1. Mortality Tables and Pension Plans
Natalie Taylor, Senior Capstone Project
December 10, 2013
Abstract
Defined benefit pension plans in America are becoming increasingly
underfunded. Among the many complex reasons for this underfunding
is the use of outdated mortality data. Defined benefit pension plans
use a formula that includes an annuity factor for each plan beneficiary.
An annuity factor is the present value of $1 of pensionable salary.
Annuity factors are calculated using mortality rates and interest rates
in their functions. We will discuss construction of a mortality table and
examine the Group Annuity Mortality (GAM) 1983 mortality table
alongside the Group Annuity Reserving (GAR) 1994 mortality table.
We will then calculate and compare the % change in mortality rates
between these two tables. Finally we will compare annuity factors that
have been calculated using the two seperate mortality table rates and
discuss the impact these changes have on the funding of our defined
benefit pension plan model.
1 Introduction
Mortality tables are an important tool used in actuarial science. Pension
actuaries price pension products and determine the value of pension reserves
using mortality rates. They do this by analyzing observed mortality data.
In section 2.2 and section 3 we are going to examine the guidelines for de-
fined benefit pension plans and construct a basic defined benefit pension
plan model. Our defined benefit model will incorporate an annuity factor.
Annuity factors are calculated using mortality rates and interest rates in
their functions. Section 5 will discuss the construction of the Group Annu-
ity Reserving (GAR) 1994 mortality table. Table 2 and table 3 will show
the calculations for the projected mortality rates to the current year, 2013,
1

2. for both the GAR 1994 and the Group Annuity Mortality (GAM) 1983 mor-
tality tables. These two tables also show the calculated percent changes in
the mortality rates for males and females seperately. Then we will calculate
annuity factors based on these two tables projected mortality rates to 2013.
When we examine table 2 and table 3 we will see that when we compare the
GAM 1983 against the GAR 1994 the mortality rates have declined for men
and increased for women at specific retirement ages. We will see how these
changing mortality rates have caused the annuity factors to increase for the
majority of retirement ages. We will show that the increase in annuity fac-
tors causes the cost of servicing our defined benefit pension plan model to
increase. Many companies are still using the GAM 1983 and this is one of
the many reasons defined benefit pension plans are underfunded today. We
will conclude there is a need for updated mortality tables.
2 Retirement and Pension Plans
In the United States people dream of and plan for retirement. One way
workers fund their retirements is via a pension plan. According to Investo-
pedia, a pension plan is defined as ”a type of retirement plan, usually tax
exempt, wherein an employer makes contributions toward a pool of funds set
aside for an employee’s future benefit. The pool of funds is then invested
on the employee’s behalf, allowing the employee to receive benefits upon
retirement.”[1]
2.1 The History of Pension Plans in America
Pensions have been in existence in the United States of America since 1875.
This date marks in history the first recorded retirement plan that was pro-
vided solely by the employer. This historical pension plan was called the
industrial pension plan and it was provided by the American Express Com-
pany. Employees did not contribute to the industrial pension plan. Em-
ployee contributions came a few years later in 1880 when Baltimore and
Ohio Railroad established the very first pension plan that was to be fi-
nanced by both employer and employee contributions. Baltimore and Ohio
Railroad’s plan covered over 77,000 workers. Fast forward to 1987 and we
see over 232,000 defined benefit pension plans in the private sector in the
United States of America. These plans were covering approximately 40 mil-
lion employees and had assets of almost $900 billion.[5] ******* editing left
off here.
2

