Fig 4.6 - all models

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

Contract expires at T = 15 years
Age of the policyholder at contract inception is 50
Both GMAB and GMDB pay the minimum amount as the roll-up of premium.
The roll-up rates is 2%
The fee is state-dependent and the base fee is 2%
The initial state-dependent barrier is the GMAB minimum amount at T = 15 indicated with G
The surrender penalty is 2% constant.

We are going to progressively increase the barrier with the formula (1 + k) * G where k varies from 0 to 1 with step 0.1

With regards to the simulation:

The number of Monte Carlo simulations will be 20000.

In the mixed approach, the regression is done only when the the account value is below the barrier

Model 4, roll-up 0% and fee 3% (mixed only)

##  [1] 104.41578 103.33255 102.25670 101.14002 100.47367  99.43514  98.97259
##  [8]  98.02593  97.78227  97.85582  97.86386  97.86896  97.80221  97.93989
## [15]  97.86584  97.83267  97.77649  97.80158  97.80797  97.83385  97.83017
## [22]  97.78651  97.81713  97.79898  97.79917  97.81982  97.83006  97.79985
## [29]  97.77974  97.80180  97.78678  97.79421  97.81529  97.78396  97.78598
## [36]  97.79204  97.78393  97.79808  97.81361  97.79467  97.81807  97.79581
## [43]  97.81244  97.81403  97.81020  97.80571  97.79934  97.78255  97.79908
## [50]  97.80474  97.79066

In the graph the dashed line corresponds to the constant fee case:

## mixed  97.80181

Model 1

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 114.6489 113.3524 112.3636 111.3350 110.5608 109.8246 109.3656
## [2,] 114.5838 113.0716 111.8815 111.0956 110.6633 110.1245 109.7374
##          [,8]     [,9]    [,10]   [,11]
## [1,] 108.8173 108.4078 108.0234 107.631
## [2,] 109.5399 109.6219 109.6404 109.570

In the graph the dashed line corresponds to the constant fee case:

## static  105.2805

## mixed  108.4422

Model 2

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 114.1988 112.7312 111.7942 110.7819 110.0073 109.4483 108.7309
## [2,] 114.1794 112.6909 111.3799 110.9317 110.6925 110.0927 109.9193
##          [,8]     [,9]    [,10]    [,11]
## [1,] 108.2584 107.8571 107.4687 107.1617
## [2,] 109.6623 109.4891 109.3646 109.4578

In the graph the dashed line corresponds to the constant fee case:

## static  105.0006

## mixed  109.3205

Model 3

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 111.7296 110.0501 108.6678 107.4463 106.6111 105.9833 105.2367
## [2,] 111.6435 109.4828 108.0673 107.0011 106.4370 106.2701 106.2157
##          [,8]     [,9]    [,10]    [,11]
## [1,] 104.7215 104.3782 104.0880 103.8476
## [2,] 106.1679 106.1109 105.9641 106.0905

In the graph the dashed line corresponds to the constant fee case:

## static  102.6379

## mixed  106.198

Model 4

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 112.2442 110.5033 109.2431 107.9601 107.0483 106.3036 105.7064
## [2,] 112.1962 110.1942 108.4788 107.5856 106.8916 106.7332 106.5082
##          [,8]     [,9]    [,10]    [,11]
## [1,] 105.2169 104.7711 104.4526 104.2258
## [2,] 106.5405 106.5856 106.5561 106.4843

In the graph the dashed line corresponds to the constant fee case:

## static  103.0778

## mixed  106.4878

Model 4 with fee 4%

##          [,1]     [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
## [1,] 104.0229 101.3585  99.33301  97.79744  96.72411  95.82361  95.21978
## [2,] 103.4059 101.6258 101.37634 101.28026 101.32531 101.30899 101.31645
##           [,8]      [,9]     [,10]    [,11]
## [1,]  94.81371  94.40213  94.14299  93.9252
## [2,] 101.30133 101.29516 101.30033 101.2907

In the graph the dashed line corresponds to the constant fee case:

## static  93.20478

## mixed  101.3039

Model 4 with fee 3%

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]      [,7]
## [1,] 108.1835 105.7843 103.8828 102.4613 101.3453 100.4824  99.80501
## [2,] 107.6903 104.9534 103.8912 103.4924 103.3356 103.3505 103.31457
##           [,8]     [,9]     [,10]     [,11]
## [1,]  99.31092  98.9407  98.60316  98.34672
## [2,] 103.33278 103.3532 103.34272 103.38613

In the graph the dashed line corresponds to the constant fee case:

## static  97.30676

## mixed  103.336

Model 5

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 104.6475 102.8488 101.3109 99.85233 98.75809 97.85550 97.11493
## [2,] 104.4950 102.4374 100.8796 99.94091 99.55432 99.47274 99.20300
##          [,8]     [,9]    [,10]    [,11]
## [1,] 96.36115 95.73888 95.21819 94.82007
## [2,] 99.27537 99.26657 99.20667 99.31455

In the graph the dashed line corresponds to the constant fee case:

## static  91.85526

## mixed  99.30101