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Section 1: Economic Factor Models and Statistical Arbitrage: For this section, w

by | Apr 28, 2022 | Investing and Financial Markets

 

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Section 1: Economic Factor Models and Statistical Arbitrage:
For this section, we are going to focus on the following six
stocks: Walmart (WMT), JP Morgan (JPM), Coca-Cola (KO), Microsoft (MSFT),
Alaska Air (ALK), Lululemon. (LULU).
We are going to estimate the alphas and factor loadings of
these stocks using the four factor model as the benchmark.
The general idea of this section is to estimate the alphas
and risks of these stocks using January 2012 – December 2016 as an in-sample
estimation period,
create a portfolio with no systematic risk based on these
estimates, and then measure the ex-post out-of-sample risk of this portfolio
during the period January 2017 – December 2021.
1.1. Retrieve monthly factor returns for the four
factor model. The data is available at:
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
Click on U.S. Research Returns” on the left side of
the webpage. Y
You will need to download the Fama-French factors (MKT-RF
HML, SMB) in one file, and then separately download the momentum factor.
Make sure you properly align the monthly return observations
on your six stocks with the returns on all four factors. (4 points)
1.2. Retrieve total return data from the Charting Tab
in FactSet for the six stocks.
Compute monthly
excess returns for each of the six stocks.
Note that the Fama-French data conveniently includes a
column for the deannualized risk-free rate expressed as a percentage. (4
points)
1.3. Using data for January 31, 2012 through December
31, 2016, estimate the alphas and factor loadings for each of the stocks.
Do any of your stocks have statistically significant
estimates for alpha, and if so, are these stocks candidates for overweight or
underweight positions? (8 points)
1.4. Using the Excel Solver and the estimates from
(1.4), compute the set of weights required to form a portfolio from the six
stocks that has zero factor risk, an expected alpha of 1% per month, and
satisfies the full investment constraint (weights sum to one). This portfolio
is an arbitrage portfolio as it is expected to generate alpha without any risk
exposure. Include a screen shot of the Solver Dialog box you used to solve for
the portfolio weights (4 points).
1.5. Using the set of weights from (1.4), compute the
monthly excess return on your portfolio during the period January 2017 –
December 2021. To maintain risk neutrality, the portfolio must be rebalanced
back to these weights each month, so the monthly portfolio return is simply rpt
= Σi wirit, where wi are the weights from (1.4). Calculate the total risk (variance)
of this supposed arbitrage portfolio. (5 points)
1.6. The portfolio defined in (1.4) is an arbitrage
portfolio that offers (in-sample) a positive alpha with no systematic risk
exposure. It is created based on information available at the end of December
2016, and we assume that you create and hold this portfolio from January 2017
through December 2021. Calculate the alpha and the factor loadings of your
portfolio using returns for January 2017 through December 2021 to measure its
out-of-sample risk. Do the alpha and factor loadings of the out of sample
arbitrage portfolio match your expectations? Are these results (alpha and
factor loadings) significant and statistically reliable? (5 points)
1.7. Calculate the alpha and the factor loadings of the
6 stocks using returns for January 2017 through December 2021. Compare the
results with your in-sample results from 1.4. How did each stock behave
differently between the two periods i.e. can you explain why your 1.6 results
are different from what out expected in 1.4? (8 points)
1.8. For each month in the out-of-sample period, use
the returns on the four factors and your estimated loadings to calculate the
systematic risk (as explained by the factor loadings i.e. factor loading from
1.6 times factor returns for each month) and idiosyncratic risk (unexplained by
the 4 factor model) on your portfolio. What percentage of the total risk
(calculated in 1.5) of your portfolio was idiosyncratic risk and what portion
was due to factor risk? Hint #1: Use variances. (5 points)
1.9. Compare the percentage of systematic risk to the
portfolio regression R-square from 1.6.
1.10.
Are they the same? Should they be? (2 points)
Section 2: Estimating a Fundamental Factor Model:
We are now going to build and backtest a fundamental factor
model based on the four variables that you choose. The first task is to
estimate the factor premia on your fundamental factors. We are going to estimate
the premia at three different points in time by using three different backtest
dates (06/30/2020, 12/31/2019 and 06/30/2019) with data accessed through
FactSet Screening. Be certain that the formulas you use are lagged properly to
avoid a look-ahead bias. One way to avoid the look-ahead bias is by using
quantitative factors specifically built by FactSet for backtesting. To estimate
the premia, you need to obtain the monthly return FOLLOWING your backtest date.
For example, use the July 2019 return for the 06/30/2019 backtest date. Use the
S&P1500 Composite as your universe.
2.1. Select 4 factors to use in your model. What are
your factors? Why did you choose these factors? What signs (exposure) do you
expect for each of the factor premia i.e. is a low value desirable or not? Each
factor must be from a different factor category (e.g. do not pick 2 valuation
factors like P/E and B/P). (8 points)
2.2. Include the Excel export of the summary page from
Universal Screener showing your factor formulas as well as your total return
formula. (2 points)
2.3. For each of the three backtest dates, estimate
the standardized fundamental factor model to obtain factor premia. Use next
month raw returns as the dependent variable. Each of the following sub-step must
be clearly identified and not combined i.e. I should be able to see the raw
data, the data (2.3.1.), with the outliers issue addressed (2.3.3) and the
standardized data (2.3.4).
2.3.1 What percentage of the universe is left after
filtering out missing data for each factor? How did you handle these missing
data points? Why did you use this approach? Hint: If you are missing more than
10% of the universe, this is probably not a good factor. (3 points)
