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					                                                         L. J. Halliwell, LLC



                                        Example 1: Independent Risk (Coin Toss for $100K)

Risk-Aversion a :          1.000E-05
Amount q                      100.0%
Price q (q)                    26,894

                    Prob          W              X          W +qX           u (W +qX )          Y   Adj Prob              E *[X ]       Var *[X ]
State 1             50%    1,000,000              0      1,000,000          4.540E-05       1.4621     73.1%                    0     7.233E+08
State 2             50%    1,000,000        100,000      1,100,000          1.670E-05       0.5379     26.9%             100,000      5.344E+09
State 3
State 4
State 5
State 6
State 7
State 8
State 9
State 10
Mean                100%   1,000,000         50,000      1,050,000              3.105E-05   1.0000   100.0%                26,894     1.966E+09

                    Cov           W             X
                     W     0.000E+00     0.000E+00
                      X    0.000E+00     2.500E+09




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                                                        L. J. Halliwell, LLC




                                    Example 1: Independent Risk (Coin Toss for $100K)

              120%


              100%

                                                                                        Actual Distribution
               80%
                                                                                        Normal Approximation

               60%


               40%


               20%
   Amount q




                0%
                      0   10,000   20,000      30,000           40,000         50,000   60,000          70,000           80,000
              -20%


              -40%


              -60%


              -80%


              -100%


              -120%
                                                                Price q




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                                                           Example 1A: Three Identical Agents with Diversification

            Agent        Rsk-Aver a              Price q       Amount q       Cov [W , X ]
            Al            1.000E-05              41,743          33.3%         0.00E+00
            Bob           1.000E-05              41,743          33.3%         0.00E+00
            Chuck         1.000E-05              41,743          33.3%         0.00E+00
            Total         3.333E-06        0.007267787           100.0%
                                              Minimized

            Three identical agents, each buys one-third.

            Al                  Prob                 W               X            W +qX           u (W +qX )          Y   Adj Prob     E *[X ]         Var *[X ]
            State 1             50%           1,000,000               0        1,000,000          4.540E-05       1.1651     58.3%           0       1.742E+09
            State 2             50%           1,000,000         100,000        1,033,333          3.253E-05       0.8349     41.7%    100,000        3.394E+09
            State 3
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean               100%           1,000,000          50,000        1,016,667              3.897E-05   1.0000   100.0%      41,743        2.432E+09

            Bob                 Prob                 W               X            W +qX           u (W +qX )          Y   Adj Prob     E *[X ]         Var *[X ]
            State 1             50%           1,000,000               0        1,000,000          4.540E-05       1.1651     58.3%           0       1.742E+09
            State 2             50%           1,000,000         100,000        1,033,333          3.253E-05       0.8349     41.7%    100,000        3.394E+09
            State 3
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean               100%           1,000,000          50,000        1,016,667              3.897E-05   1.0000   100.0%      41,743        2.432E+09

            Chuck               Prob                 W               X            W +qX           u (W +qX )          Y   Adj Prob     E *[X ]         Var *[X ]
            State 1             50%           1,000,000               0        1,000,000          4.540E-05       1.1651     58.3%           0       1.742E+09
            State 2             50%           1,000,000         100,000        1,033,333          3.253E-05       0.8349     41.7%    100,000        3.394E+09
            State 3
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean               100%           1,000,000          50,000        1,016,667              3.897E-05   1.0000   100.0%      41,743        2.432E+09



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                                                     Example 1B: Three Identical Agents with Intra-Group Risk Transfer

            Agent        Rsk-Aver a             Price q           Amount q         Cov [W , X ]
            Al            1.000E-05             41,743             -66.7%           2.50E+09
            Bob           1.000E-05             41,743              33.3%           0.00E+00
            Chuck         1.000E-05             41,743              33.3%           0.00E+00
            Total         3.333E-06        0.008636745                0.0%
                                              Minimized

            Al "owns" the risk (note Cov ), and sells some of it to Bob and Chuck. Same result as in Example 1A

