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Project 2
Due on February 3 by 5:00 pm
MGT 252 – Winter 2021
Investments and Portfolio Management
c 2021 Alexander Barinov
1
Diversification Classics (100 points)
The file Project 2.xls in the Projects folder contains the monthly return to Exxon Mobil
(XOM) and Darden Restaurants (DRI)1 in 2005-2009, as well as the market return (MKT)
and the risk-free rate (RF) in the same time period. Use these data to answer the questions
below:
i. Compute the average return, the standard deviation, and the correlation for Exxon
and Darden. (10 points)
ii. If you can only invest in the risk-free asset and either Exxon or Darden, which one
do you pick? (5 points)
iii. If you can only invest in the risk-free asset and either Exxon or Darden and your
target standard deviation is 15% per annum, what will be your best possible expected
monthly return? What weight does the risk-free asset take in your portfolio? (10
points)
iv. Compute the average return, standard deviation, and the Sharpe ratio of the portfolio
that invests 80% in Exxon and 20% in Darden (5 points)
v. Does the existence of Exxon and Darden violate the CAPM? (Hint: You will need
to use the data on the market excess returns). (5 points)
vi. Redo (iv) assuming that the correlation between Exxon and Darden is 0.6. What
does it tell you about the benefits of diversification? (5 points)
vii. What combination of Exxon and Darden creates the minimum variance portfolio
(MVP)? What are the average return, standard deviation and the Sharpe ratio of
this portfolio? (Hint: use the Solver add-in) (10 points)
1
Darden Restaurants is the holding company that owns Olive Garden, Red Lobster, and LongHorn
Steakhouse
1
viii. What combination of Exxon and Darden creates the mean-variance efficient portfolio
(MVE)? What are the average return, standard deviation and the Sharpe ratio of
this portfolio? (Hint: use the Solver add-in) (10 points)
ix. Bonus question: Estimate the market model for Exxon and the market model for
Darden. Do either Exxon or Darden violate the CAPM by a statistically significant
amount (10 points)
x. Assume that Exxon and Darden are the only two stocks in the economy. Under the
CAPM, what are the average return and standard deviation of the market portfolio?
What are the average return and standard deviation of the zero-beta portfolio? (Hint:
xi. Bonus question: Redo (iii) assuming that now you can invest in Exxon, Darden, and
the risk-free asset simultaneously. What are the weights of Exxon, Darden, and the
risk-free asset in the portfolio that delivers the best possible return for the target
standard deviation of 15% per annum? (10 points)
xii. Assume that Exxon and Darden are the only two stocks in the economy and there
is no risk-free asset. For each of the following three points – (0.83%, 5.44%), (1.03%,
6.53%), (0.95%, 5.31%) – answer two questions: (a) Will you want to choose it? (b)
Can you have it? The first number of the pair is always expected return. (Hint 1:
use Solver. Hint 2: do not forget comparing the points with MVP). (20 points)
xiii. How would your answer to (xii) change if the risk-free asset is available? (10 points)
2
Date
1/3/2005
2/1/2005
3/1/2005
4/1/2005
5/2/2005
6/1/2005
7/1/2005
8/1/2005
9/1/2005
10/3/2005
11/1/2005
12/1/2005
1/3/2006
2/1/2006
3/1/2006
4/3/2006
5/1/2006
6/1/2006
7/3/2006
8/1/2006
9/1/2006
10/2/2006
11/1/2006
12/1/2006
1/3/2007
2/1/2007
3/1/2007
4/2/2007
5/1/2007
6/1/2007
7/2/2007
8/1/2007
9/4/2007
10/1/2007
11/1/2007
12/3/2007
1/2/2008
2/1/2008
3/3/2008
4/1/2008
5/1/2008
6/2/2008
7/1/2008
8/1/2008
9/2/2008
10/1/2008
XOM
0,662983
23,29308
-5,85826
-4,31246
-0,96857
2,275449
2,224824
2,443681
6,093925
-11,6459
3,87674
-3,19617
11,70423
-4,88496
2,512095
3,648575
-2,94221
0,721761
10,40846
0,356969
-0,84074
6,440567
8,011642
-0,22692
-3,31201
-2,8521
5,266344
5,20414
5,23367
0,844046
1,49369
1,103781
7,968377
-0,61599
-2,70144
5,084135
-8,53254
1,975741
-2,79583
10,04163
