SULT

SULT#

Standard ultimate life table#

This tabulates single net premiums and basic functions (whole life and endowment insurances, whole life and temporary annuities and pure endowments) for several time periods at integer ages between 20 and 100 years. According to the SOA’s “Excel Workbook for FAM-L Tables”, this table was developed from the following assumptions:

constant interest rate \(i=0.05\)
radix of 100000 initial lives aged 20
incorporates Makeham’s Law as its survival model with \(A = 0.00022,~ B = 0.0000027, ~ c = 1.124\)

Pure endowment#

Pure endowment functions can be calculated from numbers of lives survived and compounded interest rates.

\(_tE_x = v^t ~ \dfrac{l_{x+t}}{l_x}\)

\({^2_t}E_x = v^{2t} ~ \dfrac{l_{x+t}}{l_x} = v^t ~ _tE_x\)

Term life insurance#

Term life insurance functions can be calculated from whole life insurance and pure endowment table columns.

\(A^ {1}_{x:\overline{t|}} = A_x - ~_tE_x ~ A_{x+t} = A_{x:\overline{t|}} - ~_tE_x\)

\({^2}A^ {1}_{x:\overline{t|}} = {^2}A_x - ~{^2_t}E_x ~ {^2}A_{x+t} = {^2}A_x - ~v^{t} ~ _tE_x ~ {^2}A_{x+t}\)

Methods#

The SULT class implements an instance of a LifeTable, called the standard ultimate life table, which is based on Makeham’s Law with parameters specified in SOA’s “Excel Workbook for FAM-L Tables”

import math
from actuarialmath import SULT
import describe
describe.methods(SULT)

class SULT - Generates and uses a standard ultimate life table

    Args:
      i : interest rate
      radix : initial number of lives
      minage : minimum age
      maxage : maximum age
      S : survival function, default is Makeham with SOA FAM-L parameters

    Examples:
      >>> sult = SULT()
      >>> a = sult.temporary_annuity(70, t=10)
      >>> A = sult.deferred_annuity(70, u=10)
      >>> P = sult.gross_premium(a=a, A=A, benefit=100000, initial_premium=0.75,
      >>>                        renewal_premium=0.05)

    Methods:
    --------

    frame(minage, maxage):
      Derive FAM-L exam table columns of SULT as a DataFrame

    __getitem__(col):
      Returns a column of the sult table

Examples#

SOA Question 6.52

For a fully discrete 10-payment whole life insurance of H on (45), you are given:

Expenses payable at the beginning of each year are as follows:

Expense Type	First Year	Years 2-10	Years 11+
Per policy	100	20	10
% of Premium	105%	5%	0%

Mortality follows the Standard Ultimate Life Table
i = 0.05
The gross annual premium, calculated using the equivalence principle, is of the form \(G = gH + f\), where \(g\) is the premium rate per 1 of insurance and \(f\) is the per policy fee

Calculate \(f\).

print("SOA Question 6.52:  (D) 50.80")
sult = SULT()
a = sult.temporary_annuity(45, t=10)
other_cost = 10 * sult.deferred_annuity(45, u=10)
P = sult.gross_premium(a=a, 
                       A=0, 
                       benefit=0, 
                       initial_premium=1.05, 
                       renewal_premium=0.05,
                       initial_policy=100 + other_cost, 
                       renewal_policy=20)
print(a, P)

SOA Question 6.52:  (D) 50.80
8.0750937741422 50.80135534704229

SOA Question 6.47

For a 10-year deferred whole life annuity-due with payments of 100,000 per year on (70), you are given:

Annual gross premiums of \(G\) are payable for 10 years
First year expenses are 75% of premium
Renewal expenses for years 2 and later are 5% of premium during the premium paying period
Mortality follows the Standard Ultimate Life Table
i = 0.05

Calculate \(G\) using the equivalence principle.

print("SOA Question 6.47:  (D) 66400")
sult = SULT()
a = sult.temporary_annuity(70, t=10)
A = sult.deferred_annuity(70, u=10)
P = sult.gross_premium(a=a, A=A, benefit=100000, initial_premium=0.75,
                        renewal_premium=0.05)
print(P)

SOA Question 6.47:  (D) 66400
66384.13293704337

SOA Question 6.43

For a fully discrete, 5-payment 10-year term insurance of 200,000 on (30), you are given:

Mortality follows the Standard Ultimate Life Table
The following expenses are incurred at the beginning of each respective year:

