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A multistage stochastic programming framework for cardinality constrained portfolio optimization

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This paper was prepared at the occasion of The 12th International Conference on Industrial Engineering (ICIE 2016), Tehran, Iran, January 25-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Assoc. Prof. A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey.
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  • This paper presents a multistage stochastic programming model to deal with multi-period, cardinality constrained portfolio optimization. The presented model aims to minimize investor's expected regret, while ensuring achievement of a minimum expected return. To generate scenarios of market index returns, a random walk model based on the empirical distribution of market-representative index returns is proposed. Then, a single index model is used to estimate stock returns based on market index returns. Afterward, historical returns of a number of stocks, selected from Frankfurt Stock Exchange (FSE), are used to implement the presented scenario generation method, and solve the stochastic programming model. In addition, the impact of cardinality constraints, transaction costs, minimum expected return and predetermined investor's target wealth are investigated. Results show that the inclusion of cardinality constraints and transaction costs significantly influences the investors risk-return tradeoffs. This is also the case for investors target wealth.

    Mathematics Subject Classification: 90C11, 90C15, 90C90.

    Citation:

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  • Figure 1.  The schematic representation of a scenario tree for T periods

    Figure 2.  A schematic representation of the proposed scenario tree generation method

    Figure 3.  Risk vs. expected return for portfolios with and without cardinality constraints (Target wealth = $1000000)

    Figure 4.  Risk vs. expected return obtained by setting different levels of target wealth ($1000000 and $1050000) for portfolios with and without cardinality constraints

    Figure 5.  Investor's risk for different levels of proportional transaction costs

    Table 1.  Descriptive statistics of historical CDAX returns

    Mean Standard Deviation Median Minimum Maximum Skewness Kurtosis
    0.0060 0.0569 0.0103 -0.1795 0.1745 -0.5381 1.7814
     | Show Table
    DownLoad: CSV

    Table 2.  αi and βi values of the single index model for all stocks

    Stock B & A LR81 LTEC MZA NEC1 N2X OTP
    Intercept 0.015231 0.0008692 -0.0028 0.039533 -3.1E-05 0.001772 -0.01099
    Slope 0.756845 1.211379 0.889253 1.837928 0.644086 0.971493 1.961487
    Stock SIE TAH BMW XCY O4B ZYT -
    Intercept -.00063 0.006095 0.0098 0.024254 0.001565 0.003712 -
    Slope 1.091311 0.292933 1.186136 0.592039 0.564903 1.498048 -
     | Show Table
    DownLoad: CSV

    Table 3.  Investor's expected regret considering different target wealth, minimum expected return and proportional transaction costs

    Target wealth 1000000 1050000 1100000
    Proportional transaction cost 0 0.01 0.02 0 0.01 0.02 0 0.01 0.02
    0.95 0 0 0 63429.6 85112.7 103678.9 157646.9 196749.4 223658.1
    0.99 0 0 0 63429.6 85112.7 103678.9 157646.9 196749.4 223658.1
    1.01 0 1290.1 2987.9 63429.6 85112.7 103678.9 157646.9 196749.4 223658.1
    1.03 36.1 3953.9 9018.7 63429.6 85112.7 103678.9 157646.9 196749.4 223658.1
    1.04 400.0 5567.1 12654.5 63429.6 85112.7 103678.9 157646.9 196749.4 223658.1
    1.05 1142.6 7705.6 17739.6 63429.6 85112.7 104334.1 157646.9 196749.4 223658.1
    1.06 2350.1 11121.8 26179.7 63429.6 85134.9 109571.7 157646.9 196749.4 223838.3
    1.07 3904.2 15907.5 37152.7 63429.6 87879.1 118868.3 157646.9 198064.3 226873.5
    1.08 5954.9 22562.9 49694.6 63429.6 95349.1 129104.5 157646.9 202106.9 232405.6
    1.09 8525.3 36300.4 66271.2 63694.1 106415.1 140267.7 157646.9 208788.3 240209
    1.10 12086.2 54243.7 88133.5 64838.1 120626.0 157811.5 157675.5 217600.2 251621.1
    1.11 17279.9 74827.6 - 67119.8 138148.1 - 158591.5 228534.1 -
    1.12 24358.6 98255.0 - 74520.5 159434.9 - 163337.6 242911.9 -
    1.13 52656.5 - - 104774.3 - - 185917.5 - -
    1.14 - - - - - - - - -
     | Show Table
    DownLoad: CSV

    Table 4.  Investor's expected regret considering different target wealth and minimum expected return with and without cardinality

    Cardinality Constraints No Cardinality Constraints
    Target wealth 1000000 1050000 1100000 1000000 1050000 1100000
    0.95 0 81742.31 192944.5 0 63429.62 157646.9
    0.99 0 81742.31 192944.5 0 63429.62 157646.9
    1 0 81742.31 192944.5 0 63429.62 157646.9
    1.01 0 81742.31 192944.5 0 63429.62 157646.9
    1.02 197.428 81742.31 192944.5 0 63429.62 157646.9
    1.03 893.094 81742.31 192944.5 36.097 63429.62 157646.9
    1.04 2574.389 81742.31 192944.5 399.947 63429.62 157646.9
    1.05 5261.672 81879.22 192944.5 1142.637 63429.62 157646.9
    1.06 8917.349 82803.35 192944.5 2350.142 63429.62 157646.9
    1.07 18358.44 87336.35 193443 3904.241 63429.62 157646.9
    1.08 35077.99 96174.55 198126.4 5954.918 63429.62 157646.9
    1.09 - - - 8525.318 63694.05 157646.9
    1.10 - - - 12086.15 64838.09 157675.5
    1.11 - - - 17279.88 67119.82 158591.5
    1.12 - - - 24358.63 74520.5 163337.6
    1.13 - - - 52656.51 104774.3 185917.5
    1.14 - - - - - -
     | Show Table
    DownLoad: CSV

    Table 5.  Investor's expected regret considering different proportional transaction costs and number of assets

    Number of assets 6 12
    Proportional transaction cost 0 0.01 0.02 0 0.01 0.02
    0.95 229314.2 275191.5 318875.4 192944.5 225372.1 258752.1
    0.99 229314.2 275191.5 318875.4 192944.5 225372.1 258752.1
    1.01 229314.2 275191.5 318875.4 192944.5 225372.1 258752.1
    1.03 229314.2 275191.5 318875.4 192944.5 225372.1 258752.1
    1.04 229314.2 275191.5 318875.4 192944.5 225372.1 258752.1
    1.05 229314.2 275191.5 320958.9 192944.5 225372.1 258752.1
    1.06 229314.2 277364.2 338961.2 192944.5 225372.1 258752.1
    1.07 229314.2 280367.1 - 192944.5 230553.7 263452.1
    1.08 229314.3 - - 192944.5 235638.9 -
    1.09 231175.9 - - 193443.0 239987.4 -
    1.10 - - - 198126.4 - -
    1.11 - - - - - -
     | Show Table
    DownLoad: CSV
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