# HowToDetLifeInsurance

#### 07/09/2020

DetLifeInsurance is an R package designed to provide tools for:

• Deterministic valuation of actuarial reserves and life insurance and annuities premiums (both constant and variable amounts supported).
• Deterministic valuation of actuarial life insurance and annuities premiums for a group of insurees (constant amounts).
• Applying fractional age assumptions to life tables, and generating new ones based on mortality laws.
• Calculation of equivalent interest-discount rates as well as future and present value of annuities.
• Calculation of loan amortization schedule.

In addition, 47 commonly used mortality tables are included as data.

Be aware that functions use 15 digits when performing calculations.

## Life insurance and annuities premiums

For calculating the actuarial present value:

• Pure Endowment: E(x, n, i = 0.04, data, prop = 1, assumption = "none", cap = 1) For a insuree

Em(x, n, i=0.04, data, prop = 1, assimption = "none", type = "joint", quant = 1, cap = 1) For a group of insurees.

• Life Insurance:

• Life insurance: A.(x, h, n, k = 1, i = 0.04, data, prop = 1, assumption = "none", cap = 1) For a insuree

Am.(x, h, n, k = 1, i = 0.04, data, prop = 1, assumption = "none", cap = 1)

• Varying life insurance according to a arithmetic progression: Av.(x, h, n, k = 1, r = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap = 1)
For an increasing life insurance coverage, ‘r’ must be 1.
• Varying life insurance according to a geometric progression.: Avg.(x, h, n, k = 1, r = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap = 1)
• Decreasing life insurance: AD.(x, h, n, k = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap = 1)
• Continuous life insurance: ACont.(x, h, n, i = 0.04, data, prop = 1, assumption = "UDD", cap = 1)
• Life Annuity:

• Life annuity: a(x, h, n, k = 1, i = 0.04, data, prop = 1, assumption = "none", cap = 1) For a insuree

am(x, h, n, k = 1, i = 0.04, data, prop = 1, assumption = "none", cap = 1) For a group of insurees

• Varying life annuity according to a arithmetic progression: av(x,h,n,k = 1,r = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap = 1) For an increasing life annuity coverage, ‘r’ must be 1.
• Varying life annuity according to a geometric progression: avg(x,h,n,k = 1,r = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap = 1)
• Decreasing life annuity: aD(x, h, n, k = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap = 1)
• Continuous life annuities: aCont(x, h, n, i = 0.04, data, prop = 1, assumption = "constant", cap = 1)

The mortality table in data must be a data frame with the age -an integer- in the first column (from 0 to omega), and the second column, the probability of death.

For calculating an annualized premium: PremiumFrac(px1, x, m, k, i = 0.04, data, prop = 1, effect = "yes", assumption)

## Actuarial reserves

Calculates the reserve up to the moment ‘t’:

• Pure Endowments: V_E(px, x, n, cantprem = 1, premperyear = 1, i = 0.04, data, prop = 1, assumption = "none", cap, t)

• Life annuity: V_a(px, x, h, n, k = 1, cantprem = 1, premperyear = 1, i = 0.04, data, prop = 1, assumption = "none", cap, t)

V_A.(px, x, h, n, k = 1, cantprem = 1, premperyear = 1, i = 0.04, data, prop = 1, assumption = "none", cap, t) Life insurance

• Varying life annuity according to a arithmetic progression: V_av(px, x, h, n, k = 1, r, cantprem = 1, premperyear = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap, t)

• Varying life insurance according to a arithmetic progression: V_Av.(px, x, h, n, k = 1, r, cantprem = 1, premperyear = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap, t)

• Varying life annuity according to a geometric progression: V_avg(px, x, h, n, k = 1, r, cantprem = 1, premperyear = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap, t)

• Varying life insurance according to a geometric progression:V_Avg.(px, x, h, n, k = 1, r, cantprem = 1, premperyear = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap, t)

• Decreasing life annuity: V_aD(x, h, n, k = 1, i = 0.04, data, prop = 1, assumption = "none", variation = "none", cap = 1)

• Decreasing life insurance: V_AD.(px, x, h, n, k = 1, cantprem = 1, premperyear = 1, i = 0.04, data, prop = 1, assumption = "none", variation ="none", cap, t)

