FA-63776 / Actuarial life tables / Open access

Pure endowment and survivorship accumulation: Pure endowment is not discounted · case 01

nEx equals the survival probability.

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ROOT CAUSE

The factor omits v^n.

THE FAILURE

The factor omits v^n.

Unsuccessful approach: Discounting without survival gives an annuity-certain factor.

Case contract

Input l (from issue age), n (0<=n<len(l) else 'invalid term'), i, g (premium escalation per mille), P. l_n=0 -> 'no survivors'. nEx = (l_n/l_0) v^n. Fund per survivor at n = sum_{k<n} P(1+g)^k (1+i)^(n-k) l_k/l_n (premiums at the start of each year). Return [nEx rounded 8, fund rounded half-up].

Why this case matters

Life-table arithmetic compounds across ages, so one misplaced index or assumption shifts every downstream value.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
from fractions import Fraction
N = 1
observations = []
def solve(x):
    l=x['l']; n=x['n']; i=Fraction(x['i'],1000); g=Fraction(x['g'],1000)
    if n<0 or n>=len(l): return 'invalid term'
    if l[n]==0: return 'no survivors'
    E=Fraction(l[n],l[0])
    fund=sum(x['P']*(1+g)**k*(1+i)**(n-k)*Fraction(l[k],l[n]) for k in range(n))
    return [round(float(E),8), math.floor(fund+Fraction(1,2))]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['five-year accumulation', {'l': [100000, 99627, 99178, 98677, 98090, 97396, 96614, 95732, 94714, 93602], 'n': 5, 'i': 50, 'g': 30, 'P': 310187}, [0.76312315, 1939423]], ['escalating premiums', {'l': [100000, 99499, 98915, 98211, 97439, 96539, 95498, 94341, 93034, 91551], 'n': 3, 'i': 50, 'g': 30, 'P': 473064}, [0.84838354, 1632596]], ['one year', {'l': [100000, 99657, 99256, 98789, 98220, 97539, 96746, 95842, 94806, 93669], 'n': 1, 'i': 30, 'g': 0, 'P': 301346}, [0.96754369, 311455]], ['zero term', {'l': [100000, 99466, 98873, 98210, 97456, 96581, 95573, 94407, 93118, 91686], 'n': 0, 'i': 30, 'g': 30, 'P': 127967}, [1.0, 0]], ['term at table end', {'l': [100000, 99613, 99130, 98560, 97908, 97160, 96325, 95362, 94292, 93106], 'n': 9, 'i': 30, 'g': 0, 'P': 207738}, [0.71357996, 2281783]], ['term beyond table', {'l': [100000, 99389, 98714, 97931, 97058, 96063, 94925, 93637, 92169, 90541], 'n': 10, 'i': 30, 'g': 30, 'P': 278992}, 'invalid term'], ['no survivors', {'l': [100000, 99503, 98938, 98299, 97575, 96770, 95863, 94838, 93668, 0], 'n': 9, 'i': 40, 'g': 30, 'P': 463178}, 'no survivors'], ['level premium long', {'l': [100000, 99472, 98900, 98250, 97518, 96711, 95773, 94708, 93506, 92143], 'n': 8, 'i': 40, 'g': 0, 'P': 332485}, [0.68323918, 3333117]]], [['five-year accumulation', {'l': [100000, 99419, 98767, 98017, 97178, 96239, 95176, 93966, 92603, 91035], 'n': 5, 'i': 30, 'g': 20, 'P': 