Fig 4.6 - four 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 1%
• 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.
• The financial and mortality processes are given by model 4 ( BMOP 2011)

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

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]      [,7]
## [1,] 107.6979 105.5666 103.8501 102.3045 101.1632 100.1059  99.28232
## [2,] 107.7991 105.0391 103.0704 101.8834 101.5592 101.2460 101.08661
##           [,8]      [,9]    [,10]     [,11]
## [1,]  98.52409  97.92471  97.4583  97.08107
## [2,] 101.21941 101.10002 101.1034 101.10770

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

## static  94.98361

## mixed  101.19

With model 3:

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]      [,7]
## [1,] 107.7354 105.6436 103.7885 102.3380 101.0007 100.1527  99.25351
## [2,] 107.8050 105.1122 103.0798 101.6538 101.5144 101.1466 101.13712
##          [,8]      [,9]     [,10]     [,11]
## [1,]  98.4787  97.99344  97.47482  97.12994
## [2,] 101.1312 101.05847 101.14171 101.16640

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

## static  95.0865

## mixed  101.1506

With model 2:

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 108.4823 106.9421 105.6307 104.4379 103.2962 102.5979 101.7247
## [2,] 108.3043 106.5160 105.3132 104.7083 103.9883 103.7708 103.7000
##          [,8]     [,9]    [,10]     [,11]
## [1,] 101.1649 100.6926 100.0508  99.62314
## [2,] 103.2591 103.4065 103.2751 103.32564

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

## static  96.56004

## mixed  103.2631

With model 1:

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 108.5250 107.0765 105.6672 104.5924 103.6225 102.7393 101.9616
## [2,] 108.6778 106.7090 105.1807 104.5213 103.6889 103.4193 103.2005
##          [,8]     [,9]    [,10]     [,11]
## [1,] 101.3924 100.9002 100.4529  99.83847
## [2,] 103.1891 103.0931 103.0303 102.90388

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

## static  96.58692

## mixed  102.5686