Mixed with different penalty types

With this exercise VAs with GMAB and GMDB riders are priced by means of the mixed 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 is 2%
The fee is state-dependent and the base fee varies from 1% to 10%
The state-dependent barrier is the GMAB minimum amount at T = 15
We’re going to vary the penalty type between a constant 2% penalty, a descreasing penalty with initial value 8% given by the formula

\[ 0.08\left (1-\frac{t}{T} \right )^3 \]

and exponentially descreasing with initial value 8% given by the formula:

\[ 1 - \exp\left ( -\frac{0.08}{T}\left ( T - t \right )\right ) \]

With regards to the simulation:

The number of Monte Carlo simulations will be 20000.

Model1

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]      [,7]
## [1,] 118.3837 114.6847 111.0280 107.3708 104.3498 101.9951 100.05442
## [2,] 118.2898 114.7328 110.8574 107.2463 104.1244 101.6062  99.43876
## [3,] 118.3719 114.6482 111.0806 107.4467 104.2778 101.4325  99.27603
##          [,8]     [,9]    [,10]
## [1,] 99.19017 98.56931 98.02296
## [2,] 97.70091 96.17611 95.26171
## [3,] 97.34585 95.83945 94.97799

Model 2

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]      [,7]
## [1,] 117.9172 114.2499 110.5070 106.9716 103.7827 101.3989 100.42717
## [2,] 117.8655 114.1688 110.4544 106.8498 103.5129 100.7637  98.80446
## [3,] 117.8104 114.2510 110.6234 107.1106 103.6992 101.0391  98.68481
##          [,8]     [,9]    [,10]
## [1,] 99.46740 98.72892 98.20708
## [2,] 97.29536 96.12464 95.34116
## [3,] 97.00899 95.81316 95.03352

Model 3

##          [,1]     [,2]     [,3]     [,4]      [,5]     [,6]     [,7]
## [1,] 115.8618 111.6763 107.2187 103.0103 100.54198 99.27828 98.67375
## [2,] 115.8287 111.5057 106.9372 102.7603  99.05343 96.62839 95.37518
## [3,] 115.7814 111.6719 107.3216 103.0403  99.16957 96.53218 94.96923
##          [,8]     [,9]    [,10]
## [1,] 98.19267 97.81671 97.45527
## [2,] 94.60648 94.02773 93.59514
## [3,] 94.29530 93.82390 93.40067

Model 4

##          [,1]     [,2]     [,3]     [,4]      [,5]     [,6]     [,7]
## [1,] 116.5618 112.1571 107.6758 103.3965 100.55056 99.26182 98.64001
## [2,] 116.5501 112.2464 107.4325 103.0258  99.38347 96.60126 95.39334
## [3,] 116.4413 112.2538 107.6947 103.2281  99.48796 96.41089 95.06097
##          [,8]     [,9]    [,10]
## [1,] 98.14698 97.74642 97.39982
## [2,] 94.59725 93.96188 93.51619
## [3,] 94.24233 93.74595 93.34420

Model 5

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 108.8821 104.5507 99.24021 97.63560 97.03831 96.70916 96.42809
## [2,] 108.9315 104.3379 99.10484 93.92995 92.19528 91.54218 91.15893
## [3,] 108.8423 104.4655 99.27024 94.08850 92.07375 91.45553 91.08736
##          [,8]     [,9]    [,10]
## [1,] 96.17268 95.91102 95.68083
## [2,] 90.81851 90.55733 90.29618
## [3,] 90.82045 90.55884 90.32880