Fig4.3 - four models

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 = 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 are 0%, 1% and 2%
The fee is state-dependent and the base fee varies from 1% to 13%
The state-dependent barrier is the GMAB minimum amount at T = 15

With regards to the simulation:

The number of Monte Carlo simulations will be 20000.
Underlying fund, interest rate, volatility and intensity of mortality follow the dynamics given by the systems of SDEs in BMOP2011.
The Weibull function was fitted to the 2015 Italian life table and the estimated parameters are c1=88.14778, c2=10.002.
This model will be referred to as model 4.

##          [,1]     [,2]     [,3]     [,4]      [,5]     [,6]     [,7]
## [1,] 108.0920 106.1908 104.0818 101.9802 100.02773 97.96692 95.93010
## [2,] 111.0162 107.7457 104.2747 100.8923  97.46436 94.09816 90.98326
## [3,] 115.9348 111.7223 107.5668 103.5957  99.96320 96.76929 93.97920
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 94.12606 92.38329 90.75499 89.31288 88.14248 86.82726
## [2,] 88.06379 85.41612 83.37203 81.64641 80.16610 79.03239
## [3,] 91.81163 90.13616 88.87519 88.08249 87.56808 87.09637

Here below we’re going to repeat the exercize with a different and simpler model where only the financial processes follow the dynamics of BMOP2011, while the intensity of mortality is deterministic and given by the Weibull function with the same parameters specified above.
This model will be referred to as model 3.

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 108.1199 106.2187 104.1844 102.1238 99.97217 97.95476 95.99503
## [2,] 111.1292 107.6873 104.3689 100.9405 97.44790 94.12017 90.93457
## [3,] 115.8920 111.7121 107.5960 103.6873 99.99754 96.68248 93.95735
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 94.10332 92.21594 90.63881 89.34206 88.10370 86.82634
## [2,] 88.07414 85.50759 83.38772 81.67966 80.15534 79.08481
## [3,] 91.78598 90.17765 88.88636 88.08881 87.55387 87.12385

Here below the financial model is simplified even more since just the interest rate and fund follow the dynamics of BMOP2011 while the volatility is kept constant and set to 0.2. Again the mortality is given by the deterministic Weibull function.
This model will be referred to as model 2.

##          [,1]     [,2]     [,3]     [,4]      [,5]      [,6]     [,7]
## [1,] 107.9003 105.4788 103.5389 101.2795  99.37416  97.33222 95.46662
## [2,] 111.5231 108.4815 105.4792 102.5041  99.64662  96.86869 94.50726
## [3,] 117.4332 113.6133 110.2575 107.1000 104.09094 101.39869 99.07233
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 93.66495 91.86112 90.44973 88.94973 87.53648 86.19901
## [2,] 92.13181 90.19479 88.34450 86.70446 85.28662 84.00191
## [3,] 97.00798 95.22128 93.62569 92.47694 91.45714 90.63107

Finally the fund process is the Geometric Brownian Motion with interest rate r = 0.03 and constant volatility 0.2. The intensity of mortality is again given by the Weibull function as above.
This model will be referred to as model 1.

##          [,1]     [,2]     [,3]     [,4]      [,5]      [,6]     [,7]
## [1,] 107.9279 105.8829 103.6685 101.5368  99.35985  97.41121 95.53802
## [2,] 111.8892 108.6993 105.6170 102.7058  99.90633  97.32342 94.77620
## [3,] 117.6179 113.9155 110.5553 107.2951 104.26874 101.64033 99.34948
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 93.77418 92.13610 90.63154 89.27245 87.92137 86.63103
## [2,] 92.51516 90.61942 88.76043 87.20613 85.72400 84.31663
## [3,] 97.34290 95.58565 94.06395 92.75378 91.51923 90.67779