Fig4.3 - five models

With this exercise VAs with GMAB and GMDB riders are priced by means of the static approach. The caracteristics of the contracts are the following:

• Contract expires at T = 15 years
• Age of the policyholder at contract inception in 50
• Both GMAB and GMDB pay the minimum amount as the roll-up of premium.
• The roll-up rates considered are 0%, 1% and 2%
• The fee is state-dependent and the base fee varies from 1% to 13%
• The state-dependent barrier is the GMAB minimum amount at T = 15

With regards to the simulation:

• The number of Monte Carlo simulations will be 20000.
• The Weibull function was fitted to the 2015 Italian life table and the estimated parameters are c1=88.14778, c2=10.002.

##          [,1]     [,2]     [,3]     [,4]     [,5]      [,6]      [,7]
## [1,] 109.0331 106.7115 104.5057 102.3204 100.0200  98.15608  96.27879
## [2,] 112.5827 109.4437 106.3043 103.4071 100.5693  97.90123  95.44513
## [3,] 118.3088 114.5719 111.0669 107.8782 104.9494 102.46797 100.04972
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 94.42083 92.68352 91.15798 89.78018 88.36728 87.13642
## [2,] 93.29474 91.24432 89.53735 87.88242 86.37283 85.04985
## [3,] 97.98632 96.23519 94.46933 93.29602 92.17635 91.12254

##          [,1]     [,2]     [,3]     [,4]      [,5]      [,6]     [,7]
## [1,] 108.4003 106.1043 103.9823 101.8649  99.84876  97.81884 95.78994
## [2,] 112.2402 109.0100 105.7823 102.8994 100.07052  97.24505 94.98833
## [3,] 117.8169 114.3154 110.4845 107.4115 104.43148 101.67193 99.40086
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 94.12205 92.30482 90.93897 89.41170 88.14289 86.84361
## [2,] 92.69480 90.65136 88.77794 87.03505 85.51238 84.24915
## [3,] 97.25266 95.41122 93.86882 92.56894 91.44949 90.62884

##          [,1]     [,2]     [,3]     [,4]      [,5]     [,6]     [,7]
## [1,] 107.8159 105.9380 103.8989 101.8914  99.91957 97.87714 96.09573
## [2,] 110.8759 107.7081 104.1709 100.9218  97.48535 94.29296 91.29426
## [3,] 115.8077 111.8099 107.7426 103.8342 100.13544 97.07129 94.35603
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 94.13570 92.48013 90.87205 89.35481 88.02104 86.81642
## [2,] 88.46097 86.07018 83.90019 81.86089 80.27734 79.15281
## [3,] 92.05136 90.29382 89.00157 88.09637 87.48992 87.04829

##          [,1]     [,2]     [,3]     [,4]      [,5]     [,6]     [,7]
## [1,] 108.9055 106.8613 104.8565 102.7552 100.70106 98.64793 96.63717
## [2,] 111.7138 108.3166 104.9362 101.4439  98.01645 94.59565 91.32479
## [3,] 116.4961 112.2047 107.9947 104.0252 100.38695 97.05166 94.25762
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 94.66540 92.89719 91.25185 89.71980 88.38492 87.25577
## [2,] 88.51014 85.89333 83.69727 81.72266 80.20961 78.98888
## [3,] 91.97383 90.24547 88.93893 88.02258 87.45640 87.12125

##          [,1]     [,2]      [,3]      [,4]     [,5]     [,6]     [,7]
## [1,] 106.0002 104.4575 102.46086 100.51473 98.52500 96.39945 94.18568
## [2,] 106.9303 103.8142 100.58849  96.80194 93.19097 89.44514 85.74928
## [3,] 108.7993 104.5372  99.99256  95.58238 91.01073 86.79667 82.61765
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 92.00661 90.13770 88.08915 86.18861 84.25908 82.67375
## [2,] 82.23909 78.74415 75.34210 72.44692 69.59411 67.28027
## [3,] 79.01466 75.73498 73.03827 70.74255 69.05277 67.71764