            Al                  Prob                 W                   X              W +qX           u (W +qX )          Y   Adj Prob     E *[X ]         Var *[X ]
            State 1             50%             900,000                                900,000          1.234E-04       1.1651     58.3%           0       1.742E+09
            State 2             50%           1,000,000            100,000             933,333          8.843E-05       0.8349     41.7%    100,000        3.394E+09
            State 3
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean               100%             950,000             50,000             916,667              1.059E-04   1.0000   100.0%      41,743        2.432E+09

            Bob                 Prob                 W                  X               W +qX           u (W +qX )          Y   Adj Prob     E *[X ]         Var *[X ]
            State 1             50%           1,000,000                  0           1,000,000          4.540E-05       1.1651     58.3%           0       1.742E+09
            State 2             50%           1,000,000            100,000           1,033,333          3.253E-05       0.8349     41.7%    100,000        3.394E+09
            State 3
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean               100%           1,000,000             50,000           1,016,667              3.897E-05   1.0000   100.0%      41,743        2.432E+09

            Chuck               Prob                 W                  X               W +qX           u (W +qX )          Y   Adj Prob     E *[X ]         Var *[X ]
            State 1             50%           1,000,000                  0           1,000,000          4.540E-05       1.1651     58.3%           0       1.742E+09
            State 2             50%           1,000,000            100,000           1,033,333          3.253E-05       0.8349     41.7%    100,000        3.394E+09
            State 3
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean               100%           1,000,000             50,000           1,016,667              3.897E-05   1.0000   100.0%      41,743        2.432E+09



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                                                                                    L. J. Halliwell, LLC




                                                                Example 1C: Three Agents of Differing Opinions

            Agent        Rsk-Aver a             Price q          Amount q         Cov [W , X ]
            Al            1.000E-05             41,096             36.0%           0.00E+00
            Bob           1.000E-05             41,096             36.0%           0.00E+00
            Chuck         1.000E-05             41,096             28.0%           0.00E+00
            Total         3.333E-06       0.004783507              100.0%
                                             Minimized

            Chuck thinks the coin to be unfavorably biased. The price decreases, and Chuck buys less.

            Al                 Prob                 W                  X              W +qX            u (W +qX )          Y   Adj Prob     E *[X ]         Var *[X ]
            State 1            50%           1,000,000                  0          1,000,000           4.540E-05       1.1781     58.9%           0       1.689E+09
            State 2            50%           1,000,000            100,000          1,036,001           3.167E-05       0.8219     41.1%    100,000        3.470E+09
            State 3
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean              100%           1,000,000             50,000          1,018,001               3.854E-05   1.0000   100.0%      41,096        2.421E+09

            Bob                Prob                 W                  X              W +qX            u (W +qX )          Y   Adj Prob     E *[X ]         Var *[X ]
            State 1            50%           1,000,000                  0          1,000,000           4.540E-05       1.1781     58.9%           0       1.689E+09
            State 2            50%           1,000,000            100,000          1,036,001           3.167E-05       0.8219     41.1%    100,000        3.470E+09
            State 3
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean              100%           1,000,000             50,000          1,018,001               3.854E-05   1.0000   100.0%      41,096        2.421E+09

            Chuck              Prob                 W                  X              W +qX            u (W +qX )          Y   Adj Prob     E *[X ]         Var *[X ]
            State 1            52%           1,000,000                  0          1,000,000           4.540E-05       1.1328     58.9%           0       1.689E+09
            State 2            48%           1,000,000            100,000          1,027,997           3.431E-05       0.8562     41.1%    100,000        3.470E+09
            State 3
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean              100%           1,000,000             48,000          1,013,439               4.008E-05   1.0000   100.0%      41,096        2.421E+09



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                                                          L. J. Halliwell, LLC



                                Example 2: Insurance Against $1,000,000 Medical-Malpractice Lawsuit

Risk-Aversion a :           5.000E-06
Amount q                       100.0%
Price q (q)                     90,000

