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```Factor Models
Riccardo Colacito
Diversification and Portfolio Risk
• Market risk
– Systematic or Nondiversifiable
• Firm-specific risk
– Diversifiable or nonsystematic
Foundations of Financial Markets
2
Portfolio Risk as a
Function of the Number of Stocks
Foundations of Financial Markets
3
Portfolio Risk as a
Function of Number of Securities
Foundations of Financial Markets
4
A single factor Model
Ri  ERi   i M  ei
Ri  ri  rf is excess return
ERi  is expectedexcess return
M is Market- Macroeconomic Surprise
i is sensitivity to marketsurprise
ei is unanticipated firm - specific event
Foundations of Financial Markets
5
What is M?
• Anything that can be regarded as a proxy
for macroeconomic risk
• Commonly used factor: a broad market
index like the S&P500
• Call it Rm
Foundations of Financial Markets
6
Commonly Run Regression
Ri  i  i Rm  ei
Ri  ri  rf is excess return
i is stock's excess return if the marketfactor is neutral
i Rm is componentof return due to movementsin the
overallmarket
ei is unanticipated firm - specific event
Foundations of Financial Markets
7
Scatter Diagram for Dell
Foundations of Financial Markets
8
Various Scatter Diagrams
Foundations of Financial Markets
9
Coca Cola: another example
KO vs. SP
0.5
KO
0.0
-0.5
-1.0
-1.5
-0.3
-0.2
-0.1
0.0
0.1
0.2
SP
Foundations of Financial Markets
10
Regression statistics
Dependent Variable: KO
Method: Least Squares
Sample: 1962:02 2007:10
Included observations: 549
Variable Coefficient
C
-0.005508
SP
0.816898
Std. Error
0.004206
0.098425
t-Statistic
-1.309559
8.299663
Prob.
0.1909
0.0000
R2 = 0.111846
Foundations of Financial Markets
11
A more recent sample
Dependent Variable: KO
Method: Least Squares
Sample: 1990:01 2007:10
Included observations: 214
Variable Coefficient
C
-0.004975
SP
0.523387
Std. Error
0.006378
0.158603
t-Statistic
-0.780066
3.299976
Prob.
0.4362
0.0011
R2 = 0.048858
Foundations of Financial Markets
12
Some Betas of S&P500
companies
Company
Beta
Apple
1.3
Amazon
1.6
Cisco
1.1
Coca Cola
0.8
Countrywide Financial
1.8
Goldman Sachs
1.7
Johnson & Johnson
0.5
McDonald's
0.8
Microsoft
0.9
Foundations of Financial Markets
13
Measuring Components of Risk
si2 = i2 sm2 + s2(ei)
Where:
si2 = total variance
i2 sm2 = systematic variance
s2(ei) = unsystematic variance
Foundations of Financial Markets
14
Decomposition of Risk
•
Total variability of the rate of return
depends on two components
1. The variance attributable to the
uncertainty common to the entire market
2. The variance attributable to firm specific
risk factors
Foundations of Financial Markets
15
Systematic and idiosyncratic
risk with many securities
• Two assets
R1  1  1Rm  e1
R2  2  2 Rm  e2
• Portfolio weights are w1 and (1-w1)
• What is the portfolio ?
• What is the systematic risk of the portfolio?
• What is the idiosyncratic risk of the portfolio?
Foundations of Financial Markets
16
Portfolio 
 p  w11  1 w1 2
• because
w1R1  1  w1 R2  w11  1  w1  2 
 w11  1  w1 2 Rm
 w1e1  1  w1 e2 
Foundations of Financial Markets
17
Systematic Risk
w11  1 w1 2  s
2
2
m
• A good strategy would select securities
with smallest ’s
Foundations of Financial Markets
18
Idiosyncratic Risk
Varw1e1  1 w1 e2 
• Benefits from diversification if idiosyncratic
risk is less than perfectly correlated
Foundations of Financial Markets
19
Advantages of the Single Index
Model
• Reduces the number of inputs for
diversification
• Easier for security analysts to specialize
Foundations of Financial Markets
20
What risk should be priced?
• What risk should be priced?
– Idiosyncratic risk: no
– Aggregate risk: yes
• Only aggregate/macro risk commands a
premium
Foundations of Financial Markets
21
Why?
Because:
1. idiosyncratic risk can be diversified away
2. Macro risk affects all assets and cannot
be diversified
Foundations of Financial Markets
22
Example: two assets
Twoassets:
R1  1  1Rm  e1 and R2   2  2 Rm  e2
Assume:
1  2  1
Var(e1 )  Var(e2 )  1
cov(e1, e2 )  0
Investequal portfolio sharesin the two assets
Foundations of Financial Markets
23
What is the portfolio variance?
Var(w1R1  w2 R2 )  Var(Rm ) 1/ 2
Foundations of Financial Markets
Systematic
Idiosyncratic
Risk
Risk
24
Example: three assets
Twoassets:
Ri  i  i Rm  ei , i  1,2,3
Assume:
1  2  3  1
Var(e1 )  Var(e2 )  Var(e3 )  1
cov(e1 , e2 )  cov(e1 , e3 )  cov(e2 , e3 )  0
Investequal portfolio sharesin the threeassets
Foundations of Financial Markets
25
What is the portfolio variance?
Var(w1R1  w2 R2  w3 R3 )  Var(Rm ) 1/ 3
Systematic
Idiosyncratic
Risk
Risk
•Systematic risk: unchanged
•Idiosyncratic risk: decreased
•Can you guess what would happen if we had an infinite
number of assets?
Foundations of Financial Markets
26
Infinite assets
• For a well diversified portfolio
2

 

Var  wi Ri     i wi  VarRm ,
 i
  i

or s Rp    ps m
• That is: we got rid of any idiosyncratic
shock and we are left only with systematic
risk
Foundations of Financial Markets
27
What lesson did we learn?
• The only source of risk that we are entitled
to ask a compensation for is aggregate
risk.
• Idiosyncratic risk does not entitle to any
compensation because it can be
diversified away.
Foundations of Financial Markets
28
Risk compensation
Any asset entitles to a compensation proportional
to its contribution toaggregate risk :
E( Rm  rf )
E( R1  rf ) E( R2  rf )
 ... 

1s m
2s m
sm
or, equivalently
E( R1 )  rf  1E( Rm  rf )
E( R2 )  rf  2 E( Rm  rf )
...
Foundations of Financial Markets
29
This opens up to our next topic
• The Capital Asset Pricing model!
Foundations of Financial Markets
30
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