Prem<-av(x=45,h=7,n=13,r=0.05, i=0.045, data = CSO80FALB, cap=12000) #Actuarial present value of a varying life annuity that follows an arithmetic progression
AnnualPrem<-PremiumFrac(px1 = Prem, x=45, m=7, k=1, i=0.045,data= CSO80FALB) #Annual premium to be paid at the beginning of each year of the deferral term.
V_av(px= AnnualPrem, x=45, h=7, n=13, r=0.05, cantprem = 7, premperyear = 1, i=0.045, data= CSO80FALB, cap=12000, t=20) #Reserve of the coverage up to the year 20
## Year 1  17228.44     0 1.048860  18070.22
## Year 2  17228.44     0 1.049113  37032.26
## Year 3  17228.44     0 1.049397  56941.01
## Year 4  17228.44     0 1.049703  77855.87
## Year 5  17228.44     0 1.050030  99841.34
## Year 6  17228.44     0 1.050388 122968.75
## Year 7  17228.44     0 1.050779 147316.30
## Year 8      0.00 12000 1.051223 142247.64
## Year 9      0.00 12600 1.051710 136351.71
## Year 10     0.00 13200 1.052208 129581.17
## Year 11     0.00 13800 1.052716 121884.73
## Year 12     0.00 14400 1.053215 113204.54
## Year 13     0.00 15000 1.053693 103477.43
## Year 14     0.00 15600 1.054171  92637.87
## Year 15     0.00 16200 1.054703  80619.27
## Year 16     0.00 16800 1.055342  67351.18
## Year 17     0.00 17400 1.056132  52755.02
## Year 18     0.00 18000 1.057147  36741.15
## Year 19     0.00 18600 1.058367  19200.00
## Year 20     0.00 19200 1.059752      0.00
Prem<- Avg.(x=24, h=10,n=10, k=4, r=0.06, data= CSO80MALB, i=0.03, cap= 100000, assumption = "UDD", variation = "inter") #Actuarial present value of a varying life insurance according to a geometric progression
AnnualPrem<- PremiumFrac(px1 = Prem, x=24, m=10, k=1, data= CSO80MALB, i=0.03) #Annual premium to be paid at the beginning of each year of the deferral term
V_Avg.(px =AnnualPrem, x=24, h=10, n=10, k=4, r=0.06, i=0.03, data= CSO80MALB, cantprem = 10, premperyear = 1, assumption = "UDD", variation = "inter", cap=100000, t=240) #Reserve of the coverage up to the month 240
## Month 1   272.8966  0.00000 1.002617  273.611
## Month 2     0.0000  0.00000 1.002617  274.327
## Month 3     0.0000  0.00000 1.002617  275.044
## Month 4     0.0000  0.00000 1.002617  275.764
## Month 5     0.0000  0.00000 1.002617  276.486
## Month 6     0.0000  0.00000 1.002617  277.209
## Month 7     0.0000  0.00000 1.002617  277.935
## Month 8     0.0000  0.00000 1.002617  278.662
## Month 9     0.0000  0.00000 1.002617  279.391
## Month 10    0.0000  0.00000 1.002617  280.122
## Month 11    0.0000  0.00000 1.002617  280.855
## Month 12    0.0000  0.00000 1.002617  281.590
## Month 13  272.8966  0.00000 1.002612  555.936
## Month 14    0.0000  0.00000 1.002613  557.388
## Month 15    0.0000  0.00000 1.002613  558.844
## Month 16    0.0000  0.00000 1.002613  560.304
## Month 17    0.0000  0.00000 1.002613  561.768
## Month 18    0.0000  0.00000 1.002613  563.236
## Month 19    0.0000  0.00000 1.002613  564.707
## Month 20    0.0000  0.00000 1.002613  566.183
## Month 21    0.0000  0.00000 1.002613  567.662
## Month 22    0.0000  0.00000 1.002613  569.145
## Month 23    0.0000  0.00000 1.002613  570.632
## Month 24    0.0000  0.00000 1.002613  572.123
## Month 25  272.8966  0.00000 1.002610  847.225
## Month 26    0.0000  0.00000 1.002610  849.436
## Month 27    0.0000  0.00000 1.002610  851.653
## Month 28    0.0000  0.00000 1.002610  853.876
## Month 29    0.0000  0.00000 1.002610  856.105
## Month 30    0.0000  0.00000 1.002610  858.339
## Month 31    0.0000  0.00000 1.002610  860.580
## Month 32    0.0000  0.00000 1.002610  862.826
## Month 33    0.0000  0.00000 1.002610  865.078
## Month 34    0.0000  0.00000 1.002610  867.336
## Month 35    0.0000  0.00000 1.002610  869.600
## Month 36    0.0000  0.00000 1.002610  871.870
## Month 37  272.8966  0.00000 1.002609 1147.753
## Month 38    0.0000  0.00000 1.002609 1150.748
## Month 39    0.0000  0.00000 1.002609 1153.750
## Month 40    0.0000  0.00000 1.002609 1156.761
## Month 41    0.0000  0.00000 1.002609 1159.779
## Month 42    0.0000  0.00000 1.002609 1162.805
## Month 43    0.0000  0.00000 1.002609 1165.839
## Month 44    0.0000  0.00000 1.002609 1168.881
## Month 45    0.0000  0.00000 1.002609 1171.931
## Month 46    0.0000  0.00000 1.002609 1174.989
## Month 47    0.0000  0.00000 1.002609 1178.055
## Month 48    0.0000  0.00000 1.002609 1181.129
## Month 49  272.8966  0.00000 1.002608 1457.818
## Month 50    0.0000  0.00000 1.002608 1461.621
## Month 51    0.0000  0.00000 1.002608 1465.433
## Month 52    0.0000  0.00000 1.002608 1469.255
## Month 53    0.0000  0.00000 1.002608 1473.088
## Month 54    0.0000  0.00000 1.002608 1476.930
## Month 55    0.0000  0.00000 1.002608 1480.783
## Month 56    0.0000  0.00000 1.002608 1484.645
## Month 57    0.0000  0.00000 1.002608 1488.518
## Month 58    0.0000  0.00000 1.002608 1492.401
## Month 59    0.0000  0.00000 1.002609 1496.294
## Month 60    0.0000  0.00000 1.002609 1500.197
## Month 61  272.8966  0.00000 1.002610 1777.721
## Month 62    0.0000  0.00000 1.002610 1782.361
## Month 63    0.0000  0.00000 1.002610 1787.013
## Month 64    0.0000  0.00000 1.002610 1791.677
## Month 65    0.0000  0.00000 1.002610 1796.353
## Month 66    0.0000  0.00000 1.002610 1801.042
## Month 67    0.0000  0.00000 1.002610 1805.743
## Month 68    0.0000  0.00000 1.002610 1810.456
## Month 69    0.0000  0.00000 1.002610 1815.182
## Month 70    0.0000  0.00000 1.002610 1819.920
## Month 71    0.0000  0.00000 1.002610 1824.670
## Month 72    0.0000  0.00000 1.002610 1829.433
## Month 73  272.8966  0.00000 1.002612 2107.822
## Month 74    0.0000  0.00000 1.002613 2113.328
## Month 75    0.0000  0.00000 1.002613 2118.850
## Month 76    0.0000  0.00000 1.002613 2124.385
## Month 77    0.0000  0.00000 1.002613 2129.935
## Month 78    0.0000  0.00000 1.002613 2135.500
## Month 79    0.0000  0.00000 1.002613 2141.079
## Month 80    0.0000  0.00000 1.002613 2146.673
## Month 81    0.0000  0.00000 1.002613 2152.281
## Month 82    0.0000  0.00000 1.002613 2157.905
## Month 83    0.0000  0.00000 1.002613 2163.543
## Month 84    0.0000  0.00000 1.002613 2169.195
## Month 85  272.8966  0.00000 1.002617 2448.482
## Month 86    0.0000  0.00000 1.002617 2454.889
## Month 87    0.0000  0.00000 1.002617 2461.313
## Month 88    0.0000  0.00000 1.002617 2467.753
## Month 89    0.0000  0.00000 1.002617 2474.211
## Month 90    0.0000  0.00000 1.002617 2480.685
## Month 91    0.0000  0.00000 1.002617 2487.177
## Month 92    0.0000  0.00000 1.002617 2493.685
## Month 93    0.0000  0.00000 1.002617 2500.211
## Month 94    0.0000  0.00000 1.002617 2506.754
## Month 95    0.0000  0.00000 1.002617 2513.313
## Month 96    0.0000  0.00000 1.002617 2519.891
## Month 97  272.8966  0.00000 1.002623 2800.111
## Month 98    0.0000  0.00000 1.002623 2807.455
## Month 99    0.0000  0.00000 1.002623 2814.817
## Month 100   0.0000  0.00000 1.002623 2822.200
## Month 101   0.0000  0.00000 1.002623 2829.601
## Month 102   0.0000  0.00000 1.002623 2837.022
## Month 103   0.0000  0.00000 1.002623 2844.463
## Month 104   0.0000  0.00000 1.002623 2851.923
## Month 105   0.0000  0.00000 1.002623 2859.402
## Month 106   0.0000  0.00000 1.002623 2866.902
## Month 107   0.0000  0.00000 1.002623 2874.421
## Month 108   0.0000  0.00000 1.002623 2881.960
## Month 109 272.8966  0.00000 1.002629 3163.151
## Month 110   0.0000  0.00000 1.002629 3171.468
## Month 111   