Chapter 8
Valuation Theory for Pension Plans
8.1 Valuation Theory for Pension Plans
Basic Model
Contributions are made by employees (and/or employers) in a pension plan of benefits are accrued while the employee is in active service.
Benefits are paid out upon occurrence of one of the following decrements:
1. withdrawal from service with vested benefits - 2. death in service -
3. retirement for disability - 4. retirement for age service -
Theory of pension funding is the setting aside an appropriate reserve or fund such that at any time,
actuarial present value of future benefits accumulated fund.
1. Defined contribution plan.
Contributions are made during each year of active service – either a flat rate or a percent of salary.
Pension benefit = accumulated value of contributions, which can be used to purchase a life annuity.
2. Defined benefit plan.
Benefits are accrued during each year of active services, e.g. for each year of service, accrued monthly benefit = 1% X(final monthly salary before retirement).
Terminology
actual annual salary at age x+h for employee currently age x+h and who started service at age x.
salary scale function which satisfies
expected annual salary at age x+h+t
e.g., at a 5% annual growth rate in salary,
i.e. if actual annual salary at age 40 is $50,000, expected salary at 50 is
Note
Definition of is not unique.
e.g., is also acceptable as a scale function.
8.1.1 General Rule
Most pension plans assume continuous contributions accrual of benefits. The resulting integral is approximated by a summation using this following rule:
In particular, if , then
by application of the mid-point rule
Example 8.1.1
under UDD assumption
by mid-point rule
8.1.2 Salary scale function S
yUsual assumption is that is a step rate function
i.e.,
and
We define
Formula is used in determining actuarial p.v. of benefits expressed as a percentage of the average salary over the last m years prior to retirement.
8.1.3 Defined Contribution Plans
A. Contributions as a flat rate per participant:
For a participant age x+h, paying contribution c per year continuously to age w at retirement, actuarial p.v. of future contributions at (x+h) is
by the mid-point rule
B. Contributions as a constant fraction of salary:
For a participant age (x+h), paying fraction c of salary continuously to age w and
having a salary scale function and initial salary , actuarial p.v. of future contributions at (x+h) is:
By applying the mid-point rule and assuming is a step rate function. Accumulated value of future contributions at retirement age w is
(Actuarial p.v. at x+h)/
C. Excess type plan, i.e. contributions are payable as a flat percentage of salary in excess of an earnings level.
Assume initial salary for a participant age (x+h) salary scale
c = fraction of excess earnings contributed
excess earnings level for year k+1, k=0,1,2,…….w-(x+h)-1
i.e. in year k+1, contribution is
by applying the mid-point rule and using the fact that and are step functions. Accumulated value of future contributions at retirement age w is
(Actuarial p.v. at x+h)/
Example 8.1.2 A pension plan has one participant, currently age x. You are given:
(i) Contributions are equal to 2.5% of salary. (ii) The participant’s current salary is $29,200.
(iii) The salary scale is constant within any year of age and
(iv) The actuary assumes
(v) All payments are made at mid-year (vi) i=0.08
Calculate the actuarial present value of future contributions for the participant.
Solution
………
x x+1 x+2 x+3
Let = contributions made at
Actuarial p.v. of contribution
Example 8.1.3 An employee, currently age 35, entered a defined contribution pension plan at age 30. Employee and employer contributions, each equal to 3% of salary, are made at the beginning of the year. Upon retirement, accumulated contributions are used to purchase an annual life annuity due. You are given:
(i)
(ii) , k=0,1,2,……….
(iii)
(iv) Current value of past contributions with interest equals 5000
(v) and is used to accumulate contributions and calculate annuity values.
Calculate the annual retirement benefit for this employee assuming she retires at ages 65.
Solution
Accumulated value of past contributions at
Contribution at , k=0,1,2,……..
Accumulated value at
i.e. (past contributions + future contributions) accumulated at 65
Let R = annual life annuity due payment.
Then
i.e.
Example 8.1.4 For an excess type pension plan, you are given:
(ii) Employee age = 40 (iii) Current salary = 30,000 (iv) for
(v) Contributions = 5% of salary in excess of . Contributions are payable at the beginning of each year of age.
(vi) Retirement age = 65
(vii) Salary scale for (viii)
(ix)
(x)
Calculate the present value of the current and future contributions for this employee.
Solution
………..
40 41 42 64 65
contribution at age (40+k), k=0,1,2,…………..24
actuarial p.v. of contribution
since
8.2 Defined Benefit Plans
In these plans, the annual benefit at retirement is specified and the task is to determine the level of contributions/funding to pay this benefit. The defined benefit is generally some formula relating years of service and salary levels, e.g. annual benefit at retirement = 50% of average of last five years of annual earnings prior to retirement.
Terminology
annual income at retirement age (x+h+t) for an employee currently age (x+h) who first entered service at age x.
e.g., annual income at retirement age 60 for an employee currently age 40 who first entered service at 30.
