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
Google
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