Mean | Standard Deviation | Median | Minimum | Maximum | Skewness | Kurtosis |
0.0060 | 0.0569 | 0.0103 | -0.1795 | 0.1745 | -0.5381 | 1.7814 |
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.
Citation: |
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 |
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 | - |
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 | - | - | - | - | - | - | - | - | - |
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 | - | - | - | - | - | - |
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 | - | - | - | - | - | - |
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