Unformatted Attachment Preview
Project 1
Due on January 20 by 5:00 pm
MGT 252 - Winter 2021
Investments and Portfolio Management
c 2021 Alexander Barinov
1
Price Momentum (50 points)
The file Project 1.xls in the Projects folder contains the monthly data on risk-free rate
(RF), the excess return to the market (MKT-RF), and the returns to the momentum
strategy (MOM) between January 1964 and June 2009. The momentum strategy buys the
stocks with the most positive returns in the past year and shorts the stocks with the most
negative returns in the past year. For this problem, assume that the risk-free rate RF is
not constant.
i. Compute the monthly average and the monthly standard deviation for MOM and
RF. (5 points)
ii. Based on these sample estimates, build the 95% confidence interval for the return
to the MOM portfolio in the next year. Provide a concise explanation of what this
interval means in economic terms. (10 points)
iii. Repeat (ii) for the average annual return. Explain the difference. (10 points)
iv. Assume that MOM and RF are uncorrelated. Compute the mean and standard
deviation of the daily return to strategy S, which is 70% MOM and 30% RF. Assume
22 trading days in a month. (5 points)
v. Can you reject the hypothesis that the true expected monthly return to MOM is
zero? Use all three ways to test the hypothesis we discussed in class. (10 points)
vi. Bonus question: Test the equality of the average returns to the MOM strategy and
the average excess return to the market, MKT-RF. (10 points)
vii. Compute VaR for strategy S if $1,000,000 is invested for one quarter. Assume 63
trading days per quarter. (10 points)
1
2
Event Study (50 points)
Consider a stock that responded to a certain event by gaining 0.25% on the day before
the announcement, 1.75% at the announcement date, and 0.5% on the following day. On
the same dates, the market return was 0.1%, 0.3%, and -0.3%, respectively. The risk-free
rate was constant at 0.05%. Assume that the event window includes all three days and
the standard deviation of the stock return in the pre-event window is 4.6% per month.
The market model estimated for the stock in the pre-event window yields
Rett − RFt = 0.13 + 1.18 (M KTt − RFt ) + t ,
(0.08) (0.37)
R2 = 0.31
i. Use the market model to calculate the cumulative abnormal return to the stock in
the event window. (5 points)
ii. Use the R-square of the market model to compute the daily standard deviation of
abnormal return (aka regression residual). Assume 22 trading days per month.(10
points)
iii. Did the event have a statistically significant impact on the stock price? (10 points)
iv. Use the market model to compute the daily standard deviation of the market portfolio. (10 points)
v. Was the event-window run-up for the market portfolio statistically significant?
(10 points)
vi. Assume that returns above are the average for a portfolio of 100 stocks that had
similar events and the 100 event windows do not overlap. Assume also that the
standard deviation of returns and the parameters of the market model are the same
for all 100 stocks. How does it change your answer to (iii)? (5 points)
vii. Bonus question: Suppose someone tells you that volatility should be measured in
calendar time, not in trading time, because Saturdays and Sundays have the same
amount of news as business days, even though trading on the weekend news is delayed
until Monday.. If you know that the event happened on Monday, how does it change
your answer to (iii)? Assume 30 calendar days in a month. (10 points)
2
Date
31-Jan-64
28-Feb-64
31-Mar-64
30-Apr-64
