Static approach and SD fees

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 = 5 years
Age of the policyholder at contract inception in 60
Both GMAB and GMDB pay the minimum amount as the roll-up of premium.
The roll-up rates considered are 1%, 2% and 3%
The fee is state-dependent and the base fee varies from 6% to 18%
The state-dependent barrier is the GMAB minimum amount at T = 5

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

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 105.8361 105.1459 104.4759 103.7418 103.1177 102.5476 101.9255
## [2,] 107.0095 106.1544 105.3649 104.5908 103.8362 103.1057 102.4156
## [3,] 108.8469 107.9344 107.1107 106.2875 105.5386 104.7796 104.1068
##          [,8]     [,9]    [,10]     [,11]     [,12]     [,13]
## [1,] 101.3081 100.7551 100.2314  99.74083  99.20082  98.79609
## [2,] 101.7400 101.1228 100.5435 100.00479  99.48889  99.01405
## [3,] 103.4843 102.8358 102.3032 101.75996 101.29465 100.86600

We’re going to repeat it changing the GBM and the constant interest rate with the financial and mortality model published in BMOP2011.

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 105.7035 104.9644 104.2316 103.5746 102.8758 102.2232 101.5978
## [2,] 106.0049 104.9975 104.0136 103.1158 102.2135 101.3566 100.5898
## [3,] 106.8392 105.7573 104.6575 103.6199 102.6275 101.7367 100.9180
##           [,8]      [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 101.00914 100.45646 99.97099 99.48854 99.03727 98.61510
## [2,]  99.83135  99.13004 98.48703 97.86437 97.31300 96.82758
## [3,] 100.20656  99.56610 99.00827 98.54137 98.13197 97.82150

Eventually the financial and mortality models will be the one published in BBM2010. In this case the age at contract inception is 40.

##          [,1]     [,2]     [,3]     [,4]      [,5]      [,6]     [,7]
## [1,] 103.7628 103.0553 102.2742 101.5359 100.81098 100.07626 99.36721
## [2,] 103.6281 102.6239 101.6094 100.5731  99.57107  98.50539 97.49990
## [3,] 103.9012 102.6608 101.4121 100.1903  98.94271  97.78828 96.70716
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 98.69747 98.04000 97.34235 96.69514 96.08687 95.57109
## [2,] 96.52253 95.59973 94.73625 93.92139 93.11038 92.42428
## [3,] 95.72336 94.73941 93.86892 93.03695 92.30615 91.62810