	Percent of Premium	Per Policy	Percent of Premium	Per Policy
	Year 1	Year 1	Years 2 - 10	Years 2 - 10
Taxes	5%	0	5%	0
Commissions	30%	0	10%	0
Maintenance	0%	8	0%	4

i = 0.05
\(\ddot{a}_{30:\overline{5|}} = 4.5431\)

Calculate the annual gross premium using the equivalence principle.

print("SOA Question 6.43:  (C) 170")
sult = SULT()
a = sult.temporary_annuity(30, t=5)
A = sult.term_insurance(30, t=10)
other_expenses = 4 * sult.deferred_annuity(30, u=5, t=5)
P = sult.gross_premium(a=a, A=A, benefit=200000, initial_premium=0.35,
                        initial_policy=8 + other_expenses, renewal_policy=4,
                        renewal_premium=0.15)
print(P)

SOA Question 6.43:  (C) 170
171.22371939459944

SOA Question 6.39

XYZ Insurance writes 10,000 fully discrete whole life insurance policies of 1000 on lives age 40 and an additional 10,000 fully discrete whole life policies of 1000 on lives age 80.

XYZ used the following assumptions to determine the net premiums for these policies:

Mortality follows the Standard Ultimate Life Table
i = 0.05

During the first ten years, mortality did follow the Standard Ultimate Life Table.

Calculate the average net premium per policy in force received at the beginning of the eleventh year.

print("SOA Question 6.39:  (A) 29")
sult = SULT()
P40 = sult.premium_equivalence(sult.whole_life_insurance(40), b=1000)
P80 = sult.premium_equivalence(sult.whole_life_insurance(80), b=1000)
p40 = sult.p_x(40, t=10)
p80 = sult.p_x(80, t=10)
P = (P40 * p40 + P80 * p80) / (p80 + p40)
print(P)

SOA Question 6.39:  (A) 29
29.033866427845496

SOA Question 6.37

For a fully discrete whole life insurance policy of 50,000 on (35), with premiums payable for a maximum of 10 years, you are given:

Expenses of 100 are payable at the end of each year including the year of death
Mortality follows the Standard Ultimate Life Table
i = 0.05

Calculate the annual gross premium using the equivalence principle.

print("SOA Question 6.37:  (D) 820")
sult = SULT()
benefits = sult.whole_life_insurance(35, b=50000 + 100)
expenses = sult.immediate_annuity(35, b=100)
a = sult.temporary_annuity(35, t=10)
print(benefits, expenses, a)
print((benefits + expenses) / a)

SOA Question 6.37:  (D) 820
4836.382819496279 1797.2773668474615 8.092602358383987
819.7190338249138

SOA Question 6.35

For a fully discrete whole life insurance policy of 100,000 on (35), you are given:

First year commissions are 19% of the annual gross premium
Renewal year commissions are 4% of the annual gross premium
Mortality follows the Standard Ultimate Life Table
i = 0.05

Calculate the annual gross premium for this policy using the equivalence principle.

print("SOA Question 6.35:  (D) 530")
sult = SULT()
A = sult.whole_life_insurance(35, b=100000)
a = sult.whole_life_annuity(35)
print(sult.gross_premium(a=a, A=A, initial_premium=.19, renewal_premium=.04))

SOA Question 6.35:  (D) 530
534.4072234303344

SOA Question 5.8

For an annual whole life annuity-due of 1 with a 5-year certain period on (55), you are given:

Mortality follows the Standard Ultimate Life Table
i = 0.05

Calculate the probability that the sum of the undiscounted payments actually made under this annuity will exceed the expected present value, at issue, of the annuity.

print("SOA Question 5.8: (C) 0.92118")
sult = SULT()
a = sult.certain_life_annuity(55, u=5)
print(sult.p_x(55, t=math.floor(a)))

SOA Question 5.8: (C) 0.92118
0.9211799771029529

SOA Question 5.3

You are given:

Mortality follows the Standard Ultimate Life Table
Deaths are uniformly distributed over each year of age
i = 0.05

Calculate \(\frac{d}{dt}(\overline{I}\overline{a})_{40:\overline{t|}}\) at \(t = 10.5\).

print("SOA Question 5.3:  (C) 6.239")
sult = SULT()
t = 10.5
print(t * sult.E_r(40, t=t))

SOA Question 5.3:  (C) 6.239
6.23871918627528

SOA Question 4.17

For a special whole life policy on (48), you are given:

The policy pays 5000 if the insured’s death is before the median curtate future lifetime at issue and 10,000 if death is after the median curtate future lifetime at issue
Mortality follows the Standard Ultimate Life Table
Death benefits are paid at the end of the year of death
i = 0.05

Calculate the actuarial present value of benefits for this policy.

print("SOA Question 4.17:  (A) 1126.7")
sult = SULT()
median = sult.Z_t(48, prob=0.5, discrete=False)
benefit = lambda x,t: 5000 if t < median else 10000
print(sult.A_x(48, benefit=benefit))

SOA Question 4.17:  (A) 1126.7
1126.774772894844

SOA Question 4.14

A fund is established for the benefit of 400 workers all age 60 with independent future lifetimes. When they reach age 85, the fund will be dissolved and distributed to the survivors.

The fund will earn interest at a rate of 5% per year.

The initial fund balance, \(F\), is determined so that the probability that the fund will pay at least 5000 to each survivor is 86%, using the normal approximation.

Mortality follows the Standard Ultimate Life Table.

Calculate \(F\).

print("SOA Question 4.14:  (E) 390000    ")
sult = SULT()
p = sult.p_x(60, t=85-60)
mean = sult.bernoulli(p)
var = sult.bernoulli(p, variance=True)
F = sult.portfolio_percentile(mean=mean, variance=var, prob=.86, N=400)
print(F * 5000 * sult.interest.v_t(85-60))

SOA Question 4.14:  (E) 390000    
389322.86778416135

SOA Question 4.5

For a 30-year term life insurance of 100,000 on (45), you are given:

The death benefit is payable at the moment of death
Mortality follows the Standard Ultimate Life Table
\(\delta = 0.05\)
Deaths are uniformly distributed over each year of age

Calculate the 95th percentile of the present value of benefits random variable for this insurance

print("SOA Question 4.5:  (C) 35200")
sult = SULT(udd=True).set_interest(delta=0.05)
Z = 100000 * sult.Z_from_prob(45, prob=0.95, discrete=False)
print(Z)

SOA Question 4.5:  (C) 35200
35187.952037196534

SOA Question 3.9

A father-son club has 4000 members, 2000 of which are age 20 and the other 2000 are age 45. In 25 years, the members of the club intend to hold a reunion. You are given:

All lives have independent future lifetimes.
Mortality follows the Standard Ultimate Life Table.

Using the normal approximation, without the continuity correction, calculate the 99th percentile of the number of surviving members at the time of the reunion.

print("SOA Question 3.9:  (E) 3850")
sult = SULT()
p1 = sult.p_x(20, t=25)
p2 = sult.p_x(45, t=25)
mean = sult.bernoulli(p1) * 2000 + sult.bernoulli(p2) * 2000
var = (sult.bernoulli(p1, variance=True) * 2000 
        + sult.bernoulli(p2, variance=True) * 2000)
print(sult.portfolio_percentile(mean=mean, variance=var, prob=.99))

SOA Question 3.9:  (E) 3850
3850.144345130047

SOA Question 3.8

A club is established with 2000 members, 1000 of exact age 35 and 1000 of exact age 45. You are given:

Mortality follows the Standard Ultimate Life Table
Future lifetimes are independent
N is the random variable for the number of members still alive 40 years after the club is established

Using the normal approximation, without the continuity correction, calculate the smallest \(n\) such that \(Pr( N \ge n ) \le 0.05\).

print("SOA Question 3.8:  (B) 1505")
sult = SULT()
p1 = sult.p_x(35, t=40)
p2 = sult.p_x(45, t=40)
mean = sult.bernoulli(p1) * 1000 + sult.bernoulli(p2) * 1000
var = (sult.bernoulli(p1, variance=True) * 1000 
        + sult.bernoulli(p2, variance=True) * 1000)
print(sult.portfolio_percentile(mean=mean, variance=var, prob=.95))

SOA Question 3.8:  (B) 1505
1504.8328375406456

SOA Question 3.4

The SULT Club has 4000 members all age 25 with independent future lifetimes. The mortality for each member follows the Standard Ultimate Life Table.