3. 2.2 Defined Benefit Pension Plan
Guidelines that a typical defined benefit pension plan will follow.
1. A defined benefit pension plan makes a promise to pay an employee a
certain amount of retirement income for life.
2. The amount of the employees pension is based on a formula that factors
in the employees earnings and years of service with the employer.
3. Both the employee and the employer contribute to the plan.
4. The employer is responsible for investing the contributions to ensure there
is enough money to pay the future pensions for all plan members.
5. If there is a shortfall in the money needed, the employer is obligated to
pay the difference.
It is important to note there are other types of pension plans. We are
going to focus on a model that describes a defined benefit pension plan.
2.3 Why should we care about pension plans and mortality
rates?
According to Wilshire Consulting the current, typical defined benefit pen-
sion fund is underfunded. It is lacking 23.1 percent of the assets that it
should have. There are 308 companies in Standard and Poor’s 500 Index
that have pension plans. 290 of these are underfunded. That is 94 percent![8]
We should ask ourselves. What happens to retirees when a pension fund
is not able to pay? There are many safeguards in place to help protect
employee’s from the inherent risk involved in working for a company un-
der a contract where a portion of their salary includes future payments in
the form of retirement income. The Pension Benefit Guaranty Corporation
is a government agency which attempts to safeguard employee’s pensions
from negligent and/or unscrupulous business practices. The United States
government passed the Employee Retirement Income Security Act in 1974.
This act was passed in order to assist employees by guaranteeing benefits in
private pension plans. This law established the Pension Benefit Guaranty
Corporation.[2] However, this law does not solve the problem of underfund-
ing. It merely attempts to offer support in crisis situations and help protect
employees from negligence and theft by their employers.
3

4. 3 Defined Benefit Pension Plan Contribution Model
Now we will construct a defined benefit contribution plan model. Informa-
tion for this model came from The Sustainability of Pension Schemes.[4]
The service cost is defined by the cost to a given company of servicing their
pension plan. There are many variables in a service cost equation. We start
by defining the associated financial terms. Then we will define the vari-
ables and the service cost equation. Later we will discuss annuity factors in
greater detail.
3.1 Financial Terms in the Service Cost Equation
According to Investopedia, time value of money is ”the idea that money
available at the present time is worth more than the same amount in the
future due to its potential earning capacity. This core principle of finance
holds that, provided money can earn interest, any amount of money is worth
more the sooner it is received.”[1] Based on this idea an annuity factor is
the present value of $1 of pensionable salary. An annuity factor is calcu-
lated using mortality data. Section 7 discusses annuity factors. Section 6
shows example calculations for GAM 1983 and GAR 1994 Mortality Table
rates. Accrued benefits are unpaid benefits an employee is entitled to upon
termination or retirement. Each year accrued benefits will increase for the
beneficiaries of the pension plan. Each year an employee is with the com-
pany these accrued benefits increase by a fixed percent. Most pension plans
will use an increase of 1.5% to 2.0%. A long term bond is a bond that
matures in more than 10 years.
3.2 Service Cost Equation
The following model came from The Sustainability of Pension Schemes.
The following model will explain how a company services its pension fund.
The equation is aptly named the service cost equation and denoted by SC
throughout. This equation will show us how much money a company must
add to their pension fund to ensure that there are enough funds available to
pay out the appropriate retirement incomes to the retirees participating in
the plan. It is measured during the current year and shows how much the
employees of the company have earned for the year in retirement benefits.
This equation shows the amount the company needs to contribute to the
pension fund as a percentage of all the employee’s, participating in the
pension plan, salaries. Here K is the number of employees participating in
4

5. the plan.][4]
SC =
K
i=1
Si × (1 + y + π)xi−R
× (1 + w + π)R−xi
× A(R) × ∆b (1)
.
The variables for this model are defined as follows:
Si is the pensionable salary of the ith employee.
w% is the growth rate of real wages annually.
π % is the average long term inflation rate.
R is the retirement age.
xi is the age of the ith employee .
y% is the real yield on long term bonds.
A(R) is the annuity factor.
∆b is the % change in accrued benefits from year to year.
In accounting when assets match liabilities the world is good. Assets
matching liabilities means you are breaking even. Section 3.2.1 talks more
about assets and liabilities. The goal of this service cost equation is to
tell the company what their contribution should be. When expressed as a
percent it tells the company the rate of contribution in respect to the total
of all employees salaries that are participating in the plan. The service cost
equation has a goal in accounting of having assets match liabilities. Which
means the same amount of money that is being withdrawn from the fund is
also being deposited or earned via interest.
3.2.1 Assets and Liabilities
[Since the service cost equation is a function of assets and liabilities it is
important to define these terms. The short version is that an asset is some-
thing you have and a liability is something you owe. If everything you have
equals everything you owe then you are at 0. When talking about money
as an asset and taking into account the time value of money theory, assets
equalling liabilities at all times would be ideal. Since a company will al-
ways want to have the funds available to pay it’s debts but will not want to
have an excess in the account because money today is assumed to be worth
more than money in the future. Assets is formally defined as ”all items of
property that contribute to the value of an organization, including tangible
items such as cash, stock and real estate, as well as intangible items such
as goodwill” [?] Liabilities is formally defined as ”the debts of a business,
5