0.0000  0.00000 1.002629 3179.807
## Month 112   0.0000  0.00000 1.002629 3188.167
## Month 113   0.0000  0.00000 1.002629 3196.550
## Month 114   0.0000  0.00000 1.002629 3204.955
## Month 115   0.0000  0.00000 1.002629 3213.382
## Month 116   0.0000  0.00000 1.002629 3221.831
## Month 117   0.0000  0.00000 1.002629 3230.302
## Month 118   0.0000  0.00000 1.002629 3238.796
## Month 119   0.0000  0.00000 1.002629 3247.313
## Month 120   0.0000  0.00000 1.002629 3255.851
## Month 121   0.0000 16.95756 1.002638 3247.437
## Month 122   0.0000 17.00228 1.002638 3238.955
## Month 123   0.0000 17.04713 1.002638 3230.406
## Month 124   0.0000 16.96625 1.002638 3221.915
## Month 125   0.0000 17.01100 1.002638 3213.358
## Month 126   0.0000 17.05587 1.002638 3204.733
## Month 127   0.0000 16.97496 1.002638 3196.166
## Month 128   0.0000 17.01973 1.002638 3187.532
## Month 129   0.0000 17.06463 1.002638 3178.831
## Month 130   0.0000 16.98367 1.002638 3170.188
## Month 131   0.0000 17.02847 1.002638 3161.477
## Month 132   0.0000 17.07339 1.002638 3152.698
## Month 133   0.0000 19.02721 1.002648 3141.967
## Month 134   0.0000 19.07758 1.002648 3131.158
## Month 135   0.0000 19.12809 1.002648 3120.269
## Month 136   0.0000 19.03754 1.002648 3109.443
## Month 137   0.0000 19.08794 1.002648 3098.537
## Month 138   0.0000 19.13848 1.002648 3087.552
## Month 139   0.0000 19.04787 1.002648 3076.629
## Month 140   0.0000 19.09831 1.002648 3065.627
## Month 141   0.0000 19.14888 1.002648 3054.544
## Month 142   0.0000 19.05822 1.002648 3043.524
## Month 143   0.0000 19.10869 1.002648 3032.423
## Month 144   0.0000 19.15929 1.002648 3021.243
## Month 145   0.0000 21.56300 1.002660 3007.660
## Month 146   0.0000 21.62036 1.002660 2993.983
## Month 147   0.0000 21.67787 1.002660 2980.212
## Month 148   0.0000 21.57551 1.002660 2966.507
## Month 149   0.0000 21.63291 1.002660 2952.708
## Month 150   0.0000 21.69046 1.002660 2938.815
## Month 151   0.0000 21.58804 1.002660 2924.988
## Month 152   0.0000 21.64547 1.002660 2911.066
## Month 153   0.0000 21.70306 1.002660 2897.050
## Month 154   0.0000 21.60058 1.002660 2883.099
## Month 155   0.0000 21.65805 1.002660 2869.054
## Month 156   0.0000 21.71567 1.002661 2854.914
## Month 157   0.0000 24.53163 1.002674 2837.952
## Month 158   0.0000 24.59723 1.002674 2820.878
## Month 159   0.0000 24.66302 1.002674 2803.694
## Month 160   0.0000 24.54691 1.002674 2786.579
## Month 161   0.0000 24.61256 1.002674 2769.354
## Month 162   0.0000 24.67839 1.002675 2752.016
## Month 163   0.0000 24.56221 1.002675 2734.749
## Month 164   0.0000 24.62790 1.002675 2717.369
## Month 165   0.0000 24.69377 1.002675 2699.878
## Month 166   0.0000 24.57753 1.002675 2682.456
## Month 167   0.0000 24.64327 1.002675 2664.922
## Month 168   0.0000 24.70918 1.002675 2647.274
## Month 169   0.0000 27.98773 1.002690 2626.333
## Month 170   0.0000 28.06302 1.002690 2605.260
## Month 171   0.0000 28.13852 1.002690 2584.055
## Month 172   0.0000 28.00650 1.002690 2562.925
## Month 173   0.0000 28.08184 1.002690 2541.663
## Month 174   0.0000 28.15740 1.002690 2520.268
## Month 175   0.0000 28.02529 1.002691 2498.948
## Month 176   0.0000 28.10069 1.002691 2477.495
## Month 177   0.0000 28.17629 1.002691 2455.909
## Month 178   0.0000 28.04410 1.002691 2434.398
## Month 179   0.0000 28.11956 1.002691 2412.753
## Month 180   0.0000 28.19522 1.002691 2390.974
## Month 181   0.0000 32.10235 1.002709 2365.261
## Month 182   0.0000 32.18930 1.002709 2339.391
## Month 183   0.0000 32.27649 1.002709 2313.364
## Month 184   0.0000 32.12564 1.002709 2287.417
## Month 185   0.0000 32.21266 1.002709 2261.314
## Month 186   0.0000 32.29992 1.002709 2235.052
## Month 187   0.0000 32.14896 1.002709 2208.870
## Month 188   0.0000 32.23605 1.002709 2182.531
## Month 189   0.0000 32.32338 1.002709 2156.032
## Month 190   0.0000 32.17232 1.002709 2129.614
## Month 191   0.0000 32.25948 1.002709 2103.037
## Month 192   0.0000 32.34688 1.002709 2076.300
## Month 193   0.0000 36.96198 1.002729 2044.904
## Month 194   0.0000 37.06286 1.002730 2013.322
## Month 195   0.0000 37.16403 1.002730 1981.552
## Month 196   0.0000 36.99111 1.002730 1949.869
## Month 197   0.0000 37.09208 1.002730 1917.998
## Month 198   0.0000 37.19333 1.002730 1885.939
## Month 199   0.0000 37.02028 1.002730 1853.966
## Month 200   0.0000 37.12135 1.002730 1821.805
## Month 201   0.0000 37.22269 1.002730 1789.454
## Month 202   0.0000 37.04951 1.002730 1757.189
## Month 203   0.0000 37.15066 1.002730 1724.734
## Month 204   0.0000 37.25208 1.002730 1692.089
## Month 205   0.0000 42.53796 1.002752 1654.091
## Month 206   0.0000 42.65502 1.002752 1615.871
## Month 207   0.0000 42.77241 1.002752 1577.428
## Month 208   0.0000 42.57436 1.002752 1539.078
## Month 209   0.0000 42.69153 1.002752 1500.505
## Month 210   0.0000 42.80904 1.002752 1461.708
## Month 211   0.0000 42.61082 1.002753 1423.004
## Month 212   0.0000 42.72811 1.002753 1384.075
## Month 213   0.0000 42.84572 1.002753 1344.921
## Month 214   0.0000 42.64735 1.002753 1305.859
## Month 215   0.0000 42.76474 1.002753 1266.571
## Month 216   0.0000 42.88247 1.002753 1227.057
## Month 217   0.0000 48.91367 1.002776 1181.415
## Month 218   0.0000 49.04947 1.002776 1135.509
## Month 219   0.0000 49.18565 1.002776 1089.339
## Month 220   0.0000 48.95908 1.002777 1043.269
## Month 221   0.0000 49.09502 1.002777  996.935
## Month 222   0.0000 49.23134 1.002777  950.335
## Month 223   0.0000 49.00458 1.002777  903.833
## Month 224   0.0000 49.14066 1.002777  857.066
## Month 225   0.0000 49.27712 1.002777  810.032
## Month 226   0.0000 49.05016 1.002777  763.095
## Month 227   0.0000 49.18638 1.002777  715.892
## Month 228   0.0000 49.32298 1.002777  668.420
## Month 229   0.0000 56.32060 1.002803  613.815
## Month 230   0.0000 56.47847 1.002803  558.899
## Month 231   0.0000 56.63679 1.002803  503.670
## Month 232   0.0000 56.37740 1.002803  448.547
## Month 233   0.0000 56.53545 1.002803  393.110
## Month 234   0.0000 56.69395 1.002804  337.359
## Month 235   0.0000 56.43432 1.002804  281.713
## Month 236   0.0000 56.59254 1.002804  225.751
## Month 237   0.0000 56.75122 1.002804  169.474
## Month 238   0.0000 56.49135 1.002804  113.300
## Month 239   0.0000 56.64975 1.002804   56.809
## Month 240   0.0000 56.80861 1.002804    0.000
Prem<- A.(x=40, h=0, n=15, k=1, i=0.06, data= CSO80FANB, cap=105000) #Actuarial present value of the life insurance
AnnualizedPrem<-PremiumFrac(px1 = Prem, x=40, m=15, k=4, data= CSO80FANB, i=0.06, assumption = "UDD") #Annualized value of the premium to be paid four time a year.
V_A.(px = AnnualizedPrem/4, x=40, h=0, n=15, cantprem = 60, premperyear = 4, data=CSO80FANB, i=0.06, assumption = "UDD", t= 180, cap=105000)  #Reserve of the coverage up to the month 180
