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Optimal capital allocation for individual risk model using a Mean-Variance Principle

  • *Corresponding author: Yiying Zhang, Email: zhangyy3@sustech.edu.cn, Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China

    *Corresponding author: Yiying Zhang, Email: zhangyy3@sustech.edu.cn, Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China 
Abstract Full Text(HTML) Figure(3) / Table(14) Related Papers Cited by
  • In this article, we study the problem of optimal capital allocation for the individual risk model using the Mean-Variance principle with quadratic loss function when a total amount of initial capital is granted for allocation. The determination of the optimal capital allocation strategy proceeds in two phases. First, explicit formulas for the optimal allocations are presented based on the assumption that the claim occurrence indicators are independent of the claim severities when the allocated total capital is fixed. Second, an approximating algorithm is proposed to find out the optimal value of capital used for allocation through minimizing the mean-variance loss function. As a result, the exact allocation policy can be provided by following the first phase again. Numerical examples and applications are provided to illustrate the main results and some discussions on the effect of neglecting the dependence between the claim occurrence indicators and claim severities on the optimal allocations are also given to shed light on future research.

    Mathematics Subject Classification: Primary: 60E15, 62G30; Secondary: 62D05.

    Citation:

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  • Figure 1.  Sensitivity analysis on the percentages of optimal capital allocation under different $ \beta $ and $ \mathit{\boldsymbol{\nu_{1}}} $

    Figure 2.  Sensitivity analysis on the percentages of optimal capital allocation under different $ \beta $ and $ \mathit{\boldsymbol{\nu_{2}}} $

    Figure 3.  Sensitivity analysis on the percentages of optimal capital allocation under different $ \beta $ and $ \mathit{\boldsymbol{\nu_{3}}} $

    Table 1.  Optimal capital allocations under $ {\mathit{\boldsymbol{\nu}}}_{1} $ and different values of $ \beta $

    $ \beta=0 $ $ \beta=0.001 $ $ \beta=0.01 $ $ \beta=0.05 $ $ \beta=0.1 $ $ \beta=0.4 $ $ \beta=0.7 $ $ \beta=2 $
    $ a^{\ast} $ 33.40 51.80 58.85 59.80 59.90 60.00 60.00 60.00
    $ a_{1}^{\ast} $ 15.000 25.788 29.282 29.729 29.778 29.824 29.825 29.826
    44.91% 49.78% 49.76% 49.71% 49.71% 49.71% 49.71% 49.71%
    $ a_{2}^{\ast} $ 10.600 16.119 19.013 19.428 19.474 19.517 19.518 19.519
    31.74% 31.12% 32.31% 32.49% 32.51% 32.53% 32.53% 32.53%
    $ a_{3}^{\ast} $ 7.800 9.893 10.554 10.642 10.648 10.658 10.657 10.655
    23.35% 19.10% 17.93% 17.80% 17.78% 17.76% 17.76% 17.76%
     | Show Table
    DownLoad: CSV

    Table 2.  Optimal capital allocations under $ {\mathit{\boldsymbol{\nu}}}_{2} $ and different values of $ \beta $

    $ \beta=0 $ $ \beta=0.001 $ $ \beta=0.01 $ $ \beta=0.05 $ $ \beta=0.1 $ $ \beta=0.4 $ $ \beta=0.7 $ $ \beta=2 $
    $ a^{\ast} $ 33.40 53.55 58.50 59.05 59.10 59.15 59.15 59.15
    $ a_{1}^{\ast} $ 15.000 27.086 28.817 29.008 29.032 29.050 29.053 29.055
    44.91% 50.58% 49.26% 49.12% 49.12% 49.11% 49.12% 49.12%
    $ a_{2}^{\ast} $ 10.600 18.005 21.793 22.304 22.368 22.418 22.424 22.430
    31.74% 33.62% 37.25% 37.77% 37.85% 37.90% 37.91% 37.92%
    $ a_{3}^{\ast} $ 7.800 8.459 7.890 7.738 7.700 7.681 7.673 7.665
    23.35% 15.80% 13.49% 13.10% 13.03% 12.99% 12.97% 12.97%
     | Show Table
    DownLoad: CSV

    Table 3.  Optimal capital allocations under $ {\mathit{\boldsymbol{\nu}}}_{3} $ and different values of $ \beta $

