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" Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 15 / 40 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 Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 16 / 40 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 Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 17 / 40 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? Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 18 / 40 Event Studies Price Reacts to Negative News Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 19 / 40 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% Statistics Bootcamp MGT 252 Investments 20 / 40 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 Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 21 / 40 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? Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 22 / 40 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 Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 23 / 40 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 Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 32 / 40 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% Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 33 / 40 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 Statistics Bootcamp MGT 252 Investments 34 / 40 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 ()) Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 35 / 40 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 Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 36 / 40 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 Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 37 / 40 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) Alexander Barinov (SoBA, UCR) Statistics Bootcamp MGT 252 Investments 38 / 40 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 Alexander Barinov (SoBA, UCR) Bootcamp MGT 252 Investments 39 / 40 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 Alexander Barinov (SoBA, UCR) MGT 252 Investments 40 / 40