UnIsex Rating in Insurance
An Honors Thes 1s (I D 499)
by
Anju Gupta
Thesis Director
Ball State University
Muncie, Indiana
April, 1989
May, 1989
.-.
.Introduct1on
When the Equal Rights Amendment missed ratification by three states
in 1982 CGest, 1985:40), the fight for equal rights for men and women did
not end. Various feminists groups, including the National organization of
Women (NOW), continue to lobby for equal rights for women in many
different areas. Recently, the fight for equal rights has reached the
insurance industry.
In order to calculate premiums on insurance policies, companies use
sex-distinct tables. NOW and other feminist groups argue that the use of
these tables discriminates against women. To insure the equal treatment of
men and women, NOW maintains that these sex-distinct tables should be
abolished and a unisex table should be used. The insurance industry,
however, argues gender is a primary risk factor in setting rates (Dennon,
1988: 1), and if important risk factors are eliminated, overpricing and
underpricing wi 11 occur (Carroll, 1988:56).
In the course of this paper, I will attempt to present both supporting
and opposing views of this argument as fairly as possible. However, the
main point of this paper is to present an actuarial view of the argument. I
will outline a brief history of the unisex debate, and then I will present both
views of the issue. Then, I will try to explain the actuarial viewpoint of the
issue of using gender as a basis of ratemaking for insurance products.
What Does Unisex Rating Mean?
-
When actuaries design tables for pricing insurance products, they use
actual data from past experience. They group lives according to the risks
each life represents. The risks which they use for classification have been
found to allocate cost fairly according to the degree of risk which the
insurer bears. These risks include such characteristics as the age of the
insured, the gender of the insured, whether the insured smokes or not, and
many others.
The use of unisex tables would eliminate gender as a risk
classification. Rather than establish separate male and female tables,
actuaries would design a table based on a group blended by gender. One
method of blending would involve adding the percentage of male business
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times male lives to the percentage of female business times female lives.
For example, 1f approximately seventy percent of an insurance company's
business came from males and the other thirty percent from females, the
blended group at age x would consist of seventy percent of the male I ives at
age x plus thirty percent of the female lives at age x.
Once this blended group has been established, the actuaries would
calculate, using the same methods as they would with sex-distinct rates,
the mortality tables. From these mortality tables, the net single premiums,
annuity values, and premiums would be calculated.
Using sex-distinct tables produces different premium rates for males
and females. However, premiums for females are not higher on all insurance
products. While men pay higher premiums for equal life insurance coverage
than do women, annuities used in pension programs charge women more than
men.
Unisex legislation would mandate the use of unisex mortality tables
in the annuity and premium calculations. However, the issue is not about
the use of unisex tables; the issue Is about the validity of gender as a risk
classification.
History
Over the past years, there has been a great deal of legislation which
addresses the issue of equality among men and women. Some of this
legislation has been directed at the insurance industry. The Supreme Court
made two landmark decisions, City of Los Angeles Water and Power vs.
t1anlJartand Arizona Governing Committee vs. Norris, which addressed the
issue of sex-distinction in pensions. On the state level, there have been
several states which enacted legislation requiring the use of unisex tables
for either auto insurance or all 1ines of insurance.
Title VII of the the Civil Rights Act of 1964 was used to cHe in
allegations of discrimination in both the t1anlJartand Norris cases. Each of
the titles in the Civil Rights Act of 1964 prohibits discrimination in
particular areas, and Title VII prohibits discrimination in the area of
employment (Cary and Peratis, 1978:53). Congress expressly declared that
no employer "shall discriminate against any individual on the basis of 'race,
color, religion, sex, or national origin'" (Cary and Peratis, 1978:30).
In 1978, the case of the City of Los Angeles Water and Power vs
t1anlJartcame before the Supreme Court. In this case, the employees of the
Los Angeles Department of Water and Power sued the Department for
requiring female employees to make larger monthly contributions to a
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3
pension plan than their male counterparts in order to receive the same
monthly benef1t (Cary and Peratls, 1978:71-72). The use of gender-based
tables in computing benefit amounts for a mandatory defined benefit plan
was found to be in violation of the Civi I Rights Act of 1964 (Kurlowickz and
Damiani, 1987:30).
In 1983, the case of Arizona Governing Committee vs. Norris was
brought before the Supreme Court. In this case, Nathalie NorriS, an
employee of the State of Arizona, sued her employer for violating the Civil
Rights Act of 1964 by administering a pension plan which computed benefits
using a sex-distinct table. The Supreme Court ruled in favor of Norris,
requiring plan sponsors to only use unisex tables when computing benefits.
Prior to the Supreme Court decision, a female employee would receive a
lower monthly pension benefit than her male counterpart (Gibson et aI,
1983: 1).
Since a large majority of insurance regulation is conducted on the
state level, many of the significant cases in which unisex legislation has
passed occur in the states. However, in 1983, Republican Senator Bob
Packwood of Oregon introduced a bill at the federal level, that was
eventually defeated, which would have prohibited the use of gender-based
tables in alllines of insurance (Dennon, 1988: n Several states have
enacted legislation which requires the use of unisex tables, and still others
are considering it.
There are several states which require the use of unisex tables in
setting rates on automobile insurance. As early as 1974, the use of unisex
tables for auto insurance became effective in the state of Hawaii. In 1975,
North Carolina followed with legislation requiring unisex auto insurance.
Unisex auto insurance is required in the state of Michigan, effective since
1981. A regulation requiring the use of unisex auto rates was scheduled to
take effect in September, 1986 in Pennsylvania; however, the state
legislature passed a bi 11 null ifying the regulation before it could take effect
(Dennon, 1988:2).
In 1983, Montana became the first state to enact legislation requiring
the use of un1sex tables for all lines of insurance. This law went into
effect in October, 1985 (Youngman, 1987: 1). The state legislature voted to
repeal the law in 1987; however, Governor Ted Schwinden vetoed the repeal
on the basis of constitutional guarantee against discrimination (Dennon,
1988:2).
The State of Massachusetts recently became the second state to
require the use of unisex tables for alllines of insurance. The state is the
first to move to unisex rating as a result of a regulation, rather than
legislat10n (McGhee, 1988:6). The regulation was promulgated on September
18, 1987 and became effective September 1, 1988 (Benjamin, 1988:4).
4
There are other states which are currently investigating the idea of
unisex rating. During the 1988 Iowa state legislative session, a study
committee on the elimination of discriminatory insurance practices was
establishelj, In Minnesota, a special eleven-member task force has been
established by Minnesota Insurance Commissioner Michael Hatch to stUdy
the issue (,Jones, 1988: n William E. Kingsley, an American Council of Life
Insurance (AClI) executive Vice-president, remarked that the AClI expects a
big push for unisex rating in the insurance industry. Since the defeat of the
federal b111 for un1sex rat1ng, there has been 1ncreased Interest In unIsex
rating bi lls among state legislators (Arndt, 1987:3).
Support for Untsex Rating
Support for unisex rating comes from several different directions.
The most prominent area of support is feminist organizations, such as the
National Organization of Women (NOW) as well as others. The American
Civil Liberties Union (ACLU) also actively supports unisex legislation.
Although most insurance companies have opposed unisex rating, some have
supported it. Most notably, John Hancock Financial Services has both
supported unisex rating and worked to help enact unisex legislation
(Hathaway, 1988:54).
Proponents of unisex rating have used many arguments to support
their cause; however, the most common argument used is that the use of
sex-distinct tables in insurance is unfairly discriminatory. Although gender
has been found to be a good predictor of risk, some feel it should not be used
to set rate differentials because, as gender is a characteristic that cannot
be changed, the individual should not be punished for it (Dennon, 1988:25).
While insurers continue to support the use of gender as a reliable and
significant cost factor, proponents maintain that gender-based Insurance is
offensive to basic civil rights principles (Zimmerman, 1986:24). Gender has
been found to be a socially unacceptable method of grouping people for
business purposes (Bennett, 1986: 106). According to the Senate Commerce
Committee in 1982, ..... treating men and women equally in insurance is not
so much a matter of statistics as of 'simple justice'" (Dennon, 1988:25).
Marcia Youngman, the Non-Gender Insurance Project Director for the
Women's Lobbyist Fund of Montana, is a strong supporter of unisex rating. In
her testimony to the Massachusetts Division of Insurance, Ms. Youngman
stated that her group supported the unisex legislation for economic reasons
as well as for civil rights reasons (Youngman, 1987: 1). Ms. youngman
discussed the effects of unisex legislation in Montana, including higher
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rates due to factors unrelated to the law such as 111egal pollt1cal
ratemaklng and poor rate adjustment processes (Youngman, 1987: 1-4). The
Women's Lobbyist Fund of Montana conducted a rate study to discover the
impacts of the unisex law on various types of insurance. The results of this
study showed that while some rates went up and some went down, ..... the
rate picture is generally much fairer than before, ..... (Youngman, 1987:5).
Ms. Youngman also stated in her testimony that the industry uses
gender-related statistics inconsistently, in ways that have not treated
women faIrly. She feels actuarIal data Is used In several areas In a way
which minimizes benefits to women. Ms. Youngman summarized her feelings
by stating, "It is in the best interest of companies as well as consumers to
rate people according to performance and lifestyle factors that allow
companies to reward people for safe and healthy behaviors and thus attract
low-risk customers, rather than according to the uncontrollable factor of
gender" (Youngman, 1987:7).
The ACLU supports unisex rating on the basis of the fundamental
polICy that ..... certaln protected characteristlcs--race, color, religion, sex,
marital status and national origin--should not be taken into account In
setting insurance rates ..... (ElliS, 1988:2). In her statement on the behalf of
the ACLU before the Iowa Study Committee on the Elimination of
Discriminatory Insurance Practices, Deborah A Ellis stated that insurance
is one of the few areas in this country in which intentional sex
discrimination is permitted and defended (ElliS, 1988:4). She also made the
point that insurance companies do not use either race or religion as a baSis
of rIsk class1flcation; therefore, the same policy should apply to gender
(ElliS, 1988:6). Ms. Ell is attacked the industry'S claIm that sex distInctions
are based on actuarial SCience, saying that the only "science" involved was
the taking of a group average (ElliS, 1988: 11). As Ms. Ellis stated, "The fact
of the matter is that all actuarial groupings are inevitably entirely
arbitrary, or based upon social value jUdgments, or, in the worst case, are
merely reflections of social stereotypes and prejudice" (ElliS, 1988: 12).