Calculate the largest integer N, using the normal approximation, such that the probability that there are at least N survivors at age 95 is at least 90%.

print("SOA Question 3.4:  (B) 815")
sult = SULT()
mean = sult.p_x(25, t=95-25)
var = sult.bernoulli(mean, variance=True)
print(sult.portfolio_percentile(N=4000, mean=mean, variance=var, prob=.1))

SOA Question 3.4:  (B) 815
815.0943255167722

Generate SULT Table:

import pandas as pd
print("Standard Ultimate Life Table at i=0.05")
pd.set_option('display.max_rows', None)
sult.frame()

Standard Ultimate Life Table at i=0.05

	l_x	q_x	a_x	A_x	2A_x	a_x:10	A_x:10	a_x:20	A_x:20	5_E_x	10_E_x	20_E_x
20	100000.0	0.000250	19.9664	0.04922	0.00580	8.0991	0.61433	13.0559	0.37829	0.78252	0.61224	0.37440
21	99975.0	0.000253	19.9197	0.05144	0.00614	8.0990	0.61433	13.0551	0.37833	0.78250	0.61220	0.37429
22	99949.7	0.000257	19.8707	0.05378	0.00652	8.0988	0.61434	13.0541	0.37837	0.78248	0.61215	0.37417
23	99924.0	0.000262	19.8193	0.05622	0.00694	8.0986	0.61435	13.0531	0.37842	0.78245	0.61210	0.37404
24	99897.8	0.000267	19.7655	0.05879	0.00739	8.0983	0.61437	13.0519	0.37848	0.78243	0.61205	0.37390
25	99871.1	0.000273	19.7090	0.06147	0.00788	8.0981	0.61438	13.0506	0.37854	0.78240	0.61198	0.37373
26	99843.8	0.000280	19.6499	0.06429	0.00841	8.0978	0.61439	13.0491	0.37862	0.78236	0.61191	0.37354
27	99815.9	0.000287	19.5878	0.06725	0.00900	8.0974	0.61441	13.0474	0.37869	0.78233	0.61183	0.37334
28	99787.2	0.000296	19.5228	0.07034	0.00964	8.0970	0.61443	13.0455	0.37878	0.78229	0.61174	0.37310
29	99757.7	0.000305	19.4547	0.07359	0.01033	8.0966	0.61445	13.0434	0.37888	0.78224	0.61163	0.37284
30	99727.3	0.000315	19.3834	0.07698	0.01109	8.0961	0.61447	13.0410	0.37900	0.78219	0.61152	0.37254
31	99695.8	0.000327	19.3086	0.08054	0.01192	8.0956	0.61450	13.0384	0.37913	0.78213	0.61139	0.37221
32	99663.2	0.000341	19.2303	0.08427	0.01281	8.0949	0.61453	13.0354	0.37927	0.78206	0.61124	0.37183
33	99629.3	0.000356	19.1484	0.08817	0.01379	8.0943	0.61456	13.0320	0.37943	0.78199	0.61108	0.37141
34	99593.8	0.000372	19.0626	0.09226	0.01486	8.0935	0.61460	13.0282	0.37961	0.78190	0.61090	0.37094
35	99556.7	0.000391	18.9728	0.09653	0.01601	8.0926	0.61464	13.0240	0.37981	0.78181	0.61069	0.37041
36	99517.8	0.000412	18.8788	0.10101	0.01727	8.0916	0.61468	13.0192	0.38004	0.78170	0.61046	0.36982
37	99476.7	0.000436	18.7805	0.10569	0.01863	8.0905	0.61474	13.0138	0.38029	0.78158	0.61020	0.36915
38	99433.3	0.000463	18.6777	0.11059	0.02012	8.0893	0.61480	13.0078	0.38058	0.78145	0.60990	0.36841
39	99387.3	0.000493	18.5701	0.11571	0.02173	8.0879	0.61486	13.0011	0.38090	0.78130	0.60957	0.36757
40	99338.3	0.000527	18.4578	0.12106	0.02347	8.0863	0.61494	12.9935	0.38126	0.78113	0.60920	0.36663
41	99285.9	0.000565	18.3403	0.12665	0.02536	8.0846	0.61502	12.9850	0.38167	0.78094	0.60879	0.36558
