# Statistics Question

**Part A**

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You will construct two alpha factors: V and M for the US market.

V is based on the prior 5-day returns and M is the MAX factor discussed in the lecture. The steps to construct factor V are the same as in Assignment 2. For simplicity, ranking is not used to construct V and M.

1. Consider the US market and the years 2004 to 2023 (you will also need some data in 2003 to calculate the alpha factor). The universe used for each year will be based on the universe from the start of the year (defined in the univ_h.csv file).

2. To construct factor V

a) calculate the daily volatility using the prior 21 days of daily returns (use the log return , the return is set to 0 if there is an “NA” in the adjusted prices). If obtained is less than 0.005, set it to 0.005.

b) calculate the prior 5-day return; you can use the log-return (again, the return is set to 0 if the prices are not available)

c) normalize the variable by dividing the volatility obtained in step a),

d) subtract out the industry component, , where is the average of (after step c) over all the stocks in the industry (and in the universe of time t) that the stock i belongs to

3. To construct factor M

a) Use the daily returns for the past 21 trading days, , calculated in Step 2a).

b) subtract out the corresponding industry component (calculated using a simple average):

c) get the maximum value, , of the magnitudes of the prior 21 values, . The normalization by the volatility is not applied to this factor.

d) subtract out the industry component, , where is the average of over all the stocks in the industry (and in the universe of time t) that the stock i belongs to.

4. Do a cross-sectional regression of the next day's return

on day t and get the time series of . Here is defined as ,

where ) and the industry return is the simple average of over all the stocks in the industry (and in the universe of time t) that stock i belongs to.

5. From the years 2005 to 2023, calculate the __2-year averages__ of , and the t-stat, , where T is the number of trading days in the year and the year before (for example, the average obtained for the year 2005 is over the years 2004 and 2005) and and are the standard deviation of calculated using obtained in these two-year periods. List the two-year average betas and the t-stat obtained in a table.

**Part B**

From the years 2006 to 2023, use the two-year averages calculated (in Part A) from the previous two years (for example, for the year 2006, use the 2-year average of years 2004 and 2005 obtained in Part A) and evaluate the expected returns for all trading days of the year,

Construct and evaluate the portfolio as follows,

1. On each day t, rank the stocks according to the expected returns, and long (with equal weights) the top 20% of the stocks with the largest values of and short the bottom 20% of the stocks with the smallest values (most negative values) of

2. Get the portfolio return at each time step t. The return is on the long market value of the portfolio, so it is the sum of the returns on individual positions divided by the number of long positions in the portfolio,

,

where and are the number of long and short positions (both are equal to , is the number of stocks in the universe for that year). is the stock index of the long position j. is the stock index of the short position j. Note that when calculating the portfolio return, the full return without subtracting the market return is used.

3. For each year calculate the annual return (assuming the cost of trading is 0, and for simplicity simply add up all daily portfolio returns to get the annual return) and the annualized return volatility of the portfolio. List your results in a table. Which are the best and the worst years for the strategy?

**Part C**

Assume that the percentage trading cost is 5 bps and calculate the portfolio returns, taking into account the cost. Compare the results to the case when the costs are not taken into account. For simplicity, we assume the LMV (the long market value) of the portfolio is kept the same and we ignore the cost of maintaining the constant LMV.