    $ \beta=0 $ $ \beta=0.001 $ $ \beta=0.01 $ $ \beta=0.05 $ $ \beta=0.1 $ $ \beta=0.4 $ $ \beta=0.7 $ $ \beta=2 $
    $ a^{\ast} $ 33.40 53.35 58.85 59.70 59.80 59.90 59.90 59.95
    $ a_{1}^{\ast} $ 15.000 26.378 28.942 29.238 29.274 29.305 29.308 29.315
    44.91% 50.39% 49.18% 48.97% 48.95% 48.92% 48.92% 48.90%
    $ a_{2}^{\ast} $ 10.600 16.668 20.165 20.659 20.720 20.775 20.778 20.797
    31.74% 31.84% 34.26% 34.60% 34.65% 34.68% 34.69% 34.69%
    $ a_{3}^{\ast} $ 7.800 9.304 9.743 9.804 9.805 9.820 9.814 9.837
    23.35% 17.77% 16.56% 16.42% 16.40% 16.39% 16.38% 16.41%
     | Show Table
    DownLoad: CSV

    Table 4.  Optimal capital allocations with different occurrence probabilities $ \mathit{\boldsymbol{p}} $

    $ a^{\ast} $ $ a_{1}^{\ast} $ $ a_{2}^{\ast} $ $ a_{3}^{\ast} $ $ a_{4}^{\ast} $ $ a_{5}^{\ast} $ $ a_{6}^{\ast} $ $ a_{7}^{\ast} $ $ a_{8}^{\ast} $ $ a_{9}^{\ast} $ $ a_{10}^{\ast} $
    $ {\mathit{\boldsymbol{p}}}_{1} $ 80.75 0.113 3.230 40.094 0.250 35.545 2.285 0.226 -0.540 -0.471 0.019
    $ 0.14\% $ $ 4.00\% $ $ 49.65\% $ $ 0.31\% $ $ 44.02\% $ $ 2.83\% $ $ 0.28\% $ $ -0.67\% $ $ -0.58\% $ $ 0.02\% $
    $ {\mathit{\boldsymbol{p}}}_{2} $ 76.50 0.053 0.473 39.605 0.273 34.590 0.436 0.134 -0.661 1.172 0.426
    $ 0.07\% $ $ 0.62\% $ $ 51.77\% $ $ 0.36\% $ $ 45.21\% $ $ 0.57\% $ $ 0.17\% $ $ -0.86\% $ $ 1.53\% $ $ 0.56\% $
    $ {\mathit{\boldsymbol{p}}}_{3} $ 81.75 0.189 0.954 48.726 0.095 31.451 0.272 0.095 -0.658 0.320 0.304
    $ 0.23\% $ $ 1.17\% $ $ 59.60\% $ $ 0.12\% $ $ 38.47\% $ $ 0.33\% $ $ 0.12\% $ $ -0.80\% $ $ 0.39\% $ $ 0.37\% $
    $ {\mathit{\boldsymbol{p}}}_{4} $ 77.50 0.111 0.338 38.011 0.100 33.364 0.257 0.092 -0.205 2.737 2.694
    $ 0.14\% $ $ 0.44\% $ $ 49.05\% $ $ 0.13\% $ $ 43.05\% $ $ 0.33\% $ $ 0.12\% $ $ -0.26\% $ $ 3.53\% $ $ 3.48\% $
    $ {\mathit{\boldsymbol{p}}}_{5} $ 90.75 0.229 0.736 48.046 0.234 40.364 1.133 0.038 0.293 -1.167 0.843
    $ 0.25\% $ $ 0.81\% $ $ 52.94\% $ $ 0.26\% $ $ 44.48\% $ $ 1.25\% $ $ 0.04\% $ $ 0.32\% $ $ -1.29\% $ $ 0.93\% $
    $ {\mathit{\boldsymbol{p}}}_{6} $ 160.25 25.668 38.528 16.063 12.617 5.937 25.102 14.401 5.806 5.892 10.235
    $ 16.02\% $ $ 24.04\% $ $ 10.02\% $ $ 7.87\% $ $ 3.71\% $ $ 15.66\% $ $ 8.99\% $ $ 3.62\% $ $ 3.68\% $ $ 6.39\% $
     | Show Table
    DownLoad: CSV

    Table 5.  Optimal capital allocations with different occurrence probabilities $ \mathit{\boldsymbol{\nu}} $

