library(dplyr)

\[\mu_x^{01} = a_1+b_1*{c_1}^x\] \[\mu_x^{02} = a_2+b_2*{c_2}^x\] \[\mu_x^{12} = \mu_x^{02}\] >Assumptions

a = c(0.0004 , 0.0005)
b = c(3.4674*(10^(-4)) , 7.5858*(10^(-4)))
c = c(0.238155 , 0.87498)

x = 40
ii = 0.05
Deathbenefit = 10000
Disablebenefit = 2000

\[\nu = \frac{1}{1+i}\]

\[d = \frac{i}{1+i}\]

v = 1/(1+ii)
d_ = ii/(1+ii)
t = 0:130
d = as.data.frame(t)

\[_tp_x^{ij}=exp(-\int_0^t \sum_{j=0\not=i}^n \mu_{x+s}^{ij} ds)\] >Permenant

\[_tp_x^{ij}=_tp_x^\bar{ij}\]

\[_tp_x^{00}=exp(-a_1t-\frac{b_1}{ln(c_1)}*{c_1}^x({c_1}^t-1)-a_2t-\frac{b_2}{ln(c_2)}*{c_2}^x({c_2}^t-1))\]

d = mutate(.data = d , tp00 = exp(-(a[1]*t)-((b[1]/log(c[1]))*(c[1]^x)*((c[1]^t)-1)))
                              -(a[2]*t)-((b[2]/log(c[2]))*(c[2]^x)*((c[2]^t)-1)))

\[_tp_x^{11}=exp(-a_2t-\frac{b_2}{ln(c_2)}*{c_2}^x({c_2}^t-1))\]

d = mutate(.data = d , tp11 = exp((-a[2]*t)-((b[2]/log(c[2]))*(c[2]^x)*((c[2]^t)-1))))

\[_tp_x^{12}= 1-{_tp_x^{11}}\]

d = mutate(.data = d , tp12 = 1-tp11)

\[_tp_x^{01}=\int_0^t {_sp_x}^\bar{00}*\mu_{x+s}^{01}*_{t-s}p_{x+s}^\bar{11}ds\]

\[_tp_x^{01}=\int_0^t(exp(-a_1s-\frac{b_1}{ln(c_1)}*{c_1}^x({c_1}^s-1)-a_2s-\frac{b_2}{ln(c_2)}*{c_2}^x({c_2}^s-1)))*(a_1+b_1*c_1^{x+s})*(exp(-a_2(t-s)-\frac{b_2}{ln(c_2)}*{c_2}^{x+s}({c_2}^{t-s}-1)))\]

x=130
f = function(s){
  a = (exp((-a[1]*s)-((b[1]/log(c[1]))*(c[1]^x)*((c[1]^s)-1)))+(-a[2]*s)-((b[2]/log(c[2]))*(c[2]^x)*((c[2]^s)-1)))*
      (a[1]+(b[1]*(c[1]^(x+s))))*(exp((-a[2]*(t-s))-((b[2]/log(c[2]))*(c[2]^(x+s))*((c[2]^(t-s))-1))))
  return(a)
}

i = function(t){
  p01 = integrate(f = f , lower = 0 , upper = t)
  return(p01$value)
}


for (t in 0:130) {
  print(i(t))
}
## [1] 0
## [1] 0.0003997201
## [1] 0.0007988805
## [1] 0.001197482
## [1] 0.001595524
## [1] 0.001993007
## [1] 0.002389932
## [1] 0.0027863
## [1] 0.003182109
## [1] 0.003577362
## [1] 0.003972057
## [1] 0.004366196
## [1] 0.004759779
## [1] 0.005152806
## [1] 0.005545277
## [1] 0.005937193
## [1] 0.006328554
## [1] 0.006719361
## [1] 0.007109614
## [1] 0.007499312
## [1] 0.007888457
## [1] 0.008277049
## [1] 0.008665089
## [1] 0.009052575
## [1] 0.00943951
## [1] 0.009825893
## [1] 0.01021172
## [1] 0.010597
## [1] 0.01098173
## [1] 0.01136591
## [1] 0.01174954
## [1] 0.01213262
## [1] 0.01251515
## [1] 0.01289713
## [1] 0.01327856
## [1] 0.01365945
## [1] 0.01403978
## [1] 0.01441957
## [1] 0.01479881
## [1] 0.0151775
## [1] 0.01555565
## [1] 0.01593325
## [1] 0.0163103
## [1] 0.01668681
## [1] 0.01706277
## [1] 0.01743819
## [1] 0.01781306
## [1] 0.01818739
## [1] 0.01856118
## [1] 0.01893442
## [1] 0.01930712
## [1] 0.01967927
## [1] 0.02005088
## [1] 0.02042195
## [1] 0.02079248
## [1] 0.02116246
## [1] 0.02153191
## [1] 0.02190081
## [1] 0.02226917
## [1] 0.022637
## [1] 0.02300428
## [1] 0.02337102
## [1] 0.02373722
## [1] 0.02410289
## [1] 0.02446801
## [1] 0.0248326
## [1] 0.02519665
## [1] 0.02556016
## [1] 0.02592313
## [1] 0.02628557
## [1] 0.02664747
## [1] 0.02700883
## [1] 0.02736966
## [1] 0.02772995
## [1] 0.02808971
## [1] 0.02844893
## [1] 0.02880762
## [1] 0.02916577
## [1] 0.02952339
## [1] 0.02988047
## [1] 0.03023702
## [1] 0.03059304
## [1] 0.03094852
## [1] 0.03130348
## [1] 0.0316579
## [1] 0.03201178
## [1] 0.03236514
## [1] 0.03271797
## [1] 0.03307026
## [1] 0.03342203
## [1] 0.03377326
## [1] 0.03412397
## [1] 0.03447414
## [1] 0.03482379
## [1] 0.0351729
## [1] 0.03552149
## [1] 0.03586955
## [1] 0.03621709
## [1] 0.03656409
## [1] 0.03691057
## [1] 0.03725652
## [1] 0.03760194
## [1] 0.03794684
## [1] 0.03829121
## [1] 0.03863506
## [1] 0.03897838
## [1] 0.03932118
## [1] 0.03966345
## [1] 0.0400052
## [1] 0.04034642
## [1] 0.04068712
## [1] 0.0410273
## [1] 0.04136695
## [1] 0.04170608
## [1] 0.04204469
## [1] 0.04238277
## [1] 0.04272034
## [1] 0.04305738
## [1] 0.0433939
## [1] 0.04372991
## [1] 0.04406539
## [1] 0.04440035
## [1] 0.04473479
## [1] 0.04506871
## [1] 0.04540211
## [1] 0.045735
## [1] 0.04606736
## [1] 0.04639921
## [1] 0.04673054
## [1] 0.04706135
## [1] 0.04739164
d = as.data.frame(cbind(d , tp01))

