1 Introduction
Parameters | |
---|---|
\(\psi \)
| Target funding (asset/liability) ratio |
\(c_\mathrm{b}, c_\mathrm{s}\)
| Transaction costs for buying and selling, respectively |
\(l_t\)
| Amount (liabilities) paid out at time t
|
\({\bar{g}}\)
| Guaranteed rate of return per period |
\(C_t\)
| Coupon payment at t
|
P
| Capital (principal) paid |
Decision variables | |
---|---|
\(h_t^m\)
| Holding in asset m at time t
|
\(s_t^m\)
| Amount sold of asset m at time t
|
\(b_t^m\)
| Amount bought of asset m at time t
|
Random variables | |
---|---|
\(\tilde{r}_t^m\)
| Return on asset m between time \(t-1\) and t
|
\(\tilde{L}_t\)
| Present value of the total amount of future outstanding liabilities at time t
|
\(\tilde{R}^m_t\)
| Cumulative gross return on asset m at t
|
2 Problem statement
3 Scenario-based asset–liability management model
4 Robust ALM for investment products with guarantees
4.1 Symmetric uncertainty sets
4.2 Asymmetric uncertainty sets
5 Implementation
5.1 Data
\(t = 0\)
|
\(t = 1\)
|
\(t = 2\)
|
\(t = 3\)
| |
---|---|---|---|---|
Mean return | ||||
Risky asset | 0.029 | 0.051 |
\(-\)0.003 | 0.005 |
Risk-free asset | 0.014 | 0.012 | 0.007 | 0.005 |
Standard deviation | ||||
Risky asset | 0.054 | 0.069 | 0.082 | 0.095 |
Risk-free asset | 0.005 | 0.001 | 0.004 | 0.005 |
p: Forward deviation | ||||
Risky asset | 0.054 | 0.070 | 0.082 | 0.095 |
Risk-free asset | 0.005 | 0.001 | 0.005 | 0.005 |
q: Backward deviation | ||||
Risky asset | 0.070 | 0.069 | 0.085 | 0.101 |
Risk-free asset | 0.005 | 0.002 | 0.004 | 0.005 |
\(\tilde{{\varvec{z}}}^{\alpha }:\)
| Factors | 1 | 2 | ||||
---|---|---|---|---|---|---|---|
Std dev | 0.99906 | 0.97472 | |||||
p: Forward dev | 1.02573 | 0.99454 | |||||
q: Backward dev | 0.99906 | 0.97472 |
\(\tilde{{\varvec{z}}}^{\rho }_t:\)
| Factors | 1 | 2 | 3 | |||
---|---|---|---|---|---|---|---|
\(t = 1\)
| Std dev | 1.03192 | 1.05277 | 0.95624 | |||
p: Forward dev | 1.03192 | 1.05277 | 0.95626 | ||||
q: Backward dev | 1.03398 | 1.05487 | 0.95624 | ||||
\(t = 2\)
| Std dev | 0.96412 | 0.98360 | 0.95167 | |||
p: Forward dev | 0.96428 | 0.98376 | 0.95169 | ||||
q: Backward dev | 0.96412 | 0.98360 | 0.95167 | ||||
\(t = 3\)
| Std dev | 0.93665 | 0.95557 | 0.95059 | |||
p: Forward dev | 0.93718 | 0.95611 | 0.95061 | ||||
q: Backward dev | 0.93665 | 0.95557 | 0.95059 | ||||
\(t = 4\)
| Std dev | 0.91791 | 0.93646 | 0.95750 | |||
p: Forward dev | 0.91843 | 0.93699 | 0.95752 | ||||
q: Backward dev | 0.91791 | 0.93646 | 0.95750 |
\(\tilde{{\varvec{z}}}^{\mu }_t:\)
