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
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Optimal CAL

Mean-variance with two risky assets


w in security 1, 1 – w in security 2
1  0.10  1  0.15
12  0.2
 2  0.14  2  0.20
Expected return (Mean):
 p  0.10  w  0.14  (1  w)

Variance
 2p  0.152 w2  0.202 (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 rf = 5%
What is achievable now?
Investments 10
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Eff. Frontier with Risk-Free Asset
E[r]
CAL (P)
M
M
P
P
CAL
G
F
P P&F M
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
4
Eff. Frontier with Risk-Free Asset

CAL(P) dominates other lines


Best risk and return trade-off
Steepest slope
SP 




E[rp ]  rf
p

E[rA ]  rf
A
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
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Optimal Portfolio

How much in each risky asset?
 22 ( E[r1 ]  rf )   1 2 ( E[r2 ]  rf )
w1  2
 .4584
2
 1 ( E[r2 ]  rf )   2 ( E[r1 ]  rf )   1 2 ( E[r1 ]  E[r2 ]  2rf )

The expected return and standard dev.
 p  0.10  w1  0.14  (1  w1 )  0.1217
 p  0.1394

Sharpe Ratio
SP 
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E[rp ]  rf
p
 p  rf 0.1217  0.05


 0.514
p
0.1394
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
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Optimal Portfolio Allocation
Investment Funds
y
1-y

Two Step Allocation

Step 1: Determine the
optimal risky portfolio

P
w
T-Bills

1-w

Bond
Stock
Step 2: Determine the
best complete portfolio

T - Bills Bond
Stock

1-y
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y×w
y×(1 - w)
Get the optimal mix of
stock and bond
Optimal for all investors
(market 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
Ri   i   i RM  ei


 i : stock’s excess return above market performance
i RM : stock’s return attributable to market performance
ei : 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[ Ri ]   i   i E[ RM ]  4.5%  1.4 17%



 4.5%  23.8%  28.3%
Investments 10
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Single Factor Model

Security Characteristic Line
Dell Excess
Returns (i)
Security
Characteristic
Line
.
.
.
.
.
.
. . .
.
.
.
.
23.8%
.
.
.
. .
Cov[ R , R
 
.ß = 1.4 . .

.
.
.
.
.
.
4.5%
.
.
.. . . .
17% Excess Returns
28.3%
i
i
2
M
M
on market index
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]
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
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Single Factor Model

Beta coefficients for selected firms (March 2010)
Common Stock
Citigroup
Bank of America
Adobe Systems
Apple
GE
Amazon.com
Google
Microsoft
McDonald’s
Pepsi
Exxon Mobile
Wal-Mart

Beta
2.71
2.41
1.80
1.57
1.52
1.27
1.12
0.98
0.64
0.52
0.43
0.26
Question:

What are the betas of market index and T-bills?
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Single Factor Model

Systematic Risk



Unsystematic Risk



Risk related to the macro factor or market index
Non-diversifiable/market risk
Risk related to company specific problems
Diversifiable/Firm-specific/Idiosyncratic risk
Total risk = Systematic + Unsystematic
 i2   i2 M2  Var[ei ]
2 2

2
i M
 
 i2
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% of variance explained
by the market
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. (σi= 0.25,
σM= 0.15, Cov[Ri,RM]=0.0315)
Solution
i 
Cov[ Ri , RM ]
 M2

0.0315
 1.4
2
.15
 i2 M2  1.4 2  .15 2  0.0441
Var[ei ]   i2   i2 M2  .252  0.0441  0.0184
 i2 M2 .0441
 

 .7056  70.56%
2
2
i
.25
2
Investments 10
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Diversification in a Single Factor
Security Market


A portfolio of three equally weighted assets 1,
2, and 3.
The excess return of the portfolio is
R p   p   p RM  e p
p 

1   2   3
3
p 
1   2   3
3
e1  e2  e3
ep 
3
Risk of the portfolio is
Var( R p )   Var( RM )  Var(e p )   
2
p
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2
p
2
M
 Var(e p )
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
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