42	99229.8	0.000608	18.2176	0.13249	0.02741	8.0826	0.61511	12.9754	0.38212	0.78072	0.60832	0.36440
43	99169.4	0.000656	18.0895	0.13859	0.02963	8.0804	0.61522	12.9647	0.38263	0.78048	0.60780	0.36307
44	99104.3	0.000710	17.9558	0.14496	0.03203	8.0779	0.61534	12.9526	0.38321	0.78021	0.60721	0.36159
45	99033.9	0.000771	17.8162	0.15161	0.03463	8.0751	0.61547	12.9391	0.38385	0.77991	0.60655	0.35994
46	98957.6	0.000839	17.6706	0.15854	0.03744	8.0720	0.61562	12.9240	0.38457	0.77956	0.60581	0.35809
47	98874.5	0.000916	17.5189	0.16577	0.04047	8.0684	0.61579	12.9070	0.38538	0.77918	0.60498	0.35601
48	98783.9	0.001003	17.3607	0.17330	0.04374	8.0645	0.61598	12.8880	0.38629	0.77875	0.60404	0.35370
49	98684.9	0.001100	17.1960	0.18114	0.04727	8.0600	0.61619	12.8667	0.38730	0.77827	0.60299	0.35112
50	98576.4	0.001209	17.0245	0.18931	0.05108	8.0550	0.61643	12.8428	0.38844	0.77772	0.60182	0.34824
51	98457.2	0.001331	16.8461	0.19780	0.05517	8.0494	0.61670	12.8161	0.38971	0.77711	0.60050	0.34503
52	98326.2	0.001469	16.6606	0.20664	0.05957	8.0431	0.61700	12.7862	0.39113	0.77643	0.59902	0.34146
53	98181.8	0.001623	16.4678	0.21582	0.06430	8.0360	0.61733	12.7527	0.39273	0.77566	0.59736	0.33749
54	98022.4	0.001797	16.2676	0.22535	0.06938	8.0281	0.61771	12.7154	0.39451	0.77479	0.59550	0.33308
55	97846.2	0.001993	16.0599	0.23524	0.07483	8.0192	0.61813	12.6737	0.39649	0.77382	0.59342	0.32819
56	97651.2	0.002212	15.8444	0.24550	0.08067	8.0092	0.61861	12.6271	0.39871	0.77273	0.59109	0.32279
57	97435.2	0.002459	15.6212	0.25613	0.08692	7.9980	0.61914	12.5752	0.40118	0.77151	0.58848	0.31681
58	97195.6	0.002736	15.3901	0.26714	0.09360	7.9854	0.61974	12.5174	0.40393	0.77014	0.58556	0.31024
59	96929.6	0.003048	15.1511	0.27852	0.10073	7.9713	0.62041	12.4531	0.40700	0.76860	0.58229	0.30300
60	96634.1	0.003398	14.9041	0.29028	0.10834	7.9555	0.62116	12.3816	0.41040	0.76687	0.57864	0.29508
61	96305.8	0.003792	14.6491	0.30243	0.11644	7.9379	0.62201	12.3024	0.41417	0.76493	0.57457	0.28641
62	95940.6	0.004234	14.3861	0.31495	0.12506	7.9181	0.62295	12.2145	0.41836	0.76276	0.57003	0.27698
63	95534.4	0.004730	14.1151	0.32785	0.13421	7.8960	0.62400	12.1174	0.42298	0.76033	0.56496	0.26674
64	95082.5	0.005288	13.8363	0.34113	0.14392	7.8712	0.62518	12.0101	0.42809	0.75760	0.55932	0.25569
65	94579.7	0.005915	13.5498	0.35477	0.15420	7.8435	0.62650	11.8920	0.43371	0.75455	0.55305	0.24381
66	94020.3	0.006619	13.2557	0.36878	0.16507	7.8126	0.62797	11.7622	0.43990	0.75114	0.54609	0.23112
67	93398.1	0.007409	12.9542	0.38313	0.17654	7.7781	0.62961	11.6199	0.44667	0.74732	0.53836	0.21764
68	92706.1	0.008297	12.6456	0.39783	0.18862	7.7396	0.63145	11.4643	0.45408	0.74305	0.52981	0.20343
69	91936.9	0.009294	12.3302	0.41285	0.20133	7.6968	0.63349	11.2949	0.46215	0.73828	0.52036	0.18856
70	91082.4	0.010413	12.0083	0.42818	0.21467	7.6491	0.63576	11.1109	0.47091	0.73295	0.50994	0.17313
