Download CAPM Model and Beta

By Hao Sun
Data
 Financial and Food Stocks from S&P100 Index
 Include: BAC, BK, GS, JPM, MS, NYX, WFC, HNZ, KO,
KFT, PEP
 Also used S&P100 Futures as the market index for
CAPM model
Model
 𝑅𝑉𝑡,Δ =
 𝐵𝑉𝑡,Δ =
𝜇−2
=
𝜋
2
385/Δ 2
𝑗=1 𝑟𝑡,𝑗 ,
−2 𝑀
𝜇
𝑀−1
where Δ = 5.
𝑀
𝑗=2 𝑟𝑡,𝑗−1
𝑟𝑡,𝑗 , where 𝑀 =
385
,
Δ
 CAPM: 𝑟𝑖 = 𝛽𝑖 𝑟𝑚 + 𝜖𝑖 , where 𝐶𝑜𝑣 𝜖𝑖 , 𝑟𝑚 = 0.
2.
 So, 𝐶𝑜𝑣 𝑟𝑖 , 𝑟𝑚 = 𝐶𝑜𝑣 𝛽𝑖 𝑟𝑚 + 𝜖𝑖 , 𝑟𝑚 = 𝛽𝑖 𝜎𝑚
Method
 We construct a portfolio with equal weight of 𝑟𝑖 and
𝑟𝑚 , where 𝑟𝑖 is the return on stock i, and 𝑟𝑚 is the
return on S&P100 Index.


1
1
𝑟𝑝 = 𝑟𝑖 + 𝑟𝑚
2
2
1
1
𝑅𝑉𝑟𝑝 = 𝑅𝑉𝑟𝑖 + 𝑅𝑉𝑟𝑚
4
4
 𝑅𝐶𝑜𝑣 𝑟𝑖 , 𝑟𝑚 = 2 ×
1
 𝑅𝛽𝑖 =
1
+2
𝑅𝐶𝑜𝑣 𝑟𝑖 , 𝑟𝑚
4
1
1
𝑅𝑉𝑟𝑝 − 𝑅𝑉𝑟𝑖 − 𝑅𝑉𝑟𝑚
4
4
1
2× 𝑅𝑉𝑟𝑝 −4𝑅𝑉𝑟𝑖 −4𝑅𝑉𝑟𝑚
𝑅𝑉𝑟𝑚
= 𝑅𝛽𝑖 𝑅𝑉𝑟𝑚
 Similarly,
 We have:
1
 𝐵𝛽𝑖 =
1
2× 𝐵𝑉𝑟𝑝 −4𝐵𝑉𝑟𝑖 −4𝐵𝑉𝑟𝑚
𝑅𝑉𝑟𝑚
 Since Bi-power measure is jump-robust, we can find
the contribution of jump by subtracting 𝐵𝛽𝑖 from 𝑅𝛽𝑖 .
Numerical Results
Stock
Contribution of Jump
BAC
0.0034
(0.2533)
BK
-0.0239
(0.2905)
GS
0.0232
(0.2690)
JPM
0.0253
(0.2560)
MS
0.0461
(0.3268)
NYX
0.0328
(0.3918)
WFC
-3.933e-004
(0.2532)
Calculations vs. Regression
Stock
Realized Beta
Beta from monthly Reg.
BAC
0.8717
(0.4386)
1.4797
(0.0357)
BK
0.8482
(0.3790)
1.1124
(0.0206)
GS
0.8807
(0.3895)
1.3664
(0.0282)
JPM
0.9444
(0.3702)
1.3648
(0.0251)
MS
1.2513
(0.5228)
1.7246
(0.0528)
NYX
1.0328
(0.5141)
1.6044
(0.0444)
WFC
0.7906
(0.4348)
0.9318
(0.0266)
JPM 3-mth Moving Avg. R-Beta
JPM: Realized Beta
2.5
2
Beta
1.5
1
0.5
0
Apr 09, 1997
Sep 06, 2000
Feb 20, 2004
Date
Jul 20, 2007
Dec 30, 2010
BAC: Realized Beta
GS: Realized Beta
2.5
1.5
2
1
Beta
Beta
1.5
1
0.5
0.5
0
Apr 09, 1997
Sep 07, 2000
Feb 23, 2004
Date
Jul 20, 2007
Dec 30, 2010
0
May 05, 1999
Apr 05, 2002
GS: Realized Beta
Mar 03, 2005
Date
Jan 28, 2008
Dec 30, 2010
GS: Realized Beta
2
2
1.5
1.5
Beta
2.5
Beta
2.5
1
1
0.5
0.5
0
Jan 17, 2006
Apr 10, 2007
Jul 02, 2008
Date
Sep 25, 2009
Dec 30, 2010
0
Apr 09, 1997
Sep 11, 2000
Feb 24, 2004
Date
Jul 20, 2007
Dec 30, 2010
KFT: Realized Beta
1
1
0.8
0.8
0.6
0.6
Beta
Beta
HNZ: Realized Beta
0.4
0.4
0.2
0.2
0
Apr 09, 1997
Sep 07, 2000
Feb 23, 2004
Date
Jul 20, 2007
Dec 30, 2010
0
Jun 13, 2001
Jun 04, 2003
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
Apr 09, 1997
Sep 07, 2000
Feb 20, 2004
Date
Apr 30, 2007
Apr 16, 2009
Jul 19, 2007
Dec 30, 2010
PEP: Realized Beta
Beta
Beta
KO: Realized Beta
May 17, 2005
Date
Jul 20, 2007
Dec 30, 2010
0
Apr 09, 1997
Sep 07, 2000
Feb 20, 2004
Date
Food&Beverages
Stock
Realized Beta
Bi-Power Beta
HNZ
0.4277
(0.2284)
0.4676
(0.2750)
KFT
0.3173
(0.2535)
0.3515
(0.2945)
KO
0.5645
(0.2754)
0.5715
(0.3276)
PEP
0.4974
(0.2612)
0.5461
(0.3356)
Conclusion
 Jump Contributions are not significant to beta.
 In other words, market systematic risks are not affected




jumps in prices of individual stocks.
So CAPM model will not be affected by outliers in returns
as some papers claim (Martin, Simin, 2003)
During periods of financial stress, Financial firms tend to
have a beta above 1.
On the contrary, the beta of Food&Beverage Providers
remain low during financial stress. (Some even decrease)
In general, financial firms have higher beta.