STA457 Time Series Analysis Assignment 1 (Winter 2019)
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 and monthly 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 annually at the end of each year.
Questions:
A. Technicaltradingrule
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.
Copyright © Jen-Wen Lin 2019
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B. TimeSeriesMomentum
1) Calculate the ex-ante volatility estimate 𝜎” for all 30 DJ constituents using the following formula:
and
3
𝜎# =261 )(1−𝛿)𝛿.(𝑟 −𝑟̅)#, (1) ” “010. ”
.45
𝜎 =7𝜎#, (2) “”
where the weights 𝛿.(1 − 𝛿) add up to one, and 𝑟̅ is the exponentially weighted ”
average return computed similarly.
2) Consider the predictive regression that regresses the (excess) return in month 𝑡 on
its return lagged h months, i.e.
𝑟𝑟
:,” =𝛼+𝛽>⋅:,”0>+𝜀:,”, (3)
𝜎:,”01 𝜎:,”0>
:,” =𝑎+𝑏 ⋅𝑠𝑖𝑔𝑛K𝑟 L+𝜉 . (4)
and
3) Consider a time series momentum trading strategy by constructing the following portfolios:
𝑟 𝜎:,”01
> :,”0> :,”
denotes the 𝑠-th stock in the DJ constituents and in the prediction regression, returns are scaled by their ex-ante volatilities 𝜎:,”01. Determine the optimal h for both predictive regressions for all 30 DJ constituents.
where 𝑟 :,”
>V:”0>V:” :
QRSTS 1 Y5 40%
𝑟 = )𝑠𝑖𝑔𝑛K𝑟 L⋅ 𝑟 , (5)
“,”P1 30
:,”0>V:” 𝜎:,” :,”:”P1
L is our position for the 𝑠-th constituent at time 𝑡 and
:41 :,”0>V:” :,”
where 𝑠𝑖𝑔𝑛K𝑟 L ⋅ K40%⁄𝜎
𝑟 denote the h -month lagged returns observed at time 𝑡. Summarize the
performance of the portfolio.
Hint: For simplicity, assume h: = 12 for all 30 DJ constituents.
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C. Dynamicpositionsizingfortechnicaltradingrules
1) Consider a technical indicator 𝐹 , where the technical indicator may be given by ”
_0#
𝐹=)𝑑𝑟 . (6).
>45
Suppose that our position to the trading rule is determined by the strength (or magnitude) of the signal. The h-period holding period return is then given by
”
. “0^
>01 >01 _0#
𝑅 =)𝐹 ⋅𝑟 =)a)𝑑𝑟 b⋅𝑟 . (7)
“:”P> “P.01 “P. .45
. “0^P.01 “P.
.45 ^45
Calculate the expected h-period holding period return, i.e., 𝐸(𝑅”:”P>).
Remark: In this question, we assume that our position changes linearly with the strength of the signal. We can generalize it by replacing 𝐹 with 𝑔(𝐹 ) in
Equation (7).
2) Find the optimal double MA trading rule for all 30 DJ constituents that maximize the
12-period holding period return.
Copyright © Jen-Wen Lin 2019
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“P.01 “P.01