| Factors | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|
\(t = 1\)
| Std dev | 0.99852 | 0.96851 | 1.01065 | 1.10211 | 1.14325 | 1.15469 |
p: Forward dev | 0.99852 | 0.96880 | 1.01324 | 1.10368 | 1.14500 | 1.15645 | |
q: Backward dev | 1.00034 | 0.96851 | 1.01065 | 1.10211 | 1.14325 | 1.15469 | |
\(t = 2\)
| Std dev | 0.99897 | 0.98629 | 1.09494 | 1.13022 | 1.14152 | |
p: Forward dev | 0.99914 | 0.98708 | 1.09646 | 1.13222 | 1.14354 | ||
q: Backward dev | 0.99897 | 0.98629 | 1.09494 | 1.13022 | 1.14152 | ||
\(t = 3\)
| Std dev | 0.99868 | 0.97976 | 1.13039 | 1.14170 | ||
p: Forward dev | 0.99923 | 0.98068 | 1.13155 | 1.14286 | |||
q: Backward dev | 1.01170 | 0.99253 | 1.14512 | 1.15657 |
5.2 Parameter estimation for the robust formulations
6 Computational results
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relative to the performance of the nominal and the stochastic programming models in terms of summary statistics for the realized final wealth;
-
in terms of the optimal asset allocation at the first step of the multi-period optimization problem as well as the CPU time taken to obtain an investment strategy;
-
in terms of the effect of using asymmetric versus a symmetric uncertainty set in the robust formulation; and
-
in terms of the effect of the magnitude of different parameters associated with the robust formulation, such as the budget of robustness.
ALMmodels | Problem structure | Solution time | Final wealth | Transaction cost | # Assets invested | ||
---|---|---|---|---|---|---|---|
(T, M) | Variables | Constraints | |||||
N | (4, 2) | 30 | 46 | 0.020 | 7.560 | 9.800 | 1 |
Sym-R(1) | (4, 2) | 62 | 70 | 0.030 | 4.011 | 9.540 | 2 |
Asym-R(1) | (4, 2) | 78 | 102 | 0.050 | 4.010 | 9.540 | 2 |
SP | (4, 2) | 3029 | 13,934 | 0.120 | 7.340 | 10.030 | 1 |
N | (4, 11) | 126 | 174 | 0.030 | 74.645 | 10.176 | 1 |
Sym-R(1) | (4, 11) | 222 | 262 | 0.040 |
\(-\)44.046 | 7.768 | 11 |
Asym-R(1) | (4, 11) | 302 | 422 | 0.060 |
\(-\)44.744 | 7.568 | 11 |
SP | (4, 11) | 12,725 | 26,038 | 0.340 | 68.546 | 9.870 | 2 |
N | (4, 21) | 246 | 334 | 0.054 | 76.106 | 10.161 | 1 |
Sym-R(1) | (4, 21) | 422 | 502 | 0.120 |
\(-\)24.230 | 8.528 | 17 |
Asym-R(1) | (4, 21) | 582 | 822 | 0.491 | 8.519 | 8.519 | 17 |
SP | (4, 21) | 24,845 | 41,168 | 0.410 | 69.540 | 9.177 | 3 |
N | (4, 31) | 366 | 494 | 0.101 | 80.869 | 10.122 | 1 |
Sym-R(1) | (4, 31) | 622 | 742 | 0.233 |
\(-\)15.767 | 8.526 | 24 |
Asym-R(1) | (4, 31) | 862 | 1222 | 0.751 |
\(-\)18.417 | 8.520 | 24 |
SP | (4, 31) | 36,965 | 56,298 | 1.707 | 75.392 | 10.320 | 4 |