6. including dividents owed to shareholders.”[?] The get rid of your liabilites
means to pay everything you owe.
When assets and liabilites do not match asset and liability management
is employed. This is a way that accountants manage the risks that come
about when assets and liabilities do not match. The service cost equation
is a function of a company’s pension assets and pension liabilities. ]
3.3 Annuity Factor
As we have already learned, in section 3.1 According to Investopedia, time
value of money is ”the idea that money available at the present time is
worth more than the same amount in the future due to its potential earning
capacity. This core principle of finance holds that, provided money can earn
interest, any amount of money is worth more the sooner it is received.”[1]
Based on this idea an annuity factor is the present value of $1 of pensionable
salary. It is calculated using mortality data.
In equation 4, A(R) is used in a pension scheme that takes into consider-
ation an employee is married and was the sole income provider throughout
the working age life of the couple. If the income earning partner suddenly
dies the spouse can be left financially destitute. Therefore, many pension
schemes will have an option to pay benefits to a spouse even after the em-
ployee has died. Generally the benefits to the spouse after the employee
has passed are 50% of normal benefits. When calculating A(R) we look at
whether the employee is using a single plan or a surviving spouse benefit
plan. The annuity factor Ae(R) is for the employee and the annuity factor
As(R) is for the spouse.
The annuity factor for a given employee is a function of the real yield on
long term bonds and the probability that an employee and their spouse if
applicable will survive beyond retirement age R. The probability of surviv-
ing is calculated using a mortality table. The following equations calculate
the annuity factor.
Ae (R) =
50
k=1
pe (R + k)
(1 + r)k
. (2)
When ps(R + k) gives us the survival probability for the spouse of the
deceased employee, then the annuity factor for the spouse is expressed as,
6

7. As (R) =
50
k=1
ps (R + k)
(1 + r)k
. (3)
The total annuity factor at retirement age R when granting a spouse that
survives the employee benefits is 50% of the employee pension. Equation 4
shows this,
A(R) = Ae(R) + 0.5 × As(R). (4)
4 Different Mortality Tables in Use
The 1983 Group Annuity Mortality (GAM) mortality table is frequently
used by pension actuaries. In fact, in 2003 Watson Wyatt did a survey and
showed that 75 percent of the surveyed plans used the GAM 1983 when
doing calculations to see how best to fund their pension plans. However,
the Group Annuity Reserving (GAR) 1994 mortality table is an updated
version and is a better table to use for pension valuations. As of December
31,2002 it became law that all defined benefit plans needed to adopt the
GAR 1994 for benefit starting dates that occurred on or after December 31,
2002.[3] In this section we will discuss the statistical data used in these two
mortality tables.
4.1 GAM 1983 Mortality Table
The GAM 1983 mortality table was constructed after the Society of Actuar-
ies reviewed the Group Annuity Mortality (GAM) 1971 mortality table and
based on experience by insurance companies it was found that the GAM
1971 was no longer adequate. The society of actuaries then developed a new
table. This table was named the Group Annuity Mortality (GAM) 1983
mortality table. The GAM 1983 was constructed using the same statistical
data as the GAM 1971 mortality table. The reason for this is because there
7