    $ a^{\ast} $ $ a_{1}^{\ast} $ $ a_{2}^{\ast} $ $ a_{3}^{\ast} $ $ a_{4}^{\ast} $ $ a_{5}^{\ast} $ $ a_{6}^{\ast} $ $ a_{7}^{\ast} $ $ a_{8}^{\ast} $ $ a_{9}^{\ast} $ $ a_{10}^{\ast} $
    $ {\mathit{\boldsymbol{\nu}}}_{1} $ 80.00 0.080 1.273 17.308 3.211 53.798 0.188 0.168 0.252 3.069 0.654
    $ 0.10\% $ $ 1.59\% $ $ 21.63\% $ $ 4.01\% $ $ 67.25\% $ $ 0.23\% $ $ 0.21\% $ $ 0.31\% $ $ 3.84\% $ $ 0.82\% $
    $ {\mathit{\boldsymbol{\nu}}}_{2} $ 91.00 0.289 1.307 11.946 15.503 42.413 1.073 0.176 -0.419 17.376 1.336
    $ 0.32\% $ $ 1.44\% $ $ 13.12\% $ $ 17.04\% $ $ 46.61\% $ $ 1.18\% $ $ 0.19\% $ $ -0.46\% $ $ 19.09\% $ $ 1.47\% $
    $ {\mathit{\boldsymbol{\nu}}}_{3} $ 72.75 0.285 1.261 12.005 4.500 67.447 1.354 3.403 -5.399 -8.745 -3.360
    $ 0.39\% $ $ 1.73\% $ $ 16.50\% $ $ 6.19\% $ $ 92.71\% $ $ 1.86\% $ $ 4.68\% $ $ -7.42\% $ $ -12.02\% $ $ -4.62\% $
    $ {\mathit{\boldsymbol{\nu}}}_{4} $ 88.75 0.525 1.476 7.558 3.347 72.916 0.214 0.257 0.021 1.447 0.989
    $ 0.59\% $ $ 1.66\% $ $ 8.52\% $ $ 3.77\% $ $ 82.16\% $ $ 0.24\% $ $ 0.29\% $ $ 0.02\% $ $ 1.63\% $ $ 1.11\% $
    $ {\mathit{\boldsymbol{\nu}}}_{5} $ 104.00 0.075 1.442 17.826 0.012 65.942 -0.893 0.506 -0.987 19.014 1.063
    $ 0.07\% $ $ 1.39\% $ $ 17.14\% $ $ 0.01\% $ $ 63.41\% $ $ -0.86\% $ $ 0.49\% $ $ -0.95\% $ $ 18.28\% $ $ 1.02\% $
    $ {\mathit{\boldsymbol{\nu}}}_{6} $ 72.00 0.159 1.309 29.413 1.380 40.346 0.313 0.589 -0.890 -1.878 1.259
    $ 0.22\% $ $ 1.82\% $ $ 40.85\% $ $ 1.92\% $ $ 56.04\% $ $ 0.44\% $ $ 0.82\% $ $ -1.24\% $ $ -2.61\% $ $ 1.75\% $
     | Show Table
    DownLoad: CSV

    Table 6.  Optimal capital allocations with normal distributions under $ {\mathit{\boldsymbol{p}}}_{1} $ and $ {\mathit{\boldsymbol{\nu}}}_{1} $

    $ a^{\ast} $ $ a_{1}^{\ast} $ $ a_{2}^{\ast} $ $ a_{3}^{\ast} $ $ a_{4}^{\ast} $ $ a_{5}^{\ast} $
    Independence 2.136 0.151 0.233 0.406 0.612 0.735
    $ 7.04\% $ $ 10.93\% $ $ 19.00\% $ $ 28.63\% $ $ 34.39\% $
    Positive dependence 5.312 1.103 1.014 1.280 0.981 0.934
    $ 20.76\% $ $ 19.08\% $ $ 24.10\% $ $ 18.47\% $ $ 17.59\% $
    Negative dependence 0.632 -0.196 -0.129 0.136 0.331 0.490
    $ -30.97\% $ $ -20.36\% $ $ 21.47\% $ $ 52.31\% $ $ 77.54\% $
     | Show Table
    DownLoad: CSV

