Illustrative Life Table

The following mortality table follows the Makeham law (i.e., \(\mu_x=A+Bc^x\), where \(A=0.0007\), \(B=0.00005\), and \(c=10^{0.04}\)). The commutation functions \(D_x\), \(N_x\), \(C_x\), and \(M_x\) are constructed using a \(6\%\) annual effective interest rate assumption as follows:

\[ \boxed{ \begin{aligned} \\ \quad D_x &= v^xl_x \quad \\ \\ N_x &= \displaystyle \sum_{k=0}^\infty D_{x+k} \quad \\ \\ \quad C_x &= v^{x+1}d_x \quad \\ \\ M_x &= \displaystyle \sum_{k=0}^\infty C_{x+k} \quad \\ \\ \end{aligned} } \]

\(\text{Illustrative Life Table}\)
\(\mu_x=A+Bc^x\)
\(x\)	\(l_x\)	\(d_x\)	\(q_x\)	\(\mu_x\)	\(D_x\)	\(C_x\)	\(N_x\)	\(M_x\)
0	100,000	75	0.00075	0.00075	100,000	71	1,714,781	2,937
1	99,925	76	0.00076	0.00075	94,269	67	1,614,781	2,866
2	99,849	76	0.00076	0.00076	88,865	64	1,520,512	2,799
3	99,773	77	0.00077	0.00077	83,771	61	1,431,647	2,735
4	99,696	77	0.00078	0.00077	78,969	58	1,347,875	2,674
5	99,619	78	0.00078	0.00078	74,441	55	1,268,907	2,616
6	99,541	79	0.00079	0.00079	70,172	52	1,194,466	2,561
7	99,462	80	0.00080	0.00080	66,148	50	1,124,293	2,509
8	99,383	80	0.00081	0.00080	62,354	48	1,058,145	2,459
9	99,302	81	0.00082	0.00081	58,777	45	995,791	2,411
10	99,221	82	0.00083	0.00083	55,404	43	937,014	2,366
11	99,139	84	0.00084	0.00084	52,225	42	881,610	2,322
12	99,055	85	0.00086	0.00085	49,227	40	829,385	2,281
13	98,970	86	0.00087	0.00087	46,401	38	780,158	2,241
14	98,883	88	0.00089	0.00088	43,736	37	733,757	2,203
15	98,795	90	0.00091	0.00090	41,224	35	690,020	2,166
16	98,706	92	0.00093	0.00092	38,855	34	648,796	2,131
17	98,614	94	0.00095	0.00094	36,622	33	609,941	2,097
18	98,520	96	0.00097	0.00096	34,516	32	573,319	2,064
19	98,424	99	0.00100	0.00099	32,531	31	538,803	2,032
20	98,326	101	0.00103	0.00102	30,658	30	506,273	2,002
21	98,225	104	0.00106	0.00105	28,893	29	475,614	1,972
22	98,120	108	0.00110	0.00108	27,229	28	446,721	1,943
23	98,013	111	0.00113	0.00112	25,659	27	419,492	1,915
24	97,902	115	0.00118	0.00116	24,180	27	393,833	1,887

\(\text{Illustrative Life Table}\)
\(\mu_x=A+Bc^x\)
\(x\)	\(l_x\)	\(d_x\)	\(q_x\)	\(\mu_x\)	\(D_x\)	\(C_x\)	\(N_x\)	\(M_x\)
24	97,902	115	0.00118	0.00116	24,180	27	393,833	1,887
25	97,786	120	0.00122	0.00120	22,784	26	369,653	1,860
26	97,667	124	0.00127	0.00125	21,468	26	346,869	1,834
27	97,542	130	0.00133	0.00130	20,227	25	325,401	1,808
28	97,413	135	0.00139	0.00136	19,057	25	305,174	1,783
29	97,277	142	0.00146	0.00142	17,953	25	286,117	1,758
30	97,136	149	0.00153	0.00149	16,912	24	268,164	1,733
31	96,987	156	0.00161	0.00157	15,931	24	251,251	1,709
32	96,831	164	0.00170	0.00165	15,005	24	235,321	1,685
33	96,667	173	0.00179	0.00174	14,131	24	220,316	1,661
34	96,494	183	0.00190	0.00185	13,308	24	206,185	1,637
35	96,310	194	0.00201	0.00196	12,530	24	192,877	1,613
36	96,117	206	0.00214	0.00208	11,797	24	180,347	1,589
37	95,911	219	0.00228	0.00221	11,106	24	168,549	1,565
38	95,692	233	0.00243	0.00236	10,453	24	157,443	1,541
39	95,460	248	0.00260	0.00252	9,838	24	146,990	1,517
40	95,212	265	0.00278	0.00269	9,257	24	137,152	1,493
41	94,947	283	0.00298	0.00288	8,708	24	127,896	1,469
42	94,664	303	0.00320	0.00309	8,191	25	119,187	1,445
43	94,361	325	0.00344	0.00332	7,703	25	110,996	1,420
44	94,036	349	0.00371	0.00358	7,242	25	103,294	1,395
45	93,687	374	0.00400	0.00385	6,806	26	96,052	1,369
46	93,313	403	0.00431	0.00416	6,395	26	89,246	1,344
47	92,910	433	0.00466	0.00449	6,007	26	82,850	1,318
48	92,477	466	0.00504	0.00486	5,641	27	76,843	1,291