2.3.2 Using the raw exposures to each factor extracted from
FactSet, compute the following statistics: min, max, mean, median, standard
deviation. How different are these statistics for each of the three backtest
dates? (4 points)
2.3.3 Describe how you handled it and solved the outliers
issue. If you decide to winsorize the data, use the PERCENTILE function to
obtain the 1% and 99% thresholds. (5 points)
2.3.4 Compute each stock standardized exposure (i.e.
z-scores) to each factor and provide the following summary statistics for the
distribution of standardized exposures: min, max, mean, median, standard
deviation. Hint: The monthly returns should not be standardized (9 points)
2.3.5 For each of the three backtest date, calculate the
equal weighted benchmark return and each factor premia using a cross-sectional
regression using your standardized exposures from 2.3.4 against the 1 month
holding return for the universe. What are each of the factor premia, and are
the estimates consistent with your expectations from 2.1? (9 points).
2.4. You now have three estimates for each of your
factor premia (one from estimating the standardized model for each backtest
date from 2.3.5). Average the three estimates for each factor and consider
these to be your expected factor premia. Explain what these numbers represent i.e.
how to interpret them. Be precise and specific. Hint: factor premia are not
betas. (5 points)
Section 3: Backtesting Your Fundamental Factor Model (Alpha
Testing)
Now it is time to backtest your fundamental model using
Alpha Testing (@AT) in FactSet. Import your screen and set both the universe
and the benchmark to be the same as your screening universe. Include inactive
securities, but exclude secondary listings and non-equity securities. Set the
backtest period to 12/31/2011 through 12/31/2021 and use monthly frequency for
rebalancing. Import your raw exposures as factors and then setup a multifactor
rank (MFR) based on z-scores as your signal for expected returns. The weights
assigned to each component of the multifactor ranking should be the expected
factor premia from Section 2[1]. Make sure you handle outliers for each factor
when setting up each factor and the MFR. Set the risk-free rate to be the
91-day T-Bill yield
3.1. Run your model and export the following reports
from Alpha Testing to Excel: a) Workspace, b) Summary: Single Factor, c)
Summary: All Factors, d) Periods, and e) the portion of the Constituents report
containing data for December 2016 (do not submit the entire Constituents
report). (8 points)
3.2. Using the Summary and Period output, evaluate
3.2.1.
Does you model work overall? (2 points)
3.2.2.
How much monthly return does your alpha signal generate? (2 points)
3.2.3.
How much alpha does your best quintile (F1) generate? Can this number be
trusted? (2 points)
3.2.4.
Which factor contributed the most (had the highest impact) to your MFR model?
(2 points)
3.2.5.
Is the contribution of the factor identified in 3.2.4 due to the return or the
weight of the factor? (2 points)
3.2.6.
What was the best monthly return for your best quintile (F1) and what was the
date? (1 point)
3.2.7.
What was the worst monthly return for your worst quintile (F5) and what was the
date? (1 point)
3.3. For your theorical best quintile (F1), 1)
calculate the geometric return average, 2) geometric average active return
(excess return over the benchmark/universe), 3) the total risk and 4) the
tracking error. The data needed for this step is in the Periods Report. Hint:
One risk measure is based on total return and one on active return (4 points)
3.4. Using the data you computed in Step 3.3:
3.4.1. calculate the Annual Sharpe Ratio and
Information Ratio of your F1 (theorical best) quintile. The data from Step
3.3.is expressed on a monthly so will need to convert them to be able the
calculate the data on an annualized Annualize the average monthly return
geometrically (i.e. (1+r)12−1) and annualize monthly risk measures by
multiplying by
. Hint: If you calculate a SR and IR for each month, you are
doing it wrong. (4 points)
3.4.2.
How does your Information Ratio compare to the information on the Summary Report
(single factor)? (1 point)
3.5. Using a linear regression, estimate the alpha and
beta of your F1 (theorical best) quintile relative to your benchmark/universe
(data in the Periods Report). Does this agree with your conclusion on 3.2.3? (4
points)
3.6. Using a linear regression, estimate the alpha and
beta of your F5 (theorical worst) quintile relative to your benchmark/universe
(data in the Periods Report). (3 points)
3.7. Using the data from Steps 3.5 and 3.6, calculate
the weights needed in each quintile to create a long-short market neutral
strategy that has zero exposure to the benchmark i.e. beta = 0 and a total
weight of 1. Calculate the geometric
average return and alpha on your market neutral strategy. Hint #1: geometric
average return and beta are simple weighted average. Hint #2: Don’t turn this
into a difficult problem, it is not meant to be one. If you are not doing basic
algebra computation, you are overthinking it. DO NOT USE SOLVER. (4 points)
3.8. Using the constituents report for the period
December 2016, compute the Spearman rank correlation between your MFR (alpha
signal) and realized returns (universe returns). Does your estimate for the IC
match the FactSet computed IC for the period (available for the period return)?
(6 points)
3.9. Consider the six stocks from Section 1. Compare
the alphas estimated for these stocks during the period January 2017 through
December 2021 in 1.7 to their MFR values available in the December 2016
Constituents report. Did your fundamental model correctly identify stocks with
positive or negative risk-adjusted returns? Discuss the efficacy of your model
as a predictor for future alpha. (4 points)
Section 4: Clarity and Data Presentation
4.1 Present your data and conclusions for each section
in a clear and easy way to understand. I should be able to easily identify what
question you are answering and follow your logic with ease. Data should not be
copied from one sheet to another but referenced. Conclusions should clear and
properly supported. (10 points)
[1] If the weights you computed in (2.4) are not reasonable,
you can override those estimates and use your own set of weights. Just be sure
to explain in your report why you have made this choice.

 

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