8. was not enough statistical data from enough trustworthy sources to create
an entirely new table. The SOA used mortality data from the years of 1964
to1968.[6] With this in mind it becomes we can see there is an added degree
of inaccuracy in the 1983 table.
The SOA members assigned to reseaching the statisitcal data then looked
at the United States population statistics to determine the appropriate mor-
tality improvements. They looked at mortality data from the years 1966 to
1975. Based on the improvement trends noticed from the 1966 to 1975 data,
further improvments in mortality were expected and projected. The GAM
1983 also added a 10 percent margin to their projections. They did this to
add safety to the reserves and remain as conservative as possible.
4.2 GAR 1994 Mortality Table
The following information was taken from [7]. The Society of Actuaries did
a study of annuitant experience from the years 1986 to 1990. What they
inevitably found was that there were declines in mortality rates. The largest
declines were for males and the GAM 1983 was found to be no longer valid
when valuing pension reserves. Therefore this lead to the development of
the GAR 1994 mortality table. This mortality table is designed specifically
for pension actuaries and is based on group annuitant experience that was
further projected to 1994. The years the statisical data were examined from
were 1986 to 1990.
Before the Society of Actuaries developed the GAR 1994 mortality table
they compared the experiences of pensioners who carried no insurance and
also experience from the company’s group annuitants. This comparison did
not show a significant difference in mortality. The mortality rates they
compared to come to this conclusion were from ages 66 to 95. The data
they looked at came from employees that were participating in pension plans
with the Civil Service Retirement System, the Railroad Retirement System,
twenty four private uninsured retirement plans, and one state uninsured
retirement plan.
The mortality rates from the insured versus uninsured plans were very
similar with the Railroad Retirement System showing mortality rates that
were slightly higher. After the comparison of this mortality data the GAR
1994 mortality table was constructed.
The mortality rates in the GAR 1994 mortality table at ages 0 to 24 were
taken directly from the United States life tables from 1990 that are published
in an actuarial study by the Social Security Administration (study number
107). The mortality rates for ages from 95 to 120 were also taken from
8

9. this source. The mortality rates for ages from 66 to 95 were calculated
after group annuitant experience from years 1985 to 1990 were examined.
The mortality rates for ages from 25 to 65 were calculated after data was
examined from years 1985 to 1989 for retirees and 1983 to 1986 for active
employees of the Civil Service Retirement System. The mortality rates for
ages from 51 to 65 were calculated by blending active employees and retirees
experience. The mortality rates for ages from 96 to 119 had a maximum
rate of 0.5 set on them and at age 120 the rate was simply set to 1.0. This
means there is a 100% chance you will die at the age of 120 according to the
GAR 1994.
All of the preceding mortality rates were trended to 1994 prior to inclu-
sion in the table. This trending used the Civil Service Retirement System
data because its database was very large and its data had also been used to
extend the table for active employee’s lives. It is important in the construc-
tion of mortality tables to incorporate a projection scale. The projection
scale accounts for the fact that our life expectancy continues to increase.
The GAR 1994 uses projection scale AAx. This projection scale is also
based on a blend of the Civil Service Retirement System and the Social Se-
curity Adminstration’s study number 107. The years used were from 1977
to 1993.[7]
The GAR 1994 mortality table is specifically used for valuing and reserv-
ing pensions. It incorporates the use of a generational table. Generational
tables are mortality tables that show mortality and mortality improvements
by birth cohort. A cohort is a group of people who share a common charac-
teristic or experience within a defined period. Thus a group of people who
were born on a day or in a particular period form a birth cohort.
4.3 Mortality Table Update
The Pension Pulse newsletter said changing actuarial assumptions for mor-
tality could increase pension liabilities by 2% to 5%. If a participant lives
longer, a pension plan will not realize that liability until later, but if mor-
tality tables are updated, pension accounting will see a shock to liabilities
sooner. The Society of Actuaries is currently developing a study of pension
mortality and will issue updated tables by 2014 [?]
5 Constructing the GAR 1994 Mortality Table
We reconstruct the GAR 1994 mortality table by referencing Transactions
of Society of Actuaries 1994 Vol 47.[7] The GAR 1994 Mortality Table was
9