N | (10, 11) | 222 | 432 | 0.512 | 260.832 | 17.681 | 1 |
Sym-R(1) | (10, 11) | 462 | 652 | 0.860 |
\(-\)17.359 | 9.554 | 7 |
Asym-R(1) | (10, 11) | 617 | 1052 | 1.342 |
\(-\)30.201 | 8.393 | 8 |
SP | (10, 11) | 22,421 | 52,024 | 2.084 | 217.452 | 15.790 | 2 |
N | (10, 21) | 612 | 832 | 0.609 | 318.852 | 17.687 | 1 |
Sym-R(1) | (10, 21) | 1052 | 1252 | 0.910 | 128.918 | 16.060 | 17 |
Asym-R(1) | (10, 21) | 1452 | 2052 | 1.913 | 125.504 | 16.010 | 17 |
SP | (10, 21) | 61,811 | 91,334 | 2.365 | 295.459 | 14.054 | 2 |
N | (10, 31) | 1512 | 2432 | 1.342 | 332.553 | 11.650 | 1 |
Sym-R(1) | (10, 31) | 1552 | 1852 | 1.456 | 151.101 | 8.776 | 14 |
Asym-R(1) | (10, 31) | 2152 | 3052 | 2.726 | 134.322 | 8.541 | 14 |
SP | (10, 31) | 92,111 | 130,644 | 2.908 | 296.953 | 12.948 | 2 |
Model types | Symmetric uncertainty set | Asymmetric uncertainty set | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(t=1\)
|
\(t=2\)
|
\(t=3\)
|
\(t=4\)
|
\(t=1\)
|
\(t=2\)
|
\(t=3\)
|
\(t=4\)
| |||||||||
RK | RF | RK | RF | RK | RF | RK | RF | RK | RF | RK | RF | RK | RF | RK | RF | |
GIC-N | 952.7 | 0.0 | 934.0 | 10.0 | 934.0 | 0.0 | 924.7 | 0.0 | 952.7 | 0.0 | 934.0 | 10.0 | 934.0 | 0.0 | 924.7 | 0.0 |
GIC-R(0.1) | 951.2 | 0.0 | 932.5 | 10.0 | 932.5 | 0.0 | 923.2 | 0.0 | 951.2 | 0.0 | 932.5 | 10.0 | 932.5 | 0.0 | 923.2 | 0.0 |
GIC-R(0.3) | 948.4 | 0.0 | 920.1 | 20.0 | 920.1 | 10.0 | 920.1 | 0.0 | 948.4 | 0.0 | 920.1 | 20.0 | 920.1 | 10.0 | 920.1 | 0.0 |
GIC-R(0.5) | 289.2 | 670.8 | 260.8 | 690.8 | 260.8 | 680.8 | 260.8 | 670.8 | 281.6 | 678.5 | 253.3 | 698.5 | 253.3 | 688.5 | 253.3 | 678.5 |
GIC-R(0.7) | 82.0 | 882.0 | 53.5 | 902.0 | 53.5 | 892.0 | 53.5 | 882.0 | 81.9 | 882.0 | 53.4 | 902.0 | 53.4 | 892.0 | 53.4 | 882.0 |
GIC-R(0.9) | 56.2 | 908.0 | 37.1 | 918.0 | 37.1 | 908.0 | 36.8 | 898.6 | 56.2 | 908.0 | 37.1 | 918.0 | 37.1 | 908.0 | 36.7 | 898.6 |
GIC-R(1.0) | 36.0 | 929.0 | 36.0 | 919.0 | 36.0 | 909.0 | 35.6 | 899.6 | 35.8 | 929.1 | 35.8 | 919.1 | 35.8 | 909.1 | 35.5 | 899.8 |
ELN-N | 952.7 | 0.0 | 934.0 | 10.0 | 934.0 | 0.0 | 924.7 | 0.0 | 952.7 | 0.0 | 934.0 | 10.0 | 934.0 | 0.0 | 924.7 | 0.0 |
ELN-R(0.1) | 951.2 | 0.0 | 932.5 | 10.0 | 932.5 | 0.0 | 923.2 | 0.0 | 951.2 | 0.0 | 932.5 | 10.0 | 932.5 | 0.0 | 923.2 | 0.0 |
ELN-R(0.3) | 948.4 | 0.0 | 920.1 | 20.0 | 920.1 | 10.0 | 920.1 | 0.0 | 948.4 | 0.0 | 920.1 | 20.0 | 920.1 | 10.0 | 920.1 | 0.0 |
ELN-R(0.5) | 283.9 | 676.2 | 255.5 | 696.2 | 255.5 | 686.2 | 255.5 | 676.2 | 282.2 | 677.9 | 253.8 | 697.9 | 253.8 | 687.9 | 253.8 | 677.9 |
ELN-R(0.7) | 82.7 | 881.3 | 54.2 | 901.3 | 54.2 | 891.3 | 54.2 | 881.3 | 82.0 | 881.9 | 53.5 | 901.9 | 53.5 | 891.9 | 53.5 | 881.9 |