    Table 7.  Optimal capital allocations with normal distributions under $ {\mathit{\boldsymbol{p}}}_{2} $ and $ {\mathit{\boldsymbol{\nu}}}_{2} $

    $ a^{\ast} $ $ a_{1}^{\ast} $ $ a_{2}^{\ast} $ $ a_{3}^{\ast} $ $ a_{4}^{\ast} $ $ a_{5}^{\ast} $
    Independence 1.832 0.683 0.562 0.357 0.170 0.061
    $ 37.28\% $ $ 30.67\% $ $ 19.46\% $ $ 9.26\% $ $ 3.33\% $
    Positive dependence 5.152 1.219 1.115 1.274 0.844 0.699
    $ 23.67\% $ $ 21.65\% $ $ 24.74\% $ $ 16.38\% $ $ 13.57\% $
    Positive dependence 0.704 0.470 0.323 0.135 -0.100 -0.124
    $ 66.75\% $ $ 45.88\% $ $ 19.25\% $ $ -14.19\% $ $ -17.68\% $
     | Show Table
    DownLoad: CSV

    Table 8.  Optimal capital allocations with normal distributions under $ {\boldsymbol{p}}_{3} $ and $ {\boldsymbol{\nu}}_{3} $

    $ a^{\ast} $ $ a_{1}^{\ast} $ $ a_{2}^{\ast} $ $ a_{3}^{\ast} $ $ a_{4}^{\ast} $ $ a_{5}^{\ast} $
    Independence 2.064 0.431 0.190 0.518 0.253 0.672
    $ 20.88\% $ $ 9.18\% $ $ 25.12\% $ $ 12.27\% $ $ 32.54\% $
    Positive dependence 5.216 1.055 0.835 1.344 0.773 1.209
    $ 20.22\% $ $ 16.00\% $ $ 25.76\% $ $ 14.82\% $ $ 23.18\% $
    Negative dependence 1.104 0.181 -0.035 0.420 0.068 0.469
    $ 16.36\% $ $ -3.15\% $ $ 38.03\% $ $ 6.24\% $ $ 42.51\% $
     | Show Table
    DownLoad: CSV

    Table 9.  Optimal capital allocations with gamma distributions under $ {\mathit{\boldsymbol{p}}}_{1} $ and $ {\mathit{\boldsymbol{\nu}}}_{1} $

    $ a^{\ast} $ $ a_{1}^{\ast} $ $ a_{2}^{\ast} $ $ a_{3}^{\ast} $ $ a_{4}^{\ast} $ $ a_{5}^{\ast} $
    Independence 3.208 0.217 0.337 0.608 0.934 1.112
    $ 6.77\% $ $ 10.50\% $ $ 18.95\% $ $ 29.10\% $ $ 34.68\% $
    Positive dependence 7.024 1.562 1.409 1.642 1.262 1.148
    $ 22.24\% $ $ 20.07\% $ $ 23.38\% $ $ 17.97\% $ $ 16.35\% $
    Negative dependence 1.208 0.042 0.082 0.119 0.404 0.562
    $ 3.46\% $ $ 6.74\% $ $ 9.87\% $ $ 33.43\% $ $ 46.49\% $
     | Show Table
    DownLoad: CSV

    Table 10.  Optimal capital allocations with gamma distributions under $ {\mathit{\boldsymbol{p}}}_{2} $ and $ {\mathit{\boldsymbol{\nu}}}_{2} $

    $ a^{\ast} $ $ a_{1}^{\ast} $ $ a_{2}^{\ast} $ $ a_{3}^{\ast} $ $ a_{4}^{\ast} $ $ a_{5}^{\ast} $
    Independence 2.744 1.199 0.915 0.505 0.159 -0.034
    $ 43.67\% $ $ 33.34\% $ $ 18.40\% $ $ 5.83\% $ $ -1.24\% $
    Positive dependence 6.976 1.695 1.523 1.633 1.140 0.984
    $ 24.29\% $ $ 21.83\% $ $ 23.41\% $ $ 16.35\% $ $ 14.10\% $
    Negative dependence 1.256 0.566 0.396 0.118 0.098 0.079
    $ 45.03\% $ $ 31.50\% $ $ 9.37\% $ $ 7.84\% $ $ 6.26\% $
     | Show Table
    DownLoad: CSV