28-May-64
30-Jun-64
31-Jul-64
31-Aug-64
30-Sep-64
30-Oct-64
30-Nov-64
31-Dec-64
29-Jan-65
26-Feb-65
31-Mar-65
30-Apr-65
28-May-65
30-Jun-65
30-Jul-65
31-Aug-65
30-Sep-65
29-Oct-65
30-Nov-65
31-Dec-65
31-Jan-66
28-Feb-66
31-Mar-66
29-Apr-66
31-May-66
30-Jun-66
29-Jul-66
31-Aug-66
30-Sep-66
31-Oct-66
30-Nov-66
30-Dec-66
31-Jan-67
28-Feb-67
31-Mar-67
28-Apr-67
31-May-67
30-Jun-67
31-Jul-67
31-Aug-67
29-Sep-67
31-Oct-67
MOM
1.06
0.27
0.70
-0.59
2.56
0.49
-0.32
-0.21
-0.39
0.18
1.09
-0.72
-1.40
0.31
0.08
2.57
0.53
-3.13
4.08
2.57
3.28
3.44
4.41
0.14
5.37
4.54
1.36
6.16
-4.70
3.33
-1.44
-2.09
-1.77
-5.11
5.68
1.05
-6.66
3.55
1.43
0.65
0.61
6.00
-1.12
-1.42
2.53
3.67
RF
0.3
0.26
0.31
0.29
0.26
0.3
0.3
0.28
0.28
0.29
0.29
0.31
0.28
0.3
0.36
0.31
0.31
0.35
0.31
0.33
0.31
0.31
0.35
0.33
0.38
0.35
0.38
0.34
0.41
0.38
0.35
0.41
0.4
0.45
0.4
0.4
0.43
0.36
0.39
0.32
0.33
0.27
0.32
0.31
0.32
0.39
MKT-RF
2.28
1.46
1.45
0.17
1.48
1.21
1.71
-1.41
2.77
0.6
0.02
0.06
3.58
0.39
-1.33
3.06
-0.75
-5.54
1.37
2.76
2.89
2.62
-0.04
1.02
0.83
-1.21
-2.47
2.14
-5.66
-1.41
-1.64
-7.95
-1.1
3.78
1.35
0.22
8.12
0.73
3.95
3.84
-4.26
2.42
4.6
-0.94
3.11
-3.13
30-Nov-67
29-Dec-67
31-Jan-68
29-Feb-68
29-Mar-68
30-Apr-68
31-May-68
28-Jun-68
30-Jul-68
30-Aug-68
30-Sep-68
31-Oct-68
29-Nov-68
31-Dec-68
31-Jan-69
28-Feb-69
28-Mar-69
30-Apr-69
29-May-69
30-Jun-69
31-Jul-69
29-Aug-69
30-Sep-69
31-Oct-69
28-Nov-69
31-Dec-69
30-Jan-70
27-Feb-70
31-Mar-70
30-Apr-70
29-May-70
30-Jun-70
31-Jul-70
31-Aug-70
30-Sep-70
30-Oct-70
30-Nov-70
31-Dec-70
29-Jan-71
26-Feb-71
31-Mar-71
30-Apr-71
28-May-71
30-Jun-71
30-Jul-71
31-Aug-71
30-Sep-71
1.25
3.25
-4.70
-3.38
3.21
5.09
3.71
-1.91
-0.90
1.92
-0.60
-1.42
1.77
0.29
-0.17
-2.42
3.99
1.11
1.63
-2.27
1.71
2.18
2.50
-4.28
3.64
4.97
0.60
0.20
-0.27
-0.70
-2.63
5.78
-3.08
-6.45
-8.88
9.54
2.87
-2.21
-6.60
0.77
-1.36
1.45
0.87
2.75
-2.38
3.51
2.11
0.36
0.33
0.4
0.39
0.38
0.43
0.45
0.43
0.48
0.42
0.43
0.44
0.42
0.43
0.53
0.46
0.46
0.53
0.48
0.51
0.53
0.5
0.62
0.6
0.52
0.64
0.6
0.62
0.57
0.5
0.53
0.58
0.52
0.53
0.54
0.46
0.46
0.42
0.38
0.33
0.3
0.28
0.29
0.37
0.4
0.47
0.37
0.43
3.04
-4.03
-3.75
0.13
8.98
2.25
0.72
-2.68
1.38
4.02
0.46
5.43
-3.82
-1.2
-5.82
2.59
1.52
0.02
-7.25
-7.05
4.65
-2.88
4.96
-3.74
-2.61
-7.93
5.05
-1.04
-11.03
-6.96
-5.69
6.9
4.47
4.21
-2.28
4.58
5.65
4.82
1.36
4.18
3.05
-3.93
-0.06
-4.43
3.78
-0.87
29-Oct-71
30-Nov-71
31-Dec-71
31-Jan-72
29-Feb-72
30-Mar-72
28-Apr-72
31-May-72
30-Jun-72
31-Jul-72
31-Aug-72
29-Sep-72
31-Oct-72
30-Nov-72
29-Dec-72
31-Jan-73
28-Feb-73
30-Mar-73
30-Apr-73
31-May-73
29-Jun-73
31-Jul-73
31-Aug-73
28-Sep-73
31-Oct-73
30-Nov-73
31-Dec-73
31-Jan-74
28-Feb-74
29-Mar-74
30-Apr-74
31-May-74
28-Jun-74
31-Jul-74
30-Aug-74
30-Sep-74
31-Oct-74
29-Nov-74
31-Dec-74
31-Jan-75
28-Feb-75
31-Mar-75
30-Apr-75
30-May-75
30-Jun-75
31-Jul-75
29-Aug-75
0.50
1.49
-0.61
0.27
2.54
2.90
2.70
3.28
1.94
2.76
-5.37
1.76
0.77
-5.23
4.99
3.71
2.08
3.62
6.35
7.07
4.35
-11.56
3.44
-7.07
6.86
8.65
10.33
-8.91
0.14
-0.93
2.14
-0.36
2.33
3.10
2.99
4.14
-0.76
2.18
2.93
-13.82
-0.57
-1.99
1.35
-0.51
0.05
0.43
-0.15
0.37
0.37
0.37
0.29
0.25
0.27
0.29
0.3
0.29
0.31
0.29
0.34
0.4
0.37
0.37
0.44
0.41
0.46
0.52
0.51
0.51
0.64
0.7
0.68
0.65
0.56
0.64
0.63
0.58
0.56
0.75
0.75
0.6
0.7
0.6
0.81
0.51
0.54
0.7
0.58
0.43
0.41
0.44
0.44
0.41
0.48
0.48
-4.44
-0.5
8.76
2.55
2.88
0.6
0.26
1.34
-2.38
-0.74
3.31
-1.11
0.47
4.61
0.75
-3.2
-4.86
-1.25
-5.7
-2.96
-1.38
5.07
-3.67
4.72
-0.68
-12.64
0.5
-0.19
-0.35
-2.9
-5.35
-4.95
-2.89
-7.79
-9.38
-11.78
16.05
-4.64
-3.4
13.58
5.41
2.61
4.21
5.07
4.74
-6.52
-2.84
30-Sep-75
31-Oct-75
28-Nov-75