316047}, [0.83016607, 1842455]], ['escalating premiums', {'l': [100000, 99441, 98818, 98096, 97273, 96348, 95303, 94119, 92753, 91250], 'n': 4, 'i': 40, 'g': 30, 'P': 376996}, [0.83149368, 1771442]], ['one year', {'l': [100000, 99759, 99457, 99107, 98716, 98252, 97716, 97101, 96373, 95529], 'n': 1, 'i': 30, 'g': 20, 'P': 427465}, [0.96853398, 441353]], ['zero term', {'l': [100000, 99700, 99353, 98925, 98389, 97791, 97104, 96341, 95445, 94434], 'n': 0, 'i': 30, 'g': 0, 'P': 409365}, [1.0, 0]], ['term at table end', {'l': [100000, 99579, 99072, 98516, 97875, 97124, 96284, 95355, 94273, 93076], 'n': 9, 'i': 50, 'g': 0, 'P': 263911}, [0.59997619, 3210361]], ['term beyond table', {'l': [100000, 99753, 99467, 99106, 98692, 98229, 97669, 97055, 96304, 95459], 'n': 10, 'i': 50, 'g': 20, 'P': 475728}, 'invalid term'], ['no survivors', {'l': [100000, 99778, 99462, 99065, 98590, 98022, 97414, 96703, 95907, 0], 'n': 9, 'i': 40, 'g': 0, 'P': 291350}, 'no survivors'], ['level premium long', {'l': [100000, 99664, 99270, 98810, 98263, 97627, 96880, 96004, 95006, 93923], 'n': 8, 'i': 40, 'g': 0, 'P': 156070}, [0.69419954, 1549487]]], [['five-year accumulation', {'l': [100000, 99694, 99343, 98956, 98501, 97990, 97392, 96742, 96001, 95150], 'n': 5, 'i': 50, 'g': 30, 'P': 467213}, [0.76777729, 2908794]], ['escalating premiums', {'l': [100000, 99387, 98687, 97882, 96957, 95923, 94763, 93492, 92018, 90349], 'n': 7, 'i': 40, 'g': 30, 'P': 184993}, [0.71046236, 1730047]], ['one year', {'l': [100000, 99719, 99390, 99016, 98572, 98064, 97435, 96732, 95889, 94934], 'n': 1, 'i': 50, 'g': 30, 'P': 425453}, [0.94970476, 447984]], ['zero term', {'l': [100000, 99474, 98873, 98230, 97498, 96657, 95717, 94639, 93396, 92034], 'n': 0, 'i': 40, 'g': 0, 'P': 178875}, [1.0, 0]], ['term at table end', {'l': [100000, 99481, 98896, 98219, 97413, 96468, 95420, 94270, 92939, 91465], 'n': 9, 'i': 50, 'g': 20, 'P': 382728}, [0.58959155, 5070118]], ['term beyond table', {'l': [100000, 99742, 99462, 99122, 98711, 98181, 97548, 96831, 96003, 95049], 'n': 10, 'i': 30, 'g': 20, 'P': 216948}, 'invalid term'], ['no survivors', {'l': [100000, 99486, 98893, 98221, 97466, 96612, 95647, 94525, 93256, 0], 'n': 9, 'i': 30, 'g': 30, 'P': 443695}, 'no survivors'], ['level premium long', {'l': [100000, 99463, 98869, 98187, 97370, 96463, 95422, 94269, 92993, 91535], 'n': 8, 'i': 30, 'g': 0, 'P': 279303}, [0.73409533, 2685755]]], [['five-year accumulation', {'l': [100000, 99647, 99240, 98757, 98173, 97493, 96686, 95788, 94759, 93588], 'n': 5, 'i': 30, 'g': 20, 'P': 431512}, [0.84098318, 2495359]], ['escalating premiums', {'l': [100000, 99683, 99274, 98795, 98242, 97639, 96940, 96155, 95257, 94209], 'n': 3, 'i': 30, 'g': 30, 'P': 490177}, [0.9041142, 1620833]], ['one