-4,20727
-0,70608
-8,7381
-0,01321
-2,94545
-4,55906
DRI
MKT
RF
6,558045
-2,66
-9,32722
2,27
14,46037
-1,69
-2,06262
-2,52
8,236179
3,79
1,563586
1,15
5,200137
4,33
-9,49593
-0,59
-3,30579
1,06
7,469342
-2,08
10,37344
4,04
8,677945
0,35
4,554627
4,01
3,170664
-0,16
-2,19134
1,91
-3,00546
1,3
-10,5634
-3,1
11,24409
-0,04
-14,2129
-0,19
4,752475
2,51
19,9748
1,95
-0,84034
3,71
-4,15784
2,37
0,027632
1,08
-2,54144
1,94
4,705215
-1,4
0,487277
1,29
1,266164
3,99
9,869646
3,89
-3,46247
-1,48
-2,83421
-3,18
-2,29737
1,16
0,634082
4,09
3,150433
2,58
-7,45737
-4,93
-30,363
-0,43
3,00158
-6,23
8,819018
-2,2
5,567301
-1,05
9,87984
5,12
-3,73633
2,38
-6,75292
-7,86
2,639594
-1,32
-10,089
1,1
-2,23689
-9,81
-21,943
-18,47
0,16
0,16
0,21
0,21
0,24
0,23
0,24
0,3
0,29
0,27
0,31
0,32
0,35
0,34
0,37
0,36
0,43
0,4
0,4
0,42
0,41
0,41
0,42
0,4
0,44
0,38
0,43
0,44
0,41
0,4
0,4
0,42
0,32
0,32
0,34
0,27
0,21
0,13
0,17
0,17
0,17
0,17
0,15
0,12
0,15
0,08
11/3/2008
12/1/2008
1/2/2009
2/2/2009
3/2/2009
4/1/2009
5/1/2009
6/1/2009
7/1/2009
8/3/2009
9/1/2009
10/1/2009
11/2/2009
12/1/2009
8,769428
-0,4064
-4,19903
-10,7722
0,292578
-2,08813
4,641681
0,794245
0,698781
-1,1664
-0,77682
4,456489
5,347362
-9,16678
-17,4916
54,04776
-6,3138
3,510896
26,23782
8,52378
-2,16278
-8,81326
-1,02073
1,643571
3,646164
-10,5231
3,692308
11,60567
-8,52
2,15
-7,74
-10,1
8,77
11,06
6,73
-0,28
8,25
3,19
4,52
-2,84
5,74
2,92
0,02
0,09
0
0,01
0,01
0,01
0
0
0,01
0,01
0
0
0
0
Mean
StDev
Sharpe
ExpRet
Weight
Match 6.77%
Weight
StDev
FCNTX
FCPVX
RF
6.12
9.36
16.66
23.5
0.234814 0.30434
5.730209 6.773106
0.90036 0.638298
1.166949 0.638298
19.44138
15
TgtStDev
2.208
15
Date
FHKCX
FDFFX
1/31/2005
5.49
1.02
2/28/2005
-3.29
-1.72
3/31/2005
1.56
-5.01
4/30/2005
1.60
6.73
5/31/2005
1.99
0.42
6/30/2005
4.78
5.81
7/31/2005
-2.25
-0.45
8/31/2005
3.42
2.48
9/30/2005
-5.84
-3.13
10/31/2005
5.47
5.90
11/30/2005
4.35
1.23
12/31/2005
5.52
6.29
1/31/2006
-0.41
-1.69
2/28/2006
2.68
3.24
3/31/2006
5.97
1.23
4/30/2006
-3.81
-5.04
5/31/2006
-0.89
0.71
6/30/2006
0.67
-2.48
7/31/2006
3.46
2.28
8/31/2006
2.05
0.96
9/30/2006
2.59
3.62
10/31/2006
5.62
2.96
11/30/2006
3.27
0.00
12/31/2006
0.33
1.60
1/31/2007
-2.07
-1.72
2/28/2007
0.96
2.22
3/31/2007
2.10
4.52
4/30/2007
2.94
6.10
5/31/2007
9.57
0.25
6/30/2007
8.32
-1.41
7/31/2007
3.21
1.64
8/31/2007
17.22
9.08
9/30/2007
13.14
8.13
10/31/2007
-11.45
-6.25
11/30/2007
-2.81
3.04
12/31/2007
-14.69
-8.62
1/31/2008
5.48
2.79
2/29/2008
-3.64
-3.18
3/31/2008
8.49
9.76
4/30/2008
-0.34
4.70
5/31/2008
-9.86
1.81
6/30/2008
-3.57
-10.74
7/31/2008
-4.78
-3.83
8/31/2008
-14.92
-20.63
9/30/2008
-18.87
-22.78
10/31/2008
-3.21
-8.64
11/30/2008
12/31/2008
1/31/2009
2/28/2009
3/31/2009
4/30/2009
5/31/2009
6/30/2009
7/31/2009
8/31/2009
9/30/2009
10/31/2009
11/30/2009
12/31/2009
Average
StDev
Sharpe
Correl
Weight US
Average
StDev
Sharpe
6.45
1.49
-8.64
-7.50
-1.48
-9.17
11.23
8.26
13.39
11.40
20.87
9.68
-1.20
-2.59
12.58
8.61
-6.86
4.23
8.31
7.60
1.37
-5.47
4.11
7.36
0.90
4.29
1.83
2.06
FHKCX (China) FDFFX (US) RF
1.47
0.56
7.37
6.62
0.17
0.05
0.79
0.75
0.79
6.5170
0.09
Average
StDev
Corr
Weight
StDevPort
AveRetPort
Corr
Weight
StDevPort
AveRetPort
0.23
Assume
Weight US
Average
StDev
Sharpe
0.70
0.833
6.5206
0.09
MVP with Solver
Weight US
Average
StDev
Sharpe
X
0.00
0.70
0.833
5.13
0.12
MVE with Solver
Weight US
Average
StDev
Sharpe
-2.53
3.79
16.43
0.22
Y
10.00
10.00
1.00
2.00
0.00
5.00
-1.00
0.67
0.00
11.67
15.00
20.00
Weight US
Average
StDev
Sharpe
-0.17
1.626
7.76
0.18
-1.00
0.33
(a+b)^2=a^2+b^2+2*a*b
(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc
Zero-Beta Portfolio
Weight
StdDevPort
1.36
7.10
AveRetPort
0.23
SharpePort
0.00
Asset Allocation and
Optimal Portfolio
Professor Alexander Barinov
University of California Riverside
MGT 252 Investments and Portfolio Management
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
1 / 34
Outline
1
Risky Asset Plus Risk-Free Asset
Derivation
Example
No Risk-Free Borrowing
Optimal Portfolio
2
Two Risky Assets