31-Dec-75
30-Jan-76
27-Feb-76
31-Mar-76
30-Apr-76
28-May-76
30-Jun-76
30-Jul-76
31-Aug-76
30-Sep-76
29-Oct-76
30-Nov-76
31-Dec-76
31-Jan-77
28-Feb-77
31-Mar-77
29-Apr-77
31-May-77
30-Jun-77
29-Jul-77
31-Aug-77
30-Sep-77
31-Oct-77
30-Nov-77
30-Dec-77
31-Jan-78
28-Feb-78
31-Mar-78
28-Apr-78
31-May-78
30-Jun-78
31-Jul-78
31-Aug-78
29-Sep-78
31-Oct-78
30-Nov-78
29-Dec-78
31-Jan-79
28-Feb-79
30-Mar-79
30-Apr-79
31-May-79
29-Jun-79
31-Jul-79
0.39
-0.12
-0.45
-0.12
4.46
0.38
0.18
0.49
-1.11
-0.45
-0.11
-0.86
0.21
-0.45
2.95
0.75
3.97
0.35
0.53
4.22
2.13
1.77
0.36
-1.69
2.07
-0.06
2.20
1.61
-0.70
1.97
1.31
0.82
2.87
2.77
4.18
2.87
-3.13
-8.38
5.43
3.04
-1.40
-1.09
2.90
0.73
-0.44
0.85
-1.08
0.53
0.56
0.41
0.48
0.47
0.34
0.4
0.42
0.37
0.43
0.47
0.42
0.44
0.41
0.4
0.4
0.36
0.35
0.38
0.38
0.37
0.4
0.42
0.44
0.43
0.49
0.5
0.49
0.49
0.46
0.53
0.54
0.51
0.54
0.56
0.55
0.62
0.68
0.7
0.78
0.77
0.73
0.81
0.8
0.82
0.81
0.77
-4.33
5.03
2.71
-1.58
12.13
0.39
2.28
-1.46
-1.31
4.02
-1.09
-0.56
2.01
-2.45
0.14
5.76
-3.99
-1.93
-1.3
0.12
-1.45
4.74
-1.7
-1.78
-0.27
-4.42
4.04
0.33
-6.01
-1.39
2.87
7.74
1.81
-1.62
5.11
3.69
-1.31
-11.78
2.68
0.99
4.18
-3.41
5.75
0.05
-2.18
3.88
0.73
31-Aug-79
28-Sep-79
31-Oct-79
30-Nov-79
31-Dec-79
31-Jan-80
29-Feb-80
31-Mar-80
30-Apr-80
30-May-80
30-Jun-80
31-Jul-80
29-Aug-80
30-Sep-80
31-Oct-80
28-Nov-80
31-Dec-80
30-Jan-81
27-Feb-81
31-Mar-81
30-Apr-81
29-May-81
30-Jun-81
31-Jul-81
31-Aug-81
30-Sep-81
30-Oct-81
30-Nov-81
31-Dec-81
29-Jan-82
26-Feb-82
31-Mar-82
30-Apr-82
28-May-82
30-Jun-82
30-Jul-82
31-Aug-82
30-Sep-82
29-Oct-82
30-Nov-82
31-Dec-82
31-Jan-83
28-Feb-83
31-Mar-83
29-Apr-83
31-May-83
30-Jun-83
-0.26
5.32
2.14
7.93
4.77
7.47
7.89
-9.58
-0.43
-1.13
1.57
0.39
3.26
5.44
7.31
15.21
-6.63
-7.93
-1.39
0.76
-0.92
3.75
-5.90
-2.47
-1.10
2.04
4.10
-0.28
1.33
1.70
4.88
2.93
-0.43
2.52
5.01
4.42
-3.51
4.21
0.01
5.92
0.05
-1.72
3.80
0.92
1.79
-1.58
1.78
0.77
0.83
0.87
0.99
0.95
0.8
0.89
1.21
1.26
0.81
0.61
0.53
0.64
0.75
0.95
0.96
1.31
1.04
1.07
1.21
1.08
1.15
1.35
1.24
1.28
1.24
1.21
1.07
0.87
0.8
0.92
0.98
1.13
1.06
0.96
1.05
0.76
0.51
0.59
0.63
0.67
0.69
0.62
0.63
0.71
0.69
0.67
5.7
-0.69
-8.14
5.37
1.87
5.76
-0.79
-13.23
3.97
5.2
3.16
6.41
1.72
2.2
1.06
9.53
-4.75
-5.05
0.48
3.41
-2.21
0.21
-2.37
-1.55
-6.91
-7.62
4.81
3.52
-3.68
-3.42
-6.03
-1.99
3.2
-3.88
-3.35
-3.1
11.14
1.17
11.27
4.56
0.78
3.5
2.4
2.84
6.71
0.63
3.11
29-Jul-83
31-Aug-83
30-Sep-83
31-Oct-83
30-Nov-83
30-Dec-83
31-Jan-84
29-Feb-84
30-Mar-84
30-Apr-84
31-May-84
29-Jun-84
31-Jul-84
31-Aug-84
28-Sep-84
31-Oct-84
30-Nov-84
31-Dec-84
31-Jan-85
28-Feb-85
29-Mar-85
30-Apr-85
31-May-85
28-Jun-85
31-Jul-85
30-Aug-85
30-Sep-85
31-Oct-85
29-Nov-85
31-Dec-85
31-Jan-86
28-Feb-86
31-Mar-86
30-Apr-86
30-May-86
30-Jun-86
31-Jul-86
29-Aug-86
30-Sep-86
31-Oct-86
28-Nov-86
31-Dec-86
30-Jan-87
27-Feb-87
31-Mar-87
30-Apr-87
29-May-87
-3.13
-5.44
-0.09
-4.54
-0.17
0.81
-2.47
0.20
1.04
2.11
1.65
-0.68
2.87
-5.64
3.69
3.21
1.67
1.51
-6.93
1.82
1.70
3.03
3.98
3.62
-3.93
1.77
1.48
4.87
-0.46
-0.09
2.96
2.70
2.46
-0.50
2.03
5.14
1.80
-5.02
-5.86
2.69
-0.31
0.37
2.12
-2.17
1.61
0.23
-0.73
0.74
0.76
0.76
0.76
0.7
0.73
0.76
0.71
0.73
0.81
0.78
0.75
0.82
0.83
0.86
1
0.73
0.64
0.65
0.58
0.62
0.72
0.66
0.55
0.62
0.55
0.6
0.65
0.61
0.65
0.56
0.53
0.6
0.52
0.49
0.52
0.52
0.46
0.45
0.46
0.39
0.49
0.42
0.43
0.47
0.44
0.38
-3.9
-0.41
0.85
-3.56
2.26
-1.78
-2.06
-4.62
0.61
-0.56
-6.01
1.59
-2.88
10.44
-0.82
-1.01
-1.8
1.73
7.92
1.11
-0.79
-0.94
4.92
1.16
-0.65
-1.03
-4.58
3.79
6.31
3.66
0.42
6.72
4.79
-1.31
4.59
0.9
-6.49
6.16
-8.35
4.47
1.12
-3.13
12.43
4.36
1.9
-2.14