\[_tp_x^{02}= 1-{_tp_x^{00}}-{_tp_x^{01}}\]

d = mutate(.data = d , tp02 = 1-tp00-tp01)
head(d , 10)
##    t      tp00      tp11         tp12         tp01         tp02
## 1  0 1.0000000 1.0000000 0.0000000000 0.0000000000 0.0000000000
## 2  1 0.9990967 0.9994967 0.0005032715 0.0003997187 0.0005035995
## 3  2 0.9981939 0.9989941 0.0010058653 0.0007988754 0.0010071762
## 4  3 0.9972917 0.9984922 0.0015078352 0.0011974710 0.0015107825
## 5  4 0.9963900 0.9979908 0.0020092283 0.0015955060 0.0020144642
## 6  5 0.9954888 0.9974899 0.0025100858 0.0019929810 0.0025182616
## 7  6 0.9945879 0.9969896 0.0030104436 0.0023898960 0.0030222104
## 8  7 0.9936874 0.9964897 0.0035103333 0.0027862540 0.0035263386
## 9  8 0.9927873 0.9959902 0.0040097825 0.0031820520 0.0040306767
## 10 9 0.9918875 0.9954912 0.0045088153 0.0035772940 0.0045352444

\[EPV\space of\space benefits + EPV\space of\space expenses = EPV\space of\space premium\space income\]

Premium(arithmetically increasing annuities)

\[(I\ddot{a})_x=\sum_{t=0}^\infty v^t(t+1)\space_tp_x^{00}\]

a1 = mutate(.data = d , vv = v^t , tt = t+1)

Ia = sum(a1$tp00*a1$vv*a1$tt)
Ia
## [1] 420.5795

Death benefit

\[\ddot a_x^{02}=\frac{1-A_x^{02}}{d}\] \[\ddot a_x^{02}=\sum_{t=0}^\infty v^t\space_tp_x^{02}\] \[A_x^{02}=1-d*\ddot a_x^{02}\]

a2 = sum(a1$vv*a1$tp02)
a2
## [1] 0.2109917
A_new = 1-d_*a2 
A_new
## [1] 0.9899528

\[EPV\space Death\space benefit = S *A_x^{02} \]

de_b = Deathbenefit*A_new
de_b
## [1] 9899.528

Disable benefit

\[\ddot a_x^{01}=\sum_{t=0}^\infty v^t\space_tp_x^{01}\]

a3 = sum(a1$tp00*a1$vv)
a3
## [1] 20.59249

\[EPV\space Disable\space benefit = b *\ddot a_x^{01} \]

di_b = Disablebenefit*a3
di_b
## [1] 41184.98

\[EPV\space Death\space benefit\space +\space EPV\space Disable\space benefit\space=\space EPV\space Premium\] \[S *A_x^{02}+b *\ddot a_x^{01}=p*(I\ddot{a})_x\] \[p=\frac{S *A_x^{02}+b *\ddot a_x^{01}}{(I\ddot{a})_x}\]

p = (de_b+di_b)/Ia
p
## [1] 121.4622
t = 0:130

p00_x = function(x){
    l = exp((-a[1]*t)-((b[1]/log(c[1]))*(c[1]^x)*((c[1]^t)-1))+(-a[2]*t)-((b[2]/log(c[2]))*(c[2]^x)*((c[2]^t)-1)))
    print(l)
}



p11_x = function(x){
    l = exp((-a[2]*t)-((b[2]/log(c[2]))*(c[2]^x)*((c[2]^t)-1)))
    print(l)
}

\[_tV^{(0)}=B\ddot a_{x+t}^{01}+SA_{x+t}^{02}-PI\ddot a_{x+t}^{00}\]