    Table 11.  Optimal capital allocations with gamma distributions under $ {\boldsymbol{p}}_{3} $ and $ {\boldsymbol{\nu}}_{3} $

    $ a^{\ast} $ $ a_{1}^{\ast} $ $ a_{2}^{\ast} $ $ a_{3}^{\ast} $ $ a_{4}^{\ast} $ $ a_{5}^{\ast} $
    Independence 3.144 0.684 0.200 0.799 0.279 1.181
    $ 21.75\% $ $ 6.37\% $ $ 25.43\% $ $ 8.88\% $ $ 37.57\% $
    Positive dependence 6.984 1.469 1.128 1.692 1.007 1.688
    $ 21.03\% $ $ 16.16\% $ $ 24.23\% $ $ 14.42\% $ $ 24.17\% $
    Negative dependence 1.568 0.285 0.156 0.330 0.229 0.568
    $ 18.19\% $ $ 9.94\% $ $ 21.02\% $ $ 14.60\% $ $ 36.25\% $
     | Show Table
    DownLoad: CSV

    Table 12.  Optimal capital allocations under $ {\mathit{\boldsymbol{\nu}}}_{1} $ and different values of $ \beta $

    $ \beta=0.001 $ $ \beta=0.01 $ $ \beta=0.05 $ $ \beta=0.1 $ $ \beta=0.4 $ $ \beta=0.7 $ $ \beta=2 $
    $ a^{\ast} $ 42.05 45.25 45.70 45.70 45.75 45.75 45.75
    $ a_{1}^{\ast} $ 19.325 20.925 21.150 21.150 21.175 21.175 21.175
    45.96% 46.24% 46.28% 46.28% 46.28% 46.28% 46.28%
    $ a_{2}^{\ast} $ 13.195 14.155 14.290 14.290 14.305 14.305 14.305
    31.38% 31.28% 31.27% 31.27% 31.27% 31.27% 31.27%
    $ a_{3}^{\ast} $ 9.530 10.170 10.260 10.260 10.270 10.270 10.270
    22.66% 22.48% 22.45% 22.45% 22.45% 22.45% 22.45%
     | Show Table
    DownLoad: CSV

    Table 13.  Optimal capital allocations under $ {\mathit{\boldsymbol{\nu}}}_{2} $ and different values of $ \beta $

    $ \beta=0.001 $ $ \beta=0.01 $ $ \beta=0.05 $ $ \beta=0.1 $ $ \beta=0.4 $ $ \beta=0.7 $ $ \beta=2 $
    $ a^{\ast} $ 42.05 45.25 45.65 45.70 45.75 45.75 45.75
    $ a_{1}^{\ast} $ 16.730 17.370 17.450 17.460 17.470 17.470 17.470
    39.79% 38.39% 38.23% 38.21% 38.19% 38.19% 38.18%
    $ a_{2}^{\ast} $ 13.195 14.155 14.275 14.290 14.305 14.305 14.305
    31.38% 31.28% 31.27% 31.27% 31.27% 31.27% 31.27%
    $ a_{3}^{\ast} $ 12.125 13.725 13.925 13.95 13.975 13.975 13.975
    28.83% 30.33% 30.50% 30.53% 30.55% 30.55% 30.55%
     | Show Table
    DownLoad: CSV

    Table 14.  Optimal capital allocations under $ {\mathit{\boldsymbol{\nu}}}_{3} $ and different values of $ \beta $

    $ \beta=0.001 $ $ \beta=0.01 $ $ \beta=0.05 $ $ \beta=0.1 $ $ \beta=0.4 $ $ \beta=0.7 $ $ \beta=2 $
    $ a^{\ast} $ 42.05 45.25 45.65 45.70 45.75 45.75 45.75
    $ a_{1}^{\ast} $ 17.883 18.950 19.083 19.100 19.117 19.117 19.117
    42.53% 41.88% 41.80% 41.79% 41.79% 41.79% 41.78%
    $ a_{2}^{\ast} $ 13.483 14.550 14.683 14.700 14.717 14.717 14.717
    32.06% 32.15% 32.17% 32.17% 31.17% 31.17% 31.17%
    $ a_{3}^{\ast} $ 10.683 11.750 11.883 11.900 11.917 11.917 11.917
    25.41% 25.97% 26.31% 26.40% 26.05% 26.05% 26.05%
     | Show Table
    DownLoad: CSV
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