Optimal Market Timing Strategies under Transaction Costs by sdfwerte

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Optimal Market Timing Strategies
under Transaction Costs
LI Wei
A thesis submitted in partial fulfilment of the requirements
for the degree of
Doctor of Philosophy
June 1999
Hong Kong Baptist University
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Abstract
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This thesis is the first attempt to formulate the market timing strategy
as an optimal growth investment strategy. We consider the optimal market
timing strategy when the return process follows a process other than the ran¬
dom walk and analyze the effects of transaction costs to the market timing
strategies.
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For a fictitious investor who can foresee future prices exactly, we derive
the perfect market timing strategy under transaction costs. The traditional
"perfect" timing strategy are found to be inadequate. The perfect timing-
strategy derived here is related to the filter trading rule. The optimal rule can
also be derived by solving a linear programming problem.
When the return process follows stochastic processes, assumed AR( 1)
and ARMA( 1,1) models here, we derive the optimal market timing strategies
explicitly for the finite investment horizon by the use of stochastic dynamic
programming techniques. Limiting behaviours of the optimal strategies are
analyzed. We further show that the limiting strategies are related to some
popular technical trading rules. Numerical results of the limiting stationary
trading strategies and their associated expected daily return are computed
for some realistic parameter values. For the ARMR( 1,1) model, We also try
out empirically the limiting strategy in the Hang Seng Index Futures marked
Empirical results suggest that the ARMA( 1,1) model gives rise to a reasonable
trading strategy for the Hong Kong HSIF market. Furthermore, the greater
the transaction costs, the better is the performance of the proposed strategy.
Also, we find that as the transaction costs increase, the average number of days
in the market increases, while the number of transactions decreases and the
average reward decreases, which means that the larger the transaction costs,
the less frequent the trades and the less we can gain from the strategy.
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Contents
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Declaration
Abstract
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Acknowledgements
List of Tables
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1 Introduction
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1.1 Introduction
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1.2 Organisation of this dissertation
2 Literature Review on Market Timing and Related Is-
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sues
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2.1 Introduction
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2.2 Modern asset allocation theory
2.2.1 Maximizing expected utility of multiperiod con¬
sumption 	
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2.2.2 Maximizing expected utility of terminal wealth . 10
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2.2.3 Asset allocations with transaction costs
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2.3 Market timing strategy
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2.3.1 Debates on market timing
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2.3.2 Market timers believe that the market is predictable 20
2.3.3 Growth-optimal criteria for market timers .... 23
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2.3.4 0—1 strategy for market timers
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2.4 Distinction between market timing and asset allocation . 28
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3 Problem formulation
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3.1 Introduction
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3.2 Wealth dynamics for market timers
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3.2.1 Assumptions and notations
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3.2.2 Terminal wealth with transaction costs
3.2.3 Terminal wealth for 0 — 1 market timers
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3.3 Growth-optimal investment strategy for market timers . 35
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3.4 The optimization problem in market timing
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3.4.1 Investment objectives
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3.4.2 Total excess return for market timers
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3.4.3 The case of short-selling
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3.5 Stochastic Price Models
3.5.1 AR( 1) model
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3.5.2 ARM A models
3.5.3 Background of ARMA( 1,1) model
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3.6 Stochastic dynamic programming
4 Perfect Market Timing Strategy under Transaction Costs 47
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4.1 Introduction
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4.2 Optimization problem for perfect timing
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4.3 Optimal solution and its relations to Alexander's filter
rule		
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4.3.1 Alexander's filter trading rule
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4.3.2 The optimal solution
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4.3.3 Properties of the optimal strategy
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4.4 Reduction to a Linear Programming
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4.5 Simplex solution of the LP Problem
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5 Optimal Market Timing Strategies under an AR(1) Re¬
turn Process
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5.1 Introduction
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5.2 Optimal strategy for finite time horizon
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5.2.1 Optimality equation
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5.2.2 Properties of ht(x)
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5.2.3 Optimal strategy
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5.2.4 Effect of transaction cost
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5.3 Limiting behaviour of the optimal strategy
5.3.1 Convergence of ht(x)
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5.3.2 The limiting strategy
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5.3.3 Effect of transaction cost
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5.3.4 Relationship with technical trading rule
5.3.5 Stationary distribution and its daily reward ... 71
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5.4 Numerical values specifying the limiting strategy
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5.4.1 Numerical results for // = 0
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5.4.2 Results for // ^ 0
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5.5 Conclusions
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6 Optimal Market Timing Strategies Under ARMA(l,l) 82
6.1 Introduction
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6.2 Optimal decisions for finite time horizon
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6.2.1 State space representation of ARMA(1,1) .... 83
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6.2.2 Bellman equation
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6.2.3 Optimal trading strategy
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6.2.4 Effect of transaction cost
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6.3 Limiting behaviour of the optimal strategy
6.3.1 Convergence of ut.(x)
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6.3.2 Relationship with technical trading rule
6.3.3 Stationary distribution and average return .... 91
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6.4 Numerical Results for the limiting strategy
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6.4.1 Values of a and b
. . 95
6.4.2 Expected reward of the strategy . . . .
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6.5 Conclusions
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7 Empirical Results
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7.1 Introduction
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7.2 Data description
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7.3 Parameter estimation of ARMA( 1,1) model
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7.4 Limiting strategy under ARMA( 1,1) model
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7.5 Performance of the limiting strategy . . .
. . . 104
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7.6 Conclusions
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8 Conclusion
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8.1 Contributions of this study
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8.2 Extensions for future studies
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Bibliography
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A Proofs of Lammas and Theorems
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A.l Proof of Theorem 4.3
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A.2 Proof of Lemma 5.5
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A.3 Proof of Lemma 6.2
134
A.4 Proof of Lemma 6.3
. 135
A.5 Proof of Lemma 6.6
136
B CURRICULUM VITAE
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