讲解STA457、辅导RP、MA留学生、讲解R设计、R编程语言辅导
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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 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.
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.
a.
b.
c. is defined in Equation (1) (see question B)
B.Time Series Momentum
1)Calculate the ex-ante volatility estimate for all 30 DJ constituents using the following formula:
and
where the weights add up to one, and is the exponentially weighted average return computed similarly.
Hint: Solve using
and
2)Consider the predictive regression that regresses the (excess) return in month on its return lagged months, i.e.
where denotes the -th stock in the DJ constituents and in the prediction regression, returns are scaled by their ex-ante volatilities . 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:
where is our position for the -th constituent at time and denote the -month lagged returns observed at time . Summarize the performance of the portfolio.
Hint: For simplicity, assume for all 30 DJ constituents.
C.Dynamic position sizing for technical trading rules
1)Consider a technical indicator , where the technical indicator may be given by
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
Calculate the expected -period holding period return, i.e., .
Remark: In this question, we assume that our position changes linearly with the strength of the signal. We can generalize it by replacing with in Equation (7).
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 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.
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.
a.
b.
c. is defined in Equation (1) (see question B)
B.Time Series Momentum
1)Calculate the ex-ante volatility estimate for all 30 DJ constituents using the following formula:
and
where the weights add up to one, and is the exponentially weighted average return computed similarly.
Hint: Solve using
and
2)Consider the predictive regression that regresses the (excess) return in month on its return lagged months, i.e.
where denotes the -th stock in the DJ constituents and in the prediction regression, returns are scaled by their ex-ante volatilities . 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:
where is our position for the -th constituent at time and denote the -month lagged returns observed at time . Summarize the performance of the portfolio.
Hint: For simplicity, assume for all 30 DJ constituents.
C.Dynamic position sizing for technical trading rules
1)Consider a technical indicator , where the technical indicator may be given by
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
Calculate the expected -period holding period return, i.e., .
Remark: In this question, we assume that our position changes linearly with the strength of the signal. We can generalize it by replacing with in Equation (7).
Find the optimal double MA trading rule for all 30 DJ constituents that maximize the 12-period holding period return.