                    Prob           W              X          W +qX           u (W +qX )          Y   Adj Prob              E *[X ]       Var *[X ]
State 1             91%     5,000,000              0      5,000,000          1.389E-11       1.0000     91.0%                    0     8.100E+09
State 2              9%     4,000,000      1,000,000      5,000,000          1.389E-11       1.0000      9.0%           1,000,000      8.281E+11
State 3
State 4
State 5
State 6
State 7
State 8
State 9
State 10
Mean                100%    4,910,000         90,000      5,000,000              1.389E-11   1.0000   100.0%                90,000     8.190E+10

                    Cov            W              X
                     W      8.190E+10     -8.190E+10
                      X    -8.190E+10      8.190E+10




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                                   Example 2: Insurance Against $1,000,000 Medical-Malpractice Lawsuit

              180%


              160%
                                                                                                        Actual Distribution
                                                                                                        Normal Approximation
              140%


              120%


              100%
   Amount q




              80%


              60%


              40%


              20%


               0%
                     0   100,000   200,000   300,000   400,000        500,000           600,000   700,000    800,000          900,000   1,000,000
                                                                       Price q




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                                                                      L. J. Halliwell, LLC



                                              Example 2A: Transfer of Lawsuit Risk to a Small Mutual Insurer

 Agent           Rsk-Aver a              Price q       Amount q       Cov [W , X ]
 Dr. Al           5.000E-06             118,220          93.9%         -8.19E+10
 MM Mutual        2.000E-07             118,220         -93.9%          1.53E-05


 Total            1.923E-07        0.002051284             0.0%
                                      Minimized

 Insurers' Estimated Probability of Lawsuit                10%

 Dr. Al                 Prob                 W                X           W +qX              u (W +qX )        Y   Adj Prob              E *[X ]       Var *[X ]
 State 1                91%           5,000,000                0       5,000,000             1.389E-11     0.9690     88.2%                    0     1.398E+10
 State 2                 9%           4,000,000        1,000,000       4,939,152             1.883E-11     1.3136     11.8%           1,000,000      7.775E+11
 State 3
 State 4
 State 5
 State 6
 State 7
 State 8
 State 9
 State 10
 Mean                  100%           4,910,000          90,000        4,994,524             1.433E-11     1.0000   100.0%              118,220      1.042E+11

 MM Mutual              Prob                 W                X           W +qX              u (W +qX )        Y   Adj Prob              E *[X ]       Var *[X ]
 State 1                 9%          40,000,000                0      40,000,000             3.355E-04     4.6633     42.0%                    0     1.398E+10
 State 2                72%          50,000,000                0      50,000,000             4.540E-05     0.6311     45.4%                    0     1.398E+10
 State 3                 9%          60,000,000                0      60,000,000             6.144E-06     0.0854      0.8%                    0     1.398E+10
 State 4                 1%          40,000,000        1,000,000      39,060,848             4.048E-04     5.6269      5.6%           1,000,000      7.775E+11
 State 5                 8%          50,000,000        1,000,000      49,060,848             5.478E-05     0.7615      6.1%           1,000,000      7.775E+11
 State 6                 1%          60,000,000        1,000,000      59,060,848             7.414E-06     0.1031      0.1%           1,000,000      7.775E+11
 State 7
 State 8
 State 9
 State 10
 Mean                  100%          50,000,000         100,000       49,906,085             7.194E-05     1.0000   100.0%              118,220      1.042E+11




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                                                                       Example 3A: Reinsurance Pricing

            Agent           Rsk-Aver a               Price q      Amount q      Cov [W , X ]
            Reinsurer A      5.000E-09           -5,106,791         68.3%        0.00E+00
            Reinsurer B      5.000E-09           -5,106,791         31.7%        1.00E+14
            Reinsurer C      5.000E-09           -8,000,000          0.0%        0.00E+00
            Total             1.667E-09       0.031820128           100.0%
                                                 Minimized

            Pessimistic Reinsurer C stays out of the market.

