Question

Q2 You find the following return characteristics of a stock relative to a sector ETF using 100 months of data:

1. The stock has a monthly expected return of 1.2%;
2. The stock has a monthly standard deviation of 10%;
3. The correlation between the stock and that of sector ETF is 0.5.
4. The sector ETF has a monthly mean return of 1% and monthly standard deviation of 5%.

1. If you run a regression of stock return on ETF return, what is the regression slope and regression intercept (2 points)?

return_stock = a + b * return_etf + error

2. What is the standard error, and t-stat of your b estimate? (1 point)?

  

3. What is the 95% confidence interval of your slope estimates (1 point)?

Answer 1

b = correlation_stock_etf*standard deviation_stock/standard deviation_etf
= 0.5*0.10/0.05
= 1

a = stock_mean - slope*etf_mean
= 1.2% - 1*1%
= 0.2%

standard error = sqrt [ Σ(y_i – ŷ_i)² / (n – 2) ] / sqrt [ Σ(x_i – x)² ]
= sqrt( (0.10^2/98)/0.05^2)
= 0.202
t statistic = coefficient/ standard error
= 1/0.202
= 4.95

95% confidence interval, 2 sided confidence interval with critical value 1.96
1- 1.96*0.202 , 1+ 1.96*0.202
0.604, 1.396