Correlation and Covariance
Example
The Bullet
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
2 / 34
Risky Asset Plus Risk-Free Asset
Derivation
The Setup
Consider portfolio P that invests Ãâ€° in the risky
asset (stock) X and the rest (1 Ã¢Ë†â€™ Ãâ€°) in the
risk-free asset
Assume that the return of the risk-free asset is
constant at RF
The common proxy for the risk-free asset is
3-month Treasury bill, though it is not perfectly
safe
0 Ã¢â€°Â¤ Ãâ€° < 1 - we invest both in the stock and the Treasury bill Ãâ€° > 1 – we borrow at the risk-free rate and buy
the stock on the margin
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
3 / 34
Risky Asset Plus Risk-Free Asset
Derivation
Portfolio Characteristics
Expected return of the portfolio:
E(RP ) = E(Ãâ€°RX + (1 Ã¢Ë†â€™ Ãâ€°)RF ) = Ãâ€° Ã‚Â· E(RX ) + (1 Ã¢Ë†â€™ Ãâ€°) Ã‚Â· RF
E(RP ) = RF + Ãâ€°(E(RX ) Ã¢Ë†â€™ RF )
Variance of the portfolio return:
ÃÆ’ 2 (RP ) = ÃÆ’ 2 (Ãâ€°RX + (1 Ã¢Ë†â€™ Ãâ€°)RF ) = ÃÆ’ 2 (Ãâ€°RX ) = Ãâ€° 2 Ã‚Â· ÃÆ’ 2 (RX )
Standard deviation of the portfolio return:
ÃÆ’(RP ) = Ãâ€°ÃÆ’(RX )
E(RP ) = RF +
Alexander Barinov (SoBA, UCR)
E(RX ) Ã¢Ë†â€™ RF
Ã‚Â· ÃÆ’(RP )
ÃÆ’(RX )
Asset Allocation and Optimal Portfolio
MGT 252 Investments
4 / 34
Risky Asset Plus Risk-Free Asset
Derivation
Capital Allocation Line
E(RP ) = RF +
E(RX ) Ã¢Ë†â€™ RF
Ã‚Â· ÃÆ’(RP )
ÃÆ’(RX )
If you invest in the portfolio P, your risk can be
measured by the standard deviation ÃÆ’(RP )
The reward per unit of risk, SX =
E(RX ) Ã¢Ë†â€™ RF
,
ÃÆ’(RX )
is called the Sharpe Ratio
The portfolio expected return is the risk-free rate
plus the reward for bearing risk
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
5 / 34
Risky Asset Plus Risk-Free Asset
Derivation
CAL as the Budget Constraint
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
6 / 34
Risky Asset Plus Risk-Free Asset
Derivation
Capital Market Line
E(RP ) = RF +
E(RM ) Ã¢Ë†â€™ RF
Ã‚Â· ÃÆ’(RP )
ÃÆ’M
If the risky portfolio is the market portfolio (market
index), the CAL is called the Capital Market Line
(CML)
Notice that both CAL and CML are derived from two
equations for the portfolio variance and expected
return
We did not assume that standard deviation is the right
measure of risk, or that investors care only about
mean and variance, or that mean and variance
completely explain the behavior of returns, etc.
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
7 / 34
Risky Asset Plus Risk-Free Asset
Example
Example: Fidelity Funds
Over the past 5 years, the average risk-free rate
is 2.21% (data from Ken French)
Suppose we shoot for 15% per year standard
deviation
Our investment vehicles are the money market
account that yields precisely 2.208% and one of
the two Fidelity funds: Fidelity Small Cap Value
(FCPVX) and Fidelity Contrafund (FCNTX)
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
8 / 34
Risky Asset Plus Risk-Free Asset
Example
Which One to Pick?
Small Value: average return 9.36% per year, standard deviation
23.5% per year
9.36% Ã¢Ë†â€™ 2.21%
Sharpe ratio: SFCPVX =
= 0.304
23.5%
Expected return of Small Value fund + CD:
E(RP, FCPVX ) = 2.21% + 0.304 Ã‚Â· 15% = 6.77%
Contrafund: average return 6.12% per year, standard deviation
16.66% per year
6.12% Ã¢Ë†â€™ 2.21%
Sharpe ratio: SFCNTX =
= 0.235
16.66%
Expected return of Contrafund + CD:
E(RP, FCNTX ) = 2.21% + 0.235 Ã‚Â· 15% = 5.73%
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
9 / 34
Risky Asset Plus Risk-Free Asset
Example
How to Allocate the Money?
What fraction of your money you have to invest in the Small Value
Fund to get the return of 6.77% and standard deviation of 15%?
Ãâ€°FCPVX =
15%
ÃÆ’(RP )
=
= 0.638
ÃÆ’(RFCPVX ) 23.5%
What fraction of your money do you have to invest in Contrafund
fund to match the return of Small Value?
E(RP ) = RF + Ãâ€°(E(RFCNTX ) Ã¢Ë†â€™ RF ) Ã¢â€¡â€™
Ã¢â€¡â€™ 6.77% = 2.21% + Ãâ€°FCNTX (6.12% Ã¢Ë†â€™ 2.21%) Ã¢â€¡â€™
6.77% Ã¢Ë†â€™ 2.21%
= 1.17
6.12% Ã¢Ë†â€™ 2.21%
What is the standard deviation of your portfolio in this case?
