Efficient Diversification II
Efficient Frontier with Risk-Free Asset
Optimal Capital Allocation Line
Single Factor Model
Eff. Frontier with Risk-Free Asset
With risky assets only
No portfolio with zero variance
GMVP has the lowest variance
With a risk-free asset
Zero variance if investing in risk-free asset only
How does it change the efficient frontier?
Investments 10 2
Optimal CAL
Mean-variance with two risky assets
w in security 1, 1 – w in security 2
1
2
0
0
.
10
.
14
1
2
0 .
15
0 .
20
12
0 .
2
Expected return (Mean):
p
0 .
10
w
0 .
14
( 1
w )
Variance
2 p
0 .
15
2 w
2
0 .
20
2
( 1
w )
2
2
0 .
2
0 .
15
0 .
20
w ( 1
w )
What happens when we add a risk-free asset?
A riskfree asset with r f
= 5%
What is achievable now?
Investments 10 3
Eff. Frontier with Risk-Free Asset
E[r]
M
M
CAL (P)
P
P
CAL
G
F
P P&F M
Investments 10 4
Eff. Frontier with Risk-Free Asset
CAL(P) dominates other lines
Best risk and return trade-off
Steepest slope
S
P
E [ r p
] p
r f
E [ r
A
]
A
r f
Portfolios along CAL(P) has the same highest
Sharpe ratio
No portfolio with higher Sharpe ratio is achievable
Dominance independent of risk preference
How to find portfolio (P)?
Investments 10 5
Optimal Portfolio
How much in each risky asset?
w
1
1
2
( E [ r
2
]
r f
2
)
2
( E [ r
1
]
2
2
r f
( E [ r
1
]
)
r f
)
1
2
( E [ r
2
1
2
]
r f
)
( E [ r
1
]
E [ r
2
]
2 r f
)
.4584
The expected return and standard dev.
p
0 .
10
w
1
0 .
14
( 1
w
1
)
0 .
1217
p
0 .
1394
Sharpe Ratio
S
P
E [ r p
] p
r f
p
p r f
0 .
1217
0 .
05
0 .
1394
0 .
514
Investments 10 6
Eff. Frontier with Risk-Free Asset
What’s so special about portfolio (P)?
P is the market portfolio
Mutual fund theorem: An index mutual fund
(market portfolio) and T-bills are sufficient for investors
Investors adjust the holding of index fund and T-bills according to their risk preferences
Investments 10 7
w
Optimal Portfolio Allocation
Investment Funds y 1y
P T-Bills
1w
Bond Stock
T - Bills Bond Stock
1 y y
× w y
×(1 w )
Investments 10
Two Step Allocation
Step 1: Determine the optimal risky portfolio
Get the optimal mix of stock and bond
Optimal for all investors
(market portfolio)
Step 2: Determine the best complete portfolio
Obtain the best mix of the optimal risky portfolio and T-Bills
Different investors may have different best complete portfolios
8
Single Factor Model
Quantifies idiosyncratic versus systematic risk of a stock’s rate of return
Factor is a broad market index like S&P500
The excess return is
R
R
e
i i i i i M i
: stock’s excess return above market performance
R
M
: stock’s return attributable to market performance
e i
: return component from firm-specific unexpected event
Example: a statistical analysis between the excess returns of DELL and market shows that
= 4.5%,
= 1.4. If expected market excess return is 17%, what is the expected excess return for DELL?
Solution: E [ R i
]
i
i
E [ R
M
]
4 .
5 %
1 .
4
17 %
4 .
5 %
23 .
8 %
28 .
3 %
Investments 10 9
Single Factor Model
Security Characteristic Line
Dell Excess
Returns (i)
.
.
. ..
28.3%
. .
.
.
.
.
.
.
.
. ..
.
.
.
. .
.
.
.
.
. ..
..
.
17%
.
.
Security
Characteristic
Line
23.8%
i
4.5%
Cov [
R i
2
M
, R
M
]
Excess Returns on market index
Investments 10 10
Single Factor Model
Meaning of Beta (
)
Indicator of how sensitive a security’s return is to changes in the return of the market portfolio.
A measure of the asset’s systematic risk.
Example: market portfolio’s risk premium is
+10% during a given period, and
= 0%.
= 1.50, the security’s risk premium will be +15%.
= 1.00, the security’s risk premium will be +10%
= 0.50, the security’s risk premium will be +5%
= –0.50, the security’s risk premium will be –5%
Investments 10 11
Single Factor Model
Beta coefficients for selected firms (March 2010)
Common Stock Beta
Citigroup
Bank of America
Adobe Systems
Apple
GE
Amazon.com
Microsoft
McDonald
’s
Pepsi
Exxon Mobile
Wal-Mart
Question:
2.71
2.41
1.80
1.57
1.52
1.27
1.12
0.98
0.64
0.52
0.43
0.26
What are the betas of market index and T-bills?
Investments 10 12
Single Factor Model
Systematic Risk
Risk related to the macro factor or market index
Non-diversifiable/market risk
Unsystematic Risk
Risk related to company specific problems
Diversifiable/Firm-specific/Idiosyncratic risk
Total risk = Systematic + Unsystematic
i
2
2
i
2
2
M
2
i
i
2
2
M
Var [ e i
]
% of variance explained by the market
Investments 10 13
Single Factor Model
Example
Given the following data on Microsoft, analyze the systematic risk, unsystematic risk and percentage of variance explained by systematic risk. ( σ i
σ
M
= 0.15, Cov [ R i
,R
M
]=0.0315)
= 0.25,
Solution
i
Cov [
R i
2
M
, R
M
]
0.0315
.15
2
1.4
i
2
2
M
Var [ e i
]
i
2
2
1.4
2
.15
2
2
i
i
2
2
M
i
2
2
M
0 .
0441
.
25
2
0 .
0441
0 .
0184
.0441
.7056
70 .
56 %
.25
2
Investments 10 14
A portfolio of three equally weighted assets 1,
2, and 3.
p
The excess return of the portfolio is
R p
p
p
R
M
e p
1
2
3
3
p
1
2
3
3 e p
e
1
e
2
3
e
3
Risk of the portfolio is
Var ( R p
)
2 p
Var ( R
M
)
Var ( e p
)
p
2
2
M
Var ( e p
)
Investments 10 15
Wrap-up
What does the efficient frontier look like with the presence of a risk-free asset?
What are the two steps of asset allocation?
What is a single index model?
What are the meaning of systematic and unsystematic risks?
Investments 10 16