In actuarial tables, women as a group live longer than men; however,
Indlvldual women do not always llve as long as actuarial tables predict
(Erickson, 1988: 1). Proponents of unisex rating use this argument
frequently to support their views. As Jenny A Erickson, Assistant
Legislative Counsel at John Hancock Financial Services, stated to the Iowa
Study Committee on the Elimination of Discriminatory Insurance Practices,
divergent perspectives on the same important essence of insurance loss
prediction make debate difficult (Erickson, 1988: 1) However, as Ms.
Erickson stated, "The issue is no longer an actuarial one, it's a social one"
(Erickson, 1988:2). In a poll conducted among their clients, John Hancock
Financial Services found that gender-based rates were offensive to
customers, who viewed the issue as one of fairness and unfairness. The
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company supports un1sex rat1ng as a gOOd soc1al poliCY whose time has come
(Hathaway, t 988:54-55).
Opposition to Unisex Rating
Whlle there Is strong support for unisex rating, there is also a great
deal of opposition directed towards the issue. The majority of this
opposition comes from the insurance industry sector. The American Council
of Life Insurance (ACLI), various companies, and specific actuaries have
come out in strong opposition of unisex legislation. The main argument used
by the opponents is that, under a system using unisex rates, one sex
subsidizes the other on almost all lines of business. Other arguments used
include the fact that experience has shown that males have higher mortality
rates than females and the fact that gender-based rates are actuarlally
sound.
Insurance companies use gender as a primary risk for setting rates in
several I ines of insurance. The industry defends this practice, saying
gender allocates cost fairly according to the degree of risk which the
insurer bears (Dennon, 1988: 1). Whi Ie a growing number of insurance
customers feel this system is unfair, Wi 11 iam E. Kingsley of the ACLI
suggest that there is a basic lack of understanding of risk-based rating
(Arndt, 1987:3). Barbara Lautzenhe1ser, a pract1clng actuary for over
twenty-seven years, argues that gender is valid as a risk classification,
and, if it is eliminated, a subsidy would occur as insureds with different
expected losses were placed in the same class (Lautzenheiser, 1988: 1).
William F. Carroll, president of the Life Insurance Association of
Massachusetts, says that gender is a significant risk factor in mortal ity
rates, and that if it is ignored, overpricing and underpricing wi 11 result
(Carroll, 1988:56). As Joseph 5. Diamond indicated in an editorial in the
NatIonal UnderwrIter, "'equal treatment' doesn't necessarily mean the same
treatment; ... it really means fair and equitable treatment for all" (Diamond,
1988: 18), Fairness in insurance means pricing each person according to the
risk he or she represents (Carroll, t 988:56).
In a study done in connection with her doctoral thesis, Deborah Lee
Wingard studied both the biological and behavioral factors which affect the
sex differential in mortality rates. While most studies explore the
behavioral factors which account for mortality differences between the two
sexes, they do note that the male death rate is higher even 1n prenatal 11fe,
when behavioral factors could have little effect (Wingard, 1980: 1). Ms.
Wingard discovered in her study that women have lower death rates than
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men in almost every country, at almost every age, and for most causes of
death, indicating better overall fitness among women (Wingard, 1980: 11 ).
As Ms. Lautzenheiser indicated in her testimony before the Iowa Study
Committee on the Elimination of Discriminatory Insurance Practices, the
risk classification plans and variables used to distinguish among different
classes are based on actual data collected by the insurer (Lautzenheiser,
1988: 1). Actuaries have carefully analyzed this data and constructed tables
reflecting the differences. While some object to this practice because it
treats women as a class rather than individuals, the industry defends the
practice because there are I imitations within actuarial science; the future
cannot be predicted on each individual basis.
Proponents of unisex rating frequently accuse the industry of loading
female policies with higher expense costs than male policies. Ms.
Lautzenheiser says that mortality costs, which are lower for women, are
the only differences in costs between the two sexes (Lautzenheiser,
1988:5). Competition within the industry automatically demands that
prices fairly reflect costs (Lautzenheiser, 1988: 1). If women's rates are
higher than men's rates, even under a unisex system, it is not due to a
conspiracy by the insurance industry, but rather due to actuarially sound
rates (Lautzenheiser, 1988: 13).
Actuarial Section
Given a group of lives starting at age x and the survivor experience in
the following years, actuaries can design mortality tables. Using these
tables, the actuaries can then calculate net single premiums, annUity
values, an(j premium values. These calculations are made using the same
formulas, regardless of what data was used to construct the mortal ity
tables.
Construction of Mortality Tables:
When designing mortality tables, the actuary starts with the raw data
on a group of lives starting at age O. (This can be done starting at any age
x.)
-
Let 10 = the number of 1ives in the original group,
Let Ix
= the
number of lives in the group at age x.
8
The original group,
previous year.
10,
will reduce at each age by the number of deaths in the
Let dx = the number of deaths in the year x
= Ix
- lx+l.
To make the terminology less cumbersome,
let (x) = a person aged x.
The probab il i ty that a person aged x will surv i ve to age x+ 1 equals the
number of lives at age x+ 1 divided by the number of lives at age x.
Let nPx = the probability that (x) survives n years
= the number of liyes at age x+n
the number of I ives at age X .
Px = the number of lives at age x+ 1
the number of I ives at age x
=..1:tl
Ix .
Following the basic laws of probability, the probability of a person aged x
dying before age x+ 1 equals one minus the probability of a person aged x
surviving to age x+ 1.
Let Qx = the probabi I ity that (x) dies before age x+ 1
= 1 - Px
= 1 --k1.
Ix .
.-
From this formula the probability of dying within one year is calculated for
each age x, from zero to ninety-nine, or the last year of survivorship. These
probabilities then constitute the mortality table.
To illustrate the calculations performed when designing a mortality
table, the data on the blended group of lives will be used. However, the
same formulas and steps were used to calculate the mortal1ty tables for the
male and female groups of lives. The data used in these calculations is from
9
the actual experience of the Principal Mutual Life Insurance Company,
located in Des Moines, Iowa.
Given data on males and females for age zero through age ninety-nine,
a blended group is established, assuming that eighty percent of the total
business was male and twenty percent of the total business was female.
The original number of 1ives, starting at age zero, is
10
= 501883 .
To construct the mortality table, the probabllity of dying within one year is
calculated for each age x. For example, for a person aged forty-four,
P44
=~ =
144
q44
=1-
461435
466085
P44
=1-
= 0.99582
0.99852
= 0.00418.
In the mortality tables, the probabilities of dying within one year, or the
qx's, are multiplied by one thousand, which produces 1000qx's. In
calculations involving Qx, the factor of one thousand is removed by division;
therefore, this factor does not affect the calculations, but it does make
reading the table easier.
Net Single Premiums;
Once the mortality table has been established, the net single
premiums can be calculated.
Let bt = the benefit payment for a 1ife insurance.
The benefit function, bt, is usually defined as the face value on the
insurance polley and, therefore, is a constant value. However, it can also be
a function with a value that changes over time.
Let vt = the discount function for time of policy issue.
bt x v t
= the present value
of the benefit payment at the time
of policy issue.
-
The discount function is used to discount the future benefit payment back to
the time of policy Issue.
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The net single premium represents the expected present value of the
benefit payment at the time the policy is issued. For a whole life insurance
polley for a person aged x, the net single premium equals the present value
of the benefit payment times the mortality function at age x, summed over
the span of 1ife.
Let Ax
=
the net single premium for who le 1ife insurance for a
person aged x
00
Ax
= 2:
bk+IVk.+l k PxQx+k .
k.=o
Summing over such a long period of time can become difficult. Therefore,
actuaries have developed commutation functions. These functions can be
used to calculate net single premiums in terms of stored intermediate
values. The commutation functions are defined in Actuarial Mathematics as
follows:
00
Mx
=
2:
Cx+k (Bowers et aI, 1986: 112).
1<.-0
Using these definitions of commutation functions, the net single premium
for a whole life insurance policy can be calculated.
FreQuently, net single premiums are calculated so that the benefit is
payable at the moment of death. In order to calculate these types of net
single premiums, continuous values are integrated over the span of life
rather than whole values being summed. That is,
Ax = the net single premium for whole life insurance, benefit
payable at the moment of death
-
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OD
Ax
=
fobtvttPxl.1x+tdt , where I.1x is the force of mortality,
while Ax is the net single premium for whole life insurance, benefit payable
at the end of the year in which the death occurs. However, the net single
premium with the benefit payable at the moment of death can be obtained
using the net single premium with the benefit payable at the end of the year
in which the death occurs. In order to make this calculation, several
measurements of the interest rate will be used.
Let i
= the
interest rate used to discount the benefit back to
the time of policy issue.
Let vt
.a uniform
Ax
+ i )-t .
8 = In(l + D.
Let
Assuming
= (1
=
distribution of deaths, it has been shown that
_i Ax
~
(Bowers et aI, 1986: 104).
Using commutation functions,
Ax
= bt x _i
'E>
Mx
0;.
To illustrate the necessary steps in the calculation of the whole life
net single premium, the blended mortality table is used, although the same
formulas are used to calculate the whole life net single premiums for males
and females. Let the face value on a whole life insurance policy for a person
aged forty-four be one. The assumed interest rate in the calculations will
be seven percent. The net single premium for the whole life policy with
benefits payable at the moment of death is
OC>
AM =
JovttP44Jl44+tdt.
Using commutation functions,
-
12
-A44 =
=
0.07
X
In( 1.07)
3972.30
23745.37
0.17308 .
The whole life net single premium for a male aged forty-four is
A44
= 0.18086,
and the whole life net single premium for a female aged forty-four is
~ = 0.14190.
The significant differences between these three values will be discussed in
the section on premiums.
Term insurance can also be used to illustrate premium differences for
males and females. To calculate the net single premium for a term policy,
benefit payable at the moment of death, the actuary follows the same basic
steps as for a whole life policy. Once again there is a benefit function, bt,
and a discount function, vt. However, rather than integrating from zero to
infinity, trle integration is from zero to n-1, with n being the duration of the
term policy. To calculate the net single premium for a n-year term policy,
benefit payable at the moment of death for a person aged X,
let Ax: ri1 = the net Single premium for n-year term
insurance for a person aged x
0-1
=
2:
bk+1Vk+1kPxqx+k.
k=O
Commutation funct ions can be used to calculate this. The net single
premium for a n-year term policy, benefit payable at the end of the year in
which the death occurs is, in terms of commutation functions,
Ax: ri\ = bt (M x - Mx+o)
Ox
The net single premium for a n-year term policy with benefit payable at the
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moment of death is
n
A~c 1i\ = JblvttPx Ux+t dt .
o
Under the uniform distribution of deaths,
A~: nt = _i Ax: iil
b
= bt x _i (M x - Mx+n )
~
Ox
Using the mortality rates from the blended table, the net single
premium for a ten-year term policy for a person aged thirty-five, with a
benefit of one payable at the moment of death is
10
A35:
-.
