation American express % Anheuser Busch % Apple computer %
We want a portfolio that Rp = .70
Assume we invest 90 % of our funds in Asset A (Xa = .90)
Xa = Xb = Xc = = (-2.177)
Assume want to invest 150% of our money in Asset A
Xa = Xb = Xc = = (-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. Pts are all concentric about MVP (Fig 5.5 pg 100)
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.
Investors should follow basic investment theory and rules when deciding what their holdings will be. Theory states that investors do not dislike risk. They want to be compensated for taking risk. The challenge has several hurdles to get to the point of making a decision.
What kind of returns can be expected from the asset?
What is the risk level associated with this investment?
The beginning information must then be used to develop a decision model.
Financial decision making follows the basic economic model of cost benefit analysis. The cost basis is the easier piece of information to establish. The harder information to develop is the benefit that the investor expects to receive from the investment.
The benefit is measured by utility. Utility is an economic term that transforms asset characteristics with investors traits into a value measure. This measure establishes what the investors expects to receive if they would purchase the asset.
This utility value is then compared with the set of assets to establish a priority of desirability. It is probably evident that investors want to maximize their benefits or utility.

17. Risk and Risk Aversion Speculation Gamble
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
A common statement about investing, especially during times of poor stock market activity, is that placing money in the stock market is gambling. These statements are driven by an undermining concept of investment. Investors have short-term memories. They do not view the world beyond the events that impact them in the surrounding 6 months to a year.
The basic difference between gambling and speculative investment boils down to a definition. Speculation involves taking considerable risk for commensurate gains. The key elements are 1) considerable risk for 2) a commensurate gain. Gambling involves a wager for an uncertain outcome. There is no commensurate gain that is expected. Essentially a gamble has an expected return that is negative. There is no upfront expectation of “winning”. There is definitely a desire to win, just not an expectation to win.

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?
Investors choose between several types of investments. They can choose a risk-free asset or a risky investment. Risky investments are only desirable if they provide an acceptable risk premium.
When combining assets into portfolios there are several different outcomes that are possible. First, the returns can decrease and the risk can increase. This outcome is unacceptable. The second two outcomes involve simultaneous increasing or decreasing of risk and return. The choice between assets under these circumstances is dependent on the investor’s sensitivity to risk and the utility of the investments.
The final potential outcome is the opportunity that anyone would hope to happen. It would be great if we could decrease the risk and have an increase in returns.

19. Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%)
The above portfolios represent potential investments. They range from low to high risk. Which portfolio would be best for an individual?
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
The economic concept of utility is a very complicated tenant of investing. It is the characteristic that ultimately determines whether an investment increases benefits or wealth or not.
The formula is not that complicated. Investors are aware of the expected risk and returns of their investment opportunities. These are developed through a variety of valuation techniques.
The more important variable is the coefficient of risk aversion. The coefficient measures how sensitive a person is to risk levels. The greater the sensitivity the greater the value of the coefficient.
While there is no commercial vehicle for determining a person’s coefficient of variation, the values typically range from 2.0 to 4.0.
In the formula benefits increase as expected returns increase. Benefits will decrease as the risk of the investment increases and as a person’s risk aversion increases.

21. The above tables can be developed in the Utility worksheet
The above tables can be developed in the Utility worksheet. The examples illustrate the change in utility (benefit) as sensitivity levels change. Of course for an individual this number does not change.
Utility

22. Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion
The above table then reflects the benefits for different investors. The risk aversion ranges from 2.0 to 5.0. Which of the assets should each person choose?
The mean variance criterion is a construct of investment theory. It assumes that people will attempt to gain either higher returns or lower risk. If you can obtain both at the same time, that is just that much better.
With respect to utility (benefit), a person will chose the asset that will provide the highest level of utility.
Above Investor 1 would choose the high risk investment. Investors 2 and 3 would choose the medium risk asset. They provide the highest level of benefit.

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
Let’s take a closer look at risk aversion. The higher the level, the more riskier investments will appear undesirable. A risk aversion coefficient less than zero is descriptive of gambling behavior. The benefit of a gamble develops opposite that of an investment.
This is another description of the difference between a gamble and a speculative investment.

24. Portfolio A dominates portfolio B if: And
Portfolio Dominance
What does dominance mean?
Mean Variance Criterion
Portfolio A dominates portfolio B if:
And
What does dominance mean? Another phrase is which portfolio is more desirable?
The obvious choice of portfolios is one that has higher returns and lower risk levels. If either of those characteristics is changed, the coefficient of risk aversion would determine a person’s preferences.

25. This figure shows the risk return relationship between four mutual funds, a portfolio containing those funds in equal proportion, and the S&P500 benchmark. With reference to the benchmark, there are three data points that have clear dominance to the benchmark. One that is clearly less dominant. The fourth portfolio has higher returns and higher risk. To determine if this portfolio is more desirable than the benchmark, we need to understand the persons risk aversion. In this case, since the increase in risk is fairly small and the increase in returns is fairly large, it would be hard to see how a person would not want to select this portfolio over the benchmark.