Ã¢â€¡â€™ Ãâ€°FCNTX =
ÃÆ’(RP ) = Ãâ€°ÃÆ’(RFCPVX ) = 1.17 Ã‚Â· 16.66% = 19.44%
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
10 / 34
Risky Asset Plus Risk-Free Asset
No Risk-Free Borrowing
No Risk-Free Borrowing
Ã‚ÂµP = RF +
E(RM ) Ã¢Ë†â€™ RF
Ã‚Â· ÃÆ’P
ÃÆ’M
Suppose we can still invest in the Treasury bill,
but we have to borrow at a higher rate RB
If 0 Ã¢â€°Â¤ Ãâ€° Ã¢â€°Â¤ 1, CAL/CML does not change
If Ãâ€° > 1, CAL/CML becomes flatter
Ã‚ÂµP = RF +
Alexander Barinov (SoBA, UCR)
E(RM ) Ã¢Ë†â€™ RB
Ã‚Â· ÃÆ’P
ÃÆ’M
Asset Allocation and Optimal Portfolio
MGT 252 Investments
11 / 34
Risky Asset Plus Risk-Free Asset
No Risk-Free Borrowing
No Risk-Free Borrowing
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
12 / 34
Risky Asset Plus Risk-Free Asset
No Risk-Free Borrowing
Interpretation
If we cannot borrow at the risk-free rate, we get
less reward for a unit of risk if we buy on the
margin
Part of the reward we had when risk-free
borrowing was available, now goes to the lender
It is fair, because if “we” are investment
company, the loan is risky and we are protected
by limited liability
Limited liability pushes part of the risk (severe
downturns) onto the lender
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
13 / 34
Risky Asset Plus Risk-Free Asset
Optimal Portfolio
Portfolio Choice
CAL/CML is the budget constraint
We need a utility function to figure out where on
the constraint we want to be
If we assume U = E(RP ) Ã¢Ë†â€™ 0.5 Ã‚Â· A Ã‚Â· ÃÆ’ 2 (RP ), we
will get nice curves in the E(RP ), ÃÆ’(RP ) space
The function is called mean-variance utility
It is based on the assumption that returns are
normally distributed
In reality, the return distribution has fat tails
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
14 / 34
Risky Asset Plus Risk-Free Asset
Optimal Portfolio
Utility Maximization
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
15 / 34
Risky Asset Plus Risk-Free Asset
Optimal Portfolio
Portfolio Choice
Utility function: U = E(RP ) Ã¢Ë†â€™ 0.5 Ã‚Â· A Ã‚Â· ÃÆ’ 2 (RP )
A is the risk aversion coefficient. It measures our
unwillingness to take on risk
The value of the utility function is certainty
equivalent – the minimum certain return we would
take instead of investing in the risky portfolio
Recall our expressions for the portfolio variance and
expected return:
E(RP ) = Ãâ€° Ã‚Â· E(RX ) + (1 Ã¢Ë†â€™ Ãâ€°) Ã‚Â· RF
ÃÆ’ 2 (RP ) = Ãâ€° 2 Ã‚Â· ÃÆ’ 2 (RX )
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
16 / 34
Risky Asset Plus Risk-Free Asset
Optimal Portfolio
Portfolio Choice
Just plug in the variance and the expected
return into the utility function:
U = Ãâ€° Ã‚Â· E(RX ) + (1 Ã¢Ë†â€™ Ãâ€°) Ã‚Â· RF Ã¢Ë†â€™ 0.5 Ã‚Â· A Ã‚Â· Ãâ€° 2 Ã‚Â· ÃÆ’ 2 (RX )
Now maximize the utility w.r.t Ãâ€° and obtain the
optimal weight of the risky asset, Ãâ€° Ã¢Ë†â€”
U 0 = E(RX ) Ã¢Ë†â€™ RF Ã¢Ë†â€™ A Ã‚Â· ÃÆ’ 2 (RX ) Ã‚Â· Ãâ€° Ã¢Ë†â€” = 0
Ãâ€°Ã¢Ë†â€” =
E(RX ) Ã¢Ë†â€™ RF
E(RX ) Ã¢Ë†â€™ RF
1
=
Ã‚Â·
A Ã‚Â· ÃÆ’ 2 (RX )
ÃÆ’(RX )
A Ã‚Â· ÃÆ’(RX )
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
17 / 34
Risky Asset Plus Risk-Free Asset
Optimal Portfolio
Optimal Portfolio
The weight of the risky asset in the optimal
portfolio
Increases with the Sharpe ratio (i.e. reward for
bearing risk)
Decreases with risk aversion
Decreases with portfolio volatility (the investor cares
about ÃÆ’P2 , and the Sharpe ratio measures the
compensation only for ÃÆ’P )
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
18 / 34
Two Risky Assets
Correlation and Covariance
Covariance and Its Properties
Covariance measures comovement of two
random variables
Cov (Ri , Rj ) = E[(Ri Ã¢Ë†â€™ E(Ri )) Ã‚Â· (Rj Ã¢Ë†â€™ E(Rj )]
Excel formulas: COVAR or SUMPRODUCT if
you do “by hand”
Cov (aRi , Rj ) = a Ã‚Â· Cov (Ri , Rj )
Cov (aRi , bRj ) = a Ã‚Â· b Ã‚Â· Cov (Ri , Rj )
Cov (Ri + C, Rj ) = Cov (Ri , Rj )
Cov (Ri , Ri ) = Var (Ri )
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
19 / 34
Two Risky Assets
Correlation and Covariance
Correlation and Its Properties
Correlation is a scale-free measure of
comovement
Cov (Ri , Rj )
Corr (Ri , Rj ) =
ÃÆ’(Ri ) Ã‚Â· ÃÆ’(Rj )
Correlation is bounded between -1 and 1,
covariance increases with the variances of the
variables
R-square of the market model is the squared
correlation between the stock and the market