10\ =
fo vttP35U35+tdt.
Using commutation functions,
A35:1O\ = _i (M35 - M45)
~
D35
=
0.07 x (4770.96 - 3879.45)
In( 1.07)
44796.45
= 0.02059.
The net single premium on a ten-year term policy is calculated for males
and females using the same formulas. These calculations produce
-,
A3s:i01 = 0.02146
for a·male aged thirty-five and, for a female aged thirty-five,
A3s:1Ol = 0.01708.
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The differences between these three values will be investigated in the
premium calculations.
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Anou1tles:
Annuity values can also be calculated using the newly-constructed
mortality table.
Let vt
= the discount function for the time of po I icy issue.
This discount function is the same as that used for calculating net single
premiums. There are two types of annuIties: an annUity-Immediate, wh1ch
pays at the end of the year, and an annuity-due, which pays at the beginning
of the year.
Let ax = the actuarial present value of a whole life annuity-due
of one for a person aged x
co
,-
Commutation functions can be used to calculate this actuarial present
value. A new commutation function, Nx, is introduced.
co
Let Nx =
2: Ou .
u=x
Us1ng th1s deflnit10n for the commutation function, it can be shown that
ax = Nx
Ox (Bowers et aI, 1986:140).
Given a person aged forty-four, the actuarial present value of a whole
life annuity-due of one can be calculated. To illustrate the necessary steps,
the blended mortality table will be used.
co
a44 = 2:
vkkP44.
k=O
Using commutation functions,
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15
'a44 = 302245.45
23745.37
= 12.73.
The actuarial present value of the same annulty for a male aged forty-four
is
For a female aged forty-four, the actuarial present value of a whole life
annuity-due is
a44 = 13.19.
The effect of these differences in values can be seen in the premium
calculations. Therefore, they will be investigated further in the section
regarding premiums.
The actuarial present value of a n-year temporary life annuity-due
can be calculated as well. The discount function remains the same as with a
whole life annuity-due.
Let aX:!i\ = the actuarial present value of a n-year temporary
life annuity-due of one for a person aged x
n-l
=
2:
VkkPx .
k=O
This actuarial present value can be calculated using commutation functions.
ax: iii = Nx -
Nx+n
Dx
To calculate the actuarial present value for a ten-year temporary
annuity-due of one for a person aged thirty-five, using the blended mortality
table,
9
a35:1O)
-
=
2:
k=O
Vk.k.P35.
16
-
using commutation functions,
a35: 101 = N35 - N45
D35
= 611818.28 -
278500.08
I
44796.45
7.44.
=
The same formulas are used to calculate the actuarial present values of a
temporary annuity-due for a male and for a female. For a male aged thirtyfive, the actuarial present value of a ten-year temporary annuity-due is
a35: 101 =
7.44.
The actuarial present value for a similar annuity-due for a female aged
thirty-five is
a35:161
= 7.45.
The significance of these values and theIr effect on the premiums will be
investigated later.
Calculations can also be made for annuities payable on monthly,
quarterly, or semi-annual basis. These are referred to as annuities which
are payable on a nthly basis. The actuarial present value of an annuity of
this type can be calculated using the actuarial present value of an annuitydue. Several measurements of interest rate will be used.
I
= Interest rate.
Let d = i
1+i .
Let
i(m) = m[(1+i)1/m -
Let
d(m)
1].
= m[ 1 - (1 +i)-l/m]
As shown in Actuarial MathematiCS,
~(m) =
-
id
i(m)d(m)
.
17
-
f3(m) = i - i(m)
i(m)d(m) (Bowers et aI, 1986: 137-138).
Using these definitions,
let ax(m) =O\(m)ax - J3(m) .
Commutatlon functlons can tnen be used to calculate tnls value.
To illustrate the calculation of the actuarial present value of an
annuity-due of one with rr.thly payments, the blended mortality table will be
used. For a person aged fifty-five, the actuarial present value of a whole
life annuity-due with monthly payments is
'ass(12)
=0<.( 12)ass - J3( 12).
Using an interest rate of seven percent,
i = 0.07
d = 0.07 = 0.06542056
1.07
i(12)
= 12[( 1.07)1/12
-
1]
= 0.06784974
d(12) = 12[ 1 - (1.07)-1/12] = 0.06746827
0\( 12) =
B( 12) =
(0.07)( 0.06542056)
= 1.00037888
(0.06784974)( 0.06746827)
0.07 - 0.06784974
(0.06784974)( 0.06746827)
= 0.46972346
Therefore,
a55(12)
,-
= 1.00037888a55 - 0.46972346.
18
-
Using commutation functIons,
a55(12)
= 1.00037888Ns5 - 0.46972346
D55
= 1.00037888
116006.44 - 0.46972346
10513.42
= 10.57.
The same formulas can be used to calculate the actuarial present value of a
whole life annuity-due with monthly payments for a male aged fifty-five.
The result is
a55(12)
= 10.37.
The actuarial present value of a similar annuity-due for a female aged
fifty-five is
ass(12)
= 11.34.
These vall.;,es have an impact on pension benefits and wi 11 be discussed
further in that sect ion.
Net Annua'l Premiums:
After the net single premiums and actuarial present values of
annuities have been calculated, the net annual premiums can be calculated.
Let P(Ax ) = the net annual payment, semi-continuous premium
for a whole I ife insurance for a person aged x.
Assuming the equivalence principle,
PCAx ) = Ax
ax .
-.
When premiums are calculated using separate male and female tables,
a significant difference can be seen between the two values. For example,
the net annual premium on a sixty thousand dollar whole life policy for a
female aged forty-four is
19
-
PCA44) = $60,000 A44
a44
=
$60,000 (0.1419)
( 13.19)
=
$654.54.
However, the net annual premium on a sixty thousand dollar whole life
policy for a male aged forty-four is
P(A44) = $60,000 A44
a44
= $60 000 (0.18086)
1
( 12.61)
= $860.32.
The difference between the two values is significant. The actual dollar
difference of over two hundred dollars indicates that this policy is over
thirty percent more expensive for men than for women. This difference is
not affected at all by any factors other than actual life experience.
Using a blended table to calculate the net annual premium on a sixty
thousand dollar whole life policy for a person aged forty-four
produces
P(A44 ) = $60,000 A44
;344
_
i
= $60,000 (0.17308)
( 12.73)
= $815.85
.
If this blended table were used, the premium for females would increase by
almost twenty-five percent while the premium for males would decrease by
approximately five percent.
Those opposed to unisex insurance use these figures to support their
argument that, under gender-neutral rule, the female risks would be
subsidizing the male risks by paying higher rates. The difference between
the life experience of males and females is not a small, insignificant
20
--
percentage; by blend1ng these two groups, the female group Is beIng
penalized unfairly while the male group is being given an advantage.
Term insurance is another insurance product in which male and
female premiums are significantly different. To calculate the net annual
premium for term insurance,
let peAk: nj)
= the net annual
payment, semi-continuous premium
for a n-year term life insurance for a person
aged x.
Assuming the equivalence principle,
P(A~:"ij) = AX: 1i1
ax: Ii] .
There are significant differences in the premiums for males and
females for term 1nsurance as well as for whole 11fe Insurance. For
example, the net annual premium on a twenty thousand dollar ten-year term
policy for a female aged thirty-five is
P(A35:m)
=
$20,000 A35:1o\
a35:iOl
= $20,000 (0.01708)
(7.45)
= $45.84.
The net annual premium on a twenty thousand dollar ten-year term policy
for a male aged thirty-five is
P(A35:~) = $20,000 A35:101
a35:1Ol
= $20,000 (0.02146)
(7.44)
=$57.71.
The male and female premiums once again show a remarkable difference.
The premIum for a male 1s almost twenty-s1x percent hIgher than that for a
female. H)is difference is affected by nothIng other than actual life
experience.
21
-"
using the blended table which would be mandated under unisex
legislation, the net annual premium for a twenty thousand dollar ten-year
term policy for a person aged thirty-five is
P(A~5:loP = $20,000 A~5:tQ.
a35:Tq
= $20,000 (0.02059)
(7.44)
= $55.34.
As in the case of whole life insurance, use of a blended table would
overchargE- females and undercharge males. The net annual premium for this
policy for females would increase by over twenty percent and the premium
for males would decrease by approximately four percent when an unisex
table is used. Although the differences between the male, female and
blended premiums for term insurance are not as large as for whole life, the
differences are sti 11 there. These values continue to support the arguments
of those opposing unisex insurance.
-
Pension Benefits:
Unisex legislation is already in effect in the area of pensions. In the
case of C1ty of Los Angeles Water and Power vs t1annart, the Supreme Court
outlawed the practice of requiring men and women to contribute different
amounts to a pension plan in order to obtain the same coverage. Then, in the
case of Arizona Governing Committee vs NorriS-in 1983, the Supreme Court
mandated the use of unisex tables in the calculation of pension benefits.
Before investigating the effects of using unisex tables for pension
plans, the pension benefit must be calculated. As most pensions pay
benefits on a monthly benefit,
let
ax
(12)
= the actuarial present value of the whole life
annuity-due used to calculate monthly pension
benefits for a person aged x.
Let B = the total value of the pension upon retirement.
Let
8)/12)
= the monthly pension benefit for a person aged
x.
22
The monthly penslon beneflt 1s slmply the total value of the penslon dlvlded
by the whole life annuity-due payable monthly, for an entire year.
The monthly benefit on a forty thousand dollar pension for a male
aged fifty-five is
B55(12) =
$40,000
12a55(12) .
=
$40,000
12( 10.37)
= $321.34.
The monthly benefit on a forty thousand dollar pension for a female
aged fifty-five is
B55(12)
= $40,000
12aS5(12)
= $40,000
12( 11.34)
= $294.06 .
There is a significant difference between the male and female benefits. In
the case of the pension, the female benefit is over nine percent smaller than
the male benefit. Using the blended table,
B55(12)
= $40,000
12a55(12)
=
$40,000
12( 10.57)
=
$315.40.
While using a unisex table improves the benefit to females by seven percent,
it decreases the benefit to males by almost two percent.
23
As In the case of life Insurance, the use of the unisex table is
penalizing one group unfairly and giving the other group an advantage.
However, in the case of the pension benefit, the male group is being
penalized and the female group is being given an advantage.
Opinion
.-
The actuarial calculations in this paper strongly support the
arguments of those who oppose unisex rating. In life insurance, premiums
for males Ijecrease using a unisex table while premiums for females
increase using the unisex table. In the case of penSions, the benefits for
females increase using unisex annuities and the benefits for males decrease
using unisex annuities.