ELN-R(0.9) | 56.1 | 908.1 | 37.0 | 918.1 | 37.0 | 908.1 | 36.7 | 898.7 | 56.2 | 908.0 | 37.1 | 918.0 | 37.1 | 908.0 | 36.7 | 898.7 |
ELN-R(1.0) | 35.8 | 929.2 | 35.8 | 919.2 | 35.8 | 909.2 | 35.4 | 899.8 | 35.8 | 929.2 | 35.8 | 919.2 | 35.8 | 909.2 | 35.4 | 899.8 |
Normal market regime | Symmetric uncertainty set | Asymmetric uncertainty set | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
ALM models | Mean | Sdev | VaR | CVaR | Min | Max | T-cost | Mean | Sdev | VaR | CVaR | Min | Max | T-cost |
GIC-N | 287.05 | 345.82 |
\(-\)166.95 |
\(-\)166.95 |
\(-\)258.80 | 1265.55 | 41.87 | 287.05 | 345.82 |
\(-\)166.95 |
\(-\)166.95 |
\(-\)258.80 | 1265.55 | 41.87 |
GIC-R(0.1) | 264.87 | 259.55 |
\(-\)102.70 |
\(-\)102.70 |
\(-\)235.59 | 943.39 | 38.79 | 263.92 | 257.59 |
\(-\)98.98 |
\(-\)98.98 |
\(-\)235.24 | 932.01 | 38.33 |
GIC-R(0.3) | 221.99 | 168.88 |
\(-\)14.18 |
\(-\)14.18 |
\(-\)85.46 | 624.39 | 32.61 | 218.41 | 161.89 |
\(-\)8.15 |
\(-\)8.15 |
\(-\)71.31 | 607.08 | 31.75 |
GIC-R(0.5) | 198.51 | 126.40 | 30.75 | 30.75 |
\(-\)61.39 | 525.80 | 29.25 | 195.55 | 121.86 | 41.96 | 41.96 |
\(-\)62.17 | 511.78 | 28.61 |
GIC-R(0.7) | 174.82 | 99.22 | 43.35 | 43.35 |
\(-\)48.63 | 432.86 | 25.36 | 168.72 | 92.64 | 48.77 | 48.77 |
\(-\)45.05 | 408.96 | 24.87 |
GIC-R(0.9) | 152.88 | 78.56 | 42.18 | 42.18 |
\(-\)29.10 | 352.88 | 22.71 | 149.34 | 75.10 | 45.74 | 45.74 |
\(-\)29.14 | 339.15 | 22.01 |
GIC-S | 289.23 | 331.83 |
\(-\)146.87 |
\(-\)146.87 |
\(-\)240.87 | 1235.66 | 41.78 | 289.23 | 331.83 |
\(-\)146.87 |
\(-\)146.87 |
\(-\)240.87 | 1235.66 | 41.78 |
ELN-N | 148.83 | 199.52 |
\(-\)166.95 |
\(-\)166.95 |
\(-\)258.80 | 657.78 | 41.80 | 148.83 | 199.52 |
\(-\)166.95 |
\(-\)166.95 |
\(-\)258.80 | 657.78 | 41.80 |
ELN-R(0.1) | 147.20 | 145.65 |
\(-\)102.66 |
\(-\)102.66 |
\(-\)235.63 | 491.18 | 41.09 | 146.79 | 144.47 |
\(-\)99.01 |
\(-\)99.01 |
\(-\)235.06 | 485.39 | 41.50 |
ELN-R(0.3) | 122.80 | 68.42 | 5.39 | 5.39 |
\(-\)72.53 | 261.42 | 34.73 | 121.59 | 65.64 | 12.35 | 12.35 |
\(-\)74.82 | 257.66 | 33.84 |
ELN-R(0.5) | 116.92 | 55.76 | 33.59 | 33.59 |
\(-\)60.63 | 230.54 | 32.63 | 115.99 | 54.31 | 41.32 | 41.32 |
\(-\)61.42 | 227.56 | 33.00 |
ELN-R(0.7) | 108.12 | 46.38 | 43.35 | 43.35 |
\(-\)48.65 | 193.42 | 30.89 | 105.85 | 44.00 | 48.76 | 48.76 |
\(-\)45.05 | 184.54 | 29.99 |
ELN-R(0.9) | 99.77 | 39.10 | 42.26 | 42.26 |
\(-\)29.12 | 172.60 | 28.28 | 98.30 | 37.90 | 45.80 | 45.80 |
\(-\)29.20 | 170.00 | 28.26 |
ELN-S | 144.35 | 190.79 |
\(-\)161.30 |
\(-\)161.30 |
