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 13%

• The state-dependent barrier is the 1.4 by 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 4.

##          [,1]     [,2]     [,3]      [,4]      [,5]     [,6]     [,7]
## [1,] 112.8833 106.7233 103.2326 101.31186 100.07828 99.25323 98.65653
## [2,] 112.8534 106.8438 102.4648  99.71627  97.68984 96.30904 95.27758
## [3,] 112.8094 106.2772 101.5823  98.75642  96.96110 95.73003 94.85029
##          [,8]     [,9]    [,10]
## [1,] 98.20913 97.81814 97.47070
## [2,] 94.53281 93.99610 93.59602
## [3,] 94.25582 93.81308 93.39723

With model 3:

##          [,1]     [,2]     [,3]      [,4]      [,5]     [,6]     [,7]
## [1,] 112.7896 106.7214 103.2231 101.31872 100.07430 99.26214 98.65817
## [2,] 112.9697 106.6609 102.4216  99.73469  97.68367 96.28622 95.28054
## [3,] 112.8321 106.2298 101.4491  98.73140  96.96404 95.73663 94.83440
##          [,8]     [,9]    [,10]
## [1,] 98.20006 97.80487 97.47407
## [2,] 94.52056 94.00081 93.58335
## [3,] 94.25952 93.81381 93.42872

Model 2:

##          [,1]     [,2]     [,3]     [,4]     [,5]      [,6]      [,7]
## [1,] 115.2092 110.1592 106.6823 104.2638 102.4000 101.10390 100.20216
## [2,] 115.3287 110.3238 106.0287 103.2639 100.8857  99.25122  97.98316
## [3,] 114.7335 109.7534 105.3597 102.2621 100.1667  98.58264  97.36419
##          [,8]     [,9]    [,10]
## [1,] 99.37617 98.73159 98.19963
## [2,] 96.92676 96.05264 95.36404
## [3,] 96.40515 95.66329 95.02229

With model 1

##          [,1]     [,2]     [,3]     [,4]      [,5]      [,6]     [,7]
## [1,] 115.0983 109.7295 106.2856 103.7604 102.14297 100.88531 99.90001
## [2,] 115.5482 110.1200 105.9480 102.9860 100.61862  99.01508 97.74611
## [3,] 115.1551 109.6211 105.3030 102.0438  99.82545  98.37960 97.18853
##          [,8]     [,9]    [,10]
## [1,] 99.12283 98.49381 97.98727
## [2,] 96.73361 95.89210 95.15232
## [3,] 96.23840 95.43675 94.79896