This situation indicates that one gender subsidizes the other under
unisex ratng. Although females receive higher pension benefits when
unisex rates are used, it could be considered unfair that males receive
lower pension benefits. In the case of life insurance, many females may be
upset to find that their premiums will rise drastically under unisex rating
whi Ie premiums for males drop.
Frequently, proponents of unisex rating use the argument that gender
is a characteristic which cannot be changed; therefore, a person should not
be penalized for his or her gender. However, if unisex rating were to become
mandated, this situation of penalizing an individual on the basis of gender
would not only sti 11 exist, but it would become magnified. Under the current
system using gender-based tables, the risk borne by each person due to his
or her gender is fairly allocated. If unisex rating became the law, each
person would be bearing his or her own risk as well as subsidizing the risk
of an equally situated person of the opposite sex.
The system of unisex rating will not provide equality for all. Instead,
it w111 discriminate unfairly against large segments ofpeople in almost all
lines of insurance. No unfair discrimination against women is taking place
in the insurance industry. However, unisex rating would mandate unfair
discrimination against both sexes. By saying unisex rating is right,
proponents are saying it is fair to overcharge half of the insureds and
undercharge the other half.
As the fight for unisex rating continues, the question arises about
what the consumers know. However, there seems to be very little action on
the part of unisex proponents to educate the consumers about the issue.
They wi 11 be the ones directly affected by unisex rating, and they should be
the ones to decide.
24
-
The fight for unisex rating may continue forever. Although they do
not want unisex rating to go into effect, many companies view it as
inevitable. If the mandating of unisex tables is inevitable, then the
legislation should be federal rather than from the state. I f unisex rating
represents equality for the citizens of one state, it should represent
equality for the citizens of the entire country.
Actuarially, unisex rating is not logical. It defers the risk one
individual represents to another individual. Different groups will be
overcharged whl1e others are undercharged. However, 1f unisex rat1ng
represents equality, then this situation will exist. Whether supporting or
opposing the unisex rating in insurance, it is obvious that in this case
equal ity does have a price.
2S
-
~
-
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
Male
yX
----------
Male Ix
-------
-------
1.00000000
0.93457944
0.87343873
0.81629788
0.76289618
0.71298618
0.66634222
0.62274974
0.58200910
0.54393374
0.50834929
0.47509280
0.44401196
0.41496445
0.38781724
0.36244602
0.33873460
0.31657439
0.29586392
0.27650833
0.25841900
0.24151309
0.22571317
0.21094688
0.19714662
0.18424918
0.17219549
0.16093037
0.15040221
0.14056282
0.13136712
0.12277301
0.11474113
0.10723470
0.10021934
0.09366294
0.08753546
0.08180884
0.07645686
0.07145501
0.06678038
0.06241157
0.05832857
0.05451268
0.05094643
0.04761349
0.04449859
0.04158747
0.03886679
0.03632410
371097
370121
369740
369374
369016
368673
368349
368043
367756
367480
367208
366933
366636
366299
365907
365453
364934
364354
363727
363065
362382
361693
361006
360327
359664
359017
358389
357773
357161
356554
355941
355318
354678
354015
353325
352601
351836
351020
350146
349208
348195
347098
345911
344628
343239
341739
340123
338382
336511
334502
2.63
1.03
0.99
0.97
0.93
0.88
0.83
0.78
0.75
0.74
0.75
0.81
0.92
1.07
1.24
1.42
1.59
1.72
1.82
1.88
1.90
1.90
1.88
1.84
1.80
1.75
1.72
1.71
1.70
1.72
1.75
1.80
1.87
1.95
2.05
2.17
2.32
2.49
2.68
2.90
3.15
3.42
3.71
4.03
4.37
4.73
5.12
5.53
5.97
6.46
1000qx
26
-,
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
-"
0.03394776
0.03172688
0.02965129
0.02771148
0.02589858
0.02420428
0.02262083
0.02114096
0.01975791
0.01846533
0.01725732
0.01612834
0.01507321
0.01408711
0.01316553
0.01260423
0.01149928
0.01074699
0.01004392
0.00938684
0.00877275
0.00819883
0.00766246
0.00716117
0.00669269
0.00625485
0.00584565
0.00546323
0.00510582
0.00477179
0.00445962
0.00416787
0.00389520
0.00364038
0.00340222
0.00317965
0.00297163
0.00277723
0.00259554
0.00242574
0.00226704
0.00211873
0.00198012
0.00185058
0.00172952
0.00161637
0.00151063
0.00141180
0.00131944
0.00123312
332341
330015
327497
324769
321804
318583
315091
311319
307259
302902
298231
293221
287837
282046
275816
269122
261958
254327
246242
237710
228722
219260
209299
198817
187814
176339
164480
152358
140113
127878
115760
103844
92204
80917
70087
59846
50332
41664
33926
27157
21354
16476
12454
9198
6603
4556
2953
1710
785
200
7.00
7.63
8.33
9.13
10.01
10.96
11.97
13.04
14.18
15.42
16.80
18.36
20.12
22.09
24.27
26.62
29.13
31.79
34.65
37.81
41.37
45.43
50.08
55.34
61.10
67.25
73.70
80.37
87.32
94.76
102.94
112.09
122.41
133.84
146.12
158.97
172.22
185.72
199.52
213.68
228.43
244.11
261.44
282.13
310.01
351.84
420.93
540.94
745.22
1000.00
27
Af1J
0
1
-
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
Male
Dx
---------
371097.00
345907.48
322945.24
301519.21
281520.54
262858.75
245446.49
229198.68
214037.34
199884.77
186669.93
174327.23
162790.77
152001.06
141905.04
132456.99
123615.77
115345.15
107613.69
100390.50
93646.40
87353.59
81483.81
76009.86
70906.54
66148.59
61712.97
57576.54
53717.80
50118.23
46758.94
43623.46
40696.15
37962.69
35410.00
33025.65
30798.13
28716.54
26771.06
24952.66
23252.59
21662.93
20176.49
18786.60
17486.80
16271.39
15134.99
14072.45
13079.10
12150.48
Male
ex
Male
Mx
Male
Nx
-------
--------
912.15
332.78
298.77
273.12
244.55
215.89
190.56
167.04
150.13
138.27
130.65
131.87
139.84
152.02
164.55
175.80
183.61
185.51
183.05
176.50
166.40
155.06
143.23
130.71
119.21
108.14
99.13
92.05
85.32
80.53
76.49
73.43
71.10
69.15
67.81
66.96
66.76
66.82
67.02
67.65
68.47
69.24
69.94
70.76
71.42
71.91
72.40
72.72
72.98
73.36
----------
9754.98
8842.83
8510.05
8211.28
7938.17
7693.61
7477.72
7287.15
7120.12
6969.99
6831. 72
6701.07
6569.20
6429.36
6277.33
6112.78
5936.98
5753.37
5567.86
5384.81
5208.31
5041.91
4886.84
4743.61
4612.90
4493.69
4385.55
4286.42
4194.37
4109.05
4028.52
3952.04
3878.60
3807.51
3738.35
3670.54
3603.58
3536.82
3470.00
3402.97
3335.33
3266.86
3197.62
3127.68
3056.92
2985.50
2913.59
2841.19
2768.47
2695.49
5523370.93
5152273.93
4806366.45
4483421.22
4181902.01
3900381.47
3637522.71
3392076.22
3162877.54
2948840.20
2748955.43
2562285.50
2387958.27
2225167.51
2073166.44
1931261.40
1798804.42
1675188.64
1559843.50
1452229.80
1351839.31
1258192.91
1170839.32
1089355.51
1013345.65
942439.11
876290.52
814577.55
757001.01
703283.21
653164.97
606406.03
562782.57
522086.42
484123.73
448713.73
415688.08
384889.96
356173.42
329402.35
304449.69
281197.10
259534.17
239357.67
220571.08
203084.27
186812.89
171677.89
157605.44
144526.34
28
-
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
. 95
96
97
98
99
-
11282.23
10470.35
9710.71
8999.83
8334.27
7711.07
7127.62
6581.58
6070.79
5593.19
5146.67
4729.17
4338.63
3973.21
3631.26
3311.34
3012.33
2733.25
2473.23
2231.35
2006.52
1797.68
1603.74
1423.76
1256.98
1102.97
961.49
832.37
715.39
610.21
516.25
432.81
359.15
294.57
238.45
190.29
149.57
115.71
88.06
65.88
48.41
34.91
24.66
17.02
11.42
7.36
4.46
2.41
1.04
0.25
73.80
74.66
75.60
76.79
77.96
78.99
79.74
80.22
80.45
80.61
80.80
81.15
81.58
82.02
82.36
82.38
82.01
81.21
80.09
78.85
77.58
76.33
75.06
73.64
71.77
69.32
66.23
62.52
58.38
54.04
49.66
45.34
41.09
36.85
32.56
28.27
24.07
20.08
16.42
13.16
10.34
7.96
6.03
4.49
3.31
2.42
1. 75
1.22
0.72
0.23
2622.13
2548.33
2473.67
2398.07
2321.28
2243.32
2164.33
2084.59
2004.37
1923.92
1843.31
1762.50
1681.35
1599.77
1517.75
1435.39
1353.01
1271.00
1189.79
1109.70
1030.85
953.27
876.95
801.89
728.25
656.47
587.15
520.92
458.40
400.02
345.98
296.31
250.97
209.88
173.04
140.48
112.20
88.13
68.05
51.63
38.47
28.14
20.17
14.15
9.66
6.35
3.93
2.17
0.95
0.23
132375.86
121093.63
110623.28
100912.57
91912.74
83578.47
75867.40
68739.78
62158.20