## Month 1   98.91566 19.97642 1.005070  79.339
## Month 2    0.00000 20.07770 1.005070  59.562
## Month 3    0.00000 20.17950 1.005070  39.582
## Month 4   98.91566 20.28182 1.005070 118.816
## Month 5    0.00000 20.38465 1.005070  98.930
## Month 6    0.00000 20.48801 1.005070  78.840
## Month 7   98.91566 20.59189 1.005070 157.961
## Month 8    0.00000 20.69631 1.005071 137.960
## Month 9    0.00000 20.80125 1.005071 117.753
## Month 10  98.91566 20.90672 1.005071 196.755
## Month 11   0.00000 21.01273 1.005071 176.633
## Month 12   0.00000 21.11928 1.005071 156.302
## Month 13  98.91566 21.79245 1.005089 234.613
## Month 14   0.00000 21.90335 1.005089 213.792
## Month 15   0.00000 22.01481 1.005089 192.753
## Month 16  98.91566 22.12684 1.005089 270.914
## Month 17   0.00000 22.23944 1.005089 249.940
## Month 18   0.00000 22.35261 1.005089 228.746
## Month 19  98.91566 22.46636 1.005089 306.748
## Month 20   0.00000 22.58069 1.005089 285.613
## Month 21   0.00000 22.69560 1.005089 264.256
## Month 22  98.91566 22.81110 1.005089 342.093
## Month 23   0.00000 22.92719 1.005089 320.790
## Month 24   0.00000 23.04387 1.005089 299.261
## Month 25  98.91566 23.69104 1.005108 376.399
## Month 26   0.00000 23.81205 1.005108 354.387
## Month 27   0.00000 23.93368 1.005108 332.142
## Month 28  98.91566 24.05594 1.005108 409.081
## Month 29   0.00000 24.17882 1.005108 386.868
## Month 30   0.00000 24.30233 1.005108 364.418
## Month 31  98.91566 24.42647 1.005108 441.149
## Month 32   0.00000 24.55125 1.005108 418.726
## Month 33   0.00000 24.67666 1.005108 396.062
## Month 34  98.91566 24.80272 1.005108 472.577
## Month 35   0.00000 24.92942 1.005109 449.934
## Month 36   0.00000 25.05678 1.005109 427.048
## Month 37  98.91566 25.50708 1.005126 503.022
## Month 38   0.00000 25.63783 1.005126 479.832
## Month 39   0.00000 25.76926 1.005127 456.390
## Month 40  98.91566 25.90137 1.005127 532.118
## Month 41   0.00000 26.03416 1.005127 508.679
## Month 42   0.00000 26.16762 1.005127 484.985
## Month 43  98.91566 26.30178 1.005127 560.457
## Month 44   0.00000 26.43662 1.005127 536.759
## Month 45   0.00000 26.57216 1.005127 512.802
## Month 46  98.91566 26.70839 1.005127 588.009
## Month 47   0.00000 26.84532 1.005127 564.040
## Month 48   0.00000 26.98296 1.005127 539.811
## Month 49  98.91566 27.40566 1.005146 614.467
## Month 50   0.00000 27.54668 1.005146 589.940
## Month 51   0.00000 27.68843 1.005146 565.145
## Month 52  98.91566 27.83091 1.005146 639.504
## Month 53   0.00000 27.97412 1.005146 614.676
## Month 54   0.00000 28.11807 1.005146 589.577
## Month 55  98.91566 28.26277 1.005146 663.627
## Month 56   0.00000 28.40821 1.005146 638.488
## Month 57   0.00000 28.55441 1.005146 613.072
## Month 58  98.91566 28.70136 1.005146 686.803
## Month 59   0.00000 28.84906 1.005146 661.340
## Month 60   0.00000 28.99753 1.005146 635.597
## Month 61  98.91566 29.38679 1.005166 708.768
## Month 62   0.00000 29.53860 1.005166 682.739
## Month 63   0.00000 29.69119 1.005166 656.421
## Month 64  98.91566 29.84457 1.005166 729.240
## Month 65   0.00000 29.99875 1.005166 702.854
## Month 66   0.00000 30.15373 1.005166 676.175
## Month 67  98.91566 30.30951 1.005166 748.629
## Month 68   0.00000 30.46609 1.005166 721.873
## Month 69   0.00000 30.62349 1.005166 694.821
## Month 70  98.91566 30.78171 1.005167 766.897
## Month 71   0.00000 30.94074 1.005167 739.759
## Month 72   0.00000 31.10060 1.005167 712.319
## Month 73  98.91566 31.36792 1.005186 783.911
## Month 74   0.00000 31.53059 1.005186 756.283
## Month 75   0.00000 31.69411 1.005186 728.346
## Month 76  98.91566 31.85848 1.005186 799.529
## Month 77   0.00000 32.02370 1.005186 771.485
## Month 78   0.00000 32.18978 1.005186 743.130
## Month 79  98.91566 32.35673 1.005186 813.888
## Month 80   0.00000 32.52455 1.005187 785.416
## Month 81   0.00000 32.69324 1.005187 756.627
## Month 82  98.91566 32.86281 1.005187 826.947
## Month 83   0.00000 33.03326 1.005187 798.032
## Month 84   0.00000 33.20460 1.005187 768.794
## Month 85  98.91566 33.43160 1.005207 838.622
## Month 86   0.00000 33.60568 1.005207 809.208
## Month 87   0.00000 33.78066 1.005207 779.465
## Month 88  98.91566 33.95655 1.005207 848.821
## Month 89   0.00000 34.13337 1.005207 818.930
## Month 90   0.00000 34.31111 1.005207 788.705
## Month 91  98.91566 34.48978 1.005207 857.574
## Month 92   0.00000 34.66939 1.005208 827.190
## Month 93   0.00000 34.84993 1.005208 796.466
## Month 94  98.91566 35.03142 1.005208 864.831
## Month 95   0.00000 35.21386 1.005208 833.937
## Month 96   0.00000 35.39725 1.005208 802.699
## Month 97  98.91566 35.74292 1.005230 870.401
## Month 98   0.00000 35.92987 1.005230 838.835
## Month 99   0.00000 36.11780 1.005231 806.916
## Month 100 98.91566 36.30671 1.005231 874.073
## Month 101  0.00000 36.49662 1.005231 841.958
## Month 102  0.00000 36.68753 1.005231 809.483
## Month 103 98.91566 36.87944 1.005231 876.078
## Month 104  0.00000 37.07236 1.005231 843.394
## Month 105  0.00000 37.26629 1.005231 810.345
## Month 106 98.91566 37.46124 1.005231 876.361
## Month 107  0.00000 37.65722 1.005232 843.091
## Month 108  0.00000 37.85422 1.005232 809.450
## Month 109 98.91566 38.21934 1.005255 874.719
## Month 110  0.00000 38.42020 1.005256 840.694
## Month 111  0.00000 38.62212 1.005256 806.287
## Month 112 98.91566 38.82510 1.005256 870.931
## Month 113  0.00000 39.02916 1.005256 836.275
## Month 114  0.00000 39.23430 1.005256 801.230
## Month 115 98.91566 39.44052 1.005256 865.229
## Month 116  0.00000 39.64784 1.005256 829.921
## Month 117  0.00000 39.85624 1.005257 794.218
## Month 118 98.91566 40.06575 1.005257 857.552
## Month 119  0.00000 40.27637 1.005257 821.572
## Month 120  0.00000 40.48810 1.005257 785.190
## Month 121 98.91566 40.94340 1.005283 847.617
## Month 122  0.00000 41.15970 1.005283 810.718
## Month 123  0.00000 41.37716 1.005283 773.405
## Month 124 98.91566 41.59577 1.005284 835.114
## Month 125  0.00000 41.81555 1.005284 797.491
## Month 126  0.00000 42.03649 1.005284 759.446
## Month 127 98.91566 42.25861 1.005284 820.415
## Month 128  0.00000 42.48191 1.005284 782.044
## Month 129  0.00000 42.70639 1.005284 743.245
## Month 130 98.91566 42.93207 1.005285 803.452
## Month 131  0.00000 43.15895 1.005285 764.311
## Month 132  0.00000 43.38704 1.005285 724.734
## Month 133 98.91566 43.83255 1.005312 783.960
## Month 134  0.00000 44.06540 1.005313 743.825
## Month 135  0.00000 44.29951 1.005313 703.242
## Month 136 98.91566 44.53486 1.005313 761.648
## Month 137  0.00000 44.77147 1.005313 720.686
## Month 138  0.00000 45.00935 1.005313 679.266
## Month 139 98.91566 45.24850 1.005314 736.828
## Month 140  0.00000 45.48894 1.005314 695.013
## Month 141  0.00000 45.73065 1.005314 652.732
## Month 142 98.91566 45.97367 1.005314 709.424
## Month 143  0.00000 46.21798 1.005314 666.731
## Month 144  0.00000 46.46360 1.005315 623.564
## Month 145 98.91566 47.05189 1.005345 679.038
## Month 146  0.00000 47.30338 1.005345 635.111
## Month 147  0.00000 47.55623 1.005346 590.696
## Month 148 98.91566 47.81045 1.005346 645.232
## Month 149  0.00000 48.06603 1.005346 600.358
## Month 150  0.00000 48.32299 1.005346 554.987
## Month 151 98.91566 48.58134 1.005346 608.557
## Month 152  0.00000 48.84108 1.005347 562.709
## Month 153  0.00000 49.10221 1.005347 516.353
## Month 154 98.91566 49.36476 1.005347 568.930
## Month 155  0.00000 49.62872 1.005347 522.078
## Month 156  0.00000 49.89410 1.005348 474.709
## Month 157 98.91566 50.76651 1.005383 525.672
## Month 158  0.00000 51.03978 1.005383 477.188
## Month 159  0.00000 51.31453 1.005383 428.166
## Month 160 98.91566 51.59077 1.005384 478.051
## Month 161  0.00000 51.86851 1.005384 428.477
## Month 162  0.00000 52.14777 1.005384 378.355
## Month 163 98.91566 52.42854 1.005384 427.130
## Month 164  0.00000 52.71083 1.005385 376.435
## Month 165  0.00000 52.99466 1.005385 325.182
## Month 166 98.91566 53.28004 1.005385 372.814
## Month 167  0.00000 53.56696 1.005385 320.967
## Month 168  0.00000 53.85544 1.005386 268.550
## Month 169 98.91566 54.56368 1.005421 314.598
## Month 170  0.00000 54.85949 1.005422 261.147
## Month 171  0.00000 55.15692 1.005422 207.107
## Month 172 98.91566 55.45598 1.005422 251.925
## Month 173  0.00000 55.75668 1.005423 197.232
## Month 174  0.00000 56.05902 1.005423 141.939
## Month 175 98.91566 56.36303 1.005423 185.492
## Month 176  0.00000 56.66869 1.005424 129.522
## Month 177  0.00000 56.97604 1.005424  72.940
## Month 178 98.91566 57.28507 1.005424 115.192
## Month 179  0.00000 57.59579 1.005424  57.908
## Month 180  0.00000 57.90821 1.005425   0.000
Prem<- aD(x=20, h=5, n=5, k=2, i=0.055, data= CSO80MANB, assumption = "constant", variation = "intra", cap=12000) #Actuarial present value of a decreasing life annuity