0.13
30-Jun-87
31-Jul-87
31-Aug-87
30-Sep-87
30-Oct-87
30-Nov-87
31-Dec-87
29-Jan-88
29-Feb-88
31-Mar-88
29-Apr-88
31-May-88
30-Jun-88
29-Jul-88
31-Aug-88
30-Sep-88
31-Oct-88
30-Nov-88
30-Dec-88
31-Jan-89
28-Feb-89
31-Mar-89
28-Apr-89
31-May-89
30-Jun-89
31-Jul-89
31-Aug-89
29-Sep-89
31-Oct-89
30-Nov-89
29-Dec-89
31-Jan-90
28-Feb-90
30-Mar-90
30-Apr-90
31-May-90
29-Jun-90
31-Jul-90
31-Aug-90
28-Sep-90
31-Oct-90
30-Nov-90
31-Dec-90
31-Jan-91
28-Feb-91
28-Mar-91
30-Apr-91
-0.20
2.66
-0.87
0.71
-7.89
-1.17
5.78
-7.63
-1.55
0.64
2.28
0.68
-2.93
0.61
0.31
0.27
1.31
0.42
0.42
-0.17
0.88
3.57
1.71
1.55
0.66
5.38
-0.16
3.37
1.39
2.57
2.78
-3.27
-0.54
1.66
2.42
3.04
2.42
5.87
1.79
5.52
6.74
-5.68
0.15
-6.52
-4.81
2.80
-2.37
0.48
0.46
0.47
0.45
0.6
0.35
0.39
0.29
0.46
0.44
0.46
0.51
0.49
0.51
0.59
0.62
0.61
0.57
0.63
0.55
0.61
0.67
0.67
0.79
0.71
0.7
0.74
0.65
0.68
0.69
0.61
0.57
0.57
0.64
0.69
0.68
0.63
0.68
0.66
0.6
0.68
0.57
0.6
0.52
0.48
0.44
0.53
3.89
3.96
3.24
-2.53
-23.14
-7.58
6.64
4.2
4.71
-2.1
0.64
-0.47
4.66
-1.24
-3.39
3.1
1.15
-2.21
1.48
6.06
-2.25
1.48
4.15
3.14
-1.2
7.01
1.47
-0.8
-3.61
1.09
1.22
-7.58
0.92
1.77
-3.52
8.21
-1.05
-1.62
-9.85
-5.98
-1.93
6
2.35
4.39
7.1
2.45
-0.2
31-May-91
28-Jun-91
31-Jul-91
30-Aug-91
30-Sep-91
31-Oct-91
29-Nov-91
31-Dec-91
31-Jan-92
28-Feb-92
31-Mar-92
30-Apr-92
29-May-92
30-Jun-92
31-Jul-92
31-Aug-92
30-Sep-92
30-Oct-92
30-Nov-92
31-Dec-92
29-Jan-93
26-Feb-93
31-Mar-93
30-Apr-93
28-May-93
30-Jun-93
30-Jul-93
31-Aug-93
30-Sep-93
29-Oct-93
30-Nov-93
31-Dec-93
31-Jan-94
28-Feb-94
31-Mar-94
29-Apr-94
31-May-94
30-Jun-94
29-Jul-94
31-Aug-94
30-Sep-94
31-Oct-94
30-Nov-94
30-Dec-94
31-Jan-95
28-Feb-95
31-Mar-95
-0.12
0.46
4.38
1.57
1.71
3.21
1.23
8.28
-2.44
-0.64
-0.32
-2.57
0.09
-0.60
1.42
-0.54
1.44
2.71
-0.32
4.42
4.83
3.10
3.70
0.33
0.27
4.51
3.21
2.52
3.41
-2.70
-4.70
2.28
0.00
-0.28
-1.32
0.38
-2.22
-0.83
0.18
1.54
1.32
1.47
-0.18
3.52
-1.83
-0.36
0.43
0.47
0.42
0.49
0.46
0.46
0.42
0.39
0.38
0.34
0.28
0.34
0.32
0.28
0.32
0.31
0.26
0.26
0.23
0.23
0.28
0.23
0.22
0.25
0.24
0.22
0.25
0.24
0.25
0.26
0.22
0.25
0.23
0.25
0.21
0.27
0.27
0.32
0.31
0.28
0.37
0.37
0.38
0.37
0.44
0.42
0.4
0.46
3.6
-4.82
4.19
2.22
-1.56
1.36
-4.12
10.3
-0.46
1.06
-2.71
1.02
0.36
-2.25
3.68
-2.34
0.98
0.87
3.79
1.5
1.03
0.32
2.26
-2.78
2.74
0.29
-0.32
3.7
-0.2
1.59
-2.01
1.72
2.9
-2.63
-4.85
0.68
0.62
-3.1
2.78
3.89
-2.21
1.07
-4.09
0.82
1.62
3.56
2.24
28-Apr-95
31-May-95
30-Jun-95
31-Jul-95
31-Aug-95
29-Sep-95
31-Oct-95
30-Nov-95
29-Dec-95
31-Jan-96
29-Feb-96
29-Mar-96
30-Apr-96
31-May-96
28-Jun-96
31-Jul-96
30-Aug-96
30-Sep-96
31-Oct-96
29-Nov-96
31-Dec-96
31-Jan-97
28-Feb-97
31-Mar-97
30-Apr-97
30-May-97
30-Jun-97
31-Jul-97
29-Aug-97
30-Sep-97
31-Oct-97
28-Nov-97
31-Dec-97
30-Jan-98
27-Feb-98
31-Mar-98
30-Apr-98
29-May-98
30-Jun-98
31-Jul-98
31-Aug-98
30-Sep-98
30-Oct-98
30-Nov-98
31-Dec-98
29-Jan-99
26-Feb-99
1.82
-0.43
2.91
2.57
0.11
2.77
4.16
-0.61
2.51
0.56
0.66
-1.88
-0.90
1.61
0.99
-0.17
-0.04
2.70
3.85
-2.39
0.57
1.95
-2.04
0.98
4.89
-5.19
2.58
3.81
-2.52
1.47
-0.43
0.29
3.95
0.19
-1.13
2.11
0.77
1.81
7.30
3.74
1.89
-0.64
-5.35
1.22
8.93
3.01
-0.13
0.44
0.54
0.47
0.45
0.47
0.43
0.47
0.42
0.49
0.43
0.39
0.39
0.46
0.42
0.4
0.45
0.41
0.44
0.42
0.41
0.46
0.45
0.39
0.43
0.43
0.49
0.37
0.43
0.41
0.44
0.42
0.39
0.48
0.43
0.39
0.39
0.43
0.4
0.41
0.4
0.43
0.46
0.32
0.31
0.38
0.35
0.35
2.06
2.86
2.65
3.63
0.46
3.21
-1.6
3.85
1.03
2.38
1.24
0.7
2.09
2.26
-1.23
-5.83
2.84
4.86
0.94
6.14
-1.6
4.89
-0.5
-4.91
3.8
6.67
4.04
7.22
-4.04
5.4
-3.87
2.66
1.3
0.02
6.93
4.74
0.66
-2.98
2.79
-2.73
-16.2
5.92
7.12
5.89
5.93
3.48