v_ = v^t
p00_0 = p00_x(40)
##   [1] 1.0000000 0.9990970 0.9981953 0.9972947 0.9963953 0.9954969 0.9945997
##   [8] 0.9937034 0.9928081 0.9919139 0.9910205 0.9901281 0.9892366 0.9883460
##  [15] 0.9874563 0.9865675 0.9856796 0.9847924 0.9839062 0.9830208 0.9821362
##  [22] 0.9812524 0.9803695 0.9794874 0.9786061 0.9777256 0.9768459 0.9759671
##  [29] 0.9750890 0.9742117 0.9733353 0.9724596 0.9715847 0.9707106 0.9698374
##  [36] 0.9689649 0.9680932 0.9672222 0.9663521 0.9654828 0.9646142 0.9637464
##  [43] 0.9628794 0.9620132 0.9611478 0.9602831 0.9594193 0.9585562 0.9576938
##  [50] 0.9568323 0.9559715 0.9551115 0.9542523 0.9533939 0.9525362 0.9516793
##  [57] 0.9508232 0.9499678 0.9491132 0.9482594 0.9474064 0.9465541 0.9457026
##  [64] 0.9448518 0.9440018 0.9431526 0.9423042 0.9414565 0.9406095 0.9397634
##  [71] 0.9389180 0.9380733 0.9372294 0.9363863 0.9355439 0.9347023 0.9338615
##  [78] 0.9330214 0.9321820 0.9313434 0.9305056 0.9296685 0.9288322 0.9279966
##  [85] 0.9271618 0.9263277 0.9254944 0.9246618 0.9238300 0.9229990 0.9221686
##  [92] 0.9213390 0.9205102 0.9196821 0.9188548 0.9180282 0.9172023 0.9163772
##  [99] 0.9155529 0.9147292 0.9139063 0.9130842 0.9122628 0.9114421 0.9106222
## [106] 0.9098030 0.9089846 0.9081668 0.9073498 0.9065336 0.9057181 0.9049033
## [113] 0.9040893 0.9032759 0.9024634 0.9016515 0.9008404 0.9000300 0.8992203
## [120] 0.8984114 0.8976032 0.8967957 0.8959890 0.8951829 0.8943776 0.8935731
## [127] 0.8927692 0.8919661 0.8911637 0.8903620 0.8895610
p11_0 =p11_x(40)
##   [1] 1.0000000 0.9994967 0.9989941 0.9984922 0.9979908 0.9974899 0.9969896
##   [8] 0.9964897 0.9959902 0.9954912 0.9949925 0.9944943 0.9939964 0.9934988
##  [15] 0.9930016 0.9925047 0.9920081 0.9915119 0.9910159 0.9905202 0.9900248
##  [22] 0.9895297 0.9890348 0.9885402 0.9880459 0.9875519 0.9870581 0.9865646
##  [29] 0.9860714 0.9855784 0.9850857 0.9845932 0.9841009 0.9836090 0.9831172
##  [36] 0.9826258 0.9821346 0.9816436 0.9811529 0.9806624 0.9801722 0.9796822
##  [43] 0.9791924 0.9787030 0.9782137 0.9777247 0.9772360 0.9767475 0.9762592
##  [50] 0.9757712 0.9752834 0.9747959 0.9743086 0.9738216 0.9733348 0.9728483
##  [57] 0.9723620 0.9718759 0.9713901 0.9709045 0.9704192 0.9699341 0.9694492
##  [64] 0.9689646 0.9684803 0.9679961 0.9675123 0.9670286 0.9665452 0.9660621
##  [71] 0.9655792 0.9650965 0.9646141 0.9641319 0.9636499 0.9631682 0.9626868
##  [78] 0.9622056 0.9617246 0.9612438 0.9607633 0.9602831 0.9598030 0.9593233
##  [85] 0.9588437 0.9583644 0.9578854 0.9574065 0.9569279 0.9564496 0.9559715
##  [92] 0.9554936 0.9550160 0.9545386 0.9540615 0.9535846 0.9531079 0.9526314
##  [99] 0.9521552 0.9516793 0.9512036 0.9507281 0.9502528 0.9497778 0.9493031
## [106] 0.9488285 0.9483542 0.9478802 0.9474064 0.9469328 0.9464594 0.9459863
## [113] 0.9455134 0.9450408 0.9445684 0.9440962 0.9436243 0.9431526 0.9426811
## [120] 0.9422099 0.9417389 0.9412682 0.9407977 0.9403274 0.9398573 0.9393875
## [127] 0.9389180 0.9384486 0.9379795 0.9375106 0.9370420
p12_x_0 = 1-p11_0
p01_x_0 = tp01
p02_x_0 = 1-p00_0-p01_x_0

