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 1%
• 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

• We’re going to vary the penalty type between a constant 2% penalty, a linearly descreasing penalty with initial value 4% given by the formula

\[ 0.04\left (1-\frac{t}{T} \right )^3 \]

and exponentially descreasing with initial value 4% given by the formula:

\[ 1 - \exp\left ( -\frac{0.04}{T}\left ( T - \min\left (t, T \right ) \right )\right ) \]

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 1.

##          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]
## [1,] 111.0023 107.7702 103.6560 99.53160 97.90866 97.34886 96.97445
## [2,] 111.0262 107.6461 103.4033 99.19910 96.61467 95.85731 95.39831
## [3,] 111.0148 107.7690 103.6285 99.39772 96.38356 95.75236 95.28861
##          [,8]     [,9]    [,10]    [,11]    [,12]    [,13]
## [1,] 96.59212 96.29157 96.03118 95.77532 95.49943 95.23483
## [2,] 95.01161 94.68722 94.43124 94.13241 93.86248 93.58579
## [3,] 94.93710 94.62022 94.33046 94.04790 93.77853 93.52393