year', {'l': [100000, 99746, 99410, 99036, 98581, 98074, 97498, 96854, 96107, 95270], 'n': 1, 'i': 40, 'g': 30, 'P': 187911}, [0.95909615, 195925]], ['zero term', {'l': [100000, 99732, 99391, 99018, 98550, 98024, 97402, 96653, 95801, 94842], 'n': 0, 'i': 40, 'g': 20, 'P': 383986}, [1.0, 0]], ['term at table end', {'l': [100000, 99532, 99019, 98411, 97699, 96870, 95914, 94809, 93568, 92171], 'n': 9, 'i': 40, 'g': 30, 'P': 374272}, [0.64758122, 4874021]], ['term beyond table', {'l': [100000, 99395, 98677, 97899, 96989, 95965, 94826, 93546, 92113, 90505], 'n': 10, 'i': 40, 'g': 20, 'P': 391759}, 'invalid term'], ['no survivors', {'l': [100000, 99436, 98759, 97991, 97084, 96058, 94874, 93546, 92075, 0], 'n': 9, 'i': 40, 'g': 0, 'P': 258512}, 'no survivors'], ['level premium long', {'l': [100000, 99575, 99078, 98515, 97855, 97101, 96216, 95203, 94075, 92820], 'n': 8, 'i': 40, 'g': 0, 'P': 274632}, [0.68739681, 2743824]]], [['five-year accumulation', {'l': [100000, 99618, 99173, 98667, 98087, 97408, 96641, 95782, 94815, 93766], 'n': 5, 'i': 30, 'g': 20, 'P': 427312}, [0.84024996, 2471882]], ['escalating premiums', {'l': [100000, 99483, 98900, 98235, 97459, 96563, 95554, 94423, 93132, 91708], 'n': 8, 'i': 40, 'g': 30, 'P': 364091}, [0.6805064, 4040305]], ['one year', {'l': [100000, 99741, 99435, 99053, 98614, 98098, 97514, 96814, 96053, 95155], 'n': 1, 'i': 30, 'g': 30, 'P': 382445}, [0.96835922, 394941]], ['zero term', {'l': [100000, 99625, 99217, 98716, 98122, 97448, 96665, 95754, 94700, 93479], 'n': 0, 'i': 40, 'g': 20, 'P': 150292}, [1.0, 0]], ['term at table end', {'l': [100000, 99624, 99187, 98665, 98044, 97353, 96572, 95690, 94680, 93547], 'n': 9, 'i': 40, 'g': 30, 'P': 414508}, [0.65724881, 5342367]], ['term beyond table', {'l': [100000, 99720, 99377, 98960, 98499, 97952, 97305, 96596, 95793, 94879], 'n': 10, 'i': 40, 'g': 30, 'P': 308034}, 'invalid term'], ['no survivors', {'l': [100000, 99649, 99279, 98844, 98335, 97740, 97075, 96329, 95486, 0], 'n': 9, 'i': 50, 'g': 30, 'P': 170315}, 'no survivors'], ['level premium long', {'l': [100000, 99549, 99017, 98394, 97709, 96899, 95973, 94973, 93856, 92607], 'n': 8, 'i': 40, 'g': 0, 'P': 100799}, [0.6857966, 1008179]]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
five-year accumulation	`[0.97396, 1939423]`	`[0.76312315, 1939423]`	Failed
escalating premiums	`[0.98211, 1632596]`	`[0.84838354, 1632596]`	Failed
one year	`[0.99657, 311455]`	`[0.96754369, 311455]`	Failed
zero term	`[1.0, 0]`	`[1.0, 0]`	Passed
term at table end	`[0.93106, 2281783]`	`[0.71357996, 2281783]`	Failed
term beyond table	`invalid term`	`invalid term`	Passed
no survivors	`no survivors`	`no survivors`	Passed
level premium long	`[0.93506, 3333117]`	`[0.68323918, 3333117]`	Failed