Actuarial present value at retirement age of annual income is
where = life annuity of one per year payable continuously for a retiree age x.
Actuarial p.v. of retirement benefit at is:
If retirement occurs continuously between ages to , actuarial p.v. of retirement benefit at (x+h) is
under UDD assumptions and mid-point rule.
8.2.1 R(x,h,t) Independent Salary
where g is some function involving years of service until retirement age .
Then actuarial p.v. of retirement benefit at (x+h)
for continuous retirement.
under UDD and mid-point rule.
Example 8.2.1 Retirement occurs at the beginning of ages 60 to 65. Retirement benefit is $1000 per year for every year of service. Consider an individual age 40 who entered service at 35.
actuarial p.v.
If retirement occurs continuously between ages 60 to 65,
actuarial p.v.
R(x,h,t) Dependent on Final Salary
General: fraction g of average of last m years of salary prior to retirement for a
given salary scale function .
Case 1:
t
m
x x+h x+h+t-m x+h+t
In particular, for t integer, average retirement benefit between and
If we assume is a step rate function,
where
Case 2:
x x+h+t-m x+h x+h+t
Example 8.2.2 Individual age 50, entered service at 30.
Retirement benefit = (1%) x (# of years of service until retirement).
Given salary at 50 = $50,000 per year;
Then annual retirement benefit at 60 is
For a continuous salary scale,
If retirement benefit is same as before, expect that it used average of final 12 years of
service and .
then
8.2.3 Step-rate Plan
Retirement benefit = (# years of service) x [(1% x(final 10 year average salary 50,000)
+ 2% x (final 10 year average salary > 50,000)].
Then
8.2.4 Offset Plan
Retirement benefit = {(# of years of service) X (factor 1) X (final m-year average salary) – offset}
where offset = (factor 2) X (estimated initial retirement income from social security)
Example 8.2.4 Same as before
Assume for offset, offset factor = 0.5; estimated initial retirement income from social security = 10,000.
Then
8.2.5 Add-on Plan
Retirement benefit = basic benefit + supplement for early retirement
Basic retirement benefit =(factor 1) x (# of years of service) x (final m-year average salary).
Basic benefit payable at retirement until time of death of participant.
Supplement for early retirement = (factor 2) X (# of years of service) X (final m-year average salary).
Example 8.2.5 Individual age 50, entered service at 30. Normal retirement age = 60; early retirement age = 50
Basic retirement benefit = (# of years of service) x (final 5 year average salary) x 2%
Early retirement benefit = (# of years of service) x (final 5 year average salary) x 1%
Calculate and
Solution
For
basic retirement benefit
early retirement benefit
i.e.
For , only benefit is basic retirement benefit.
i.e.
Example 8.2.6 In a step rate retirement plan, the income benefit rate for retirement in year (k+1) equals the total number of years of service, including any final fraction, multiplied by the sum of:
2. 2.25% of the three year final average salary in excess of .
You are given:
(i) and is a step function, constant over each year of age. (ii)
(iii)
Calculate
Solution
retirement benefit at 65.5 for individual age 30 who entered service at 25.
30 31 32……….62 63 64 65 66
Total years of service at retirement age 65.5 = 40.5
Portion of retirement benefit below excess earnings level
3 year final average salary
Solving
Example 8.2.7 For a defined benefit pension plan, you are given:
(i) The retirement benefit is $15 per month per year of service. (ii) Retirement is allowed beginning at age 63.
(iii) There is one employee who is currently exact age 45 and was hired at exact age 40.
(iv)
(v) Probabilities of retirement and retirement annuities:
Age
63 0.2 10.0
64 0.3 9.5
65 1.0 9.0
Assume:
(i) Employees retire only on their birthdays. (ii) There are no pre-retirement decrements.
Calculate the actuarial p.v. of the projected retirement benefit for this employee.
Solution
40 45 63 64 65
actuarial p.v. at 45
Substituting actuarial p.v.
8.2.6 Career Average Plans
= fraction of entire career earnings
=f x (total salary over entire career)
=f x (# of year of service) x(average salary over entire career)
=f x (total past salary + estimated total future service salary)
Notation:
total of past salaries for plan participant currently age .
Actuarial p.v. at of past services portion of retirement benefit at retirement age
If retirement can occur over a range of ages to , then actuarial p.v. of past service
portion of retirement benefit
If retirement occurs continuously between ages and , actuarial p.v. of past service
retirement benefit
Actuarial p.v. at of future service salary portion of retirement benefit at
For retirement over ages to , actuarial p.v. of future service retirement benefit
For continuous retirement between and , actuarial p.v. of future service retirement benefit
under UDD and the mid-point rule
where
Example 8.2.8 Retirement benefit is 2% of aggregate salary during a participant’s years of service. Participant is currently age 40, entered service at 30 with $200,000 of total past
salaries. Current salary is $25,000 per year. Salary scale function is . Retirement benefit commences at the beginning of the year of retirement.
a. Determine retirement benefit at the beginning of age 60.
b. What is actuarial p.v. at 40 of retirement benefit at 60?