56087.41
50494.22
45347.55
40618.38
36279.76
32306.54
28675.28
25363.94
22351.61
19618.36
17145.13
14913.78
12907.26
11109.59
9505.84
8082.08
6825.10
5722.13
4760.63
3928.27
3212.88
2602.67
2086.42
1653.62
1294.46
999.89
761.44
571.15
421.58
305.87
217.82
151.94
103.53
68.62
43.96
26.94
15.52
8.16
3.70
1.28
0.25
29
,-
Male
Aq3
°
1
-
-.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
1000Ax
Male
ax
-------
-------
27.20
26.45
27.26
28.18
29.17
30.28
31.52
32.89
34.42
36.08
37.86
39.77
41.75
43.76
45.77
47.75
49.69
51.61
53.53
55.49
57.54
59.72
62.05
64.57
67.31
70.28
73.52
77.02
80.78
84.82
89.14
93.73
98.60
103.77
109.23
114.99
121.06
127.43
134.10
141.10
148.40
156.02
163.97
172.25
180.86
189.83
199.17
208.88
219.00
229.52
14.88
14.89
14.88
14.87
14.85
14.84
14.82
14.80
14.78
14.75
14.73
14.70
14.67
14.64
14.61
14.58
14.55
14.52
14.49
14.47
14.44
14.40
14.37
14.33
14.29
14.25
14.20
14.15
14.09
14.03
13.97
13.90
13.83
13.75
13.67
13.59
13.50
13.40
13.30
13.20
13.09
12.98
12.86
12.74
12.61
12.48
12.34
12.20
12.05
11.89
Male
$60,000P(Ax )
----------109.63
106.54
109.91
113.69
117.83
122.45
127.61
133.36
139.74
146.73
154.27
162.35
170.77
179.36
187.96
196.48
204.88
213.20
221.58
230.18
239.17
248.76
259.09
270.31
282.58
295.99
310.67
326.65
343.95
362.69
382.87
404.56
427.82
452.71
479.35
507.79
538.14
570.43
604.77
641.29
680.06
721.18
764.82
811.15
860.32
912.57
968.16
1027.33
1090.42
1157.76
30
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
-
240.46
251.81
263.55
275.68
288.16
300.99
314.16
327.69
341.59
355.88
370.55
385.59
400.94
416.57
432.43
448.48
464.70
481.10
497.71
514.53
531.53
548.63
565.74
582.71
599.41
615.78
631.80
647.49
662.94
678.23
693.37
708.32
722.97
737.17
750.79
763.77
776.14
788.00
799.49
810.81
822.17
833.86
846.26
859.79
874.93
891.95
910.83
930.97
950.80
966.92
11.73
11.57
11.39
11.21
11.03
10.84
10.64
10.44
10.24
10.03
9.81
9.59
9.36
9.13
8.90
8.66
8.42
8.18
7.93
7.68
7.43
7.18
6.93
6.68
6.43
6.19
5.95
5.72
5.49
5.27
5.04
4.82
4.60
4.39
4.19
4.00
3.82
3.64
3.47
3.31
3.14
2.97
2.78
2.58
2.36
2.11
1.83
1.53
1.24
1.00
1229.62
1306.35
1388.10
1475.17
1567.76
1666.19
1770.90
1882.51
2001.73
2129.35
2266.12
2412.69
2569.58
2737.28
2916.32
3107.33
3311.38
3529.89
3764.73
4017.83
4290.76
4584.69
4900.07
5236.58
5593.46
5970.80
6369.66
6792.58
7243.88
7728.81
8251.92
8816.06
9421.46
10065.05
10742.70
11452.19
12194.90
12976.72
13809.69
14712.95
15717.10
16869.42
18246.52
19973.58
22251.80
25390.83
29885.23
36480.19
46076.93
58015.25
31
-"
Male
~
1000Ax:~
--------
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
8.15
6.41
6.22
6.11
6.10
6.22
6.49
6.92
7.50
8.20
9.00
9.85
10.69
11.47
12.14
12.65
12.98
13.16
13.20
13.15
13.03
12.91
12.80
12.74
12.76
12.87
13.11
13.47
13.95
14.58
15.34
16.25
17.31
18.53
19.91
21.46
23.18
25.06
27.11
29.33
31.73
34.32
37.12
40.18
43.52
47.19
51.22
55.63
60.44
65.70
Male
ax: 10)
7.48
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.48
7.48
7.48
7.47
7.47
7.47
7.46
7.46
7.46
7.46
7.46
7.46
7.46
7.46
7.46
7.46
7.46
7.46
7.46
7.46
7.46
7.45
7.45
7.45
7.44
7.44
7.43
7.42
7.42
7.41
7.40
7.39
7.38
7.37
7.36
7.34
7.33
7.31
7.30
7.28
Male
$20 ,000P(Ak TOJ)
-------------21.80
17.11
16.61
16.33
16.30
16.61
17.34
18.49
20.04
21.92
24.04
26.33
28.60
30.71
32.50
33.88
34.80
35.27
35.40
35.25
34.94
34.60
34.31
34.15
34.19
34.50
35.13
36.10
37.39
39.08
41.13
43.60
46.47
49.75
53.50
57.71
62.38
67.51
73.10
79.18
85.76
92.86
100.60
109.05
118.31
128.51
139.74
152.09
165.65
180.53
32
--
-,
,-
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
71.42
77.65
84.42
91.77
99.75
108.40
117.77
127.89
138.82
150.61
163.32
177.04
191.82
207.77
224.94
243.37
263.04
283.92
305.95
329.06
353.14
378.10
403.83
430.19
456.99
484.01
511.06
537.95
564.54
590.70
616.27
641.07
664.87
687.49
708.88
729.25
748.97
768.58
788.31
807.19
822.17
833.86
846.26
859.79
874.93
891.95
910.83
930.97
950.80
966.92
7.26
7.23
7.21
7.18
7.15
7.12
7.09
7.05
7.01
6.96
6.91
6.86
6.80
6.74
6.67
6.60
6.52
6.44
6.34
6.24
6.14
6.02
5.90
5.77
5.63
5.50
5.36
5.21
5.06
4.91
4.75
4.58
4.41
4.25
4.08
3.92
3.76
3.61
3.01
3.30
3.14
2.97
2.78
2.58
2.36
2.11
1.83
1.53
1.24
1.00
196.81
214.67
234.19
255.58
278.94
304.94
332.41
362.91
396.23
432.64
472.49
516.17
564.06
616.64
674.38
737.63
806.81
882.30
964.55
1054.02
1151.11
1256.27
1369.80
1491.80
1622.16
1760.83
1907.91
2063.91
2229.82
2406.96
2596.36
2798.52
3013.12
3238.90
3474.67
3720.72
3979.53
4256.27
4557.95
4887.96
5239.03
5623.14
6082.17
6657.86
7417.27
8463.61
9961.74
12160.06
15358.98
19338.42
-
33
Af¥J
0
1
-
-.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
ax(Male12)
Male
Monthly Benefit on $40,000 Pension
------
---------------------------
14.42
14.43
14.42
14.41
14.39
14.37
14.36
14.34
14.31
14.29
14.26
14.23
14.20
14.17
14.15
14.12
14.09
14.06
14.03
14.00
13.97
13.94
13.90
13.87
13.83
13.78
13.74
13.68
13.63
13.57
13.50
13.44
13.36
13.29
13.21
13.12
13.03
12.94
12.84
12.74
12.63
12.52
12.40
12.28
12.15
12.02
11.88
231.16
230.99
231.18
231.40
231.63
231.90
232.19
232.52
232.89
233.29
233.72
234.18
234.66
235.16
235.65
236.14
236.62
237.10
237.58
238.07
238.58
239.13
239.73
240.37
241.07
241.84
242.69
243.60
244.60
245.67
246.83
248.08
249.42
250.85
252.38
254.02
255.77
251.63
259.61
261.72
263.96
266.33
268.85
271.53
274.38
277.41
280.63
11.73
284.06
11.59
11.43
287.73
291.64
34
-
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
.-
11.27
11. 10
10.93
10.75
10.56
10.37
10.18
9.98
9.77
9.56
9.35
9.12
8.90
8.66
8.43
8.19
7.95
7.71
7.47
7.22
6.97
6.71
6.46
6.21
5.96
5.72
5.48
5.25
5.02
4.80
4.57
4.35
4.14
3.93
3.73
3.53
3.35
3.18
3.01
2.84
2.67
2.50
2.31
2.11
1.89
1.64
1.36
1.06
0.77
0.53
295.83
300.30
305.07
310.16
315.57
321.34
327.49
334.05
341.07
348.61
356.70
365.38
374.71
·384.70
395.39
406.84
419.10
432.28
446.50
461.88
478.53
496.55
515.98
536.82
559.05
582.70
607.85
634.70
663.56
694.81
728.80
765.80
805.89
848.96
894.82
943.41
994.91
1049.83
1109.18
1174.52
1248.40
1334.81
1440.46
1576.76
1763.36
2034.02
2451.64
3138.60
4335.45
6281.54
3S
AtfJ
.-
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
yx
Female
----------
Female Ix
--------
1.00000000
0.93457944
0.87343873
0.81629788
0.76289618
0.71298618
0.66634222
0.62274974
0.58200910
0.54393374
0.50834929
0.47509280
0.44401196
0.41496445
0.38781724
0.36244602
0.33873460
0.31657439
0.29586392
0.27650833
0.25841900
0.24151309
0.22571317
0.21094688
0.19714662
0.18424918
0.17219549
0.16093037
0.15040221
0.14056282
0.13136712
0.12277301
0.11474113
0.10723470
0.10021934
0.09366294
0.08753546
0.08180884
0.07645686
0.07145501
0.06678038
0.06241157
0.05832857
0.05451268
0.05094643
0.04761349
0.04449859
0.04158747
0.03886679
0.03632410
130109
129864
129755
129651
129550
129450
129353
129259
129167
129077
128988
128900
128810
128716
128617
128512
128400
128282
128159
128031
127899
127763
127625
127485
127342
127196
127047
126895
126738
126576
126409
126236
126057
125872
125678
125476
125263
125035
124790
124524
124235
123921
123580
123212
122818
122396
121946
121468
120959
120417
1000Qx
------1.88
0.84
0.80
0.78
0.77
0.75
0.73
0.71
0.70
0.69
0.68
0.70
0.73
0.77
0.82
0.87
0.92
0.96
1.00
1.03
1.06
1.08
1.10
1.12
1.15
1.17
1.20
1.24
1.28
1.32
1.37
1.42
1.47
1.54
1.61
1.70
1.82
1.96
2.13
2.32
2.53
2.75
2.98
3.20
3.44
3.68
3.92
4.19
4.48
4.79
36
-
-
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
0.03394776
0.03172688
0.02965129
0.02771148
0.02589858
0.02420428
0.02262083
0.02114096
0.01975791
0.01846533
0.01725732
0.01612834
0.01507321
0.01408711
0.01316553
0.01260423
0.01149928
0.01074699
0.01004392
0.00938684
0.00877275
0.00819883
0.00766246
0.00716117
0.00669269
0.00625485
0.00584565
0.00546323
0.00510582
0.00477179
0.00445962
0.00416787
0.00389520
0.00364038
0.00340222
0.00317965
0.00297163
0.00277723
0.00259554
0.00242574
0.00226704
0.00211873
0.00198012
0.00185058
0.00172952
0.00161637
0.00151063
0.00141180
0.00131944
0.00123312
119840
119225
118569
117867
117115
116313
115460
114559
113614
112626
111590
110496
109331
108075
106710
105225
103616
101885
100038
98078
95996
93773
91379
88777
85937
82839
79479
75867
72022
67966
63713
59274
54661
49892
45008
40074