-4.16
31-Mar-99
30-Apr-99
28-May-99
30-Jun-99
30-Jul-99
31-Aug-99
30-Sep-99
29-Oct-99
30-Nov-99
31-Dec-99
31-Jan-00
29-Feb-00
31-Mar-00
28-Apr-00
31-May-00
30-Jun-00
31-Jul-00
31-Aug-00
29-Sep-00
31-Oct-00
30-Nov-00
29-Dec-00
31-Jan-01
28-Feb-01
30-Mar-01
30-Apr-01
31-May-01
29-Jun-01
31-Jul-01
31-Aug-01
28-Sep-01
31-Oct-01
30-Nov-01
31-Dec-01
31-Jan-02
28-Feb-02
28-Mar-02
30-Apr-02
31-May-02
28-Jun-02
31-Jul-02
30-Aug-02
30-Sep-02
31-Oct-02
29-Nov-02
31-Dec-02
31-Jan-03
-1.36
-9.12
-5.24
4.99
1.64
3.06
6.50
5.52
5.62
13.02
1.82
18.39
-6.78
-8.48
-9.13
16.41
-0.06
5.73
2.15
-4.72
-2.49
6.85
-25.06
12.59
8.39
-8.10
2.16
0.29
5.59
5.54
11.53
-8.41
-8.59
0.00
3.70
6.84
-1.70
7.92
3.05
6.17
3.41
1.68
9.09
-5.17
-16.26
9.63
1.53
0.43
0.37
0.34
0.4
0.38
0.39
0.39
0.39
0.36
0.44
0.41
0.43
0.47
0.46
0.5
0.4
0.48
0.5
0.51
0.56
0.51
0.5
0.54
0.39
0.44
0.39
0.32
0.28
0.3
0.31
0.28
0.22
0.17
0.15
0.14
0.13
0.13
0.15
0.14
0.13
0.15
0.14
0.14
0.14
0.12
0.11
0.1
3.36
4.53
-2.41
4.7
-3.44
-1.39
-2.68
5.81
3.33
7.95
-4.39
2.75
4.89
-6.41
-4.4
4.76
-2.19
7.08
-5.62
-3.02
-10.76
1.53
3.41
-10.32
-7.47
8
0.74
-2.03
-2.13
-6.22
-9.43
2.58
7.7
1.63
-1.75
-2.3
4.34
-5.11
-1.19
-7.15
-8.26
0.66
-10.14
7.36
6.01
-5.44
-2.44
28-Feb-03
31-Mar-03
30-Apr-03
30-May-03
30-Jun-03
31-Jul-03
29-Aug-03
30-Sep-03
31-Oct-03
28-Nov-03
31-Dec-03
30-Jan-04
27-Feb-04
31-Mar-04
30-Apr-04
28-May-04
30-Jun-04
30-Jul-04
31-Aug-04
30-Sep-04
29-Oct-04
30-Nov-04
31-Dec-04
31-Jan-05
28-Feb-05
31-Mar-05
29-Apr-05
31-May-05
30-Jun-05
29-Jul-05
31-Aug-05
30-Sep-05
31-Oct-05
30-Nov-05
30-Dec-05
31-Jan-06
28-Feb-06
31-Mar-06
28-Apr-06
31-May-06
30-Jun-06
31-Jul-06
31-Aug-06
29-Sep-06
31-Oct-06
30-Nov-06
29-Dec-06
1.29
1.50
-9.48
-10.79
-1.06
-0.35
-0.55
-0.07
3.70
1.63
-5.67
2.58
-1.14
0.20
-5.33
1.64
2.08
-2.32
-1.54
5.28
-1.54
3.24
-2.82
3.12
3.19
0.93
-0.84
0.46
2.10
0.05
2.24
3.50
-1.37
0.39
0.77
2.77
-1.80
1.22
0.65
-3.66
1.52
-2.24
-3.48
-0.98
-0.18
-1.00
0.81
0.09
0.1
0.1
0.09
0.1
0.07
0.07
0.08
0.07
0.07
0.08
0.07
0.06
0.09
0.08
0.06
0.08
0.1
0.11
0.11
0.11
0.15
0.16
0.16
0.16
0.21
0.21
0.24
0.23
0.24
0.3
0.29
0.27
0.31
0.32
0.35
0.34
0.37
0.36
0.43
0.4
0.4
0.42
0.41
0.41
0.42
0.4
-1.63
0.93
8.18
6.26
1.53
2.24
2.42
-0.99
5.96
1.59
4.47
2.24
1.49
-1.16
-2.5
1.35
2.08
-3.87
0.16
1.95
1.67
4.67
3.36
-2.82
2.11
-1.9
-2.73
3.55
0.92
4.09
-0.89
0.77
-2.35
3.73
0.03
3.66
-0.5
1.54
0.94
-3.53
-0.44
-0.59
2.09
1.54
3.3
1.95
0.68
31-Jan-07
28-Feb-07
30-Mar-07
30-Apr-07
31-May-07
29-Jun-07
31-Jul-07
31-Aug-07
28-Sep-07
31-Oct-07
30-Nov-07
31-Dec-07
31-Jan-08
29-Feb-08
31-Mar-08
30-Apr-08
30-May-08
30-Jun-08
31-Jul-08
29-Aug-08
30-Sep-08
31-Oct-08
28-Nov-08
31-Dec-08
30-Jan-09
27-Feb-09
31-Mar-09
30-Apr-09
29-May-09
30-Jun-09
0.22
-1.32
2.48
-0.14
-0.33
0.40
2.80
0.14
4.64
4.86
0.93
6.48
-7.89
6.23
4.12
-0.38
3.20
12.45
-5.12
-3.82
0.36
7.91
7.20
-5.03
-2.00
4.31
-11.60
-34.75
-12.45
5.36
0.44
0.38
0.43
0.44
0.41
0.4
0.4
0.42
0.32
0.32
0.34
0.27
0.21
0.13
0.17
0.17
0.17
0.17
0.15
0.12
0.15
0.08
0.02
0.09
0
0.01
0.01
0.01
0
0
1.5
-1.78
0.86
3.55
3.48
-1.88
-3.58
0.74
3.77
2.26
-5.27
-0.7
-6.44
-2.33
-1.22
4.94
2.21
-8.03
-1.46
0.98
-9.96
-18.55
-8.54
2.06
-7.74
-10.11
8.76
11.05
6.73
-0.28
Statistics Bootcamp
Professor Alexander Barinov
School of Business Administration
University of California Riverside
MGT 252 Investments and Portfolio Management
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
1 / 40
Outline
1
Averages
2
Variance and Standard Deviation
3
Confidence Intervals
4
Hypothesis Testing
5
Event Studies
6
Market Model
7
Event Study Revisited
8
Idiosyncratic Risk