v_0 = (Disablebenefit*sum(p01_x_0*v_))+(Deathbenefit*(1-d_*(sum(p02_x_0*v_))))-p*(sum(v_*(t+1)*p00_0))
p00_1 = p00_x(41)
##   [1] 1.0000000 0.9990974 0.9981961 0.9972958 0.9963967 0.9954986 0.9946015
##   [8] 0.9937054 0.9928104 0.9919162 0.9910230 0.9901307 0.9892393 0.9883488
##  [15] 0.9874592 0.9865704 0.9856825 0.9847954 0.9839092 0.9830238 0.9821393
##  [22] 0.9812556 0.9803727 0.9794906 0.9786093 0.9777288 0.9768491 0.9759703
##  [29] 0.9750922 0.9742150 0.9733385 0.9724629 0.9715880 0.9707139 0.9698406
##  [36] 0.9689681 0.9680964 0.9672255 0.9663554 0.9654860 0.9646175 0.9637497
##  [43] 0.9628827 0.9620165 0.9611510 0.9602864 0.9594225 0.9585594 0.9576971
##  [50] 0.9568355 0.9559748 0.9551148 0.9542556 0.9533971 0.9525394 0.9516825
##  [57] 0.9508264 0.9499710 0.9491165 0.9482626 0.9474096 0.9465573 0.9457058
##  [64] 0.9448550 0.9440050 0.9431558 0.9423074 0.9414597 0.9406127 0.9397666
##  [71] 0.9389211 0.9380765 0.9372326 0.9363895 0.9355471 0.9347055 0.9338646
##  [78] 0.9330245 0.9321852 0.9313466 0.9305088 0.9296717 0.9288354 0.9279998
##  [85] 0.9271650 0.9263309 0.9254976 0.9246650 0.9238332 0.9230021 0.9221718
##  [92] 0.9213422 0.9205133 0.9196853 0.9188579 0.9180313 0.9172055 0.9163803
##  [99] 0.9155560 0.9147323 0.9139094 0.9130873 0.9122659 0.9114452 0.9106253
## [106] 0.9098061 0.9089876 0.9081699 0.9073529 0.9065367 0.9057212 0.9049064
## [113] 0.9040923 0.9032790 0.9024664 0.9016546 0.9008435 0.9000331 0.8992234
## [120] 0.8984145 0.8976062 0.8967988 0.8959920 0.8951860 0.8943807 0.8935761
## [127] 0.8927722 0.8919691 0.8911667 0.8903650 0.8895640
p11_1 =p11_x(41)
##   [1] 1.0000000 0.9994972 0.9989949 0.9984933 0.9979922 0.9974916 0.9969914
##   [8] 0.9964917 0.9959924 0.9954936 0.9949950 0.9944969 0.9939991 0.9935016
##  [15] 0.9930045 0.9925076 0.9920111 0.9915149 0.9910189 0.9905233 0.9900279
##  [22] 0.9895328 0.9890380 0.9885435 0.9880492 0.9875551 0.9870614 0.9865679
##  [29] 0.9860746 0.9855817 0.9850889 0.9845965 0.9841042 0.9836123 0.9831206
##  [36] 0.9826291 0.9821379 0.9816469 0.9811562 0.9806657 0.9801755 0.9796855
##  [43] 0.9791958 0.9787063 0.9782170 0.9777280 0.9772393 0.9767508 0.9762625
##  [50] 0.9757745 0.9752867 0.9747992 0.9743119 0.9738249 0.9733381 0.9728516
##  [57] 0.9723653 0.9718792 0.9713934 0.9709078 0.9704225 0.9699374 0.9694525
##  [64] 0.9689679 0.9684836 0.9679994 0.9675156 0.9670319 0.9665485 0.9660654
##  [71] 0.9655825 0.9650998 0.9646174 0.9641352 0.9636532 0.9631715 0.9626900
##  [78] 0.9622088 0.9617278 0.9612471 0.9607666 0.9602863 0.9598063 0.9593265
##  [85] 0.9588470 0.9583677 0.9578886 0.9574098 0.9569312 0.9564529 0.9559747
##  [92] 0.9554969 0.9550192 0.9545419 0.9540647 0.9535878 0.9531111 0.9526347
##  [99] 0.9521585 0.9516825 0.9512068 0.9507313 0.9502561 0.9497811 0.9493063
## [106] 0.9488318 0.9483575 0.9478834 0.9474096 0.9469360 0.9464626 0.9459895
## [113] 0.9455166 0.9450440 0.9445716 0.9440994 0.9436275 0.9431558 0.9426843
## [120] 0.9422131 0.9417421 0.9412714 0.9408009 0.9403306 0.9398605 0.9393907
## [127] 0.9389211 0.9384518 0.9379827 0.9375138 0.9370452
p12_x_1 = 1-p11_1
p01_x_1 = c(0,  0.0003997189,   0.000798876,    0.001197472,    0.001595508,    0.001992984,    0.002389901,    0.002786259,    0.00318206, 0.003577302,    0.003971987,    0.004366116,    0.004759688,    0.005152704,    0.005545165,    0.00593707, 0.006328421,    0.006719217,    0.007109459,    0.007499147,    0.007888281,    0.008276863,    0.008664892,    0.009052368,    0.009439292,    0.009825665,    0.01021149, 0.01059676, 0.01098148, 0.01136564, 0.01174926, 0.01213233, 0.01251485, 0.01289682, 0.01327825, 0.01365912, 0.01403944, 0.01441922, 0.01479845, 0.01517714, 0.01555527, 0.01593286, 0.01630991, 0.01668641, 0.01706236, 0.01743777, 0.01781263, 0.01818695, 0.01856073, 0.01893396, 0.01930665, 0.01967879, 0.02005039, 0.02042145, 0.02079197, 0.02116195, 0.02153138, 0.02190028, 0.02226863, 0.02263644, 0.02300372, 0.02337045, 0.02373664, 0.0241023,  0.02446741, 0.02483199, 0.02519603, 0.02555953, 0.0259225,  0.02628492, 0.02664682, 0.02700817, 0.02736899, 0.02772927, 0.02808902, 0.02844823, 0.02880691, 0.02916505, 0.02952266, 0.02987973, 0.03023627, 0.03059228, 0.03094776, 0.0313027,  0.03165711, 0.03201099, 0.03236434, 0.03271716, 0.03306944, 0.0334212,  0.03377242, 0.03412312, 0.03447329, 0.03482292, 0.03517203, 0.03552061, 0.03586866, 0.03621619, 0.03656318, 0.03690965, 0.03725559, 0.03760101, 0.0379459,  0.03829026, 0.0386341,  0.03897741, 0.0393202,  0.03966246, 0.0400042,  0.04034541, 0.0406861,  0.04102627, 0.04136591, 0.04170504, 0.04204363, 0.04238171, 0.04271927, 0.0430563,  0.04339282, 0.04372881, 0.04406428, 0.04439923, 0.04473366, 0.04506758, 0.04540097, 0.04573384, 0.0460662,  0.04639804, 0.04672936, 0.04706016, 0.04739045)
p02_x_1 = 1-p00_1-p01_x_1

