Static and mixed approach with SD fees while increasing the barrier

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 = 5 years
Age of the policyholder at contract inception is 60
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 8%
The initial state-dependent barrier is the GMAB minimum amount at T = 5 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 2 with step 0.1

With regards to the simulation:

The number of Monte Carlo simulations will be 20000.
Constant interest rate r = 0.03
Underlying fund is a GBM with volatility 0.2
Intensity of mortality is given by the Weibull mortality function with c1=90.43, c2=10.36

In the mixed approach, the regression is done only when the GMAB guarantee is in the money

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 107.1008 104.6876 103.0690 101.9494 101.2275 100.6937 100.3046
## [2,] 107.1291 104.6970 103.0372 102.0242 101.3063 100.9868 100.8182
##          [,8]      [,9]     [,10]     [,11]     [,12]     [,13]     [,14]
## [1,] 100.0700  99.90808  99.76501  99.71036  99.64234  99.61792  99.60595
## [2,] 100.6909 100.59840 100.54105 100.46813 100.45476 100.40725 100.41703
##         [,15]     [,16]     [,17]     [,18]     [,19]    [,20]     [,21]
## [1,]  99.5772  99.53462  99.56056  99.55577  99.52864  99.5363  99.53028
## [2,] 100.4071 100.39872 100.38888 100.39857 100.39267 100.3918 100.38682
##          [,22]
## [1,]  99.53519
## [2,] 100.38959

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

We’re going to repeat it changing the GBM and the constant interest rate with the financial and mortality model published in BMOP2011. Again in the mixed approach, the regression is done only when the GMAB guarantee is in the money

##          [,1]     [,2]      [,3]      [,4]     [,5]     [,6]     [,7]
## [1,] 104.6409 100.9464  99.11352  98.24378 97.82551 97.65288 97.57401
## [2,] 104.3815 101.2347 100.45356 100.11094 99.97093 99.90849 99.88196
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]    [,14]
## [1,] 97.54018 97.52746 97.51179 97.52320 97.51674 97.52010 97.51517
## [2,] 99.86574 99.86427 99.86040 99.85787 99.85896 99.85743 99.85638
##         [,15]    [,16]    [,17]    [,18]    [,19]    [,20]    [,21]
## [1,] 97.51554 97.51865 97.50922 97.51783 97.51141 97.51098 97.51668
## [2,] 99.85931 99.85888 99.85852 99.85290 99.85640 99.85121 99.85817
##         [,22]
## [1,] 97.52306
## [2,] 99.85537

Eventually the financial and mortality models will be the ones published in BBM2010. In this case the age at contract inception is 40 and the fee is 5%. In addition, during the mixed approach the regression is done regardless the moneyness of the guarantee.

##          [,1]     [,2]     [,3]      [,4]      [,5]      [,6]      [,7]
## [1,] 105.1084 102.1917 100.0797  98.64447  97.66836  97.07092  96.69854
## [2,] 103.5041 100.8495 100.1164 100.02685 100.02807 100.03147 100.03419
##           [,8]      [,9]     [,10]     [,11]     [,12]     [,13]     [,14]
## [1,]  96.46175  96.32902  96.24521  96.19128  96.15433  96.13897  96.14049
## [2,] 100.04183 100.03675 100.03784 100.04908 100.04845 100.04335 100.04401
##          [,15]     [,16]     [,17]     [,18]     [,19]     [,20]     [,21]
## [1,]  96.12562  96.11131  96.12612  96.10919  96.12263  96.12023  96.12829
## [2,] 100.03908 100.03977 100.03950 100.04488 100.04523 100.05463 100.05487
##         [,22]
## [1,]  96.1227
## [2,] 100.0372