9
Forecasting
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
2 / 40
Averages
Notation
E(R) is the notation for mean return (population
average)
R is the notation for average return (sample
average)
We will never know what the true mean is, but
we will approximate it by the sample average
On average, we will make no mistake, i.e.
E(R − R) = 0
The longer is the sample, the closer we are to
the population average
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
3 / 40
Averages
Average is Linear
You can always multiply average by a number
If you invest $1 into the market for one year, your
expected gain is E($1 · R) = $1 · E(R) = $0.1139
If you invest $100 into the market for one year, your
expected gain is E($100 · R) = $100 · E(R) = $11.39
You can always add two (or more) averages
If you invest $1 into the market and $1 into Treasury bills
for one year, your expected gain is E(RMKT + RF ) =
E(RMKT ) + E(RF ) = $0.1139 + $0.0376 = $0.1515
If you invest $100 for one year, 70% in the market and
the rest in Treasury bills, what is your expected gain?
E($70RMKT + $30RF ) = $70 · E(RMKT ) + $30 · E(RF ) =
$70 · 0.1139 + $30 · 0.0376 = $9.10
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
4 / 40
Variance and Standard Deviation
Multiplying by a Constant
If you multiply the random variable by a number,
multiply the variance by this number squared
Because standard deviation is the square root of
the variance, if you multiply the random variable
by a number, multiply the standard deviation by
this number (do not square)
The standard deviation of monthly market return is
5.45%
What is the standard deviation of the expected gain
on the $100 invested for a month?
σ($100 · RMKT ) = $100 · σ(RMKT ) = $100 · 0.0545 =
$5.45
Interpretation: it is not unusual to gain or lose $11 on
a $100 bet within a month
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
5 / 40
Variance and Standard Deviation
Adding Variances
For independent (or at least uncorrelated variables)
we can add up variances (but not standard
deviations!)
Standard deviation of daily market return is 0.95%
According to Efficient Market Hypothesis, return today and
return tomorrow are independent
Assume returns on different weekdays have the same
standard deviation
σ 2 (Rweekly ) = σ 2 (RMo + RTu + RW + RTh + RFr ) =
σ 2 (RMo ) + σ 2 (RTu ) + σ 2 (RW ) + σ 2 (RTh ) + σ 2 (RFr ) =
5σ 2 (Rdaily ) = 5 · 0.95%2 = 4.5125%
p
√
√
σ(Rweekly ) = 5σ 2 (Rdaily ) = 5 · σ(Rdaily ) = 5 · 0.95% =
2.124%
If we add constant to a random variable, the variance
does not change: σ2 (MKT − RF ) = σ2 (MKT )
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
6 / 40
Variance and Standard Deviation
Time Aggregation
The example in the previous slide is generalizable to
other frequencies
If the standard deviation of daily market return is 0.95%,
what is the standard deviation of monthly market return
(assume 22 trading days per month)?
σ 2 (Rmonthly ) =
√
22 · σ 2 (Rdaily ) =
√
22 · 0.95% = 4.46%
If the standard deviation of annual market return is
20.75%, what is the standard deviation of monthly market
return?
σ 2 (Rmonthly ) =
σ 2 (Rannual )
20.75%
√
= √
= 5.99%
12
12
Warning: these estimates are approximate and will differ
from what you get if you estimate the standard deviation
of monthly return directly from monthly data (5.45%)
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
7 / 40
Variance and Standard Deviation
Variance of a Sum: Example #1
Consider portfolio P that is 70% MKT and 30%
RF
We know that σ(RMKT ) = 20.75%
Assume that RF is constant
What is the annual σ(RP )?