v_1 = (Disablebenefit*sum(p01_x_1*v_))+(Deathbenefit*(1-d_*(sum(p02_x_1*v_))))-p*(sum(v_*(t+1)*p00_1))
p00_2 = p00_x(42)
##   [1] 1.0000000 0.9990978 0.9981967 0.9972968 0.9963979 0.9955000 0.9946031
##   [8] 0.9937072 0.9928123 0.9919183 0.9910252 0.9901330 0.9892417 0.9883512
##  [15] 0.9874617 0.9865730 0.9856851 0.9847981 0.9839119 0.9830265 0.9821420
##  [22] 0.9812583 0.9803754 0.9794933 0.9786121 0.9777316 0.9768520 0.9759731
##  [29] 0.9750951 0.9742178 0.9733414 0.9724657 0.9715908 0.9707168 0.9698435
##  [36] 0.9689710 0.9680993 0.9672284 0.9663582 0.9654889 0.9646203 0.9637525
##  [43] 0.9628855 0.9620193 0.9611539 0.9602892 0.9594254 0.9585623 0.9576999
##  [50] 0.9568384 0.9559776 0.9551176 0.9542584 0.9533999 0.9525423 0.9516854
##  [57] 0.9508292 0.9499739 0.9491193 0.9482655 0.9474124 0.9465601 0.9457086
##  [64] 0.9448578 0.9440078 0.9431586 0.9423102 0.9414625 0.9406155 0.9397693
##  [71] 0.9389239 0.9380793 0.9372354 0.9363923 0.9355499 0.9347083 0.9338674
##  [78] 0.9330273 0.9321880 0.9313494 0.9305115 0.9296744 0.9288381 0.9280025
##  [85] 0.9271677 0.9263336 0.9255003 0.9246677 0.9238359 0.9230048 0.9221745
##  [92] 0.9213449 0.9205161 0.9196880 0.9188606 0.9180340 0.9172082 0.9163831
##  [99] 0.9155587 0.9147351 0.9139122 0.9130900 0.9122686 0.9114479 0.9106280
## [106] 0.9098088 0.9089903 0.9081726 0.9073556 0.9065394 0.9057239 0.9049091
## [113] 0.9040950 0.9032817 0.9024691 0.9016573 0.9008461 0.9000357 0.8992261
## [120] 0.8984171 0.8976089 0.8968014 0.8959947 0.8951886 0.8943833 0.8935788
## [127] 0.8927749 0.8919718 0.8911693 0.8903677 0.8895667
p11_2 =p11_x(42)
##   [1] 1.0000000 0.9994975 0.9989956 0.9984943 0.9979934 0.9974930 0.9969931
##   [8] 0.9964935 0.9959944 0.9954956 0.9949972 0.9944992 0.9940014 0.9935040
##  [15] 0.9930070 0.9925102 0.9920137 0.9915175 0.9910216 0.9905260 0.9900307
##  [22] 0.9895356 0.9890408 0.9885463 0.9880520 0.9875580 0.9870642 0.9865707
##  [29] 0.9860775 0.9855845 0.9850918 0.9845993 0.9841071 0.9836152 0.9831234
##  [36] 0.9826320 0.9821408 0.9816498 0.9811591 0.9806686 0.9801784 0.9796884
##  [43] 0.9791987 0.9787092 0.9782199 0.9777309 0.9772422 0.9767537 0.9762654
##  [50] 0.9757774 0.9752896 0.9748021 0.9743148 0.9738278 0.9733410 0.9728545
##  [57] 0.9723681 0.9718821 0.9713963 0.9709107 0.9704253 0.9699403 0.9694554
##  [64] 0.9689708 0.9684864 0.9680023 0.9675184 0.9670348 0.9665514 0.9660682
##  [71] 0.9655853 0.9651027 0.9646202 0.9641380 0.9636561 0.9631744 0.9626929
##  [78] 0.9622117 0.9617307 0.9612500 0.9607694 0.9602892 0.9598092 0.9593294
##  [85] 0.9588498 0.9583705 0.9578915 0.9574126 0.9569340 0.9564557 0.9559776
##  [92] 0.9554997 0.9550221 0.9545447 0.9540675 0.9535906 0.9531140 0.9526375
##  [99] 0.9521613 0.9516854 0.9512096 0.9507341 0.9502589 0.9497839 0.9493091
## [106] 0.9488346 0.9483603 0.9478862 0.9474124 0.9469388 0.9464655 0.9459923
## [113] 0.9455195 0.9450468 0.9445744 0.9441022 0.9436303 0.9431586 0.9426872
## [120] 0.9422159 0.9417449 0.9412742 0.9408037 0.9403334 0.9398633 0.9393935
## [127] 0.9389239 0.9384546 0.9379855 0.9375166 0.9370480
p12_x_2 = 1-p11_2
p01_x_2 = c(0,  0.000399719,    0.0007988766,   0.001197473,    0.00159551, 0.001992987,    0.002389905,    0.002786264,    0.003182066,    0.00357731, 0.003971996,    0.004366126,    0.004759699,    0.005152717,    0.005545179,    0.005937086,    0.006328437,    0.006719235,    0.007109478,    0.007499167,    0.007888303,    0.008276886,    0.008664916,    0.009052394,    0.00943932, 0.009825693,    0.01021152, 0.01059679, 0.01098151, 0.01136568, 0.0117493,  0.01213237, 0.01251489, 0.01289686, 0.01327829, 0.01365916, 0.01403949, 0.01441927, 0.0147985,  0.01517718, 0.01555532, 0.01593291, 0.01630996, 0.01668646, 0.01706241, 0.01743782, 0.01781269, 0.01818701, 0.01856078, 0.01893402, 0.01930671, 0.01967885, 0.02005046, 0.02042152, 0.02079204, 0.02116201, 0.02153145, 0.02190034, 0.0222687,  0.02263651, 0.02300379, 0.02337052, 0.02373672, 0.02410237, 0.02446749, 0.02483207, 0.02519611, 0.02555961, 0.02592258, 0.02628501, 0.0266469,  0.02700825, 0.02736907, 0.02772936, 0.0280891,  0.02844832, 0.028807,   0.02916514, 0.02952275, 0.02987983, 0.03023637, 0.03059238, 0.03094785, 0.0313028,  0.03165721, 0.03201109, 0.03236444, 0.03271726, 0.03306955, 0.0334213,  0.03377253, 0.03412323, 0.03447339, 0.03482303, 0.03517214, 0.03552072, 0.03586877, 0.0362163,  0.03656329, 0.03690976, 0.03725571, 0.03760112, 0.03794601, 0.03829038, 0.03863422, 0.03897753, 0.03932032, 0.03966258, 0.04000432, 0.04034554, 0.04068623, 0.0410264,  0.04136604, 0.04170517, 0.04204377, 0.04238185, 0.0427194,  0.04305644, 0.04339295, 0.04372894, 0.04406442, 0.04439937, 0.0447338,  0.04506772, 0.04540111, 0.04573399, 0.04606635, 0.04639818, 0.04672951, 0.04706031, 0.0473906)
p02_x_2 = 1-p00_2-p01_x_2