\(-\)269.32 | 627.93 | 40.69 | 144.35 | 190.79 |
\(-\)161.30 |
\(-\)161.30 |
\(-\)269.32 | 627.93 | 40.69 |
Unfavorable market regime | Symmetric uncertainty set | Asymmetric uncertainty set | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
ALM models | Mean | Sdev | VaR | CVaR | Min | Max | T-cost | Mean | Sdev | VaR | CVaR | Min | Max | T-cost |
GIC-N |
\(-\)253.13 | 206.11 |
\(-\)524.47 |
\(-\)524.47 |
\(-\)583.74 | 327.78 | 43.58 |
\(-\)253.13 | 206.11 |
\(-\)524.47 |
\(-\)524.47 |
\(-\)583.74 | 327.78 | 43.58 |
GIC-R(0.1) |
\(-\)257.41 | 155.94 |
\(-\)479.71 |
\(-\)479.71 |
\(-\)556.02 | 147.31 | 39.94 |
\(-\)257.53 | 154.80 |
\(-\)477.06 |
\(-\)477.06 |
\(-\)555.51 | 141.02 | 38.95 |
GIC-R(0.3) |
\(-\)254.90 | 102.88 |
\(-\)400.04 |
\(-\)400.04 |
\(-\)442.71 |
\(-\)10.71 | 34.33 |
\(-\)254.84 | 98.84 |
\(-\)394.13 |
\(-\)394.13 |
\(-\)434.53 |
\(-\)18.20 | 32.41 |
GIC-R(0.5) |
\(-\)253.14 | 77.89 |
\(-\)353.38 |
\(-\)353.38 |
\(-\)416.06 |
\(-\)51.29 | 30.71 |
\(-\)252.99 | 75.27 |
\(-\)347.40 |
\(-\)347.40 |
\(-\)414.82 |
\(-\)57.44 | 28.81 |
GIC-R(0.7) |
\(-\)241.37 | 62.03 |
\(-\)322.99 |
\(-\)322.99 |
\(-\)383.29 |
\(-\)80.42 | 26.29 |
\(-\)230.14 | 57.88 |
\(-\)304.58 |
\(-\)304.58 |
\(-\)365.67 |
\(-\)80.46 | 26.34 |
GIC-R(0.9) |
\(-\)215.53 | 49.40 |
\(-\)285.26 |
\(-\)285.26 |
\(-\)331.43 |
\(-\)90.76 | 23.14 |
\(-\)214.92 | 47.37 |
\(-\)280.26 |
\(-\)280.26 |
\(-\)328.92 |
\(-\)96.25 | 22.10 |
GIC-S |
\(-\)252.25 | 197.04 |
\(-\)515.81 |
\(-\)515.81 |
\(-\)582.69 | 329.92 | 42.01 |
\(-\)252.25 | 197.04 |
\(-\)515.81 |
\(-\)515.81 |
\(-\)582.69 | 329.92 | 42.01 |
ELN-N |
\(-\)259.09 | 191.76 |
\(-\)524.47 |
\(-\)524.47 |
\(-\)583.74 | 188.89 | 42.37 |
\(-\)259.09 | 191.76 |
\(-\)524.47 |
\(-\)524.47 |
\(-\)583.74 | 188.89 | 42.37 |
ELN-R(0.1) |
\(-\)258.30 | 153.41 |
\(-\)479.65 |
\(-\)479.65 |
\(-\)556.04 | 98.32 | 42.36 |
\(-\)258.39 | 152.30 |
\(-\)477.01 |
\(-\)477.01 |
\(-\)555.39 | 95.11 | 42.25 |
ELN-R(0.3) |
\(-\)254.53 | 93.84 |
\(-\)382.82 |
\(-\)382.82 |
\(-\)432.85 |
\(-\)23.02 | 35.27 |
\(-\)254.43 | 90.36 |
\(-\)376.33 |
\(-\)376.33 |
\(-\)432.46 |
\(-\)30.17 | 34.88 |
ELN-R(0.5) |
\(-\)253.09 | 77.17 |
\(-\)351.51 |
\(-\)351.51 |
\(-\)415.01 |
\(-\)52.89 | 33.16 |
\(-\)252.96 | 74.63 |
\(-\)347.41 |
\(-\)347.41 |
\(-\)413.78 |
\(-\)59.02 | 33.64 |
ELN-R(0.7) |
\(-\)241.37 | 62.03 |
\(-\)322.99 |
\(-\)322.99 |
\(-\)383.30 |
\(-\)80.46 | 32.76 |
\(-\)230.16 | 57.89 |
\(-\)304.61 |
\(-\)304.61 |
\(-\)365.70 |
\(-\)80.47 | 30.61 |
ELN-R(0.9) |
\(-\)215.53 | 49.36 |
\(-\)285.16 |
\(-\)285.16 |
