STA457辅导、Series Analysis讲解、辅导R设计、讲解R语言
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Jen-Wen Lin, PhD, CFA
Date: February 07, 2019
Please check in Quercus regularly for the update of the assignment.
Background reading:
1. Assignment and solution (Fall 2018)
2. Moskowitz et al. (2012), “Time series momentum”, Journal of Financial Economics
General instruction
§ Download daily data of 30 constituents in the Dow Jones (DJ) index from 1999 December to
2018 December. Please see https://money.cnn.com/data/dow30/ for the list of DJ
constituents.
§ Calculate the performance based on a 60-month rolling window and rebalance the portfolio
monthly but calibrate/estimate parameters () at the end of each year.
§ Performance: Annualized expected return, annualized volatility (standard deviation), and
Annualized Sharpe ratio. Annualization is done using the squared root of time. Use Sharpe
ratio as example
where assume that annual risk free rate , = 0.02 and ) is the sample mean of monthly
strategy returns and ./ is the monthly volatility.
Questions:
A. Technical trading rule
1) Find the optimal double moving average (MA) trading rules for all 30 DJ constituents
(stocks) using monthly data.
Hint: see Assignment (Fall 2018) for more details.Copyright Jen-Wen Lin 2019
2) Construct the equally weighted (EW) and risk-parity (RP) weighted portfolio using all
30 DJ constituents. Summarize the performances of EW and RP portfolios (trading
strategies).
Hint: For simplicity, assume the correlations among stocks are zero when
constructing the risk-parity portfolio.
#BCD #D3B#E
/G = ∑ H IJKL
∑ IM NO KL M; is defined in Equation (1) (see question B)
B. Time Series Momentum
1) Calculate the ex-ante volatility estimate 3 for all 30 DJ constituents using the
following formula:
R = 261 T(1)X(2)
where the weights X
(1) add up to one, and
;,3 is the exponentially weighted
average return computed similarly.
Hint: Solve using
T(1XR\8XF8= 1and;,3 = T(1)XR\8XF8
;,3=6=X.Copyright Jen-Wen Lin 201932) Consider the predictive regression that regresses the (excess) return in month on
its return lagged months, i.e.
(4)
where ;,3 denotes the -th stock in the DJ constituents and in the prediction
regression, returns are scaled by their ex-ante volatilities ;,3=6. Determine the
optimal for both predictive regressions for all 30 DJ constituents.
Remark: For simplicity, students only need to consider Equation (4) in this question
and use R-squared to evaluate the predictive regression.
3) Consider a time series momentum trading strategy by constructing the following
portfolios:(5)
where ,3=cJ:3[ Y40%;,3[ is our position for the -th constituent at time and
cJ:3=cJ:3 denote the ;-month lagged returns observed at time. Summarize the
performance of the portfolio.
Hint: For simplicity, assume ; = 12 for all 30 DJ constituents.Copyright Jen-Wen Lin 2019
C. Dynamic position sizing for technical trading rules
1) Consider a technical indicator 3, where the technical indicator may be given by(6).
Suppose that our position to the trading rule is determined by the strength (or
magnitude) of the signal. The -period holding period return is then given by
(7
Calculate the expected -period holding period return, i.e.,(3:3qc).
Remark: In this question, we assume that our position changes linearly with the
strength of the signal. We can generalize it by replacing ?3qX=6 with (3qX=6) in
Equation (7).
2) Find the optimal double MA trading rule for all 30 DJ constituents that maximize the
12-period holding period return.
Jen-Wen Lin, PhD, CFA
Date: February 07, 2019
Please check in Quercus regularly for the update of the assignment.
Background reading:
1. Assignment and solution (Fall 2018)
2. Moskowitz et al. (2012), “Time series momentum”, Journal of Financial Economics
General instruction
§ Download daily data of 30 constituents in the Dow Jones (DJ) index from 1999 December to
2018 December. Please see https://money.cnn.com/data/dow30/ for the list of DJ
constituents.
§ Calculate the performance based on a 60-month rolling window and rebalance the portfolio
monthly but calibrate/estimate parameters () at the end of each year.
§ Performance: Annualized expected return, annualized volatility (standard deviation), and
Annualized Sharpe ratio. Annualization is done using the squared root of time. Use Sharpe
ratio as example
where assume that annual risk free rate , = 0.02 and ) is the sample mean of monthly
strategy returns and ./ is the monthly volatility.
Questions:
A. Technical trading rule
1) Find the optimal double moving average (MA) trading rules for all 30 DJ constituents
(stocks) using monthly data.
Hint: see Assignment (Fall 2018) for more details.Copyright Jen-Wen Lin 2019
2) Construct the equally weighted (EW) and risk-parity (RP) weighted portfolio using all
30 DJ constituents. Summarize the performances of EW and RP portfolios (trading
strategies).
Hint: For simplicity, assume the correlations among stocks are zero when
constructing the risk-parity portfolio.
#BCD #D3B#E
/G = ∑ H IJKL
∑ IM NO KL M; is defined in Equation (1) (see question B)
B. Time Series Momentum
1) Calculate the ex-ante volatility estimate 3 for all 30 DJ constituents using the
following formula:
R = 261 T(1)X(2)
where the weights X
(1) add up to one, and
;,3 is the exponentially weighted
average return computed similarly.
Hint: Solve using
T(1XR\8XF8= 1and;,3 = T(1)XR\8XF8
;,3=6=X.Copyright Jen-Wen Lin 201932) Consider the predictive regression that regresses the (excess) return in month on
its return lagged months, i.e.
(4)
where ;,3 denotes the -th stock in the DJ constituents and in the prediction
regression, returns are scaled by their ex-ante volatilities ;,3=6. Determine the
optimal for both predictive regressions for all 30 DJ constituents.
Remark: For simplicity, students only need to consider Equation (4) in this question
and use R-squared to evaluate the predictive regression.
3) Consider a time series momentum trading strategy by constructing the following
portfolios:(5)
where ,3=cJ:3[ Y40%;,3[ is our position for the -th constituent at time and
cJ:3=cJ:3 denote the ;-month lagged returns observed at time. Summarize the
performance of the portfolio.
Hint: For simplicity, assume ; = 12 for all 30 DJ constituents.Copyright Jen-Wen Lin 2019
C. Dynamic position sizing for technical trading rules
1) Consider a technical indicator 3, where the technical indicator may be given by(6).
Suppose that our position to the trading rule is determined by the strength (or
magnitude) of the signal. The -period holding period return is then given by
(7
Calculate the expected -period holding period return, i.e.,(3:3qc).
Remark: In this question, we assume that our position changes linearly with the
strength of the signal. We can generalize it by replacing ?3qX=6 with (3qX=6) in
Equation (7).
2) Find the optimal double MA trading rule for all 30 DJ constituents that maximize the
12-period holding period return.