1 of 59 chapter 5 finding the efficient set. 2 of 59 attainable portfolios in the last chapter we...
Post on 28-Dec-2015
214 Views
Preview:
TRANSCRIPT
2 of 59
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 returnB) maximize return for given risk
Efficient set
3 of 59
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 of 59
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 of 59
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
CCBBAAp R*XR*XR*X)R(E
CBABBAAp R*)XX1(R*XR*X)R(E
CBCACBBAAp R*XR*XRR*XR*X)R(E
ABC
CA
BC
pCB X*
RR
RR
RR
RRX
6 of 59
Returns Variance Standard deviationAmerican express .04% .0071 .0852Anheuser 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
AAB X*785.571.1X*54.26.
26.04.
54.26.
70.26.X
277.290.*785.571.1XB
748.250.1*785.571.1XB
8 of 59
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 of 59
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
)X10(.XX1X XX 90.X bbacbba
bccbacca
abba2cc
22b
2b
2a
2a
2p
COV*X*X*2COV*X*X*2
COV*X*X*2*X*X*X
2bb X*0138.X*00352.006169.02.
0cbXaX2
a2
ac4bbX,X
2
21
10 of 59
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
12 of 59
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 of 59
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 of 59
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 of 59
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 of 59
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 of 59
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 of 59
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 of 59
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 of 59
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
= variance of returns
½ = a scaling factor
21( )
2U E r A
21 of 59
Risky InvestmentStandard Deviation 19.70%Expected Returns 12.75%Risk Aversion 2.75Utility 0.074138
Risky InvestmentStandard Deviation 19.70%Expected Returns 12.75%Risk Aversion 3.5Utility 0.059584
Risky InvestmentStandard Deviation 19.70%Expected Returns 12.75%Risk Aversion 4.5Utility 0.04018
Risk Free AssetStandard Deviation 0.00%Expected Returns 4.00%Risk Aversion 4.5Utility 0.04
Utility
22 of 59
Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion
BA rErE
BA
23 of 59
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 of 59
Portfolio Dominance
What does dominance mean? Mean Variance Criterion
• Portfolio A dominates portfolio B if:
• And
BA rErE
BA
top related