            Reinsurer A            Prob                 W                X           W +qX             u (W +qX )        Y   Adj Prob          E *[X ]      Var *[X ]
            State 1                94%       1,000,000,000                0   1,000,000,000            6.738E-03     0.9847     92.6%                0    2.608E+13
            State 2                 4%       1,000,000,000      -50,000,000     965,852,707            7.992E-03     1.1680      4.7%     -50,000,000     2.015E+15
            State 3                 2%       1,000,000,000     -100,000,000     931,705,414            9.480E-03     1.3854      2.8%    -100,000,000     9.005E+15
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean                  100%       1,000,000,000       -4,000,000    997,268,217             6.843E-03     1.0000   100.0%       -5,106,791     3.678E+14

            Reinsurer B            Prob                 W                X           W +qX             u (W +qX )        Y   Adj Prob          E *[X ]      Var *[X ]
            State 1               23.5%      1,100,000,000                0   1,100,000,000            4.087E-03     0.5658     13.3%                0    2.608E+13
            State 2                  1%      1,100,000,000      -25,000,000   1,092,073,647            4.252E-03     0.5887      0.6%     -25,000,000     3.957E+14
            State 3                0.5%      1,100,000,000      -50,000,000   1,084,147,293            4.424E-03     0.6125      0.3%     -50,000,000     2.015E+15
            State 4                 47%      1,000,000,000                0   1,000,000,000            6.738E-03     0.9329     43.8%                0    2.608E+13
            State 5                  2%      1,000,000,000      -50,000,000     984,147,293            7.294E-03     1.0099      2.0%     -50,000,000     2.015E+15
            State 6                  1%      1,000,000,000     -100,000,000     968,294,586            7.895E-03     1.0932      1.1%    -100,000,000     9.005E+15
            State 7               23.5%        900,000,000                0     900,000,000            1.111E-02     1.5381     36.1%                0    2.608E+13
            State 8                  1%        900,000,000     -100,000,000     868,294,586            1.302E-02     1.8023      1.8%    -100,000,000     9.005E+15
            State 9                0.5%        900,000,000     -100,000,000     868,294,586            1.302E-02     1.8023      0.9%    -100,000,000     9.005E+15
            State 10
            Mean                  100%       1,000,000,000       -4,000,000    998,731,783             7.223E-03     1.0000   100.0%       -5,106,791     4.154E+14

            Reinsurer C            Prob                 W                X           W +qX             u (W +qX )        Y   Adj Prob          E *[X ]      Var *[X ]
            State 1                87%       1,000,000,000                0   1,000,000,000            6.738E-03     1.0000     87.0%                0    6.400E+13
            State 2                10%       1,000,000,000      -50,000,000   1,000,000,000            6.738E-03     1.0000     10.0%     -50,000,000     1.764E+15
            State 3                 3%       1,000,000,000     -100,000,000   1,000,000,000            6.738E-03     1.0000      3.0%    -100,000,000     8.464E+15
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean                  100%       1,000,000,000       -8,000,000   1,000,000,000            6.738E-03     1.0000   100.0%       -8,000,000     4.860E+14



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                                                                                 L. J. Halliwell, LLC




                                                              Example 3B: Reinsurance Pricing with a Short Position

            Agent           Rsk-Aver a              Price q        Amount q      Cov [W , X ]
            Reinsurer A      5.000E-09          -6,077,898          116.5%        0.00E+00
            Reinsurer B      5.000E-09          -6,077,896           74.5%        1.00E+14
            Reinsurer C      5.000E-09          -6,077,898          -90.9%        0.00E+00
            Total            1.667E-09        1.541155325            100.0%
                                                 Minimized

            Reinsurer C allowed to buy a negative share.

            Reinsurer A            Prob                 W                 X           W +qX             u (W +qX )        Y   Adj Prob          E *[X ]      Var *[X ]
            State 1                94%       1,000,000,000                 0   1,000,000,000            6.738E-03     0.9715     91.3%                0    3.694E+13
            State 2                 4%       1,000,000,000       -50,000,000     941,772,667            9.015E-03     1.2998      5.2%     -50,000,000     1.929E+15
            State 3                 2%       1,000,000,000      -100,000,000     883,545,333            1.206E-02     1.7391      3.5%    -100,000,000     8.821E+15
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean                  100%       1,000,000,000        -4,000,000     995,341,813            6.936E-03     1.0000   100.0%       -6,077,898     4.409E+14