\(-\)331.40 |
\(-\)90.86 | 29.56 |
\(-\)214.92 | 47.37 |
\(-\)280.20 |
\(-\)280.20 |
\(-\)328.95 |
\(-\)96.21 | 29.40 |
ELN-S |
\(-\)258.68 | 187.35 |
\(-\)517.76 |
\(-\)517.76 |
\(-\)569.56 | 175.43 | 42.22 |
\(-\)258.68 | 187.35 |
\(-\)517.76 |
\(-\)517.76 |
\(-\)569.56 | 175.43 | 42.22 |
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The investment strategies obtained from the stochastic and nominal ALM models provide higher expected wealth (as well as higher average transaction costs) than the investment strategies obtained from the robust ALM models using symmetric and asymmetric uncertainty sets under the normal market regime. This is not surprising because stochastic programming maximizes the expected profit over known discrete scenarios whereas robust optimization uses a worst-case decision-making criterion. Moreover, under normal market regime, the future return scenarios are realized as expected.
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If the future asset returns follow the same distribution as the distribution used as input to the robust formulation, then the expected terminal wealth and the variance of terminal wealth obtained by all robust models decrease when the robustness budget increases (see, for example, the results under normal market regime). In other words, there is a trade-off between the average performance and the amount of protection desired. The nominal and stochastic investment strategies provide higher average wealth than the robust strategy for a high budget of robustness.
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On the other hand, the robust investment strategies at (high) budget of robustness appear to perform better than those obtained with the stochastic and nominal formulations in unfavorable market regimes. Recall that in the unfavorable market regime, the future return realizations are generated by the expected value \(\mu - k\sigma \) (that is k standard deviation lower than the estimated expected value from the data) and the standard deviation (\(\sigma \)). As shown in Table 6, the terminal wealth obtained by all robust models when future realized asset returns are worse than expected increases as the robustness budget increases, and the variance of final wealth decreases. Both the nominal and the stochastic investment strategies result in lower expected wealth than the robust strategy for symmetric and asymmetric uncertainty sets at high value of the price of robustness. This argues for using robust optimization in unfavorable market scenarios.