35173
30396
25831
21559
17645
14136
11059
8419
6201
4370
2880
1688
781
200
5.13
5.50
5.92
6.38
6.85
7.33
7.80
8.25
8.70
9.20
9.80
10.54
11.46
12.63
13.92
15.29
16.71
18.13
19.59
21.23
23.16
25.53
28.47
31.99
36.05
40.56
45.45
50.68
56.32
62.58
69.67
77.83
87.25
97.89
109.62
122.30
135.81
150.18
165.38
181.55
198.87
217.67
238.72
263.45
295.27
340.96
413.89
537.32
743.92
1000.00
37
ACIJ
0
1
-
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
"
Female
Female
Female
Female
Dx
---------
ex
-------
Mx
--------
Nx
----------
130109.00
121368.22
113333.04
105883.84
98833.07
92296.06
86193.37
80496.01
75176.37
70209.34
65570.96
61239.46
57193.18
53412.56
49879.89
46578.66
43493.52
40610.80
37917.62
35401.64
33051.53
30856.44
28806.64
26892.56
25105.04
23435.76
21876.92
20421.26
19061.68
17791.88
16605.99
15498.37
14463.92
13497.85
12595.37
11752.45
10964.95
10228.97
9541.05
8897.86
8296.46
7734.10
7208.24
6716.62
6257.14
5827.70
5426.42
5051.55
4701.29
4374.04
228.97
95.20
84.89
77.05
71.30
64.64
58.54
53.54
48.95
45.24
41.81
39.96
39.01
38.39
38.06
37.94
37.36
36.39
35.39
34.11
32.85
31.15
29.53
28.19
26.90
25.66
24.46
23.61
22.77
21.94
21.24
20.54
19.84
19.44
18.92
18.65
18.65
18.73
19.01
19.30
19.60
19.89
20.06
20.07
20.09
20.02
19.88
19.78
19.69
19.59
2605.91
2376.99
2281.78
2196.89
2119.84
2048.54
1983.90
1925.36
1871.82
1822.86
1777.62
1735.81
1695.85
1656.85
1618.45
1580.40
1542.46
1505.10
1468.71
1433.32
1399.21
1366.36
1335.21
1305.68
1277.49
1250.59
1224.93
1200.47
1176.85
1154.08
1132.15
1110.91
1090.37
1070.53
1051.09
1032.17
1013.52
994.87
976.14
957.13
937.83
918.23
898.34
878.28
858.21
838.12
818.09
798.21
778.43
758.74
1948975.05
1818866.04
1697497.81
1584164.77
1478330.93
1379497.86
1287201.80
1201008.43
1120512.42
1045336.05
975126.72
909555.76
848316.30
791123.12
737710.55
687830.66
641252.00
597758.48
557147.68
519230.06
483828.42
450776.89
419920.45
391113.81
364221.25
339116.20
315680.44
293808.52
273382.26
254320.59
236528.71
219922.72
204424.35
189960.43
176462.58
163867.21
152114.76
141149.81
130920.84
121379.79
112481.93
104185.46
96451.36
89243.12
82526.50
76269.36
70441.66
65015.23
59963.69
55262.40
38
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
.95
96
97
98
99
-
4068.30
3782.64
3515.72
3266.27
3033.11
2815.27
2611.80
2421.89
2244.77
2079.68
1925.74
1782.12
1647.97
1522.46
1404.89
1294.71
1191.51
1094.96
1004.77
920.64
842.15
768.83
700.19
635.75
575.15
518.15
464.61
414.48
367.73
324.32
284.14
247.05
212.92
181.63
153.13
127.42
104.52
84.42
67.05
52.30
40.00
29.95
21.90
15.58
10.72
7.06
4.35
2.38
1.03
0.25
19.51
19.45
19.45
19.48
19.41
19.30
19.05
18.67
18.24
17.88
17.64
17.56
17.69
17.97
18.27
18.50
18.60
18.55
18.40
18.26
18.23
18.34
18.63
19.01
19.38
19.64
19.73
19.63
19.35
18.97
18.50
17.97
17.36
16.62
15.69
14.56
13.27
11.85
10.36
8.87
7.43
6.09
4.89
3.84
2.96
2.25
1.68
1.20
0.72
0.23
739.15
719.64
700.19
680.74
661.26
641.85
622.55
603.51
584.84
566.59
548.71
531.07
516.51
495.81
477.84
459.57
441.07
422.47
403.92
385.52
367.25
349.03
330.68
312.05
293.04
273.66
254.02
234.29
214.66
195.30
176.34
157.84
139.87
122.51
105.89
90.20
75.64
62.37
50.52
40.16
31.29
23.85
17.76
12.87
9.04
6.08
3.83
2.14
0.95
0.23
50888.36
46820.06
43037.42
39521.70
36255.43
33222.32
30407.05
27795.25
25373.36
23129.58
21048.91
19123.16
17341.05
15693.08
14171.61
12765.72
11471.01
10279.50
9184.54
8179.77
7259.12
6416.98
5648.15
4947.96
4312.21
3737.06
3218.92
2754.31
2339.83
1972.10
1647.78
1363.65
1116.60
903.68
722.06
568.93
441.51
336.99
252.57
185.53
133.23
93.23
63.28
41.38
25.80
15.07
8.01
3.66
1.28
0.25
39
-
Aq3
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
Female
Female
Female
1000Ax
ax
-------
-------
$60,000P(Ax )
20.72
20.26
20.83
21.48
22.19
22.96
23.81
24.75
25.76
26.86
28.05
29.33
30.68
32.09
33.57
35.10
36.69
38.34
40.07
41.89
43.80
45.81
47.95
50.23
52.65
55.21
57.93
60.82
63.88
67.11
70.54
74.16
77.99
82.06
86.34
90.86
95.63
100.63
105.85
111.29
116.95
122.83
128.94
135.29
141.90
148.79
155.98
163.48
171.31
179.47
14.98
14.99
14.98
14.97
14.96
14.95
14.93
14.92
14.91
14.89
14.87
14.85
14.83
14.81
14.79
14.77
14.74
14.72
14.69
14.67
14.64
14.61
14.58
14.54
14.51
14.47
14.43
14.39
14.34
14.29
14.24
14.19
14.13
14.07
14.01
13.94
13.87
13.80
13.72
13.64
13.56
13.47
13.38
13.29
13.19
13.09
12.98
12.87
12.75
12.63
----------83.00
81.12
83.44
86.09
89.01
92.18
95.68
99.52
103.70
108.25
113.16
118.47
124.10
130.01
136.19
142.63
149.32
156.30
163.64
171.36
179.52
188.16
197.38
207.23
217.73
228.92
240.87
253.64
267.23
281.70
297.13
313.57
331. 11
349.83
369.75
391.01
413.61
437.53
462.84
489.50
517.57
547.11
578.18
610.92
645.54
682.15
720.94
762.13
805.86
852.30
40
-
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
187.97
196.83
206.05
215.63
255.56
235.88
246.61
257.81
269.55
281.87
294.80
308.31
322.38
336.94
351.90
367.24
382.99
399.18
415.91
433.24
451.18
469.68
488.62
507.82
527.14
546.44
565.67
584.83
603.94
623.03
642.08
661.00
679.65
697.84
715.45
732.40
748.70
764.41
779.63
794.49
809.18
823.92
839.02
854.84
871.79
890.16
909.98
930.65
950.72
966.92
12.51
12.38
12.24
12.10
11.95
11.80
11.64
11.48
11.30
11.12
10.93
10.73
10.52
10.31
10.09
9.86
9.63
9.39
9.14
8.88
8.62
8.35
8.07
7.78
7.50
7.21
6.93
6.65
6.36
6.08
5.80
5.52
5.24
4.98
4.72
4.46
4.22
3.99
3.77
3.55
3.33
3.11
2.89
2.66
2.41
2.13
1.84
1.54
1.24
1.00
901.66
954.14
1009.94
1069.23
1132.21
1199.30
1270.95
1347.84
1430.81
1520.71
1618.24
1723.92
1838.22
1961.27
2093.26
2234.78
2386.89
2551.21
2729.98
2925.70
3140.56
3376.40
3634.39
3914.92
4218.48
4545.84
4989.80
5280.40
5694.93
6147.62
6643.08
7185.05
7775.79
8415.27
9103.49
9841.92
10634.64
11489.24
12417.15
13437.12
14577.23
15881.61
17421.48
19311.87
21744.17
25026.62
29561.98
36355.91
46027.59
58015.25
-
41
Ar¥J
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
Female
Female
1OOoAx:
--------6.59
5.47
5.35
5.28
5.25
5.25
5.30
5.40
5.55
5.74
5.97
6.24
6.52
6.80
7.07
7.33
7.55
7.76
7.96
8.16
8.36
8.57
8.79
9.05
9.33
9.64
10.00
10.42
10.89
11.45
12.11
12.86
13.74
14.74
15.84
17.08
18.44
19.89
21.44
23.07
24.78
26.57
28.44
30.43
32.56
34.84
37.28
39.88
42.60
45.45
aX:1q
f6l
Female
$20,000P(Ai fOP
------
--------------
7.48
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.48
7.48
7.48
7.48
7.48
7.48
7.48
7.48
7.48
7.48
7.48
7.47
7.47
7.47
7.47
7.46
7.46
7.46
7.45
7.45
7.44
7.44
7.43
7.42
7.42
7.41
7.40
7.39
7.39
7.38
7.37
7.36
7.35
17.60
14.59
14.28
14.09
14.01
14.01
14.14
14.42
14.81
15.32
15.94
16.66
17.42
18.17
18.89
19.57
20.18
20.74
21.28
21.81
22.35
22.90
23.51
24.19
24.95
25.79
26.74
27.87
29.15
30.66
32.41
34.45
36.80
39.50
42.49
45.84
49.51
53.45
57.65
62.09
66.74
71.63
76.76
82.21
88.07
94.34
101.06
108.25
115.81
123.73
--
42
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
-,
48.43
51.58
54.94
58.58
62.56
66.99
71.89
77.34
83.38
90.08
97.49
105.68
114.78
124.88
136.09
148.56
162.42
177.80
194.88
213.76
234.55
257.28
281.95
308.46
336.66
366.33
397.24
429.14
461.79
494.92
528.16
561.11
593.35
624.51
654.39
683.05
710.82
738.14
765.01
789.93
809.18
823.92
839.02
854.84
871.79
890.16
909.98
930.65
950.72
966.92
7.33
7.32
7.31
7.30
7.28
7.27
7.25
7.23
7.21
7.19
7.16
7.13
7.10
7.06
7.02
6.97
6.93
6.87
6.81
6.74
6.66
6.57
6.47
6.36
6.24
6.11
5.98
5.83
5.68
5.51
5.33
5.14
4.95
4.75
4.55
4.35
4.15
3.95
3.75
3.54
3.33
3.11
2.89
2.66
2.41
2.13
1.84
1.54
1.24
1.00
132.06
140.88
150.33
160.58
171.85
184.38
198.32
213.87
231.25
250.64
272.29
296.46
323.53
353.88
387.89
426.07
469.02
517.43
572.04
634.04
704.01
782.88
871.31
969.78
1078.67
1198.26
1329.00
1471.62
1627.17
1796.88
1981.71
2182.29
2398.78
2630.78
2878.37
3142.88
3427.74
3738.73
4082.11
4459.26
4859.08
5293.87
5807.16
6437.29
7248.06
8342.21
9883.99
12118.64
15342.53
19338.42
43
-
Female
ArjJ
0
1
-
-
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
Female
'aX< 12)
Month ly Benefit on $40,000 Pension
------
---------------------------
14.52
14.52
14.51
14.50
14.49
14.48
14.47
14.46
14.44
14.42
14.41
14.39
14.37
14.35
14.33
14.30
14.28
14.26
14.23
14.20
14.17
14.14
14.11
14.08
14.04
14.01
13.97
13.92
13.88
13.83
13.78
13.73
13.67
13.61
13.55
13.48
13.41
13.33
13.26
