1. The mortality tables

My calculations use:

- the age of retirement
- an annuity rate R for this age
- an interest rate Y
- mortality tables PMA92 (C=2010) ^x (dated February 2004)

We are assuming that someone buying a pension annuity has mortality given by the mortality tables.

The ONS expectation of life is calculated using the current mortality at different ages. It gives the expectation of life for a man age 60 was 16.3 years in 1981 and 19.8 years in 2001. ^x 19.8 - 16.3 = 3.5 years. Continuing this increase in the expectation of life at the same rate from 20.0 years gives 23.3 years which is almost the 23.5 years in the mortality tables. Therefore the mortality tables take into account future increases in mortality.

I have had a quotation for a man aged 60 for a pension annuity (which is compulsory) and a purchased life annuity (which is voluntary). The difference in values is 5%. This is the result of a selection effect. Annuities tend to be bought particularly by people who are healthier. But my calculations and PMA92 is based on compulsory pension annuities, so that this phenomenon hardly applies. Is there any such selection effect left for pension annuities? Healthier people may tend to buy personal pensions.

2. Example

An annuity rate of 5.412% ( = 451 x 12) for a pension annuity, for a man age 50 results from an interest rate of 3.604% (assuming no charges).

Retirement age = 50
Annuity yield Y = 3.604%
Annuity rate R = 5.412%

Suppose we start with an annuity capital of £100. The following table shows how the capital decreases with age until it has all gone.

1	2	3	4
Age	probability of dying during year when alive at the start	probability still alive at start of year	capital at start of year
50	0.000729	1	100
51	0.000842	0.999	98.2
52	0.000976	0.998	96.3
53	0.00113	0.988	94.4
54	0.00131	0.997	92.5
55	0.00153	0.996	90.3
56	0.00178	0.995	88.2
57	0.00208	0.993	86.1
58	0.00242	0.991	83.8
59	0.00282	0.989	81.4
60	0.003277	0.987	79
61	0.003892	0.984	76
62	0.004612	0.981	74
63	0.005451	0.977	71
64	0.006425	0.972	68
65	0.007552	0.961	66
66	0.008851	0.954	63
67	0.010343	0.961	60
68	0.012051	0.951	59
69	0.013999	0.939	57
70	0.016213	0.926	54
71	0.018718	0.911	51
72	0.021544	0.894	48
73	0.024720	0.874	45
74	0.028274	0.853	42
75	0.032238	0.829	39
76	0.036640	0.802	36
77	0.041513	0.772	34
78	0.046882	0.741	31
79	0.052777	0.706	28
80	0.059223	0.669	25
81	0.066241	0.629	22
82	0.073854	0.588	20
83	0.082075	0.544	18
84	0.090916	0.499	16
85	0.100385	0.454	15
86	0.110484	0.408	13
87	0.121206	0.363	11
88	0.132541	0.319	10
89	0.144472	0.277	8
90	0.156976	0.237	7
91	0.170020	0.200	5
92	0.183568	0.166	4
93	0.197576	0.135	3
94	0.211992	0.109	2
95	0.226764	0.085	1
96	0.241828	0.066	1
97	0.257120	0.050	1
98	0.272571	0.037	1
99	0.288112	0.027	0
100	0.303666	0.019	0
101	0.319160	0.013	0
102	0.334518	0.009	0
103	0.349666	0.006	0
104	0.364532	0.004	0
105	0.379044	0.003	0
106	0.393133	0.002	0
107	0.406734	0.001	0
108	0.419786	0.000	0
109	0.432231	0.000	0
110	0.444014	0.000	0

Note about the columns

1	age x
2	probability P(x) of dying at this age given that you are alive at the beginning of the year from PMA92.
3	probability Q(x) still being alive at the beginning of the year Q(x+1)= (1 - P(x)).Q(x)
4	Annuity capital C(x) at the beginning of the year x C(x+1) = C(x) + C(x).(1 + Y/100) - Q(x+1) x (R = 5.412)

May 2005