            Reinsurer B            Prob                 W                 X           W +qX             u (W +qX )        Y   Adj Prob          E *[X ]      Var *[X ]
            State 1                24%       1,100,000,000                 0   1,100,000,000            4.087E-03     0.5591     13.1%                0    3.694E+13
            State 2                 1%       1,100,000,000       -25,000,000   1,081,386,517            4.485E-03     0.6137      0.6%     -25,000,000     3.580E+14
            State 3                 1%       1,100,000,000       -50,000,000   1,062,773,033            4.923E-03     0.6735      0.3%     -50,000,000     1.929E+15
            State 4                47%       1,000,000,000                 0   1,000,000,000            6.738E-03     0.9218     43.3%                0    3.694E+13
            State 5                 2%       1,000,000,000       -50,000,000     962,773,033            8.116E-03     1.1104      2.2%     -50,000,000     1.929E+15
            State 6                 1%       1,000,000,000      -100,000,000     925,546,066            9.777E-03     1.3376      1.3%    -100,000,000     8.821E+15
            State 7                24%         900,000,000                 0     900,000,000            1.111E-02     1.5199     35.7%                0    3.694E+13
            State 8                 1%         900,000,000      -100,000,000     825,546,066            1.612E-02     2.2054      2.2%    -100,000,000     8.821E+15
            State 9                 1%         900,000,000      -100,000,000     825,546,066            1.612E-02     2.2054      1.1%    -100,000,000     8.821E+15
            State 10
            Mean                  100%       1,000,000,000        -4,000,000     997,021,843            7.309E-03     1.0000   100.0%       -6,077,896     4.954E+14

            Reinsurer C            Prob                 W                 X           W +qX             u (W +qX )        Y   Adj Prob          E *[X ]      Var *[X ]
            State 1                87%       1,000,000,000                 0   1,000,000,000            6.738E-03     1.0323     89.8%                0    3.694E+13
            State 2                10%       1,000,000,000       -50,000,000   1,045,454,300            5.368E-03     0.8224      8.2%     -50,000,000     1.929E+15
            State 3                 3%       1,000,000,000      -100,000,000   1,090,908,600            4.277E-03     0.6552      2.0%    -100,000,000     8.821E+15
            State 4
            State 5
            State 6
            State 7
            State 8
            State 9
            State 10
            Mean                  100%       1,000,000,000        -8,000,000   1,007,272,688            6.527E-03     1.0000   100.0%       -6,077,898     3.652E+14



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                                                           L. J. Halliwell, LLC



                                             Example 4: Capital-Consumption Example

Risk-Aversion a :             5.000E-09
Amount q                         100.0%
Price q (q)                    -431,194

                    Prob             W1            X           W +qX          u (W +qX )          Y   Adj Prob              E *[X ]       Var *[X ]
State 1              20%   1,000,000,000            0   1,000,000,000         6.738E-03       0.9978    19.96%                    0     1.859E+11
State 2              15%   1,000,000,000     -100,000     999,900,000         6.741E-03       0.9983    14.98%            -100,000      1.097E+11
State 3              15%   1,000,000,000     -250,000     999,750,000         6.746E-03       0.9991    14.99%            -250,000      3.283E+10
State 4              15%   1,000,000,000     -350,000     999,650,000         6.750E-03       0.9996    14.99%            -350,000      6.592E+09
State 5              10%   1,000,000,000     -500,000     999,500,000         6.755E-03       1.0003    10.00%            -500,000      4.734E+09
State 6              10%   1,000,000,000     -750,000     999,250,000         6.763E-03       1.0016    10.02%            -750,000      1.016E+11
State 7              10%   1,000,000,000   -1,000,000     999,000,000         6.772E-03       1.0029    10.03%          -1,000,000      3.235E+11
State 8             2.5%   1,000,000,000   -1,500,000     998,500,000         6.789E-03       1.0054     2.51%          -1,500,000      1.142E+12
State 9             2.5%   1,000,000,000   -2,500,000     997,500,000         6.823E-03       1.0104     2.53%          -2,500,000      4.280E+12
State 10
Mean                100%   1,000,000,000    -430,000     999,570,000              6.752E-03   1.0000   100.00%            -431,194      2.394E+11