Corr (aRi , Rj ) = Corr (Ri , Rj )
Corr (Ri + C, Rj ) = Corr (Ri , Rj )
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
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Two Risky Assets
Correlation and Covariance
Variance of the Sum
ÃÆ’ 2 (Ri + Rj ) = ÃÆ’ 2 (Ri ) + ÃÆ’ 2 (Rj )
For dependent variables, the covariance term appears:
ÃÆ’ 2 (Ri + Rj ) = ÃÆ’ 2 (Ri ) + 2Cov (Ri , Rj ) + ÃÆ’ 2 (Rj ) =
= ÃÆ’ 2 (Ri ) + 2Corr (Ri , Rj )ÃÆ’(Ri )ÃÆ’(Rj ) + ÃÆ’ 2 (Rj )
Which explains why we had ÃÆ’2 (RMKT Ã¢Ë†â€™ RF ) = ÃÆ’2 (RMKT )covariance of the constant with anything is zero
Economic interpretation: the smaller is the correlation, the
larger is the diversification benefit
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
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Two Risky Assets
Correlation and Covariance
Variance of the Weighted Sum
Suppose we invest Ãâ€° and 1 Ã¢Ë†â€™ Ãâ€° in the two stocks
Recall that ÃÆ’ 2 (Ãâ€°Ri ) = Ãâ€° 2 ÃÆ’ 2 (Ri ) and
Cov (Ãâ€°Ri ; (1 Ã¢Ë†â€™ Ãâ€°)Rj ) = Ãâ€°(1 Ã¢Ë†â€™ Ãâ€°)Cov (Ri ; Rj )
ÃÆ’ 2 (Ãâ€°Ri +(1Ã¢Ë†â€™Ãâ€°)Rj ) = ÃÆ’ 2 (Ãâ€°Ri )+2Cov (Ãâ€°Ri ; (1Ã¢Ë†â€™Ãâ€°)Rj )+ÃÆ’ 2 ((1Ã¢Ë†â€™Ãâ€°)Rj ) =
= Ãâ€° 2 ÃÆ’ 2 (Ri ) + 2Ãâ€°(1 Ã¢Ë†â€™ Ãâ€°)Cov (Ri ; Rj ) + (1 Ã¢Ë†â€™ Ãâ€°)2 ÃÆ’ 2 (Rj ) =
= Ãâ€° 2 ÃÆ’ 2 (Ri )+2Corr (Ri ; Rj )Ã‚Â·(Ãâ€°ÃÆ’(Ri ))Ã‚Â·((1Ã¢Ë†â€™Ãâ€°)ÃÆ’(Rj ))+(1Ã¢Ë†â€™Ãâ€°)2 ÃÆ’ 2 (Rj )
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
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Two Risky Assets
Example
Benefits of Diversification
Consider two Fidelity funds:
Fidelity Independence (ticker FDFFX, US stocks,
large growth)
Fidelity China (ticker FHKCX, Chinese stocks, large
value)
In 2005-2009, ÃÆ’(RFDFFX ) = 6.62%, ÃÆ’(RFHKCX ) = 7.37%
Average returns: R FDFFX = 0.56%, R FHKCX = 1.47%
Correlation of the two returns: 0.79
What is the standard deviation of the portfolio
that is 70% FDFFX and 30% FKHCX?
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Asset Allocation and Optimal Portfolio
MGT 252 Investments
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Two Risky Assets
Example
Benefits of Diversification
ÃÆ’ 2 (0.7RFDFFX + 0.3RFHKCX ) = 0.72 ÃÆ’ 2 (RFDFFX )+
+2Ã‚Â·0.7Ã‚Â·0.3Ã‚Â·ÃÆ’(RFDFFX )ÃÆ’(RFHKCX )Ã‚Â·Corr (RFDFFX ; RFHKCX )+0.32 ÃÆ’ 2 (RFHKCX ) =
= 0.49Ã‚Â·6.622 +2Ã‚Â·0.21Ã‚Â·6.62Ã‚Â·7.37Ã‚Â·0.79+0.09Ã‚Â·7.372 = 42.52
ÃÆ’(0.7RFDFFX + 0.3RFHKCX ) =
Ã¢Ë†Å¡
42.52 = 6.52%
The standard deviation of the portfolio is smaller than
the standard deviations of the funds
Verify that the standard deviation of the portfolio would
be 5.13% if the correlation was 0
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
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Two Risky Assets
Example
International Diversification
The example above suggests that you can get
some extra mileage diversifying your portfolio
internationally
LetÃ¢â‚¬â„¢s do the three Sharpe ratios (the average
risk-free rate is 0.23%)
R port = 0.7Ã‚Â·R FDFFX +0.3Ã‚Â·R FHKCX = 0.7Ã‚Â·0.56%+0.3Ã‚Â·1.47% = 0.83%
SFDFFX =
0.56% Ã¢Ë†â€™ 0.23%
0.9% Ã¢Ë†â€™ 0.23%
= 0.05; SFHKCX =
= 0.17
6.62%
7.37%
Sport =
Alexander Barinov (SoBA, UCR)
0.83% Ã¢Ë†â€™ 0.23%
= 0.09
6.52%
Asset Allocation and Optimal Portfolio
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Two Risky Assets
Example
Goal Seek
Goal Seek is a useful Excel tool that lets you tweak the
parameters to get the desired value of the formula
Data tab – What-If Analysis – Goal Seek
We want to change the weight on FDFFX so that the
Sharpe ratio of the portfolio beats the Sharpe ratio of
FHKCX
If we want Sport = 0.18, Ãâ€° = Ã¢Ë†â€™0.17 – we have to short sell
FDFFX and buy 117% of FHKCX
Note that we would never get the Sharpe ratio of 0.18
playing with FHKCX and RF
Also, bringing FHKCX into play allows us to easily beat
FDFFX on return, standard deviation and the Sharpe ratio
simultaneously
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
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Two Risky Assets
Example
Problems with Diversification
Consider two more funds: Fidelity Large Cap
(FLCSX, large US stocks) and Fidelity Latin
America (FLATX, large Brazilean stocks)
In 2005-2009, what do you think was the
correlation between the returns to FDFFX and
FLCSX?
And what do you think was the correlation
between the returns to FDFFX and FLATX?