σ(RP ) = σ(0.7 · MKT + 0.3 · RF )
We can throw out (+0.3 · RF ), because RF is
constant and it does not add to the variance of P
Then σ(RP ) = σ(0.7 · MKT ) = 0.7 · σ(RMKT ) =
0.7 · 20.75% = 14.525%
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
8 / 40
Variance and Standard Deviation
Variance of a Sum: Example #2
We do see in the data that RF varies, so let’s not consider it
constant - in the data, annual σ(RF ) = 3.1%
For simplicity, let’s consider it independent of MKT
2
Then σ 2 (R
pP ) = σ (0.7 · MKT + 0.3 · RF ),
2
σ(R
pP ) = σ (0.7 · MKT + 0.3 · RF ) =
= 0.72 · σ 2 (RMKT ) + 0.32 · σ 2 (RF )
√
σ(RP ) = 0.72 · 20.75%2 + 0.32 · 3.1%2 = 14.55% - small
change compared to when we assumed that RF is constant,
because the weight on RF and σ(RF ) = 3.1% are both small
If we construct P and compute σ(RP ) from the data, we will
find that it is 14.54% - the small difference comes from the
fact that RF and MKT are a little bit related
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
9 / 40
Variance and Standard Deviation
Expected Return
EMH implies that returns are unpredictable and the
best expectation of future return is the long-term
average
But how sure we are about this prediction?
T
X
T
1 2 X
2
2 1
Rt =
Rt = 2 σ
σ (R) = σ
T
T
t=1
t=1
T
1 X 2
1
σ 2 (R)
2
σ
(R
)
=
·
T
σ
(R)
=
t
T2
T2
T
t=1
σ(R)
σ(R) = √ - we are much more sure about
T
expectations than about actual realizations
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
10 / 40
Confidence Intervals
Confidence Interval for the
Next-Period Return
Assume that returns are normally distributed
Prob[E(R)−1.96σ(R) ≤ Rt ≤ E(R)+1.96σ(R)] = 95%
Example: Between 1927 and 2008 (82 years) annual
market return has mean E(RMKT ) = 11.4% and
standard deviation σ(RMKT ) = 20.75%
Prob[11.4%−1.96·20.75% ≤ RMKT ≤ 11.4%+1.96·20.75%] = 95%
⇒ Prob[−29.27% ≤ RMKT ≤ 52.06%] = 95%
With probability 95%, next year market return falls
between -29.27% and 52.05%
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
11 / 40
Confidence Intervals
Where Did the Numbers Come From?
The average and the standard deviation are
from Excel functions AVERAGE and STDEV
For these, select the data you want to average
or compute standard deviation for
1.96 are from the inverse of the standard normal
distribution: Excel function NORMSINV
You need 2.5% on each tail - you ask for
NORMSINV(0.025)
What if we need 5% on each tail? Use
NORMSINV(0.05)!
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
12 / 40
Confidence Intervals
Application: Value-at-Risk (VaR)
If one assumes that events with 5% or 10% probability
are unlikely, one can compute the maximum likely
loss, which is usually called value-at-risk
Suppose you invested $1,000,000 in the market
portfolio for a year
Prob[E(R) − 1.65σ(R) ≤ Rt ] = 5% ⇒
Prob[11.4% − 1.65 · 20.75% ≤ Rt ] ⇒ Prob[−22.8% ≤ Rt ]
So, your 5% VaR is 22.8% of the initial investment, i.e.
$228,000
"The only problem is that lately we see too many
events that should occur once every thousand years"
c Wall Street traders
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
13 / 40
Confidence Intervals
Confidence Interval for the
Expected Return
In the previous slide we assume that if nothing unexpected
happens, the next year return will be precisely at the long-run
average
There is a second layer of uncertainty: how sure we are that
11.4% is the true population average?
20.75%
σ(RMKT ) = 20.75% ⇒ σ(RMKT ) = √
= 2.29%
82
Prob[11.4% − 1.96 · 2.29% ≤ RMKT ≤ 11.4% + 1.96 · 2.29%] = 95%
⇒ Prob[6.9% ≤ RMKT ≤ 15.9%] = 95%
With probability 95%, the true population average for annual
market return (aka expected market return) can be anywhere
between 6.9% and 15.9% per annum
Alexander Barinov (SoBA, UCR)
Statistics Bootcamp
MGT 252 Investments
14 / 40
Hypothesis Testing
Using Confidence Interval
H0 : Expected annual market return is equal to the
risk-free rate, 3.76%, E(RMKT ) = 3.76%
Ha : Expected annual market return is not equal to the
risk-free rate, E(RMKT ) 6= 3.76%
Prob[6.9% ≤ RMKT ≤ 15.9%] = 95% - 3.76% is not in
the confidence interval, i.e. is not among plausible
events
Therefore, we reject the null and accept the
alternative: "Expected annual market return is not
equal to the risk-free rate"
Usual wording: "Expected annual market return is
significantly different from the risk-free rate"
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Hypothesis Testing
Using t-statistic
RMKT − RF
11.4% − 3.76%
= 3.33
=
2.29%
σ(RMKT )
Since the absolute value of the t-statistic is
larger than 1.96, we reject the null and conclude
that expected annual market return is significant
"the absolute value of t-statistic is larger than
1.96" is equivalent to "the value under the null is
outside of the 95% confidence interval"
Important: in all three examples
t=
σ(RMKT − RF ) = σ(RMKT )
because RF is assumed to be constant
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Hypothesis Testing
Using p-value
p-value: under the null hypothesis, what is the
probability to see what we observe in the
sample?
Two equivalent ways:
NORMDIST (3.76%; 11.4%; 2.29%; 1)
1 − NORMDIST (11.4%; 3.76%; 2.29%; 1)
Both give you the same answer p = 4.3E−4 way less than 0.025, our tail threshold of
plausibility, so we reject the null
Almost always, the p-value above would be
doubled and compared to 0.05, to reflect the fact
that it is a two-tailed test
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Event Studies
Example: Ken Lewis’ Departure
On October 1, 2009, Ken Lewis announced his
resignation as Bank of America CEO
Was the market reaction positive, negative, or
neutral?
On October 1, BofA stock price dropped by
-4.2%
But could it have been just random variation?