SHA-256 / 57ceb96c7ee22d73b6dd90203cf235034a709101848c30b58c087013404ec354

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
from fractions import Fraction
N = 1
observations = []
def solve(x):
    l=x['l']; n=x['n']; i=Fraction(x['i'],1000); g=Fraction(x['g'],1000)
    if n<0 or n>=len(l): return 'invalid term'
    if l[n]==0: return 'no survivors'
    E=1/(1+i)**n
    fund=sum(x['P']*(1+g)**k*(1+i)**(n-k)*Fraction(l[k],l[n]) for k in range(n))
    return [round(float(E),8), math.floor(fund+Fraction(1,2))]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['five-year accumulation', {'l': [100000, 99627, 99178, 98677, 98090, 97396, 96614, 95732, 94714, 93602], 'n': 5, 'i': 50, 'g': 30, 'P': 310187}, [0.76312315, 1939423]], ['escalating premiums', {'l': [100000, 99499, 98915, 98211, 97439, 96539, 95498, 94341, 93034, 91551], 'n': 3, 'i': 50, 'g': 30, 'P': 473064}, [0.84838354, 1632596]], ['one year', {'l': [100000, 99657, 99256, 98789, 98220, 97539, 96746, 95842, 94806, 93669], 'n': 1, 'i': 30, 'g': 0, 'P': 301346}, [0.96754369, 311455]], ['zero term', {'l': [100000, 99466, 98873, 98210, 97456, 96581, 95573, 94407, 93118, 91686], 'n': 0, 'i': 30, 'g': 30, 'P': 127967}, [1.0, 0]], ['term at table end', {'l': [100000, 99613, 99130, 98560, 97908, 97160, 96325, 95362, 94292, 93106], 'n': 9, 'i': 30, 'g': 0, 'P': 207738}, [0.71357996, 2281783]], ['term beyond table', {'l': [100000, 99389, 98714, 97931, 97058, 96063, 94925, 93637, 92169, 90541], 'n': 10, 'i': 30, 'g': 30, 'P': 278992}, 'invalid term'], ['no survivors', {'l': [100000, 99503, 98938, 98299, 97575, 96770, 95863, 94838, 93668, 0], 'n': 9, 'i': 40, 'g': 30, 'P': 463178}, 'no survivors'], ['level premium long', {'l': [100000, 99472, 98900, 98250, 97518, 96711, 95773, 94708, 93506, 92143], 'n': 8, 'i': 40, 'g': 0, 'P': 332485}, [0.68323918, 3333117]]], [['five-year accumulation', {'l': [100000, 99419, 98767, 98017, 97178, 96239, 95176, 93966, 92603, 91035], 'n': 5, 'i': 30, 'g': 20, 'P': 316047}, [0.83016607, 1842455]], ['escalating premiums', {'l': [100000, 99441, 98818, 98096, 97273, 96348, 95303, 94119, 92753, 91250], 'n': 4, 'i': 40, 'g': 30, 'P': 376996}, [0.83149368, 1771442]], ['one year', {'l': [100000, 99759, 99457, 99107, 98716, 98252, 97716, 97101, 96373, 95529], 'n': 1, 'i': 30, 'g': 20, 'P': 427465}, [0.96853398, 441353]], ['zero term', {'l': [100000, 99700, 99353, 98925, 98389, 97791, 97104, 96341, 95445, 94434], 'n': 0, 'i': 30, 'g': 0, 'P': 409365}, [1.0, 0]], ['term at table end', {'l': [100000, 99579, 99072, 98516, 97875, 97124, 96284, 95355, 94273, 93076], 'n': 9, 'i': 50, 'g': 0, 'P': 263911}, [0.59997619, 3210361]], ['term beyond table', {'l': [100000, 99753, 99467, 99106, 98692, 98229, 97669, 97055, 96304, 95459], 'n': 10, 'i': 50, 'g': 20, 'P': 475728}, 'invalid term'], ['no survivors', {'l': [100000, 99778, 99462, 99065, 98590, 98022, 97414, 96703, 95907, 0], 'n': 9, 'i': 40, 'g': 0, 'P': 291350}, 'no survivors'], ['level premium long', {'l': [100000, 99664, 99270, 98810, 98263, 97627, 96880, 96004, 95006, 93923], 'n': 8, 'i': 40, 'g': 0, 'P': 156070}, [0.69419954, 1549487]]], [['five-year accumulation', {'l': [100000, 99694, 99343, 98956, 98501, 97990, 97392, 96742, 96001, 95150], 'n': 5, 'i': 50, 'g': 30, 'P': 467213}, [0.76777729, 2908794]], ['escalating premiums', {'l': [100000, 99387, 98687, 97882, 96957, 95923, 94763, 93492, 92018, 90349], 'n': 7, 'i': 40, 'g': 30, 'P': 184993}, [0.71046236, 1730047]], ['one year', {'l': [100000, 99719, 99390, 99016, 98572, 98064, 97435, 96732, 95889, 94934], 'n': 1, 'i': 50, 'g': 30, 'P': 425453}, [0.94970476, 447984]], ['zero term', {'l': [100000, 99474, 98873, 98230, 97498, 96657, 95717, 94639, 93396, 92034], 'n': 0, 'i': 40, 'g': 0, 'P': 178875}, [1.0, 0]], ['term at table end', {'l': [100000, 99481, 98896, 98219, 97413, 96468, 95420, 94270, 92939, 91465], 'n': 9, 'i': 50, 'g': 20, 'P': 382728}, [0.58959155, 5070118]], ['term beyond table', {'l': [100000, 99742, 99462, 99122, 98711, 98181, 97548, 96831, 96003, 95049], 'n': 10, 'i': 30, 'g': 20, 