Annualized_Prem<- PremiumFrac(px1 = Prem, x=20, m=5, k=12, assumption = "constant", data=CSO80MANB, i=0.055) #Annualized value of the premium to be paid monthly
V_aD(px= Annualized_Prem/12 ,x=20,h=5,n=5, k=2, premperyear = 12, cantprem = 60, i=0.055, data= CSO80MANB, assumption = "constant", variation = "intra", cap=12000, t=120) #Reserve up to the month 120
## Month 1   4392.886     0 1.004515   4412.718
## Month 2   4392.886     0 1.004535   8845.539
## Month 3   4392.886     0 1.004556  13298.738
## Month 4   4392.886     0 1.004577  17772.594
## Month 5   4392.886     0 1.004598  22267.392
## Month 6   4392.886     0 1.004619  26783.422
## Month 7   4392.886     0 1.004640  31320.980
## Month 8   4392.886     0 1.004662  35880.367
## Month 9   4392.886     0 1.004684  40461.890
## Month 10  4392.886     0 1.004706  45065.861
## Month 11  4392.886     0 1.004728  49692.599
## Month 12  4392.886     0 1.004751  54342.428
## Month 13  4392.886     0 1.004516  59000.536
## Month 14  4392.886     0 1.004536  63680.977
## Month 15  4392.886     0 1.004557  68384.055
## Month 16  4392.886     0 1.004578  73110.082
## Month 17  4392.886     0 1.004599  77859.375
## Month 18  4392.886     0 1.004620  82632.254
## Month 19  4392.886     0 1.004641  87429.050
## Month 20  4392.886     0 1.004663  92250.097
## Month 21  4392.886     0 1.004685  97095.735
## Month 22  4392.886     0 1.004707 101966.311
## Month 23  4392.886     0 1.004729 106862.180
## Month 24  4392.886     0 1.004752 111783.703
## Month 25  4392.886     0 1.004514 116701.005
## Month 26  4392.886     0 1.004534 121642.982
## Month 27  4392.886     0 1.004555 126609.972
## Month 28  4392.886     0 1.004576 131602.317
## Month 29  4392.886     0 1.004597 136620.368
## Month 30  4392.886     0 1.004618 141664.480
## Month 31  4392.886     0 1.004640 146735.016
## Month 32  4392.886     0 1.004661 151832.346
## Month 33  4392.886     0 1.004683 156956.847
## Month 34  4392.886     0 1.004705 162108.900
## Month 35  4392.886     0 1.004727 167288.899
## Month 36  4392.886     0 1.004750 172497.240
## Month 37  4392.886     0 1.004512 177688.179
## Month 38  4392.886     0 1.004532 182906.260
## Month 39  4392.886     0 1.004553 188151.854
## Month 40  4392.886     0 1.004573 193425.338
## Month 41  4392.886     0 1.004594 198727.098
## Month 42  4392.886     0 1.004616 204057.524
## Month 43  4392.886     0 1.004637 209417.015
## Month 44  4392.886     0 1.004659 214805.977
## Month 45  4392.886     0 1.004681 220224.825
## Month 46  4392.886     0 1.004703 225673.979
## Month 47  4392.886     0 1.004725 231153.870
## Month 48  4392.886     0 1.004747 236664.935
## Month 49  4392.886     0 1.004508 242144.603
## Month 50  4392.886     0 1.004529 247654.008
## Month 51  4392.886     0 1.004549 253193.558
## Month 52  4392.886     0 1.004570 258763.665
## Month 53  4392.886     0 1.004591 264364.750
## Month 54  4392.886     0 1.004612 269997.242
## Month 55  4392.886     0 1.004634 275661.578
## Month 56  4392.886     0 1.004655 281358.201
## Month 57  4392.886     0 1.004677 287087.566
## Month 58  4392.886     0 1.004699 292850.134
## Month 59  4392.886     0 1.004721 298646.374
## Month 60  4392.886     0 1.004744 304476.767
## Month 61     0.000 60000 1.004504 245577.989
## Month 62     0.000     0 1.004525 246689.177
## Month 63     0.000     0 1.004545 247810.466
## Month 64     0.000     0 1.004566 248941.995
## Month 65     0.000     0 1.004587 250083.905
## Month 66     0.000     0 1.004608 251236.338
## Month 67     0.000 54000 1.004630 198149.449
## Month 68     0.000     0 1.004651 199071.053
## Month 69     0.000     0 1.004673 200001.269
## Month 70     0.000     0 1.004695 200940.221
## Month 71     0.000     0 1.004717 201888.029
## Month 72     0.000     0 1.004739 202844.822
## Month 73     0.000 48000 1.004501 155541.812
## Month 74     0.000     0 1.004522 156245.105
## Month 75     0.000     0 1.004542 156954.787
## Month 76     0.000     0 1.004563 157670.945
## Month 77     0.000     0 1.004584 158393.669
## Month 78     0.000     0 1.004605 159123.048
## Month 79     0.000 42000 1.004626 117664.878
## Month 80     0.000     0 1.004648 118211.744
## Month 81     0.000     0 1.004669 118763.717
## Month 82     0.000     0 1.004691 119320.869
## Month 83     0.000     0 1.004713 119883.273
## Month 84     0.000     0 1.004736 120451.004
## Month 85     0.000 36000 1.004500  84831.001
## Month 86     0.000     0 1.004520  85214.433
## Month 87     0.000     0 1.004540  85601.348
## Month 88     0.000     0 1.004561  85991.792
## Month 89     0.000     0 1.004582  86385.814
## Month 90     0.000     0 1.004603  86783.463
## Month 91     0.000 30000 1.004624  57046.056
## Month 92     0.000     0 1.004646  57311.090
## Month 93     0.000     0 1.004668  57578.597
## Month 94     0.000     0 1.004690  57848.613
## Month 95     0.000     0 1.004712  58121.174
## Month 96     0.000     0 1.004734  58396.316
## Month 97     0.000 24000 1.004499  34551.059
## Month 98     0.000     0 1.004519  34707.200
## Month 99     0.000     0 1.004540  34864.759
## Month 100    0.000     0 1.004560  35023.756
## Month 101    0.000     0 1.004581  35184.209
## Month 102    0.000     0 1.004602  35346.139
## Month 103    0.000 18000 1.004624  17426.341
## Month 104    0.000     0 1.004645  17507.288
## Month 105    0.000     0 1.004667  17588.991
## Month 106    0.000     0 1.004689  17671.460
## Month 107    0.000     0 1.004711  17754.705
## Month 108    0.000     0 1.004733  17838.739
## Month 109    0.000 12000 1.004500   5865.011
## Month 110    0.000     0 1.004520   5891.521
## Month 111    0.000     0 1.004540   5918.271
## Month 112    0.000     0 1.004561   5945.266
## Month 113    0.000     0 1.004582   5972.507
## Month 114    0.000     0 1.004603   6000.000
## Month 115    0.000  6000 1.004624      0.000
## Month 116    0.000     0 1.004646      0.000
## Month 117    0.000     0 1.004668      0.000
## Month 118    0.000     0 1.004690      0.000
## Month 119    0.000     0 1.004712      0.000
## Month 120    0.000     0 1.004734      0.000
ages<-c(20,34,36,50)
Px<-Am.(x=ages,h=5,n=30,i=0.06,data=CSO80MANB,ndeath=2,cap=200000) #Actuarial present value of a life insurnace for a group to be paid when the second death occur.
Px/am(x=ages,h=0,n=5,i=0.06,data=CSO80MANB,type="atleast",quant=3) #Annual premium to be paid when at least 3 of the group are still alive
## [1] 5065.443