13.18
13.09
13.01
12.92
12.82
12.72
12.62
12.52
12.41
12.29
12.17
229.64
229.53
229.66
229.82
229.98
230.16
230.36
230.58
230.82
231.08
231.37
231.67
231.99
232.33
232.68
233.05
233.44
233.84
234.26
234.70
235.17
235.66
236.19
236.75
237.35
238.00
238.68
239.41
240.19
241.02
241.91
242.85
243.86
244.94
246.08
247.30
248.60
249.98
251.43
252.97
254.58
256.29
258.08
259.97
261.96
264.08
266.32
268.70
271.23
273.91
--
.-
44
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
12.04
11.91
11.78
11.63
11.49
11.34
11.18
11.01
10.84
10.66
10.46
10.26
10.06
9.84
9.62
9.39
9.16
8.92
8.67
8.42
8.15
7.88
7.60
7.32
7.03
6.75
6.46
6.18
5.90
5.61
5.33
5.05
4.78
4.51
4.25
4.00
3.76
3.52
3.30
3.08
2.86
2.64
2.42
2.19
1.94
1.67
1.37
1.07
0.77
0.53
276.77
279.82
283.05
286.50
290.16
294.06
298.24
302.72
307.56
312.82
318.53
324.73
331.45
338.69
346.47
354.84
363.85
373.61
384.26
395.95
408.83
423.02
438.60
455.61
474.11
494.16
515.90
539.55
565.40
593.83
625.19
659.78
697.85
739.48
784.78
833.97
887.47
945.96
1010.44
1082. 53
1164.64
1260.62
1376.85
1524.03
1721.09
2001.77
2429.03
3124.74
4328.58
6281.54
-
.-
-
4S
Blended
~
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
yX
Blended Ix
1000Qx
----------
--------
-------
1.00000000
0.93457944
0.87343873
0.81629788
0.76289618
0.71298618
0.66634222
0.62274974
0.58200910
0.54393374
0.50834929
0.47509280
0.44401196
0.41496445
0.38781724
0.36244602
0.33873460
0.31657439
0.29586392
0.27650833
0.25841900
0.24151309
0.22571317
0.21094688
0.19714662
0.18424918
0.17219549
0.16093037
0.15040221
0.14056282
0.13136712
0.12277301
0.11474113
0.10723470
0.10021934
0.09366294
0.08753546
0.08180884
0.07645686
0.07145501
0.06678038
0.06241157
0.05832857
0.05451268
0.05094643
0.04761349
0.04449859
0.04158747
0.03886679
0.03632410
501883
500637
500140
499663
499198
498749
498323
497920
497538
497170
496807
496442
496050
495613
495112
494539
493891
493171
492397
491580
490738
489886
489035
488191
487362
486548
485753
484968
484184
483402
482610
481801
480971
480109
479212
478273
477280
476220
475085
473864
472545
471115
469566
467893
466085
464135
462038
459784
457365
454771
2.48
0.99
0.95
0.93
0.90
0.85
0.81
0.77
0.74
0.73
0.73
0.79
0.88
1.01
1.16
1.31
1.46
1.57
1.66
1.71
1.74
1.74
1.73
1.70
1.67
1.63
1.62
1.62
1.62
1.64
1.68
1.72
1.79
1.87
1:96
2.08
2.22
2.38
2.57
2.78
3.03
3.29
3.56
3.86
4.18
4.52
4.88
5.26
5.67
6.13
-,
.-
-
46
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
0.03394776
0.03172668
0.02965129
0.02771148
0.02569858
0.02420428
0.02262083
0.02114096
0.01975791
0.01846533
0.01725732
0.01612834
0.01507321
0.01408711
0.01316553
0.01260423
0.01149928
0.01074699
0.01004392
0.00938684
0.00877275
0.00819883
0.00766246
0.00716117
0.00669269
0.00625485
0.00584565
0.00546323
0.00510582
0.00477179
0.00445962
0.00416787
0.00389520
0.00364038
0.00340222
0.00317965
0.00297163
0.00277723
0.00259554
0.00242574
0.00226704
0.00211873
0.00198012
0.00185058
0.00172952
0.00161637
0.00151063
0.00141180
0.00131944
0.00123312
451965
448992
445759
442262
438470
434362
429921
425139
420012
414528
406667
402394
395660
388416
380611
372212
363208
353604
343418
332662
321317
309350
296712
283350
269241
254423
238990
223080
206859
190489
174097
157783
141638
125757
110286
95417
81363
68322
56452
45858
36584
28623
21919
16379
11877
8264
5393
3138
1445
369
6.62
7.20
7.65
8.57
9.37
10.22
11.12
12.06
13.06
14.14
15.35
16.73
18.31
20.09
22.07
24.19
26.44
28.81
31.32
34.10
37.24
40.85
45.03
49.79
55.04
60.66
66.57
72.71
79.14
86.05
93.71
102.32
112.12
123.02
134.82
147.29
160.28
173.74
187.66
202.23
217.61
234.22
252.75
274.86
304.20
347.41
416.13
539.52
744.64
1000.00
--
47
Aq3
Blended
Blended
Blended
Dx
ex
------1164.49
434.10
389.37
354.75
320.13
283.86
250.97
222.33
200.17
184.53
173.41
174.05
181.34
194.30
207.68
219.50
227.93
229.00
225.91
217.59
205.77
192.08
178.04
163.43
149.98
136.90
126.33
117.92
109.92
104.04
99.32
95.24
92.44
89.90
87.95
86.92
86.72
86.78
87.25
88.08
89.25
90.35
91.20
92.11
92.85
93.31
93.74
94.02
94.22
94.58
Mx
--------0
1
-
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
501883.00
467885.05
436841.65
407873.85
380835.76
355601.14
332053.66
310079.55
289571.65
270427.54
252551.49
235856.02
220252.13
205661.77
192012.97
179243.69
167297.97
156125.31
145682.50
135925.97
126816.02
118313.88
110381.64
102982.37
96081.77
89464.07
83644.48
78046.08
72822.34
67948.35
63399.08
59152.16
55187.15
51484.34
48026.31
44796.45
41778.92
38959.00
36323.51
33859.96
31556.74
29403.03
27389.11
25506.10
23745.37
22099.09
20560.04
19121.25
17776.31
16519.15
Blended
Nx
--------
----------
12575.61
11411.12
10977.02
10587.65
10232.90
9912.77
9628.91
9377.94
9155.61
8955.45
8770.92
8597.51
8423.45
8242.12
8047.82
7840.14
7620.64
7392.70
7163.70
6937.80
6720.21
6514.44
6322.36
6144.32
5980.88
5830.91
5694.01
5567.68
5449.76
5339.84
5235.80
5136.48
5041.24
4948.81
4858.91
4770.96
4684.04
4597.32
4510.54
4423.29
4335.21
4245.96
4155.61
4064.11
3972.30
3879.45
3786.14
3692.40
3598.38
3504.16
7479412.99
6977529.99
6509644.94
6072803.30
5664929.45
5284093.69
4928492.54
4596438.89
4286359.34
3396787.69
3726360.15
3473808.66
3237952.65
3017700.51
2812038.74
2620025.77
2440782.08
2273484.11
2117358.80
1971676.29
1835750.33
1708934.30
1590620.42
1480238.79
1377256.42
1281174.64
1191528.58
1107884.10
1029838.02
957015.68
889067.33
825668.24
766516.09
711328.93
659844.59
611818.28
567021.82
525242.90
486283.89
449960.39
416100.43
384543.70
355140.67
327751.55
302245.45
278500.08
256401.00
235840.96
216719.71
198943.40
,-
.-
-
48
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
15343.88
14245.11
13217.33
12255.74
11355.75
10513.42
9725.17
8987.85
8298.56
7654.40
7052.50
6489.95
5963,87
5471.66
5010.94
4579.78
4176.63
3800.18
3449.26
3122.64
2818.83
2536.31
2273.54
2029.12
1801.95
1591.38
1397.05
1218.74
1056.18
908.97
776.41
657.62
551.71
457.80
375.22
303.39
241. 78
189.75
146.52
111.24
82.94
60.64
43.40
30.31
20.54
13.36
8.15
4.43
1.91
0.46
94.96
95.86
96.91
98.21
99.43
100.46
101.10
101.30
101.26
101.15
101.17
101.50
102.05
102.76
103.34
103.54
103.21
102.31
100.96
99.53
98.12
96.84
95.69
94.43
92.68
90.22
86.92
82.82
78.11
73.10
67.99
62.89
57.81
52.64
47.28
41.76
36.22
30.81
25.70
21.02
16.87
13.27
10.25
7.79
5.84
4.34
3.18
2.23
1.33
0.43
3409.58
3314.62
3218.76
3121.85
3023.65
2924.21
282376
3722.66
2621.36
2520,10
2418.95
2317.78
2216.28
2114.23
2011.47
1908.13
1804.59
1701.37
1599.07
1498.10
1398.58
1300.46
1203.62
1107.93
1013.51
920,82
830.61
743.69
660.87
582.75
509.65
441.65
378.77
320.95
268.32
221.04
179.28
143.06
112.25
86.55
65.53
48.66
35.38
25.13
17.35
11.51
7.17
3.99
1.75
0.43
182424.25
167080.37
152835.26
139617.93
127362.19
116006.44
105493.02
95767.85
86780.01
78481.45
70827.05
63774.55
57284.61
51320.74
45849.08
40838.14
36258.36
32081.73
28281.55
24832.28
21709.64
18890.81
16354.50
14080.96
12051.84
10249,89
8658.52
7261.46
6042.73
4986.54
4077.57
3301.16
2643.54
2091.84
1634.03
1258.81
955.42
713.64
523.90
377.37
266.13
183.20
122.55
79.15
48.84
28.30
14.94
6.79
2.36
0.46
-
49
At}3
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
Blended
BlenOOd
1000Ax
ax
-------
-------
25.92
25.23
26.00
26.86
27.80
28.84
30.00
31.29
32.71
34.26
35.93
37.71
39.57
41.46
43.36
45.25
47.13
48.99
50.88
52.81
54.88
56.97
59.26
61.73
64.40
67.29
70.43
73.81
77.43
81.31
85.44
89.84
94.51
99.45
104.67
110.19
115.19
122.09
128.47
135.16
142.13
149.40
156.98
164.86
173.08
181.62
190.52
199.79
209.43
219.47
14.90
14.91
14.90
14.42
14.87
14.86
14.84
14.82
14.80
14.78
14.75
14.73
14.70
14.67
14.65
14.62
14.59
14.56
14.53
14.51
14.48
14.44
14.41
14.37
14.33
14.29
14.25
14.20
14.14
14.08
14.02
13.96
13.89
13.82
13.74
13.66
13.57
13.48
13.39
13.29
13.19
13.08
12.97
12.85
12.73
12.60
12.47
12.33
12.19
12.04
Blended
$60 ,000P(Ax )
----------104.37
101.52
104.68
108.23
112.13
116.45
121.28
126.65
132.59
139.09
146.11
153.64
161.49
169.55
177.66
185.76
193.82
201.85
210.02
218.43
227.25
236.63
246.74
257.67
269.57
282.52
296.65
311.96
328.50
346.37
365.57
386.18
408.27
431.87
457.11
484.07
512.80
543.34
575.79
610.24
646.75
685.42
726.37
769.80
815.85
864.71
916.65
971.89
1030.71
1093.40
so
.-.