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
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Two Risky Assets
Example
Problems with Diversification
Country funds can have high Sharpe ratio and
you may capture some of it without really
shifting away from the US stocks
Country funds usually hold huge multinationals,
and therefore do not offer much diversification
benefit
The real diversification probably comes from
smaller foreign firms, which are hard to get
Even with those, most likely you have to short
something to beat all components on the
Sharpe ratio
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
MGT 252 Investments
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Two Risky Assets
The Bullet
Minimum Variance Frontier
The set of all possible combinations of average
returns and standard deviations one can construct
from two (or more) assets is called the Minimum
Variance Frontier (MVF)
If the two assets have perfect positive correlation,
MVF is a straight line joining the two assets
The straight line intersects with the vertical axis,
meaning that we can reach zero standard deviation if
we short the top asset
If the two assets have perfect negative correlation,
MVF is a kinked line and we can achieve zero
standard deviation with two positive weights
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
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Two Risky Assets
The Bullet
Numerical Example
Consider two stocks X and Y: E(RX ) = 10%,
ÃÆ’(RX ) = 10% and E(RY ) = 15%, ÃÆ’(RY ) = 20%
If the correlation between their returns is +1, buy
2 stocks X and short 1 stock Y
Verify that this portfolio has E(RP ) = 5%,
ÃÆ’(RP ) = 0%
If the correlation is -1, buy 2 stocks X and 1
stock Y
Verify that this portfolio has E(RP ) = 11.67%,
ÃÆ’(RP ) = 0%
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
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Two Risky Assets
The Bullet
Minimum Variance Frontier
If the correlation is between -1 and +1, the MVF
is a curved line usually called the bullet
Play with portfolio weights in the Excel file and
draw the bullet
With more than two assets, the solution is very
complicated, but we still have the bullet, as
Markowitz has shown
The more negative is the correlation, the greater
is the space inside the bullet, and we like it
Alexander Barinov (SoBA, UCR)
Asset Allocation and Optimal Portfolio
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Two Risky Assets
The Bullet
Minimum Variance Frontier
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Two Risky Assets
The Bullet
Efficient Frontier
Minimum Variance Portfolio (MVP) – where
the bullet touches the vertical line
MVF is investorÃ¢â‚¬â„¢s “budget constraint”
The points inside the bullet are inefficient: you
take too much risk for too little reward
The points outside should be unattainable
The points on the frontier, but above MVP are
the efficient frontier
The points on MVF, but below MVP are also
inefficient
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Asset Allocation and Optimal Portfolio
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Two Risky Assets
The Bullet
Efficient Frontier
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Capital Asset Pricing Model
Professor Alexander Barinov
University of California Riverside
MGT 252 Investments and Portfolio Management
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Capital Asset Pricing Model
MGT 252 Investments
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Outline
1
Two-Fund Separation
2
CAPM and the Market Portfolio
3
Security Market Line
4
Zero-Beta CAPM
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Capital Asset Pricing Model
MGT 252 Investments
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Two-Fund Separation
Minimum Variance Frontier
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Capital Asset Pricing Model
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Two-Fund Separation
Suppose we can invest in any point on the bullet
and the risk-free asset
Essentially, we can draw a Capital Allocation
Line through the risk-free rate and any point on
the bullet
Any point on the CAL would be a portfolio
combining the RF asset and the risky portfolio
(the point at which the CAL crosses the bullet)
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Capital Asset Pricing Model
MGT 252 Investments
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Two-Fund Separation
Two-Fund Separation
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Capital Asset Pricing Model
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Two-Fund Separation
Two-Fund Separation
Recall that the slope of CAL is the Sharpe ratio, our
reward-for-volatility ratio
Hence, we would be making the CAL as steep as
possible, until it is tangent to the bullet
The tangency point is the Mean-Variance Efficient
(MVE) portfolio
Irrespective of risk aversion, all investors will hold this
portfolio
Risk aversion will only determine how they split their
investment between the MVE portfolio and the RF asset
Essentially, we are back to CAL and optimal portfolio
given CAL
Alexander Barinov (SoBA, UCR)
Capital Asset Pricing Model
MGT 252 Investments
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Two-Fund Separation
Solver add-in is an extended version of Goal Seek,
which lets you minimize and maximize the output
of the formulas instead of setting it to a number
To solve for MVP, minimize the standard
deviation of the portfolio by changing the
portfolio weight
Fidelity example: if we combine Fidelity
Independence and Fidelity China, the minimum
possible standard deviation (which is the standard
deviation of the MVP) is 6.52%, attained by the
portfolio that invests 75% in Fidelity Independence
and 25% in Fidelity China
Alexander Barinov (SoBA, UCR)
Capital Asset Pricing Model
MGT 252 Investments
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Two-Fund Separation
To solve for MVE, maximize the Sharpe ratio
of the portfolio by changing the weight
Fidelity example: if we combine Fidelity
Independence and Fidelity China, the MVE has
the Sharpe ratio of 0.22, attained by the weights
of -253% on Fidelity Independence and 353%
on Fidelity China
Weights interpretation: if you have \$100 to
invest, short sell Fidelity Independence for \$253
and invest \$353 in Fidelity China – it will give you
the best possible Sharpe ratio
Alexander Barinov (SoBA, UCR)
Capital Asset Pricing Model
MGT 252 Investments
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CAPM and the Market Portfolio
Two-Fund Separation
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Capital Asset Pricing Model
MGT 252 Investments
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CAPM and the Market Portfolio
CAPM and Two-Fund Separation
Two-Fund Separation says that everyone holds the MVE
portfolio and the RF asset
By definition, if we sum everyoneÃ¢â‚¬â„¢s holdings of risky
assets, we will get the market portfolio (i.e. everything in
the market of risky assets)
Therefore, everyone holds the market portfolio and the
market portfolio is mean-variance efficient
That is, in the long term you should not be able to beat
the market on the Sharpe ratio
If you can beat the Sharpe ratio of the market, the
CAPM is not valid
This is the first part of the CAPM (due to Sharpe, Lintner,
and Mossin)
Alexander Barinov (SoBA, UCR)
Capital Asset Pricing Model
MGT 252 Investments
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CAPM and the Market Portfolio
Market Portfolio in the CAPM
By market portfolio, the CAPM means all risky
assets in the economy, not just S&P500 or
exchange-listed stocks
Market portfolio in the CAPM includes other
securities, durable goods, human capital, etc.