v_2 = (Disablebenefit*sum(p01_x_2*v_))+(Deathbenefit*(1-d_*(sum(p02_x_2*v_))))-p*(sum(v_*(t+1)*p00_2))
p00_3 = p00_x(43)
##   [1] 1.0000000 0.9990981 0.9981974 0.9972976 0.9963990 0.9955013 0.9946046
##   [8] 0.9937088 0.9928140 0.9919201 0.9910271 0.9901350 0.9892437 0.9883534
##  [15] 0.9874638 0.9865752 0.9856874 0.9848004 0.9839142 0.9830289 0.9821444
##  [22] 0.9812607 0.9803778 0.9794958 0.9786145 0.9777341 0.9768544 0.9759756
##  [29] 0.9750975 0.9742203 0.9733438 0.9724682 0.9715933 0.9707193 0.9698460
##  [36] 0.9689735 0.9681018 0.9672309 0.9663607 0.9654914 0.9646228 0.9637550
##  [43] 0.9628880 0.9620218 0.9611564 0.9602917 0.9594278 0.9585647 0.9577024
##  [50] 0.9568409 0.9559801 0.9551201 0.9542609 0.9534024 0.9525447 0.9516878
##  [57] 0.9508317 0.9499763 0.9491217 0.9482679 0.9474149 0.9465626 0.9457110
##  [64] 0.9448603 0.9440103 0.9431611 0.9423126 0.9414649 0.9406180 0.9397718
##  [71] 0.9389264 0.9380817 0.9372378 0.9363947 0.9355523 0.9347107 0.9338698
##  [78] 0.9330297 0.9321904 0.9313518 0.9305140 0.9296769 0.9288405 0.9280050
##  [85] 0.9271701 0.9263360 0.9255027 0.9246701 0.9238383 0.9230072 0.9221769
##  [92] 0.9213473 0.9205185 0.9196904 0.9188630 0.9180364 0.9172106 0.9163854
##  [99] 0.9155611 0.9147374 0.9139145 0.9130924 0.9122710 0.9114503 0.9106304
## [106] 0.9098112 0.9089927 0.9081750 0.9073580 0.9065417 0.9057262 0.9049114
## [113] 0.9040974 0.9032841 0.9024715 0.9016596 0.9008485 0.9000381 0.8992284
## [120] 0.8984195 0.8976113 0.8968038 0.8959970 0.8951910 0.8943857 0.8935811
## [127] 0.8927772 0.8919741 0.8911717 0.8903700 0.8895690
p11_3 =p11_x(43)
##   [1] 1.0000000 0.9994978 0.9989962 0.9984951 0.9979945 0.9974943 0.9969945
##   [8] 0.9964951 0.9959961 0.9954974 0.9949991 0.9945012 0.9940035 0.9935062
##  [15] 0.9930091 0.9925124 0.9920160 0.9915198 0.9910240 0.9905284 0.9900331
##  [22] 0.9895380 0.9890432 0.9885487 0.9880545 0.9875605 0.9870667 0.9865732
##  [29] 0.9860800 0.9855870 0.9850943 0.9846019 0.9841097 0.9836177 0.9831260
##  [36] 0.9826345 0.9821433 0.9816523 0.9811616 0.9806711 0.9801809 0.9796909
##  [43] 0.9792012 0.9787117 0.9782225 0.9777335 0.9772447 0.9767562 0.9762680
##  [50] 0.9757799 0.9752922 0.9748046 0.9743174 0.9738303 0.9733435 0.9728570
##  [57] 0.9723707 0.9718846 0.9713988 0.9709132 0.9704279 0.9699428 0.9694579
##  [64] 0.9689733 0.9684890 0.9680048 0.9675209 0.9670373 0.9665539 0.9660708
##  [71] 0.9655878 0.9651052 0.9646227 0.9641405 0.9636586 0.9631769 0.9626954
##  [78] 0.9622142 0.9617332 0.9612525 0.9607719 0.9602917 0.9598117 0.9593319
##  [85] 0.9588523 0.9583730 0.9578939 0.9574151 0.9569365 0.9564582 0.9559801
##  [92] 0.9555022 0.9550246 0.9545472 0.9540700 0.9535931 0.9531164 0.9526400
##  [99] 0.9521638 0.9516878 0.9512121 0.9507366 0.9502614 0.9497864 0.9493116
## [106] 0.9488370 0.9483627 0.9478887 0.9474149 0.9469413 0.9464679 0.9459948
## [113] 0.9455219 0.9450493 0.9445769 0.9441047 0.9436328 0.9431611 0.9426896
## [120] 0.9422184 0.9417474 0.9412766 0.9408061 0.9403358 0.9398658 0.9393960
## [127] 0.9389264 0.9384570 0.9379879 0.9375190 0.9370504
p12_x_3 = 1-p11_3
p01_x_3 = c(0,  0.0003997191,   0.000798877,    0.001197474,    0.001595512,    0.001992989,    0.002389908,    0.002786269,    0.003182071,    0.003577316,    0.003972004,    0.004366135,    0.004759709,    0.005152728,    0.005545191,    0.005937099,    0.006328452,    0.006719251,    0.007109495,    0.007499186,    0.007888323,    0.008276907,    0.008664938,    0.009052417,    0.009439343,    0.009825718,    0.01021154, 0.01059681, 0.01098154, 0.01136571, 0.01174933, 0.0121324,  0.01251492, 0.0128969,  0.01327832, 0.0136592,  0.01403952, 0.0144193,  0.01479854, 0.01517722, 0.01555536, 0.01593295, 0.01631,    0.0166865,  0.01706246, 0.01743787, 0.01781273, 0.01818706, 0.01856083, 0.01893407, 0.01930676, 0.0196789,  0.02005051, 0.02042157, 0.02079209, 0.02116207, 0.02153151, 0.0219004,  0.02226876, 0.02263657, 0.02300385, 0.02337058, 0.02373678, 0.02410244, 0.02446755, 0.02483213, 0.02519618, 0.02555968, 0.02592265, 0.02628508, 0.02664697, 0.02700832, 0.02736915, 0.02772943, 0.02808918, 0.02844839, 0.02880707, 0.02916522, 0.02952283, 0.02987991, 0.03023645, 0.03059246, 0.03094794, 0.03130288, 0.0316573,  0.03201118, 0.03236453, 0.03271735, 0.03306964, 0.03342139, 0.03377262, 0.03412332, 0.03447349, 0.03482313, 0.03517224, 0.03552082, 0.03586887, 0.0362164,  0.03656339, 0.03690987, 0.03725581, 0.03760123, 0.03794612, 0.03829048, 0.03863432, 0.03897764, 0.03932043, 0.03966269, 0.04000443, 0.04034565, 0.04068634, 0.04102651, 0.04136616, 0.04170528, 0.04204388, 0.04238196, 0.04271952, 0.04305656, 0.04339307, 0.04372907, 0.04406454, 0.04439949, 0.04473393, 0.04506784, 0.04540124, 0.04573411, 0.04606647, 0.04639831, 0.04672963, 0.04706044, 0.04739073)
p02_x_3 = 1-p00_3-p01_x_3