## Life tables

Creates a mortality table under a certain law of mortality:

• Table_Moivre(x0, omega) de Moivre’s law
• Table_Dormoy(x0, omega, a) Dormoy’s law
• Table_Makeham(x0, omega, A, B, C) Makeham’s law
• Table_Gompertz(x0, omega, B, C) Gompertz’s law

Creates a fractional mortality table for a given mortality table:

Fractional_table(data, frac, i = 0.04, assumption = "UDD")

#### Examples

##      x            q
## 1    0 8.276500e-05
## 2    1 8.855830e-05
## 3    2 9.475708e-05
## 4    3 1.013897e-04
## 5    4 1.084866e-04
## 6    5 1.160803e-04
## 7    6 1.242054e-04
## 8    7 1.328992e-04
## 9    8 1.422015e-04
## 10   9 1.521548e-04
## 11  10 1.628048e-04
## 12  11 1.742001e-04
## 13  12 1.863930e-04
## 14  13 1.994392e-04
## 15  14 2.133984e-04
## 16  15 2.283346e-04
## 17  16 2.443161e-04
## 18  17 2.614160e-04
## 19  18 2.797126e-04
## 20  19 2.992895e-04
## 21  20 3.202364e-04
## 22  21 3.426491e-04
## 23  22 3.666302e-04
## 24  23 3.922892e-04
## 25  24 4.197437e-04
## 26  25 4.491192e-04
## 27  26 4.805500e-04
## 28  27 5.141798e-04
## 29  28 5.501625e-04
## 30  29 5.886625e-04
## 31  30 6.298559e-04
## 32  31 6.739310e-04
## 33  32 7.210892e-04
## 34  33 7.715459e-04
## 35  34 8.255318e-04
## 36  35 8.832935e-04
## 37  36 9.450949e-04
## 38  37 1.011218e-03
## 39  38 1.081965e-03
## 40  39 1.157659e-03
## 41  40 1.238645e-03
## 42  41 1.325292e-03
## 43  42 1.417997e-03
## 44  43 1.517181e-03
## 45  44 1.623298e-03
## 46  45 1.736830e-03
## 47  46 1.858295e-03
## 48  47 1.988246e-03
## 49  48 2.127275e-03
## 50  49 2.276015e-03
## 51  50 2.435142e-03
## 52  51 2.605380e-03
## 53  52 2.787502e-03
## 54  53 2.982336e-03
## 55  54 3.190766e-03
## 56  55 3.413738e-03
## 57  56 3.652262e-03
## 58  57 3.907421e-03
## 59  58 4.180368e-03
## 60  59 4.472338e-03
## 61  60 4.784652e-03
## 62  61 5.118719e-03
## 63  62 5.476046e-03
## 64  63 5.858244e-03
## 65  64 6.267034e-03
## 66  65 6.704253e-03
## 67  66 7.171863e-03
## 68  67 7.671963e-03
## 69  68 8.206791e-03
## 70  69 8.778738e-03
## 71  70 9.390355e-03
## 72  71 1.004437e-02
## 73  72 1.074368e-02
## 74  73 1.149140e-02
## 75  74 1.229084e-02
## 76  75 1.314552e-02
## 77  76 1.405921e-02
## 78  77 1.503592e-02
## 79  78 1.607993e-02
## 80  79 1.719579e-02
## 81  80 1.838836e-02
## 82  81 1.966281e-02
## 83  82 2.102464e-02
## 84  83 2.247970e-02
## 85  84 2.403422e-02
## 86  85 2.569482e-02
## 87  86 2.746853e-02
## 88  87 2.936283e-02
## 89  88 3.138564e-02
## 90  89 3.354538e-02
## 91  90 3.585097e-02
## 92  91 3.831186e-02
## 93  92 4.093805e-02
## 94  93 4.374014e-02
## 95  94 4.672931e-02
## 96  95 4.991737e-02
## 97  96 5.331679e-02
## 98  97 5.694070e-02
## 99  98 6.080293e-02
## 100 99 1.000000e+00
## [1] 132.2875
##              X            Q
## 1    0.0000000 0.0023600000
## 2    0.3333333 0.0023655828
## 3    0.6666667 0.0023711920
## 4    1.0000000 0.0005866667
## 5    1.3333333 0.0005870110
## 6    1.6666667 0.0005873558
## 7    2.0000000 0.0005066667
## 8    2.3333333 0.0005069235
## 9    2.6666667 0.0005071806
## 10   3.0000000 0.0004866667
## 11   3.3333333 0.0004869036
## 12   3.6666667 0.0004871408
## 13   4.0000000 0.0004666667
## 14   4.3333333 0.0004668845
## 15   4.6666667 0.0004671026
## 16   5.0000000 0.0004500000
## 17   5.3333333 0.0004502026
## 18   5.6666667 0.0004504054
## 19   6.0000000 0.0004333333
## 20   6.3333333 0.0004335212
## 21   6.6666667 0.0004337092
## 22   7.0000000 0.0004200000
## 23   7.3333333 0.0004201765
## 24   7.6666667 0.0004203531
## 25   8.0000000 0.0004100000
## 26   8.3333333 0.0004101682
## 27   8.6666667 0.0004103365
## 28   9.0000000 0.0004033333
## 29   9.3333333 0.0004034961
## 30   9.6666667 0.0004036590
## 31  10.0000000 0.0004033333
## 32  10.3333333 0.0004034961
## 33  10.6666667 0.0004036590
## 34  11.0000000 0.0004100000
## 35  11.3333333 0.0004101682
## 36  11.6666667 0.0004103365
## 37  12.0000000 0.0004200000
## 38  12.3333333 0.0004201765
## 39  12.6666667 0.0004203531
## 40  13.0000000 0.0004400000
## 41  13.3333333 0.0004401937
## 42  13.6666667 0.0004403875
## 43  14.0000000 0.0004633333
## 44  14.3333333 0.0004635481
## 45  14.6666667 0.0004637631
## 46  15.0000000 0.0004866667
## 47  15.3333333 0.0004869036
## 48  15.6666667 0.0004871408
## 49  16.0000000 0.0005133333
## 50  16.3333333 0.0005135970
## 51  16.6666667 0.0005138609
## 52  17.0000000 0.0005400000
## 53  17.3333333 0.0005402918
## 54  17.6666667 0.0005405838
## 55  18.0000000 0.0005633333
## 56  18.3333333 0.0005636509
## 57  18.6666667 0.0005639687
## 58  19.0000000 0.0005800000
## 59  19.3333333 0.0005803366
## 60  19.6666667 0.0005806736
## 61  20.0000000 0.0005966667
## 62  20.3333333 0.0005970229
## 63  20.6666667 0.0005973795
## 64  21.0000000 0.0006100000
## 65  21.3333333 0.0006103723
## 66  21.6666667 0.0006107451
## 67  22.0000000 0.0006200000
## 68  22.3333333 0.0006203846
## 69  22.6666667 0.0006207698
## 70  23.0000000 0.0006300000
## 71  23.3333333 0.0006303972
## 72  23.6666667 0.0006307948
## 73  24.0000000 0.0006366667
## 74  24.3333333 0.0006370723
## 75  24.6666667 0.0006374784
## 76  25.0000000 0.0006433333
## 77  25.3333333 0.0006437475
## 78  25.6666667 0.0006441622
## 79  26.0000000 0.0006533333
## 80  26.3333333 0.0006537605
## 81  26.6666667 0.0006541881
## 82  27.0000000 0.0006633333
## 83  27.3333333 0.0006637736
## 84  27.6666667 0.0006642145
## 85  28.0000000 0.0006766667
## 86  28.3333333 0.0006771249
## 87  28.6666667 0.0006775837
## 88  29.0000000 0.0006933333
## 89  29.3333333 0.0006938144
## 90  29.6666667 0.0006942961
## 91  30.0000000 0.0007100000
## 92  30.3333333 0.0007105045
## 93  30.6666667 0.0007110096
## 94  31.0000000 0.0007300000
## 95  31.3333333 0.0007305333
## 96  31.6666667 0.0007310674
## 97  32.0000000 0.0007500000
## 98  32.3333333 0.0007505629
## 99  32.6666667 0.0007511267
## 100 33.0000000 0.0007733333
## 101 33.3333333 0.0007739318
## 102 33.6666667 0.0007745313
## 103 34.0000000 0.0008000000
## 104 34.3333333 0.0008006405
## 105 34.6666667 0.0008012821
## 106 35.0000000 0.0008366667
## 107 35.3333333 0.0008373673
## 108 35.6666667 0.0008380690
## 109 36.0000000 0.0008800000
## 110 36.3333333 0.0008807751
## 111 36.6666667 0.0008815515
## 112 37.0000000 0.0009333333
## 113 37.3333333 0.0009342053
## 114 37.6666667 0.0009350788
## 115 38.0000000 0.0010033333
## 116 38.3333333 0.0010043410
## 117 38.6666667 0.0010053507
## 118 39.0000000 0.0010833333
## 119 39.3333333 0.0010845082
## 120 39.6666667 0.0010856857
## 121 40.0000000 0.0011766667
## 122 40.3333333 0.0011780528
## 123 40.6666667 0.0011794423
## 124 41.0000000 0.0012800000
## 125 41.3333333 0.0012816405
## 126 41.6666667 0.0012832852
## 127 42.0000000 0.0013900000
## 128 42.3333333 0.0013919348
## 129 42.6666667 0.0013938750
## 130 43.0000000 0.0015100000
## 131 43.3333333 0.0015122835
## 132 43.6666667 0.0015145740
## 133 44.0000000 0.0016400000
## 134 44.3333333 0.0016426940
## 135 44.6666667 0.0016453969
## 136 45.0000000 0.0017833333
## 137 45.3333333 0.0017865193
## 138 45.6666667 0.0017897167
## 139 46.0000000 0.0019433333
## 140 46.3333333 0.0019471172
## 141 46.6666667 0.0019509159
## 142 47.0000000 0.0021200000
## 143 47.3333333 0.0021245039
## 144 47.6666667 0.0021290271
## 145 48.0000000 0.0023166667
## 146 48.3333333 0.0023220461
## 147 48.6666667 0.0023274505
## 148 49.0000000 0.0025333333
## 149 49.3333333 0.0025397674
## 150 49.6666667 0.0025462343
## 151 50.0000000 0.0027733333
## 152 50.3333333 0.0027810461
## 153 50.6666667 0.0027888019
## 154 51.0000000 0.0030366667
## 155 51.3333333 0.0030459161
## 156 51.6666667 0.0030552220
## 157 52.0000000 0.0033200000
## 158 52.3333333 0.0033310591
## 159 52.6666667 0.0033421922
## 160 53.0000000 0.0036300000
## 161 53.3333333 0.0036432249
## 162 53.6666667 0.0036565465
## 163 54.0000000 0.0039666667
## 164 54.3333333 0.0039824638
## 165 54.6666667 0.0039983872
## 166 55.0000000 0.0043333333
## 167 55.3333333 0.0043521928
## 168 55.6666667 0.0043712172
## 169 56.0000000 0.0047366667
## 170 56.3333333 0.0047592095
## 171 56.6666667 0.0047819678
## 172 57.0000000 0.0051800000
## 173 57.3333333 0.0052069721
## 174 57.6666667 0.0052342266
## 175 58.0000000 0.0056666667
## 176 58.3333333 0.0056989608
## 177 58.6666667 0.0057316251
## 178 59.0000000 0.0061966667
## 179 59.3333333 0.0062353048
## 180 59.6666667 0.0062744277
## 181 60.0000000 0.0067800000
## 182 60.3333333 0.0068262822
## 183 60.6666667 0.0068732006
## 184 61.0000000 0.0074133333
## 185 61.3333333 0.0074687013
## 186 61.6666667 0.0075249026
## 187 62.0000000 0.0081033333
## 188 62.3333333 0.0081695338
## 189 62.6666667 0.0082368248
## 190 63.0000000 0.0088566667
## 191 63.3333333 0.0089358081
## 192 63.6666667 0.0090163768
## 193 64.0000000 0.0096800000
## 194 64.3333333 0.0097746183
## 195 64.6666667 0.0098711046
## 196 65.0000000 0.0105833333
## 197 65.3333333 0.0106965384
## 198 65.6666667 0.0108121914
## 199 66.0000000 0.0115800000
## 200 66.3333333 0.0117156674
## 201 66.6666667 0.0118545514
## 202 67.0000000 0.0126800000
## 203 67.3333333 0.0128428473
## 204 67.6666667 0.0130099319
## 205 68.0000000 0.0138933333
## 206 68.3333333 0.0140890776
## 207 68.6666667 0.0142904164
## 208 69.0000000 0.0152033333
## 209 69.3333333 0.0154380430
## 210 69.6666667 0.0156801133
## 211 70.0000000 0.0165966667
## 212 70.3333333 0.0168767647
## 213 70.6666667 0.0171664793
## 214 71.0000000 0.0180500000
## 215 71.3333333 0.0183817913
## 216 71.6666667 0.0187260089
## 217 72.0000000 0.0195500000
## 218 72.3333333 0.0199398236
## 219 72.6666667 0.0203455094
## 220 73.0000000 0.0210866667
## 221 73.3333333 0.0215408923
## 222 73.6666667 0.0220151176
## 223 74.0000000 0.0227066667
## 224 74.3333333 0.0232342388
## 225 74.6666667 0.0237869095
## 226 75.0000000 0.0244566667
## 227 75.3333333 0.0250697902
## 228 75.6666667 0.0257144459
## 229 76.0000000 0.0263933333
## 230 76.3333333 0.0271088256
## 231 76.6666667 0.0278641911
## 232 77.0000000 0.0285666667
## 233 77.3333333 0.0294067186
## 234 77.6666667 0.0302976738
## 235 78.0000000 0.0310200000
## 236 78.3333333 0.0320130446
## 237 78.6666667 0.0330717728
## 238 79.0000000 0.0337300000
## 239 79.3333333 0.0349074275
## 240 79.6666667 0.0361700302
## 241 80.0000000 0.0366600000
## 242 80.3333333 0.0380551000
## 243 80.6666667 0.0395605819
## 244 81.0000000 0.0397833333
## 245 81.3333333 0.0414316213
## 246 81.6666667 0.0432223952
## 247 82.0000000 0.0430566667
## 248 82.3333333 0.0449939565
## 249 82.6666667 0.0471137924
## 250 83.0000000 0.0464600000
## 251 83.3333333 0.0487237033
## 252 83.6666667 0.0512192971
## 253 84.0000000 0.0500033333
## 254 84.3333333 0.0526352724
## 255 84.6666667 0.0555596708
## 256 85.0000000 0.0537133333
## 257 85.3333333 0.0567622215
## 258 85.6666667 0.0601780619
## 259 86.0000000 0.0576066667
## 260 86.3333333 0.0611280499
## 261 86.6666667 0.0651079733
## 262 87.0000000 0.0617100000
## 263 87.3333333 0.0657685790
## 264 87.6666667 0.0703985945
## 265 88.0000000 0.0660833333
## 266 88.3333333 0.0707593468
## 267 88.6666667 0.0761474938
## 268 89.0000000 0.0708200000
## 269 89.3333333 0.0762177404
## 270 89.6666667 0.0825061746
## 271 90.0000000 0.0760466667
## 272 90.3333333 0.0823057442
## 273 90.6666667 0.0896875442
## 274 91.0000000 0.0819233333
## 275 91.3333333 0.0892336515
## 276 91.6666667 0.0979764477
## 277 92.0000000 0.0886433333
## 278 92.3333333 0.0972652492
## 279 92.6666667 0.1077451036
## 280 93.0000000 0.0964333333
## 281 93.3333333 0.1067252001
## 282 93.6666667 0.1194763360
## 283 94.0000000 0.1055533333
## 284 94.3333333 0.1180096447
## 285 94.6666667 0.1337992462
## 286 95.0000000 0.1170800000
## 287 95.3333333 0.1326054456
## 288 95.6666667 0.1528778857
## 289 96.0000000 0.1335200000
## 290 96.3333333 0.1540947281
## 291 96.6666667 0.1821654661
## 292 97.0000000 0.1628066667
## 293 97.3333333 0.1944672278
## 294 97.6666667 0.2414144210
## 295 98.0000000 0.2227166667
## 296 98.3333333 0.2865321526
## 297 98.6666667 0.4016048566
## 298 99.0000000 0.3333333333
## 299 99.3333333 0.5000000000
## 300 99.6666667 1.0000000000