10. constructed using a parametric model with nine parameters. qy
x is the mor-
tality rate where y is the calendar year and x is the person’s age.
The parametric formula for the male rates is
qy
x (θ) = ψ1 × rx (m1, σ1, ) + ψ2 + sx (m2, σ2) + ψ3 × tx (m3, σ3) . (5)
The parametric formula for the female rates is
qy
x (θ) = ψ1 × rx (m1, σ1, ) + ψ2 + rx (m2, σ2) + ψ3 × tx (m3, σ3) . (6)
5.1 Parameter Estimates
There are nine different parameters that go into these parametric models
for men and women. θ are these parameters. Table 1 shows the estimates
used.
θ = (ψ1, m1, σ1, m2, σ2, ψ3, m3, σ3)
The statistical data used to estimate θ was compiled from Life Tables
from the U.S. census, 25 large insurance companies and the Civil Service
Retirement System for retired and active annuitants.
5.2 Exposure Formulas
An exposure is a risk you take. The very act of being alive puts you at risk
of dying. Therefore to calculate mortality rates we use exposure formulas.
Defined here the exposure is the risk you take of dying.
The exposure formulas used to construct the GAR 1994 Mortality Table
are as follows:[7]
rx (m, σ) = exp −
x
m
m
σ
, (7)
sx (m, σ) = 1 − exp −
x
m
−m
σ
, (8)
tx (m, σ) =
1
σ
x
m
m
σ
−1
exp −
x
m
m
σ
, (9)
10

11. 5.3 Mortality Improvement Factors
Life expectancies continue to increase thus when a mortality table is con-
structed there is also a need to project mortality rates into the future. To
do this we use a mortality improvement factor. The GAR 1994 Mortality
Table improvement factor was obtained by using a blend of the Civil Service
Retirement System and the Social Security Administration’s study number
107. The mortality reduction trends based upon experience between years
1977 through 1993, with adjustments. [7]
The formula used to calculate a projected mortality rate for the GAR
1994 is
q1994+n
x = q1994
x × (1 − AAx)(n)
. (10)
The formula used to calculate a projected mortality rate for the GAM
1983 is
q1983+n
x = q1983
x × (1 − AAx)(n)
. (11)
qy
x is the mortality rate where y is the calendar year and x is the person’s
age.
AAx is the annual mortality improvement factor where x is the person’s age.
6 GAM 1983 and GAR 1994 Tables
We took the mortality rates from GAM 1983 and using the projection for-
mula we established the new mortality rates for our current year 2013. Table
blank shows the GAM 1983 mortality rates for males at ages 55, 62, 67, and
72, the GAM 1983 projection factor AAx and the corresponding 2013 mor-
tality rates. Table blank also shows GAR 1994 mortality rates for males
at ages 55, 62, 67, and 72, the GAR 1994 projection factor AAx and the
corresponding 2013 mortality rates.
Table blank shows the GAM 1983 mortality rates for females at ages 55,
62, 67, and 72, the GAM 1983 projection factor AAx and the corresponding
2013 mortality rates. Table blank also shows GAR 1994 mortality rates for
females at ages 55, 62, 67, and 72, the GAR 1994 projection factor AAx and
the corresponding 2013 mortality rates.
11

12. %change =
1994q2013
x 1983q2013
x
1983q2013
x
. (12)
%change = 1994q2013
x 1983q2013
x /1983q2013
x . (13)
7 Annuity Factors
As we have already learned, in section 3.1 According to Investopedia, time
value of money is ”the idea that money available at the present time is
worth more than the same amount in the future due to its potential earning
capacity. This core principle of finance holds that, provided money can earn
interest, any amount of money is worth more the sooner it is received.”[1]
Based on this idea an annuity factor is the present value of $1 of pensionable
salary. It is calculated using mortality data.
Table 4 shows the calculated annuity factors for male employees and
female employees at ages 55, 62, 67, and 72 using the projected mortality
rates to 2013 from the GAM 1983 and GAR 1994 mortality tables and also
shows the percent change. Which we calculated with % change equation
here
%change =
GAM1994Ae(R)usingq2013
x − GAM1983Ae(R)usingq2013
x
GAM1983Ae(R)usingq2013
x
(14)
8 Comparison Mortality Tables
Table 1: GAR 1994 Estimates of Parameters used
ψ1 m1 σ1 ψ2 m2 σ2 ψ3 m3 σ3
Male 9.800 82.090 22.150 -2.428 75.880 11.670 13.820 11.200 4.882
Female 188.300 96.590 59.350 -180.900 97.840 62.850 12.100 9.973 4.459
References
[1] Investopedia Dictionary. Retrieved from
http://www.investopedia.com/dictionary/ on October 3, 2013
12