v_3 = (Disablebenefit*sum(p01_x_3*v_))+(Deathbenefit*(1-d_*(sum(p02_x_3*v_))))-p*(sum(v_*(t+1)*p00_3))
p00_60 = p00_x(130)
##   [1] 1.0000000 0.9991004 0.9982016 0.9973036 0.9964065 0.9955101 0.9946146
##   [8] 0.9937198 0.9928259 0.9919327 0.9910404 0.9901488 0.9892581 0.9883682
##  [15] 0.9874790 0.9865907 0.9857032 0.9848165 0.9839305 0.9830454 0.9821610
##  [22] 0.9812775 0.9803947 0.9795128 0.9786316 0.9777512 0.9768717 0.9759929
##  [29] 0.9751149 0.9742377 0.9733612 0.9724856 0.9716108 0.9707367 0.9698634
##  [36] 0.9689910 0.9681193 0.9672483 0.9663782 0.9655089 0.9646403 0.9637725
##  [43] 0.9629055 0.9620393 0.9611738 0.9603092 0.9594453 0.9585822 0.9577198
##  [50] 0.9568583 0.9559975 0.9551375 0.9542782 0.9534198 0.9525621 0.9517052
##  [57] 0.9508490 0.9499936 0.9491390 0.9482852 0.9474321 0.9465798 0.9457283
##  [64] 0.9448775 0.9440275 0.9431782 0.9423298 0.9414820 0.9406351 0.9397889
##  [71] 0.9389435 0.9380988 0.9372549 0.9364117 0.9355694 0.9347277 0.9338868
##  [78] 0.9330467 0.9322074 0.9313688 0.9305309 0.9296938 0.9288574 0.9280219
##  [85] 0.9271870 0.9263529 0.9255196 0.9246870 0.9238551 0.9230240 0.9221937
##  [92] 0.9213641 0.9205352 0.9197071 0.9188798 0.9180531 0.9172273 0.9164021
##  [99] 0.9155777 0.9147541 0.9139312 0.9131090 0.9122876 0.9114669 0.9106469
## [106] 0.9098277 0.9090093 0.9081915 0.9073745 0.9065582 0.9057427 0.9049279
## [113] 0.9041138 0.9033005 0.9024879 0.9016760 0.9008649 0.9000545 0.8992448
## [120] 0.8984358 0.8976276 0.8968201 0.8960133 0.8952073 0.8944019 0.8935973
## [127] 0.8927935 0.8919903 0.8911879 0.8903862 0.8895852
p11_60 =p11_x(130)
##   [1] 1.0000000 0.9995001 0.9990005 0.9985011 0.9980020 0.9975031 0.9970045
##   [8] 0.9965061 0.9960080 0.9955101 0.9950125 0.9945151 0.9940180 0.9935211
##  [15] 0.9930244 0.9925281 0.9920319 0.9915360 0.9910404 0.9905450 0.9900498
##  [22] 0.9895549 0.9890603 0.9885659 0.9880717 0.9875778 0.9870841 0.9865907
##  [29] 0.9860975 0.9856046 0.9851119 0.9846195 0.9841273 0.9836354 0.9831437
##  [36] 0.9826522 0.9821610 0.9816701 0.9811794 0.9806889 0.9801987 0.9797087
##  [43] 0.9792190 0.9787295 0.9782402 0.9777512 0.9772625 0.9767740 0.9762857
##  [50] 0.9757977 0.9753099 0.9748224 0.9743351 0.9738480 0.9733612 0.9728747
##  [57] 0.9723884 0.9719023 0.9714165 0.9709309 0.9704455 0.9699604 0.9694756
##  [64] 0.9689910 0.9685066 0.9680224 0.9675386 0.9670549 0.9665715 0.9660883
##  [71] 0.9656054 0.9651227 0.9646403 0.9641581 