## Loan Amortization

Loan_amortization(V0, n, i, i2 = 0, alic = 0, ins = 0, method) For calculating the loan amortization schedule.

Loan_amortization(V0=130000,n=15,i=0.04,method = "interest_only")
##    Period Payment  Int TAX  Amort Insurance Outstanding_debt
## 1       0       0    0   0      0         0           130000
## 2       1    5200 5200   0      0         0           130000
## 3       2    5200 5200   0      0         0           130000
## 4       3    5200 5200   0      0         0           130000
## 5       4    5200 5200   0      0         0           130000
## 6       5    5200 5200   0      0         0           130000
## 7       6    5200 5200   0      0         0           130000
## 8       7    5200 5200   0      0         0           130000
## 9       8    5200 5200   0      0         0           130000
## 10      9    5200 5200   0      0         0           130000
## 11     10    5200 5200   0      0         0           130000
## 12     11    5200 5200   0      0         0           130000
## 13     12    5200 5200   0      0         0           130000
## 14     13    5200 5200   0      0         0           130000
## 15     14    5200 5200   0      0         0           130000
## 16     15  135200 5200   0 130000         0                0
Loan_amortization(V0=123000,n=12,i=0.06,alic=0.21,method="constant_installment")
##    Period  Payment Pure_Payment       Int     Amort       TAX Insurance
## 1       0     0.00         0.00    0.0000     0.000    0.0000         0
## 2       1 16220.87     14671.07 7380.0000  7291.075 1549.8000         0
## 3       2 16129.01     14671.07 6942.5355  7728.539 1457.9325         0
## 4       3 16031.63     14671.07 6478.8232  8192.251 1360.5529         0
## 5       4 15928.41     14671.07 5987.2881  8683.787 1257.3305         0
## 6       5 15818.99     14671.07 5466.2609  9204.814 1147.9148         0
## 7       6 15703.01     14671.07 4913.9721  9757.103 1031.9341         0
## 8       7 15580.07     14671.07 4328.5459 10342.529  908.9946         0
## 9       8 15449.75     14671.07 3707.9942 10963.080  778.6788         0
## 10      9 15311.62     14671.07 3050.2094 11620.865  640.5440         0
## 11     10 15165.20     14671.07 2352.9575 12318.117  494.1211         0
## 12     11 15009.99     14671.07 1613.8704 13057.204  338.9128         0
## 13     12 14845.47     14671.07  830.4382 13840.636  174.3920         0
##    Outstanding_debt
## 1      1.230000e+05
## 2      1.157089e+05
## 3      1.079804e+05
## 4      9.978813e+04
## 5      9.110435e+04
## 6      8.189953e+04
## 7      7.214243e+04
## 8      6.179990e+04
## 9      5.083682e+04
## 10     3.921596e+04
## 11     2.689784e+04
## 12     1.384064e+04
## 13     3.074092e-10
Loan_amortization(V0=300000,n=10,i=0.03,ins=0.01,method = "constant_principal")
##    Period Payment Pure_Payment  Int Amort TAX Insurance Outstanding_debt
## 1       0       0            0    0     0   0         0           300000
## 2       1   42000        39000 9000 30000   0      3000           270000
## 3       2   40800        38100 8100 30000   0      2700           240000
## 4       3   39600        37200 7200 30000   0      2400           210000
## 5       4   38400        36300 6300 30000   0      2100           180000
## 6       5   37200        35400 5400 30000   0      1800           150000
## 7       6   36000        34500 4500 30000   0      1500           120000
## 8       7   34800        33600 3600 30000   0      1200            90000
## 9       8   33600        32700 2700 30000   0       900            60000
## 10      9   32400        31800 1800 30000   0       600            30000
## 11     10   31200        30900  900 30000   0       300                0

Payment_Protection(x, n, k = 1, V0, i = 0.04, ip = 0.04, data, prop = 1, type = "outstanding_debt", method = "interest_only") For calculating the actuarial present value of a loan insurance.

If “outstanding_debt” is chosen, the insurance company pays the outstanding debt at the moment of the next theoretical payment. If “payments” is selected, the company takes responsibility for the payment of each installment at each accorded moment.