Market portfolio is unobservable, therefore,
strictly speaking, the CAPM is not testable (Roll
critique)
Shanken (JF 1986, JFE 1987) argues that the
stock market portfolio can be a good proxy for
the market portfolio meant by the CAPM
Alexander Barinov (SoBA, UCR)
Capital Asset Pricing Model
MGT 252 Investments
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CAPM and the Market Portfolio
CAPM Assumptions
All changes in our wealth come from our investment
decisions (there are no other payments like wages, or
The CAPM is one-period model – in the CAPM,
investors do not try to shift wealth around because
some periods are better than the others
We assume that investors only care about mean and
variance
Investors have the same investment opportunity set
and understand that it is the same
It is possible to diversify away all idiosyncratic risks
Alexander Barinov (SoBA, UCR)
Capital Asset Pricing Model
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Security Market Line
Stock Contribution to Portfolio Variance
ÃÆ’ 2 (Ãâ€°A RA + Ãâ€°B RB ) = Ãâ€°A2 ÃÆ’ 2 (RA ) + 2Ãâ€°A Ãâ€°B Cov (RA , RB ) + Ãâ€°B2 ÃÆ’ 2 (RB )
Stock A contribution to the portfolio variance
Ãâ€°A2 ÃÆ’ 2 (RA ) + Ãâ€°A Ãâ€°B Cov (RA , RB )
Stock B contribution to the portfolio variance
Ãâ€°B2 ÃÆ’ 2 (RB ) + Ãâ€°A Ãâ€°B Cov (RA , RB )
If you throw in a third stock C, it will contribute
Ãâ€°C2 ÃÆ’ 2 (RC )+Ãâ€°A Ãâ€°C Cov (RA , RC )+Ãâ€°B Ãâ€°C Cov (RB , RC )
Alexander Barinov (SoBA, UCR)
Capital Asset Pricing Model
MGT 252 Investments
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Security Market Line
Stock Contribution to Portfolio Variance
Now, recall three facts about covariance: ÃÆ’ 2 (RC ) = Cov (RC , RC ),
Cov (RC , Ãâ€°A RA ) = Ãâ€°A Cov (RC , RA ) and
Cov (RC , RA + RB ) = Cov (RC , RA ) + Cov (RC , RB )
If you add stock J to a portfolio of N stocks, its contribution to the
variance will be
Ãâ€°J
N
X
Ãâ€°i Cov (RJ , Ri ) = Ãâ€°J
i=1
If we assume that
N
X
Cov (RJ , Ãâ€°i Ri ) = Ãâ€°J Cov (RJ ,
i=1
N
X
N
X
Ãâ€° i Ri )
i=1
Ãâ€°i Ri = RMKT , then adding more of stock J to
i=1
the market portfolio increases the variance by Ãâ€°J Cov (RJ , RMKT )
The contribution of stock J to expected return is easy: it is just
Ãâ€°J E(RJ )
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Capital Asset Pricing Model
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Security Market Line
Two Bullets
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Security Market Line
Two Bullets
Small bullet cannot go outside of the big one,
because big one combines everything in it,
including MKT and J, in the optimal fashion
But MKT is on both: it is on the big one,
because MKT is MVE, and it is on the small
one, because if you mix MKT and J, you can
always set Ãâ€°MKT = 1 and Ãâ€°J = 0
Hence, at MKT the big bullet and the small bullet
have to be tangent, i.e. have the same slope
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Capital Asset Pricing Model
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Security Market Line
From CML to Beta
Because what we know is the contribution of J to MKT expected
return and variance, instead of slopes we will use
return-to-variance ratios
For the MKT, it is
E(RMKT ) Ã¢Ë†â€™ RF
ÃÆ’ 2 (RMKT )
For J, it is its contribution to the expected return over its
Ãâ€°J (E(RJ ) Ã¢Ë†â€™ RF )
contribution to variance
Ãâ€°J Cov (RJ , RMKT )
Set those two ratios equal:
E(RJ ) Ã¢Ë†â€™ RF
E(RMKT ) Ã¢Ë†â€™ RF
=
Cov (RJ , RMKT )
ÃÆ’ 2 (RMKT )
E(RJ ) Ã¢Ë†â€™ RF =
Alexander Barinov (SoBA, UCR)
Cov (RJ , RMKT )
(E(RM ) Ã¢Ë†â€™ RF )
ÃÆ’ 2 (RMKT )
Capital Asset Pricing Model
MGT 252 Investments
17 / 28
Security Market Line
Security Market Line
E(RJ ) Ã¢Ë†â€™ RF = ÃŽÂ²J (E(RM ) Ã¢Ë†â€™ RF )
Cov (RJ , RMKT )
ÃŽÂ²J =
ÃÆ’ 2 (RMKT )
ÃŽÂ²J happens to be the slope of the regression of
the stock excess return on the market excess
return
Excess return is stock return minus the risk-free rate
The equation at the top of the slide is the main
result of the CAPM
It is called the Security Market Line (SML)
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Capital Asset Pricing Model
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Security Market Line
Interpreting Beta
Beta as the regression coefficient: if the market
goes up by 1%, the stock goes up, on average,
by ÃŽÂ²%
Beta does not measure how closely the asset
tracks the market (we have R-squared for this)
It is possible to have high beta and low
R-squared, and vice versa
By definition, beta measures the comovement of
stock and the market
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Capital Asset Pricing Model
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Security Market Line
How the Notion of Risk Has Changed
We started with standard deviation as a
measure of risk (CAL)
When we mixed together two risky assets, we
realized that covariance also matters
When we looked at the portfolio of N stocks, we
concluded that the numerous covariance terms
are much more important than one single
variance term