71	90134.0	0.011670	11.6803	0.44379	0.22864	7.5961	0.63828	10.9118	0.48039	0.72701	0.49848	0.15730
72	89082.1	0.013081	11.3468	0.45968	0.24324	7.5373	0.64108	10.6974	0.49060	0.72039	0.48590	0.14122
73	87916.8	0.014664	11.0081	0.47580	0.25847	7.4721	0.64419	10.4675	0.50155	0.71303	0.47215	0.12511
74	86627.6	0.016440	10.6649	0.49215	0.27433	7.3999	0.64762	10.2221	0.51323	0.70483	0.45715	0.10918
75	85203.5	0.018433	10.3178	0.50868	0.29079	7.3203	0.65142	9.9616	0.52564	0.69574	0.44085	0.09368
76	83632.9	0.020668	9.9674	0.52536	0.30783	7.2325	0.65560	9.6866	0.53873	0.68566	0.42323	0.07887
77	81904.3	0.023175	9.6145	0.54217	0.32544	7.1360	0.66019	9.3980	0.55247	0.67450	0.40427	0.06500
78	80006.2	0.025984	9.2598	0.55906	0.34359	7.0302	0.66523	9.0970	0.56681	0.66217	0.38396	0.05230
79	77927.4	0.029132	8.9042	0.57599	0.36224	6.9146	0.67074	8.7850	0.58166	0.64859	0.36235	0.04096
80	75657.2	0.032658	8.5484	0.59293	0.38134	6.7885	0.67674	8.4639	0.59696	0.63365	0.33952	0.03113
81	73186.3	0.036607	8.1934	0.60984	0.40086	6.6517	0.68325	8.1354	0.61260	0.61727	0.31556	0.02286
82	70507.2	0.041025	7.8401	0.62666	0.42075	6.5037	0.69030	7.8018	0.62848	0.59936	0.29064	0.01616
83	67614.6	0.045968	7.4893	0.64336	0.44094	6.3443	0.69789	7.4651	0.64452	0.57985	0.26498	0.01094
84	64506.5	0.051493	7.1421	0.65990	0.46137	6.1735	0.70602	7.1275	0.66059	0.55868	0.23882	0.00706
85	61184.9	0.057665	6.7993	0.67622	0.48199	5.9915	0.71469	6.7910	0.67662	0.53581	0.21250	0.00431
86	57656.7	0.064554	6.4619	0.69229	0.50272	5.7986	0.72388	6.4574	0.69250	0.51122	0.18635	0.00248
87	53934.7	0.072237	6.1308	0.70806	0.52349	5.5954	0.73355	6.1285	0.70817	0.48492	0.16079	0.00133
88	50038.6	0.080798	5.8068	0.72349	0.54422	5.3828	0.74368	5.8057	0.72354	0.45697	0.13621	0.00066
89	45995.6	0.090326	5.4908	0.73853	0.56484	5.1620	0.75419	5.4903	0.73856	0.42748	0.11305	0.00030
90	41841.1	0.100917	5.1835	0.75317	0.58528	4.9346	0.76502	5.1833	0.75317	0.39659	0.09168	0.00012
91	37618.6	0.112675	4.8858	0.76735	0.60545	4.7021	0.77609	4.8857	0.76735	0.36453	0.07244	0.00005
92	33379.9	0.125708	4.5981	0.78104	0.62529	4.4665	0.78731	4.5981	0.78104	0.33158	0.05559	0.00002
93	29183.8	0.140128	4.3213	0.79423	0.64472	4.2299	0.79858	4.3213	0.79423	0.29808	0.04128	0.00000
94	25094.3	0.156052	4.0556	0.80688	0.66368	3.9945	0.80979	4.0556	0.80688	0.26445	0.02955	0.00000
95	21178.3	0.173599	3.8017	0.81897	0.68209	3.7624	0.82084	3.8017	0.81897	0.23116	0.02029	0.00000
96	17501.8	0.192887	3.5597	0.83049	0.69991	3.5356	0.83164	3.5597	0.83049	0.19872	0.01330	0.00000
97	14125.9	0.214030	3.3300	0.84143	0.71708	3.3159	0.84210	3.3300	0.84143	0.16765	0.00827	0.00000
98	11102.5	0.237134	3.1127	0.85177	0.73356	3.1050	0.85214	3.1127	0.85177	0.13850	0.00485	0.00000
99	8469.7	0.262294	2.9079	0.86153	0.74930	2.9039	0.86172	2.9079	0.86153	0.11173	0.00266	0.00000
100	6248.2	0.289584	2.7156	0.87068	0.76427	2.7137	0.87078	2.7156	0.87068	0.08777	0.00136	0.00000