                    Cov              W            X                                                               Total Premium             606,194
                     W        0.000E+00    0.000E+00                                                              E [PV [Loss]]             430,000
                      X       0.000E+00    2.381E+11                                                              Profit Load                 1,194
                                                                                                                  Expenses                  175,000
                    Corr             W            X
                     W           #DIV/0!     #DIV/0!
                      X          #DIV/0!      100%




8/23/2011                                                        11 of 12                                     9e004790-4b84-4fbc-a3d5-88294d7e826f.xls
                                                               L. J. Halliwell, LLC



                                                     Alternative Solutions of Example 4

                                    Parameter    Premium      Desired            Actual     Abs(Diff)      Risk Load
            Up/Down (R2R)               8.000   1,067,683      8.000             8.000        0.000          462,683
            CapConsumption              2.000     739,000           0                0             0         134,000
            Wang Transform              0.750   1,009,099           0                0             0         404,099

            R2R              Prob    Premium    Expenses     PV[Loss]           Profit     Downside          Upside
            State 1           20%   1,067,683    175,000            0         892,683                       892,683
            State 2           15%   1,067,683    175,000      100,000         792,683                       792,683
            State 3           15%   1,067,683    175,000      250,000         642,683                       642,683
            State 4           15%   1,067,683    175,000      350,000         542,683                       542,683
            State 5           10%   1,067,683    175,000      500,000         392,683                       392,683
            State 6           10%   1,067,683    175,000      750,000         142,683                       142,683
            State 7           10%   1,067,683    175,000    1,000,000        -107,317       -107,317
            State 8          2.5%   1,067,683    175,000    1,500,000        -607,317       -607,317
            State 9          2.5%   1,067,683    175,000    2,500,000      -1,607,317     -1,607,317
            Mean             100%   1,067,683    175,000     430,000            462,683      -66,098        528,780

            CapCon           Prob    Premium    Expenses     PV[Loss]           Profit       AdjProfit
            State 1           20%     739,000    175,000            0         564,000        564,000
            State 2           15%     739,000    175,000      100,000         464,000        464,000
            State 3           15%     739,000    175,000      250,000         314,000        314,000
            State 4           15%     739,000    175,000      350,000         214,000        214,000
            State 5           10%     739,000    175,000      500,000          64,000         64,000
            State 6           10%     739,000    175,000      750,000        -186,000       -372,000
            State 7           10%     739,000    175,000    1,000,000        -436,000       -872,000
            State 8          2.5%     739,000    175,000    1,500,000        -936,000     -1,872,000
            State 9          2.5%     739,000    175,000    2,500,000      -1,936,000     -3,872,000
            Mean             100%     739,000    175,000     430,000            134,000             0

            Wang             Prob    Premium    Expenses     PV[Loss]             CDF      NormsInv         AdjCDF     AdjProb         Profit
            State 1           20%   1,009,099    175,000            0              20%       -0.842           5.6%       5.6%        834,099
            State 2           15%   1,009,099    175,000      100,000              35%       -0.385          12.8%       7.2%        734,099
            State 3           15%   1,009,099    175,000      250,000              50%        0.000          22.7%       9.9%        584,099
            State 4           15%   1,009,099    175,000      350,000              65%        0.385          35.8%      13.1%        484,099
            State 5           10%   1,009,099    175,000      500,000              75%        0.674          47.0%      11.2%        334,099
            State 6           10%   1,009,099    175,000      750,000              85%        1.036          61.3%      14.3%         84,099
            State 7           10%   1,009,099    175,000    1,000,000              95%        1.645          81.5%      20.2%       -165,901
            State 8          2.5%   1,009,099    175,000    1,500,000            97.5%        1.960          88.7%       7.2%       -665,901
            State 9          2.5%   1,009,099    175,000    2,500,000           100.0%          +     100.0%      11.3%     -1,665,901
            Mean             100%   1,009,099    175,000     430,000                                                   100.0%                0



8/23/2011                                                            12 of 12                                              9e004790-4b84-4fbc-a3d5-88294d7e826f.xls

				
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