The CAPM proclaimed that all what matters is
the covariance with the market (recall what the
CAPM means by “market portfolio”)
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Capital Asset Pricing Model
MGT 252 Investments
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Zero-Beta CAPM
Zero-Beta CAPM
Even Treasuries have some risk (inflation, small
risk of default)
Availability of borrowing at the risk-free rate is a
dubious assumption
Instead of the risk-free rate, we can take the rate of
return of any asset that has zero correlation with
the market
However, there is only one line that is tangent to
the bullet at M
Black (1972) shows that if we draw a horizontal line
from the CML intercept to the MVF, we will hit the
portfolio that has zero correlation with the market
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Capital Asset Pricing Model
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Zero-Beta CAPM
Zero-Beta CAPM
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Capital Asset Pricing Model
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Zero-Beta CAPM
Making It Work
You need to know the mean, the standard
deviation, and the Sharpe ratio of the market
portfolio
Then you take the CML and solve for “RF”, which
is now the zero-beta rate
This is the average return of the zero-beta portfolio
Now use Solver to figure out the standard deviation
the CML for by changing the weight” and watch the
standard deviation cell to update automatically
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Capital Asset Pricing Model
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Zero-Beta CAPM
Zero-Beta CAPM: Example
Assume that the Fidelity China fund and the Fidelity
Independence fund are the only two assets in the
economy
We have already solved for MVE: mean 3.79%, standard
deviation 16.43%, Sharpe 0.22
CML :
E(RM ) = ZB + SM Ã‚Â· ÃÆ’(RM )
Substitute and solve for ZB – it will be 0.23%, just as the
risk-free rate, because you used 0.23 in the Sharpe ratio
when you maximized it
If you could just observe the market portfolio instead of
solving for it, you would figure out the slope of MVF at the
market portfolio and ZB will not be RF when you solve for
it from the CML
Alexander Barinov (SoBA, UCR)
Capital Asset Pricing Model
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Zero-Beta CAPM
Zero-Beta CAPM: Example
Now, what risky portfolio has the expected return of
0.23%?
changing the weight”
Solver says the weight is 136% in Fidelity
Independence (-36% in Fidelity China)
This risky portfolio is a substitute for the risk-free
asset, because it has zero beta and thus the CAPM
says it is as risky as the risk-free asset – adding it to
your portfolio will not increase the portfolioÃ¢â‚¬â„¢s variance
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Capital Asset Pricing Model
MGT 252 Investments
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Zero-Beta CAPM
Do We Want Inefficient Portfolios?
CAPM says we have to hold MVE (the market),
but it can price anything else
SML :
E(RP ) = RF + ÃŽÂ²P Ã‚Â· (RM Ã¢Ë†â€™ RF )
For example, zero-beta portfolio is from the
inefficient part of MVF, so you do not want to put
there all the money you plan to put into the stock
market
But if this portfolio has a zero alpha, it is a fair
deal on the margin: we can add a small bit of it
to our diversified portfolio, it will not hurt
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Capital Asset Pricing Model
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Zero-Beta CAPM
Alpha and Sharpe Ratio
Diversified portfolios with positive alpha will beat
the market portfolio on the Sharpe ratio and vice
versa
An individual stock can have a positive alpha
and lose to the market portfolio on the Sharpe
ratio, because one stock is very volatile
CAPM says you can diversify this risk away adding the positive alpha stock to your portfolio
from CML will create a portfolio that beats the
market on the Sharpe ratio
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Capital Asset Pricing Model
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Zero-Beta CAPM
Zero-Beta CAPM
Suppose that the zero-beta CAPM holds and the true
SML is E(Ret Ã¢Ë†â€™ RZB ) = ÃŽÂ²i Ã‚Â· E(RM Ã¢Ë†â€™ RZB )
But instead we estimate the CAPM with the risk-free rate
Deduct RF from both sides of the SML and collect the
terms:
E(Ret) Ã¢Ë†â€™ RF = E(RZB ) + ÃŽÂ²i E(RM ) Ã¢Ë†â€™ ÃŽÂ²i E(RZB ) Ã¢Ë†â€™ ÃŽÂ²i RF + ÃŽÂ²i RF Ã¢Ë†â€™ RF
E(Ret)Ã¢Ë†â€™RF = (1Ã¢Ë†â€™ÃŽÂ²i )E(RZB )+ÃŽÂ²i (E(RM )Ã¢Ë†â€™RF )Ã¢Ë†â€™(1Ã¢Ë†â€™ÃŽÂ²i )RF
E(Ret) Ã¢Ë†â€™ RF = (1 Ã¢Ë†â€™ ÃŽÂ²i )(E(RZB ) Ã¢Ë†â€™ RF ) + ÃŽÂ²i (E(RM ) Ã¢Ë†â€™ RF )
So, if you ignore the inability to borrow and lend at the
risk-free rate, you will underestimate the performance of
aggressive stocks (ÃŽÂ²i > 1) and overestimate the
performance of conservative stocks (ÃŽÂ²i < 1) Alexander Barinov (SoBA, UCR) Capital Asset Pricing Model MGT 252 Investments 28 / 28 Purchase answer to see full attachment

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