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Event Studies
Price Reacts to Negative News
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Event Studies
Event Window
BofA average return between January 1, 2009,
and September 30, 2009 - 0.49% per day
Daily standard deviation of BofA for the same
period - 8.95%
Abnormal return (return absent any news):
−4.2% − 0.49% = −4.69%
t-test: t =
Alexander Barinov (SoBA, UCR)
−4.2% − 0.49%
= −0.52 > −1.96
8.95%
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Event Studies
Event Window
We conclude that the price drop of BofA stock
on the departure of Ken Lewis was not
statistically significant
We may say that the change of the CEO did not
have an impact on BofA stock
Given the magnitude of the change though, I
would say that the departure of Ken Lewis does
not stand out among other BofA events in the
first three quarters of 2009
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Event Studies
Post-Event Window
After the market digests the information in the
event window, under EMH it should not drift in
the same direction or bounce back in the
post-event window
The cumulative return (sum of daily returns) in
the 10 trading days following Ken Lewis’
departure is 11.3%
Do we take the bounce back as the evidence
that the market first thought it is bad Ken Lewis
retires, and then thought it is actually good?
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Event Studies
Post-Event Window
10-day standard
deviation:
√
8.95% · 10 = 28.29%
11.3% − 4.9%
= 0.226 < 1.96
28.29%
The bounce back after the resignation of Ken
Lewis is not statistically significant
t-test: t =
We do not have any evidence to claim that the
market did not process the information in his
resignation correctly
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Event Studies
Thought Experiment
What if we had "25 Banks of America", i.e. the
event-window return was an average across 25 firms
with the same average return and standard deviation?
Then
the standard deviation of the average would be
√
25 = 5 times smaller, and the t-statistic would be
-2.6 −1.96
6.68%
Still no significant market reaction
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Event Study Revisited
Event Study: Two Days
What if we include September 30, when Ken Lewis told
the board he would step down?
On September 30, BofA stock lost -1.4%
S&P500, however, gained 1.78%, and Treasury bill
yielded 0.14%
Absent the news, BofA should have gained
0.14% + 3.12 · (1.78% − 0.14%) = 5.26%
Standard
√deviation of residuals during two days is
6.68% · 2 = 9.45%
Cumulative abnormal return:
−4.2% − 1.4% − (5.26% − 0.9%) = −9.96%
−9.96%
t-test:
= −1.05 > −1.96
9.45%
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Idiosyncratic Risk
R-square
Notice that the standard deviation of the
residuals (6.68% per day) is smaller than the
standard deviation of BofA returns (8.95%)
This is because part of variation in the BofA
returns is explained by what happens to
S&P500
R-square of the market model for BofA stock is
44% - "44% of the variance of the BofA returns
is explained by S&P500"
0.44 = 1 −
Alexander Barinov (SoBA, UCR)
6.682
8.952
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Idiosyncratic Risk
Idiosyncratic Risk
Market model: Rett − RFt = α + β · (MKTt − RFt ) + t
Assume that the risk-free rate, RF , is constant:
σ 2 (Ret − RF ) = σ 2 (Ret)
Assume that t and MKTt are uncorrelated - if we
could say: "our model usually underestimates returns
when market goes up", we would have used the
information to improve the model (increase the beta)
σ 2 (Ret − RF ) = σ 2 (α + β · (MKT − RF ) + )
σ 2 (Ret) = σ 2 (β · (MKT )) + σ 2 () = β 2 · σ 2 (MKT ) + σ 2 ()
Total risk (σ 2 (Ret)) is systematic risk (β 2 · σ 2 (MKT ))
plus idiosyncratic risk (σ 2 ())
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Idiosyncratic Risk
Diversification
Suppose we invest equal sums into N stocks
What we get is the average return: Ret =
N
1X
Reti
N
i=1
For each stock, Reti − RF = αi + βi · (MKT − RF ) + i
N
1X
(αi + βi · (MKT − RF ) + i ) =
N
i=1
X
N
1
2
·
σ
(α
+
β
·
(MKT
−
RF
)
+
)
i
i
i
N2
σ 2 (Ret) = σ 2
i=1
Suppose the residuals, i , are uncorrelated across stocks
2
σ (Ret) =
PN
2
i=1 βi
N2
2
PN
· σ (MKT ) +
σ 2 (i )
σ 2 ()
2
2
=
β
·
σ
(MKT
)
+
N2
N
i=1
The last term will become very close to zero if N is large
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Idiosyncratic Risk
Diversification
If we hold many stocks, all idiosyncratic risk will
disappear
Systematic risk will remain: our portfolio will be
riskier if it has higher beta or if the market is
being volatile
Remember about the assumption of
uncorrelated i - the stocks should be truly
unrelated
A portfolio of 50 airline companies is NOT
well-diversified
A portfolio of 50 stocks from different industries IS
well-diversified
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Forecasting
Forecasting Daily Market Return
MKTt = 0.036 + 0.085 MKTt−1 + t
(0.009) (0.009)
The slope is significant: we can indeed predict
tomorrow market return from today’s return
If today market goes up by 1%, tomorrow it will
go up by another 8.5 bp (0.085%)
8.5 bp is not a large effect, given that the
standard deviation of the daily market return is
0.95% (returns exceeding 2% should be very
rare)
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Forecasting
R-square and Forecasting Ability
R-square is 0.7% - very unreliable model, on
average we will anticipate less than 1 bp of a
1% return
Standard deviation of residuals is very close to
the standard deviation of returns, 0.95% consequence of the small R-square
Even if return today is 2%, the confidence
interval for return tomorrow will be
[0.206%-1.9%; 0.206%+1.9%], i.e. from
-1.694% to 2.106%
The low R-square means that you will never get
a clear buy or sellStatistics
signal
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Forecasting
More on R-square
With that said, there is no absolute scale of
R-square
If you replace return with price in the regression
above, the R-square will change from 1% to
98%, but it will be the same bad model of
beating EMH
R-squared for market model will be quite low for
individual firms (10-15%) and quite high for
large portfolios (80-90%)
It only makes sense to compare the R-square of
two models of the same thing (i.e., two models
of market return) Statistics Bootcamp
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