'P': 216948}, 'invalid term'], ['no survivors', {'l': [100000, 99486, 98893, 98221, 97466, 96612, 95647, 94525, 93256, 0], 'n': 9, 'i': 30, 'g': 30, 'P': 443695}, 'no survivors'], ['level premium long', {'l': [100000, 99463, 98869, 98187, 97370, 96463, 95422, 94269, 92993, 91535], 'n': 8, 'i': 30, 'g': 0, 'P': 279303}, [0.73409533, 2685755]]], [['five-year accumulation', {'l': [100000, 99647, 99240, 98757, 98173, 97493, 96686, 95788, 94759, 93588], 'n': 5, 'i': 30, 'g': 20, 'P': 431512}, [0.84098318, 2495359]], ['escalating premiums', {'l': [100000, 99683, 99274, 98795, 98242, 97639, 96940, 96155, 95257, 94209], 'n': 3, 'i': 30, 'g': 30, 'P': 490177}, [0.9041142, 1620833]], ['one year', {'l': [100000, 99746, 99410, 99036, 98581, 98074, 97498, 96854, 96107, 95270], 'n': 1, 'i': 40, 'g': 30, 'P': 187911}, [0.95909615, 195925]], ['zero term', {'l': [100000, 99732, 99391, 99018, 98550, 98024, 97402, 96653, 95801, 94842], 'n': 0, 'i': 40, 'g': 20, 'P': 383986}, [1.0, 0]], ['term at table end', {'l': [100000, 99532, 99019, 98411, 97699, 96870, 95914, 94809, 93568, 92171], 'n': 9, 'i': 40, 'g': 30, 'P': 374272}, [0.64758122, 4874021]], ['term beyond table', {'l': [100000, 99395, 98677, 97899, 96989, 95965, 94826, 93546, 92113, 90505], 'n': 10, 'i': 40, 'g': 20, 'P': 391759}, 'invalid term'], ['no survivors', {'l': [100000, 99436, 98759, 97991, 97084, 96058, 94874, 93546, 92075, 0], 'n': 9, 'i': 40, 'g': 0, 'P': 258512}, 'no survivors'], ['level premium long', {'l': [100000, 99575, 99078, 98515, 97855, 97101, 96216, 95203, 94075, 92820], 'n': 8, 'i': 40, 'g': 0, 'P': 274632}, [0.68739681, 2743824]]], [['five-year accumulation', {'l': [100000, 99618, 99173, 98667, 98087, 97408, 96641, 95782, 94815, 93766], 'n': 5, 'i': 30, 'g': 20, 'P': 427312}, [0.84024996, 2471882]], ['escalating premiums', {'l': [100000, 99483, 98900, 98235, 97459, 96563, 95554, 94423, 93132, 91708], 'n': 8, 'i': 40, 'g': 30, 'P': 364091}, [0.6805064, 4040305]], ['one year', {'l': [100000, 99741, 99435, 99053, 98614, 98098, 97514, 96814, 96053, 95155], 'n': 1, 'i': 30, 'g': 30, 'P': 382445}, [0.96835922, 394941]], ['zero term', {'l': [100000, 99625, 99217, 98716, 98122, 97448, 96665, 95754, 94700, 93479], 'n': 0, 'i': 40, 'g': 20, 'P': 150292}, [1.0, 0]], ['term at table end', {'l': [100000, 99624, 99187, 98665, 98044, 97353, 96572, 95690, 94680, 93547], 'n': 9, 'i': 40, 'g': 30, 'P': 414508}, [0.65724881, 5342367]], ['term beyond table', {'l': [100000, 99720, 99377, 98960, 98499, 97952, 97305, 96596, 95793, 94879], 'n': 10, 'i': 40, 'g': 30, 'P': 308034}, 'invalid term'], ['no survivors', {'l': [100000, 99649, 99279, 98844, 98335, 97740, 97075, 96329, 95486, 0], 'n': 9, 'i': 50, 'g': 30, 'P': 170315}, 'no survivors'], ['level premium long', {'l': [100000, 99549, 99017, 98394, 97709, 96899, 95973, 94973, 93856, 92607], 'n': 8, 'i': 40, 'g': 0, 'P': 100799}, [0.6857966, 1008179]]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
five-year accumulation	`[0.78352617, 1939423]`	`[0.76312315, 1939423]`	Failed
escalating premiums	`[0.8638376, 1632596]`	`[0.84838354, 1632596]`	Failed
one year	`[0.97087379, 311455]`	`[0.96754369, 311455]`	Failed
zero term	`[1.0, 0]`	`[1.0, 0]`	Passed
term at table end	`[0.76641673, 2281783]`	`[0.71357996, 2281783]`	Failed
term beyond table	`invalid term`	`invalid term`	Passed
no survivors	`no survivors`	`no survivors`	Passed
level premium long	`[0.73069021, 3333117]`	`[0.68323918, 3333117]`	Failed

SHA-256 / 02a769044fb8259dd4b6a83d5dc805e4c74524e849dcbbf24e953afd50fa9382

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

Verification & scope

A deterministic bounded teaching model of a stipulated toy actuarial contract. Rates are small synthetic tables, exact rationals are used where practical and floats are rounded at the output; it makes no claim of conformance with any published table, standard of practice or regulation. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:47:17.793837+00:00.

Case digest / b5f652f5e442c21d1e9a534af9e74466ed73f97388cdbb1b77da483581127de6