13. Table 2: Comparison of Male Mortality Rates
GAM 1983 GAR 1994
Age(x) q1983
x AAx q2013
x q1994
x AAx q2013
x % change
55 0.006131 0.015 0.004600655 0.004425 0.019 0.003073438 -33.2
62 0.011133 0.015 0.008354118 0.010147 0.015 0.007614231 -8.86
67 0.019804 0.015 0.014860769 0.018034 0.013 0.014064295 -5.36
72 0.03337 0.0125 0.026276117 0.028481 0.015 0.021371924 -18.66
Table 3: Comparison of Female Mortality Rates
GAM 1983 GAR 1994
Age(x) q1983
x AAx q2013
x q1994
x AAx q2013
x % change
55 0.002541 0.0175 0.001816868 0.002294 0.008 0.001969315 8.39
62 0.00521 0.0175 0.003725258 0.005832 0.005 0.005302199 42.33
67 0.008681 0.0175 0.006207095 0.010764 0.005 0.009786158 57.66
72 0.01616 0.0175 0.011554735 0.016506 0.006 0.014722554 27.42
[2] Pension Benefit Guaranty Corporation History of Pension Benefit Guar-
anty Corporation Retrieved from http://www.pbgc.gov/about/who-we-
are/pg/history-of-pbgc.html on November 15, 2013.
[3] Prudential Retirement. ”IRS Updates Rules for Defined Benefit Plans.”
Pension Analyst. August, 2002: 1-7. Print.
[4] Ramaswamy, Srichander, The Sustainability of Pension Schemes. Bank
for International Settlements (BIS) Working Papers No 368, 2012.
[5] Seburn, Patrick W., Evolution of employer-provided defined benefit pen-
sions Monthly Labor Review, Dec. 1991.
[6] Society of Actuaries Committee on Annuities, Transactions
of Society of Actuaries 1983 Vol 35. 1983. Retrieved as pdf
document from Society of Actuaries (SOA) library website
http://www.soa.org/library/research/transactions-of-society-of-
actuaries/1983/january/tsa83v3527.pdf on September 15, 2013.
[7] Society of Actuaries Group Annuity Valuation Table Task Force,
Transactions of Society of Actuaries 1994 Vol 47. 1994. Re-
trieved as pdf document from Society of Actuaries (SOA) library
website http://www.soa.org/library/research/transactions-of-society-of-
actuaries/1990-94/1994/january/tsa95v4722.pdf on September 15, 2013.
13

14. Table 4: Comparison of Male and Female Annuity Factors
GAM 1983 Ae(R) using q2013
x GAM 1994 Ae(R) using q2013
x
Age(R) Male Female Male Female % ch M % ch F
55 14.273 14.45085 14.31173 14.42482 0.27 -0.18
62 13.78203 13.99002 13.92107 14.09163 1.01 0.73
67 13.24088 13.49643 13.52993 13.75086 2.18 1.89
72 12.53127 12.82446 12.9674 13.2465 3.48 3.29
[8] Wilshire Associates Incorporated. Wilshire Consulting Re-
port Shows 94 Percent of Corporate Pension Plans Remain
Underfunded. GlobeNewswire. Globe News Wire, 8 Apr.
2013. Retrieved from website http://globenewswire.com/news-
release/2013/04/08/536575/10027692/en/Wilshire-Consulting-
Report-Shows-94-Percent-of-Corporate-Pension-Plans-Remain-
Underfunded.html in Fall 2013.
14