Px1<-Payment_Protection(x=35,n=15,V0=1050000,i=0.05,ip=0.055,data=CSO58MALB,type="outstanding_debt",method="constant_principal") #Actuarial present value of the coverage
V_Payment_Protection(px=Annual_Px1,x=35,n=15,k=1,cantprem=5,premperyear=1,i=0.05,ip=0.055,data =CSO58MALB, type="outstanding_debt",method="constant_principal",V0=1050000,t=15) #Reserve uo to the year 15
## Year 1  5339.352 2716.6250 1.052711  2760.973
## Year 2  5339.352 2678.1949 1.052864  5708.765
## Year 3  5339.352 2656.0469 1.053059  8837.345
## Year 4  5339.352 2641.4668 1.053297 12150.018
## Year 5  5339.352 2622.5753 1.053571 15663.230
## Year 6     0.000 2591.6427 1.053883 13775.928
## Year 7     0.000 2535.0384 1.054222 11850.393
## Year 8     0.000 2447.3749 1.054587  9916.301
## Year 9     0.000 2326.0288 1.054984  8007.618
## Year 10    0.000 2166.7590 1.055419  6164.553
## Year 11    0.000 1965.6057 1.055902  4433.677
## Year 12    0.000 1714.4735 1.056438  2872.670
## Year 13    0.000 1404.0151 1.057034  1552.417
## Year 14    0.000 1023.1953 1.057694   559.755
## Year 15    0.000  559.7549 1.058424     0.000
Px1 <- Payment_Protection(x=30,n=10,k=2,V0=1000000,i=0.035,ip=0.06,data=CSO80FANB,type="payments",method="constant_instalment") #Actuarial present value of the coverage
Annual_Px1 <- PremiumFrac(px1=Px1,x=30,m=3, k = 6,i=0.035,data=CSO80FANB, assumption = "UDD") #Annualized value of the premium to be paid every two months
V_Payment_Protection(px=Annual_Px1/6,x=30,n=10,k=2,cantprem=18,premperyear=6,i=0.035,ip=0.06,data =CSO80FANB, type="payments",method="constant_instalment",V0=1000000,t=120) #Reserve up to the month 120
## Month 1   496.049 126.35553 1.002984  370.796
## Month 2     0.000 126.73255 1.002984  244.792
## Month 3   496.049 127.11068 1.002984  615.562
## Month 4     0.000 127.48995 1.002984  489.528
## Month 5   496.049 127.87035 1.002984  860.266
## Month 6     0.000 128.25189 1.002984  734.198
## Month 7   496.049 121.09823 1.002984 1112.458
## Month 8     0.000 121.45957 1.002984  993.956
## Month 9   496.049 121.82198 1.002984 1372.265
## Month 10    0.000 122.18548 1.002984 1253.810
## Month 11  496.049 122.55006 1.002984 1632.164
## Month 12    0.000 122.91573 1.002984 1513.752
## Month 13  496.049 119.86573 1.002988 1895.582
## Month 14    0.000 120.22388 1.002988 1780.663
## Month 15  496.049 120.58310 1.002988 2162.571
## Month 16    0.000 120.94340 1.002988 2047.728
## Month 17  496.049 121.30477 1.002988 2429.710
## Month 18    0.000 121.66723 1.002988 2314.940
## Month 19  496.049 114.21511 1.002988 2704.831
## Month 20    0.000 114.55639 1.002988 2598.015
## Month 21  496.049 114.89868 1.002988 2988.067
## Month 22    0.000 115.24200 1.002988 2881.409
## Month 23  496.049 115.58635 1.002988 3271.618
## Month 24    0.000 115.93173 1.002988 3165.116
## Month 25  496.049 112.17320 1.002992 3559.611
## Month 26    0.000 112.50884 1.002992 3457.416
## Month 27  496.049 112.84548 1.002992 3852.111
## Month 28    0.000 113.18312 1.002992 3750.115
## Month 29  496.049 113.52178 1.002992 4145.008
## Month 30    0.000 113.86146 1.002992 4043.208
## Month 31  496.049 106.10716 1.002992 4446.415
## Month 32    0.000 106.42465 1.002992 4352.976
## Month 33  496.049 106.74310 1.002992 4756.472
## Month 34    0.000 107.06249 1.002992 4663.322
## Month 35  496.049 107.38285 1.002992 5067.104
## Month 36    0.000 107.70417 1.002992 4974.240
## Month 37    0.000 103.22140 1.002996 4885.614
## Month 38    0.000 103.53068 1.002996 4796.411
## Month 39    0.000 103.84089 1.002996 4706.631
## Month 40    0.000 104.15202 1.002996 4616.269
## Month 41    0.000 104.46410 1.002996 4525.324
## Month 42    0.000 104.77711 1.002996 4433.793
## Month 43    0.000  96.71672 1.002996 4350.071
## Month 44    0.000  97.00652 1.002996 4265.809
## Month 45    0.000  97.29719 1.002996 4181.002
## Month 46    0.000  97.58873 1.002996 4095.649
## Month 47    0.000  97.88115 1.002996 4009.747
## Month 48    0.000  98.17444 1.002996 3923.293
## Month 49    0.000  94.75034 1.003003 3840.040
## Month 50    0.000  95.03487 1.003003 3756.251
## Month 51    0.000  95.32026 1.003003 3671.924
## Month 52    0.000  95.60651 1.003003 3587.058
## Month 53    0.000  95.89362 1.003003 3501.648
## Month 54    0.000  96.18159 1.003003 3415.693
## Month 55    0.000  87.64910 1.003003 3338.039
## Month 56    0.000  87.91232 1.003003 3259.887
## Month 57    0.000  88.17633 1.003003 3181.235
## Month 58    0.000  88.44113 1.003003 3102.082
## Month 59    0.000  88.70673 1.003003 3022.425
## Month 60    0.000  88.97313 1.003003 2942.262
## Month 61    0.000  83.84191 1.003009 2867.020
## Month 62    0.000  84.09417 1.003009 2791.299
## Month 63    0.000  84.34720 1.003009 2715.097
## Month 64    0.000  84.60098 1.003009 2638.411
## Month 65    0.000  84.85554 1.003009 2561.239
## Month 66    0.000  85.11086 1.003009 2483.578
## Month 67    0.000  76.15449 1.003009 2414.667
## Month 68    0.000  76.38363 1.003009 2345.320
## Month 69    0.000  76.61347 1.003009 2275.533
## Month 70    0.000  76.84399 1.003009 2205.304
## Month 71    0.000  77.07522 1.003009 2134.633
## Month 72    0.000  77.30713 1.003009 2063.516
## Month 73    0.000  72.75408 1.003018 1996.770
## Month 74    0.000  72.97366 1.003018 1929.603
## Month 75    0.000  73.19389 1.003018 1862.012
## Month 76    0.000  73.41479 1.003018 1793.995
## Month 77    0.000  73.63637 1.003018 1725.551
## Month 78    0.000  73.85861 1.003018 1656.677
## Month 79    0.000  64.25436 1.003018 1597.229
## Month 80    0.000  64.44828 1.003018 1537.407
## Month 81    0.000  64.64280 1.003018 1477.209
## Month 82    0.000  64.83790 1.003018 1416.634
## Month 83    0.000  65.03360 1.003018 1355.680
## Month 84    0.000  65.22988 1.003018 1294.345
## Month 85    0.000  59.59202 1.003029 1238.493
## Month 86    0.000  59.77251 1.003029 1182.291
## Month 87    0.000  59.95356 1.003029 1125.736
## Month 88    0.000  60.13515 1.003029 1068.829
## Month 89    0.000  60.31730 1.003029 1011.566
## Month 90    0.000  60.50000 1.003029  953.947
## Month 91    0.000  50.12953 1.003029  906.555
## Month 92    0.000  50.28137 1.003029  858.868
## Month 93    0.000  50.43368 1.003029  810.883
## Month 94    0.000  50.58645 1.003029  762.599
## Month 95    0.000  50.73968 1.003029  714.016
## Month 96    0.000  50.89337 1.003029  665.131
## Month 97    0.000  43.61424 1.003041  623.407
## Month 98    0.000  43.74689 1.003041  581.424
## Month 99    0.000  43.87994 1.003041  539.179
## Month 100   0.000  44.01340 1.003042  496.671
## Month 101   0.000  44.14727 1.003042  453.900
## Month 102   0.000  44.28154 1.003042  410.865
## Month 103   0.000  33.02405 1.003042  378.990
## Month 104   0.000  33.12450 1.003042  346.917
## Month 105   0.000  33.22525 1.003042  314.646
## Month 106   0.000  33.32631 1.003042  282.176
## Month 107   0.000  33.42768 1.003042  249.504
## Month 108   0.000  33.52936 1.003042  216.632
## Month 109   0.000  24.13943 1.003056  193.081
## Month 110   0.000  24.21321 1.003056  169.384
## Month 111   0.000  24.28722 1.003057  145.540
## Month 112   0.000  24.36145 1.003057  121.549
## Month 113   0.000  24.43592 1.003057   97.410
## Month 114   0.000  24.51061 1.003057   73.122
## Month 115   0.000  12.18704 1.003057   61.121
## Month 116   0.000  12.22430 1.003057   49.047
## Month 117   0.000  12.26166 1.003057   36.897
## Month 118   0.000  12.29914 1.003057   24.673
## Month 119   0.000  12.33674 1.003057   12.374
## Month 120   0.000  12.37445 1.003057    0.000

## Annuities and Interest Rates

sf(l = 0, n, i) For calculating the final value of an annuity

af(l = 0, n, i) For calculating the present value of an annuity

Rate_converter(num, rate1, m, rate2, k, type = "days") For converting nominal and effective interest and discount rates.

i<-Rate_converter(0.4,"j",60,"i",365, "days") #Convert the annual nominal rate of 2 periods to a effective annual interest rate
af(0,20,i)*1200 # Calculate the present value of an annuity-due of amount $1200 paid annually for 20 years at the rate of interest of i. ## [1] 3734.614 i<-Rate_converter(0.07,"f",4,"i",1, "frac") #Convert the annual nominal discount rate of 4 periods to a effective annual interest rate sf(1,10,i)*3000 # Calculate the final value of an annuity-immediate of amount$3000 paid annually for 20 years at the rate of interest of i.
## [1] 42075.79