\(\text{Illustrative Life Table}\)
\(\mu_x=A+Bc^x\)
\(x\)	\(l_x\)	\(d_x\)	\(q_x\)	\(\mu_x\)	\(D_x\)	\(C_x\)	\(N_x\)	\(M_x\)
48	92,477	466	0.00504	0.00486	5,641	27	76,843	1,291
49	92,011	503	0.00546	0.00526	5,295	27	71,202	1,265
50	91,508	542	0.00592	0.00570	4,968	28	65,907	1,237
51	90,966	584	0.00642	0.00618	4,659	28	60,939	1,209
52	90,382	630	0.00697	0.00671	4,367	29	56,280	1,181
53	89,752	680	0.00758	0.00729	4,091	29	51,914	1,153
54	89,072	734	0.00824	0.00793	3,830	30	47,823	1,123
55	88,338	792	0.00896	0.00862	3,584	30	43,992	1,094
56	87,547	854	0.00975	0.00939	3,351	31	40,409	1,063
57	86,693	921	0.01062	0.01023	3,130	31	37,058	1,032
58	85,772	993	0.01158	0.01115	2,921	32	33,928	1,001
59	84,779	1,070	0.01262	0.01215	2,724	32	31,007	969
60	83,709	1,152	0.01376	0.01326	2,538	33	28,282	937
61	82,557	1,239	0.01501	0.01447	2,361	33	25,745	904
62	81,318	1,332	0.01638	0.01580	2,194	34	23,384	870
63	79,986	1,430	0.01788	0.01726	2,036	34	21,190	836
64	78,556	1,534	0.01952	0.01885	1,886	35	19,154	802
65	77,022	1,642	0.02132	0.02061	1,745	35	17,268	767
66	75,380	1,755	0.02329	0.02253	1,611	35	15,523	732
67	73,625	1,873	0.02544	0.02463	1,484	36	13,912	697
68	71,752	1,994	0.02779	0.02694	1,365	36	12,428	661
69	69,758	2,118	0.03037	0.02947	1,252	36	11,063	625
70	67,639	2,244	0.03318	0.03225	1,145	36	9,811	590
71	65,395	2,371	0.03626	0.03529	1,044	36	8,666	554
72	63,023	2,497	0.03962	0.03863	949	35	7,622	518

\(\text{Illustrative Life Table}\)
\(\mu_x=A+Bc^x\)
\(x\)	\(l_x\)	\(d_x\)	\(q_x\)	\(\mu_x\)	\(D_x\)	\(C_x\)	\(N_x\)	\(M_x\)
73	60,526	2,621	0.04330	0.04229	860	35	6,673	483
74	57,905	2,740	0.04731	0.04630	776	35	5,812	447
75	55,166	2,852	0.05169	0.05070	698	34	5,036	413
76	52,314	2,954	0.05647	0.05552	624	33	4,338	379
77	49,360	3,045	0.06168	0.06081	556	32	3,714	345
78	46,315	3,120	0.06737	0.06661	492	31	3,158	313
79	43,195	3,177	0.07356	0.07297	433	30	2,666	282
80	40,018	3,213	0.08030	0.07994	378	29	2,234	252
81	36,804	3,225	0.08764	0.08759	328	27	1,855	223
82	33,579	3,211	0.09561	0.09597	282	25	1,527	196
83	30,368	3,167	0.10428	0.10516	241	24	1,245	171
84	27,202	3,092	0.11369	0.11524	204	22	1,004	147
85	24,109	2,987	0.12389	0.12629	170	20	800	125
86	21,122	2,850	0.13494	0.13841	141	18	630	105
87	18,272	2,684	0.14689	0.15170	115	16	489	87
88	15,588	2,491	0.15981	0.16627	92	14	374	71
89	13,097	2,276	0.17375	0.18224	73	12	282	57
90	10,821	2,043	0.18877	0.19975	57	10	208	45
91	8,779	1,799	0.20493	0.21896	44	8	151	35
92	6,980	1,551	0.22227	0.24002	33	7	108	27
93	5,428	1,307	0.24086	0.26310	24	5	75	20
94	4,121	1,074	0.26073	0.28842	17	4	51	14
95	3,046	859	0.28191	0.31618	12	3	34	10
96	2,188	666	0.30445	0.34662	8	2	21	7
97	1,522	500	0.32834	0.37999	5	2	13	5