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
-
229.90
240.74
251.95
263.54
275.48
287.77
300.40
313.41
326.81
340.63
354.86
369.49
384.48
399.77
415.31
431.06
447.02
463.20
479.64
496.36
513.32
530.48
547.72
564.91
581.92
598.66
615.12
631.33
647.36
663.29
679.14
694.84
710.29
725.34
739.85
753.77
767.14
780.04
792.60
804.99
817.42
830.14
843.49
857.86
873.68
891.23
910.49
930.84
950.77
966.92
11.89
11.73
11.56
11.39
11.22
11.03
10.85
10.66
10.46
10.25
10.04
9.83
9.61
9.38
9.15
8.92
8.68
8.44
8.20
7.95
7.70
7.45
7.19
6.94
6.69
6.44
6.20
5.96
5.72
5.49
5.25
5.02
4.79
4.57
4.35
4.15
3.95
3.76
3.58
3.39
3.21
3.02
2.82
2.61
2.38
2.12
1.83
1.53
1.24
1.00
1160.23
1231.50
1307.35
1388.02
1473.72
1564.78
1661.61
1764.82
1875.14
1993.32
2120.09
2256.06
2401.66
2557.32
2723.39
2900.46
3089.56
3292.06
3509.86
3744.99
3999.07
4273.39
4568.56
4884.36
5220.35
5576.77
5954.95
6357.58
6789.01
7254.54
7758.83
8305.05
8894.29
9524.48
10193.31
10900.21
11648.08
12444.02 .
13300.46
14237.40
15284.37
16488.51
17923.74
19711.67
22048.60
25242.94
29792.32
36430.51
46054.75
58015.25
51
.-
Blenood
AgJ
1OOOA
x:fOJ
BlendOO
ax: fOl
------
Blended
nw
7.84
6.22
6.05
5.95
5.94
6.03
6.26
6.62
7.12
7.72
8.40
9.14
9.87
10.55
11.14
11.60
11.91
12.09
12.17
12.16
12.11
12.05
12.01
12.01
12.08
12.23
12.49
12.86
13.34
13.96
14.70
15.58
16.60
17.77
19.10
20.59
22.24
24.03
25.98
28.08
30.35
32.77
35.39
38.23
41.33
44.72
46
7.48
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.49
7.48
7.48
7.48
7.47
7.47
7.47
7.47
7.47
7.46
7.47
7.47
7.47
7.47
7.47
7.47
7.47
7.47
7.46
7.46
7.46
7.46
7.45
7.45
7.45
7.44
7.43
7.43
7.42
7.41
7.40
7.40
7.39
7.38
7.36
7.35
48.43
$20,OOOP(A*:
-------------20.98
16.62
16.15
15.89
15.85
16.10
16.70
17.68
19.00
20.62
22.44
24.42
26.39
28.23
29.81
31.05
31.91
32.40
32.61
32.59
32.45
32.28
32.17
32.17
32.36
32.77
33.46
34.46
35.75
37.40
39.40
41.77
44.55
47.71
51.30
55.34
59.81
64.70
70.02
75.77
81.96
88.62
95.62
103.67
112.24
121.64
7.34
47
48
49
131.96
52.47
56.86
61.63
7.33
7.31
7.29
143.25
155.59
169.04
--------
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
S2
-
-
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
66.80
72.40
78.47
7.27
7.25
7.23
85.06
7.20
92.22
99.99
7.18
7.15
7.12
108.42
117.56
127.45
138.14
149.69
162.18
175.67
190.27
206.05
223.04
241.27
260.73
281.41
303.28
326.27
350.32
375.36
401.27
427.86
454.95
482.35
509.88
537.41
564.78
591.82
618.28
643.93
668.54
692.02
714.53
736.46
758.31
780.23
801.04
817.42
830.14
843.49
857.86
873.68
891.23
910.49
930.84
950.77
966.92
7.09
7.05
7.01
6.96
6.92
6.86
6.81
6.74
6.68
6.61
6.53
6.45
6.36
6.26
6.15
6.03
5.91
5.78
5.65
5.51
5.37
5.23
5.07
4.91
4.74
4.57
183.68
199.67
217.09
236.13
256.94
279.71
304.60
331.82
361.61
394.18
429.86
469.00
511.94
559.14
611.00
667.88
730.21
798.40
872.95
954.39
1043.20
1139.89
1244.84
1358.26
1480.11
1610.48
1749.61
1898.08
2056.96
2227.61
2411.12
2608.08
2818.45
4.40
3041.29
4.22
4.06
3.89
3.73
3.56
3.39
3.21
3.02
2.82
2.61
2.38
2.12
1.83
1.53
1.24
3276.02
3523.47
3786.63
1.00
4071.18
4384.05
4728.18
5094.79
5496.17
5974.58
6570.56
7349.53
8414.31
9930.77
12143.50
15351.58
19338.42
S3
Af1j
0
1
-.
-
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
Blended
ax( 12)
Blended
Monthly Benefit on $40,000 Pension
------
---------------------------
14.44
14.45
14.44
14.42
14.41
14.40
14.38
14.36
14.34
14.32
14.29
14.26
14.24
14.21
14.18
14.15
14.13
14.10
14.07
14.04
14.01
13.98
13.95
13.91
13.87
13.83
13.78
13.73
13.68
13.62
13.56
13.49
13.42
13.35
13.27
13.19
13.11
13.02
12.92
12.82
12.72
12.61
12.50
12.39
12.26
12.14
12.01
11.87
11.73
11.58
230.86
230.70
230.88
231.08
231.31
231.55
231.83
232.14
232.48
232.85
233.25
233.68
234.13
234.59
235.06
235.52
235.99
236.44
236.91
237.40
237.90
238.44
239.02
239.65
240.33
241.07
241.88
242.76
243.71
244.74
245.84
247.02
248.29
249.65
251.10
252.66
254.31
256.07
257.94
259.93
262.03
264.26
266.63
269.14
271.80
274.63
277.64
280.85
284.26
287.90
-
54
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
11.42
11.26
11.10
10.93
10.75
10.57
10.38
10.19
9.99
9.79
9.58
9.36
9.14
8.91
8.68
8.45
8.21
7.98
7.73
7.49
7.23
6.98
6.73
6.49
6.22
5.97
5.73
5.49
5.25
5.02
4.78
4.55
4.32
4.10
3.89
3.68
3.48
3.29
3.11
2.92
2.74
2.55
2.35
2.14
1.91
1.65
1.36
1.06
0.77
0.53
291.79
295.94
300.36
305.06
310.07
315.40
321.07
327.13
333.62
340.58
348.06
356.10
364.73
373.98
383.87
394.44
405.77
417.94
431.07
445.30
460.73
477.47
495.56
515.01
535.82
558.01
581. 70
607.08
634.47
664.24
696.75
732.27
770.95
812.75
857.60
905.56
956.93
1012.33
1072.80
1139,.99
1216.40
1306.05
1415.46
1555.81
1746.39
2020.89
2442.62
3133.05
4332.36
6281.54
-.
B1bl10graphy
Arndt, ShE'ri!. "ACLI Predicts Big Push for Unisex Rating." National
Underwriter - Life & Health/Financial Services Edition, NO.8
Feb. 23, 1987, pp. 3-50.
Benjamin, Deborah. "Affidavit Presented in Superior Court of the
Commonwealth of Massachusetts in Civil Action No. 88-0221,
the American Council of Life Insurance, et al. vs Roger Singer,
Commissioner of Insurance." May 26, 1988.
Bennett, Andrea. "Setting the Unisex Pace." Best's Review -Life/Health
Insurance Edition, 86 Jan. 1986, pp. 22-108.
Bowers, Newton L. Jr., et al. Actuarial Mathematics. Itasca: The Society of
Actuaries, 1986.
Carroll, William F. "Let Consumers Choose." Best's Review - Life/Health
Insurance Edition, 89 June 1988, p. 56.
Cary, Eve and Kathleen Wi llert Peratis. Woman and the Law. Skokie:
National Textbook Company, 1978.
Dennon, A. R "The Facts About Unisex Insurance." Consumer Research
Magazine, 71 Feb. 1988, pp.25-27.
Diamond, .Joseph F. "Editorial Comment: Why Not Unisex Rating?" National
Underwriter - Life & Health/Financial Services Edition, No. 22
Ma~' 30, 1988, p. 18.
Ellis, Deborah A. "Statement on Behalf of the American Civil Liberties Union
Before the Elimination of Discrimination Insurance Practices Study
Committee." Nov. 17, 1988.
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