Saturday, July 19, 2014

Female live expectancy and annuities.

Introduction Females have longer life expectancy than males in virtually all countries.   Gender related differences in life expectancy make it more likely that females will out live their retirement resources than will males.  This post considers how differences in the expected future life of a 75-year-old male and a 75 year-old female might impact annuity payments.

Sources:  The analysis presented here pertains to males and females in the United States.   The source of information is the United States Life Tables, 2008 published on September 24, 2012 by the National Center for Health Statistics of the Centers for Disease Control and Prevention


Table Two of the report has life statistics for males and Table Three of the report has life statistics for females.   Both tables can be downloaded directly into an EXCEL Spreadsheet.

Questions:   A 75 year-old person wants to set up a monthly annuity so that he or she has only a 10% chance of living past the final payment.  How long is the annuity in  number of months if the person is a U.S. male or a U.S. female? 

The person has $100,000 to spend on this annuity. What is the payment on the annuity for the male and the female at an interest rate of 0%, 3% and 6%? 

A CAUTION:  The annuity that I am experimenting with here is not a market-based annuity, which would be based on expected future life rather than the 10% chance of outliving resources rule.  

Answer:   The data in Table One is from the CDC life tables.  Based on this cohort of 100,000 females, 73,974 females have survived to page 75.   Around 90% of these females are still alive until somewhere between age 95 and 96.   (This seems really high but perhaps not because we are looking at the probability of survival given that the woman lived until age 75.)

In a cohort of 100,000 men 61,980 are still alive at age 75 and the 90% survival mark for these 75-year olds is reached somewhere between age 94 and 95. 



Age
Total number of females alive at age x
Proportion of 75-year-old females who survive to age X
Total number of Males Alive at age X
Proportion of 75-year Males surviving until age X
75
73,974
61,980
76
71,973
97.3%
59,531
96.0%
77
69,831
94.4%
56,962
91.9%
78
67,539
91.3%
54,268
87.6%
79
65,080
88.0%
51,437
83.0%
80
62,448
84.4%
48,469
78.2%
81
59,647
80.6%
45,390
73.2%
82
56,688
76.6%
42,227
68.1%
83
53,563
72.4%
38,994
62.9%
84
50,253
67.9%
35,694
57.6%
85
46,782
63.2%
32,360
52.2%
86
43,166
58.4%
28,996
46.8%
87
39,414
53.3%
25,650
41.4%
88
35,567
48.1%
22,372
36.1%
89
31,677
42.8%
19,213
31.0%
90
27,805
37.6%
16,223
26.2%
91
24,017
32.5%
13,448
21.7%
92
20,380
27.6%
10,928
17.6%
93
16,962
22.9%
8,691
14.0%
94
13,821
18.7%
6,754
10.9%
95
11,006
14.9%
5,122
8.3%
96
8,549
11.6%
3,783
6.1%
97
6,467
8.7%
2,719
4.4%
98
4,754
6.4%
1,898
3.1%
99
3,392
4.6%
1,286
2.1%
100
2,345
3.2%
844
1.4%


Let’s interpolate to get an exact number of months for our annuity formula. 

For females we get 96-75 (21) years plus (11.6-10)/(11.6-8.7) x 12 or (7) months.  The 75-year old female must buy an annuity of 259 months in order to reduce the probability that she will outlive the annuity to 10 percent.

For males we get 94-75 (19) years plus (10.9-10)/(10.9-8.3) x 12 or 5 months (I am rounding up to fulfill the contract.)   The 75-year old male must buy an annuity of 233 months in order to reduce the probability that he will outlive the annuity to 10 percent.

So now we calculate the annuity payments for the female and the male with the PMT function.   The only input that differs is the duration of the contract – 259 months for females and 233 months for males.
   
The annuity payment calculations were obtained from the PMT function in Excel.


Female
Male

Rate 0%
0
0

Rate 3%
0.03
0.03
Rate 6%
0.06
0.06

NPER
259
233
PV
$100,000
$100,000

Type
1
1
DIFFERENCE
% DIFFERENCE WITH FEMALE AS BASE
PMT rate=0
$386.10
$429.18
($43.08)
-11.2%
PMT rate =3%
$524.96
$566.77
($41.81)
-8.0%
PMT rate=6%
$689.45
$727.62
($38.17)
-5.5%




In order to reduce their greater longevity risk females must buy a longer-term annuity and will therefore receive a lower payment.  The potential reduction in annuity payments is greatest when interest rates are low. 


This annuity calculator on the web confirms that females receive lower annuity payments.  I am not familiar with the specific formulas used by this calculator or the product that it pertains to.   My sole interest here is to provide some insight on how gender determines longevity risk.







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