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The robust strategies with asymmetric uncertainty sets result in slightly higher wealth (and/or lower variance) than the wealth obtained for robust strategies generated with the symmetric uncertainty sets formulation for any degree of robustness (apart from the lowest value 0.1) under unfavorable market regimes. This is because the future return realizations for the generated data have asymmetric characteristics. It appears that the distribution of various price realizations and the choice of uncertainty set impact the performance of the robust investment strategy.
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In both market regimes, the robust GIC models result in a slightly higher expected value and variance of wealth than the robust ELN models regardless of the type of uncertainty sets. This is because the extra return on the terminal wealth is paid to the holder of ELNs. On the other hand, we observe that the ALM models for GIC products produce lower expected transaction cost than those provided by the ALM models using ELN regardless the market conditions.
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Number of risky assets and diversification: We first extend the simulation experiments conducted for the 2-asset real market case to a set of generated data with a larger number of assets (10, 20 and 30). The average wealth obtained with the stochastic, nominal and robust (at fixed budget of robustness of 0.3, 0.7 and 1.0) strategies is displayed in Fig. 2. As the market conditions deteriorate, the robust strategies with high budget of robustness outperform both the nominal and the stochastic ones. However, the gap between the expected wealth obtained with the stochastic programming and the robust optimization models diminishes as the number of assets increases due to the impact of diversification. Diversification appears to help the stochastic programming strategies do well even in unfavorable market conditions. For instance, while the deteriorating point for stochastic programming strategies is at \(k\approx 0.15\) for the 2-asset case, it is at \(k \approx 0.5\) for the 10-asset case.×
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Performance evaluation approaches: We consider static and dynamic performance evaluation approaches using cases with 10 and 30 assets. As explained before, the rolling horizon method updates the strategy as more info is received whereas the fixed horizon applies the prior strategy over time. Their results are shown in Fig. 3. We see that the average wealth obtained by the robust models is higher than the one obtained by the nominal model for smaller values of k (at lower levels of the unfavorable regime) if the number of assets is large. This is because diversification has a stronger effect when there are more investment alternatives.×
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Uncertainty sets: In order to understand how the size and the shape of the symmetric and asymmetric uncertainty sets affect the robust investment strategies, we vary the budget of robustness for the uncertainty sets associated with the uncertain coefficients in the objective function and the balance constraints. The numerical experiments indicate that the budget of robustness for the funding ratio constraints does not have a substantial effect on the investment decisions; hence those results on are not included here. The simulation results presented in terms of average final wealth in Fig. 4 illustrate the effect of the budget of robustness (\(\theta ^o\), \(\theta ^h_t,\theta ^f_t\)) and (\(\Omega ^o\), \(\Omega ^h_t,\Omega ^f_t \)) for symmetric and asymmetric uncertainty sets, respectively, on the robust decisions under normal and unfavorable market regimes. For instance, in order to understand the impact of \(\theta ^o\), we vary \(\theta ^o\) within the interval [0, 1] and fix \(\theta ^h_t\) and \(\theta ^f_t\) at 0.1. Similarly, for \(\theta ^h_t\) we select the same values from the interval [0, 1] while \(\theta ^o =\theta ^f_t=0.1\). These two cases are abbreviated as (1, 0.1, 0.1) and (0.1, 1, 0.1), respectively, in Fig. 4. The computational results indicate that the budget of robustness (for both symmetric and asymmetric uncertainty sets) plays an important role for the performance of robust investment strategies in terms of the realized final expected wealth. The budget of robustness associated with the uncertain coefficients in the balance constraints has higher impact on the final expected wealth than the budget of robustness for the objective function. This may be because of the cumulative effect of multiple time periods. Note that there is a balance constraint at each time period whereas the objective function involves only the terminal date. Finally, we observe that the setting of (1, 0.1, 0.1) produces the highest expected wealth under any market regime regardless the choice of model. This may be due to the high impact of the budget of robustness (the first parameter) related to the original objective function on the final wealth. Similarly, when the budget of robustness of the asymmetric uncertainty set is fixed at (1, 1, 1), it leads to almost the same expected wealth for any market regime for the GIC and ELN types of ALM models.