0.9636761 0.9631944 0.9627129
##  [78] 0.9622317 0.9617507 0.9612700 0.9607894 0.9603092 0.9598291 0.9593493
##  [85] 0.9588698 0.9583905 0.9579114 0.9574326 0.9569540 0.9564756 0.9559975
##  [92] 0.9555196 0.9550420 0.9545646 0.9540874 0.9536105 0.9531338 0.9526573
##  [99] 0.9521811 0.9517052 0.9512294 0.9507539 0.9502787 0.9498036 0.9493289
## [106] 0.9488543 0.9483800 0.9479059 0.9474321 0.9469585 0.9464851 0.9460120
## [113] 0.9455391 0.9450665 0.9445941 0.9441219 0.9436499 0.9431782 0.9427068
## [120] 0.9422355 0.9417645 0.9412938 0.9408232 0.9403529 0.9398829 0.9394131
## [127] 0.9389435 0.9384741 0.9380050 0.9375361 0.9370675
p12_x_60 = 1-p11_60
p01_x_60 = c(0, 0.0003997201,   0.0007988805,   0.001197482,    0.001595524,    0.001993007,    0.002389932,    0.0027863,  0.003182109,    0.003577362,    0.003972057,    0.004366196,    0.004759779,    0.005152806,    0.005545277,    0.005937193,    0.006328554,    0.006719361,    0.007109614,    0.007499312,    0.007888457,    0.008277049,    0.008665089,    0.009052575,    0.00943951, 0.009825893,    0.01021172, 0.010597,   0.01098173, 0.01136591, 0.01174954, 0.01213262, 0.01251515, 0.01289713, 0.01327856, 0.01365945, 0.01403978, 0.01441957, 0.01479881, 0.0151775,  0.01555565, 0.01593325, 0.0163103,  0.01668681, 0.01706277, 0.01743819, 0.01781306, 0.01818739, 0.01856118, 0.01893442, 0.01930712, 0.01967927, 0.02005088, 0.02042195, 0.02079248, 0.02116246, 0.02153191, 0.02190081, 0.02226917, 0.022637,   0.02300428, 0.02337102, 0.02373722, 0.02410289, 0.02446801, 0.0248326,  0.02519665, 0.02556016, 0.02592313, 0.02628557, 0.02664747, 0.02700883, 0.02736966, 0.02772995, 0.02808971, 0.02844893, 0.02880762, 0.02916577, 0.02952339, 0.02988047, 0.03023702, 0.03059304, 0.03094852, 0.03130348, 0.0316579,  0.03201178, 0.03236514, 0.03271797, 0.03307026, 0.03342203, 0.03377326, 0.03412397, 0.03447414, 0.03482379, 0.0351729,  0.03552149, 0.03586955, 0.03621709, 0.03656409, 0.03691057, 0.03725652, 0.03760194, 0.03794684, 0.03829121, 0.03863506, 0.03897838, 0.03932118, 0.03966345, 0.0400052,  0.04034642, 0.04068712, 0.0410273,  0.04136695, 0.04170608, 0.04204469, 0.04238277, 0.04272034, 0.04305738, 0.0433939,  0.04372991, 0.04406539, 0.04440035, 0.04473479, 0.04506871, 0.04540211, 0.045735,   0.04606736, 0.04639921, 0.04673054, 0.04706135, 0.04739164)
p02_x_60 = 1-p00_60-p01_x_60

v_60 = (Disablebenefit*sum(p01_x_60*v_))+(Deathbenefit*(1-d_*(sum(p02_x_60*v_))))-p*(sum(v_*(t+1)*p00_60))