Actuarial p.v.
c. What is the actuarial p.v. at 40 of retirement benefit if retirement benefit if retirement can occur at the beginning of ages 60 to 64?
Actuarial p.v.
where
d. What is the actuarial p.v. at 40 of retirement benefit if retirement benefit if retirement can occur continuously between ages 60 and 65?
Actuarial p.v.
where
Example 8.2.9 A pension plan provides an income benefit rate equal to 25% of the average salary rate over the last 10 years prior to retirement 1.5% of career average salary for each year of service.
Salaries increase continuously. The salary scale function is:
,
For a participant now earning $12,000, calculate
Total past service salary
Total future salary
Total salary over last 10 years
Example 8.2.10 A new participant, age 45, in a pension plan has a choice of two retirement income benefit rates:
1. Under Option 1, the retirement benefit rate is 1.4% of the 5-year final average salary times the number of years of service.
2. Under Option2, the retirement income benefit rate is 2.0% of the career average salary times the number of years of service.
You are given:
(i) ,
is a step function constant over each year of age. (ii) Retirement may occur on any birthday on or after age 60.
Determine the youngest age at which the benefit rate under Option 1 will exceed the benefit rate under Option 2.
Solution
……….. ………
45 46 47 59 60 61 45+t-1 45+t
i.e.
Total future salaries to retirement at 45+t
i.e.
Want
i.e.
i.e.
i.e.
i.e.
i.e.
,
,
i.e. earliest retirement age when Option 1 retirement > Option 2 retirement is 45 + 22 = 67.
8.3 Disability Benefits in Pension Plans
Typical benefit is
Example 8.3.1 Consider a participant currently age 40 who entered service at 38. To qualify for disability benefits, the participant must have a minimum of seven years of completed service and disability must occur before normal retirement at 65. The disability benefit is a continuous life annuity of 10% of the annual salary at the time of disability times the larger of {10 year, actual full years of service}.
Assume the following:
(i) Current salary per year.
(ii) and is a step function.
(iii) Disability benefits commence at the end of the year of disability.
a. Determine disability income benefit .
for
38 39 40 41 42……….45 46…………63 64 65
for
for , integer and where for , integer
b. Determine the actuarial p.v. of the disability income benefit at 40.
Actuarial p.v.
8.3.1 Withdrawal Benefits in Pension Plans
2. Accumulated value of past contribution in a lump sum.
Example 8.3.2 Participant age 40; entered service at 30. Normal retirement is at age 60. Withdrawal benefit for withdrawals prior to age 60 is:
(i) accumulated value of participant’s past contributions paid at moment of withdrawal, plus
(ii) deferred continuous life annuity at 60 equal to 5% X (# of years of service) X (salary at withdrawal)
Assume:
1. Past contributions . No contributions made after 40.
2. Salary at 40 per year;
3. Interest rate used for accumulating and discounting =5%
What is the actuarial p.v. at 40 of the withdrawal benefit?
Solution
actuarial p.v. = actuarial p.v. of past contributions + actuarial p.v. of past contributions
actuarial p.v. of past contributions
under UDD
actuarial p.v. of deferred annuity
under UDD and mid-point rule.
Example 8.3.3 A pension plan provides a disability income benefit payable for life, of 50% of the annual rate of salary at the moment of disability. To qualify for the benefit, a
participant must serve at least three years and then become disabled before age 65.
An employee age 40, has just been hired at a starting salary of $15,000.
You are given:
(i) , . Salary increases continuously (ii)
(iii) (iv)
(v)
Determine the actuarial p.v. of the disability benefit.
Solution
annual disability benefit at
actuarial p.v. at 40 of disability benefit
Example 8.3.4 A pension plan provides a withdrawal benefit of a deferred annuity beginning at age 60 if a participant has at least 5 years of service. The income benefit rate equals 2% of salary at time of withdrawal times the number of completed years of service.
An employee age 25, has just been hired with starting salary of $2,000.
You are given:
(i) ; is constant over each year of age. (ii)
(iii) for all integral (iv) for all integral
(v)
Assume withdrawals occur midyear and there are no withdrawals after age 60. Calculate the actuarial p.v. of the withdrawal benefit.
Solution
Actuarial p.v.
Example 8.3.5 A pension plan provides an annual retirement benefit equal to 1,000 times the total number of years of service, including any final fraction. No more than 25 years of service may count for the benefit.
You are given the following extract from the service table:
46 32,349 --- ---
---60 23,856 313 --- 12
61 --- 298 19,693 10
62 0 --- ---
---You are also given:
(i) Death and retirement are the only decrements. Death and retirement are assumed to occur mid-year.
(ii)
(iii) No retirements occur prior to age 60.
(iv) The plan has a single participant currently age 46, who entered at age 36.
Calculate the current actuarial p.v. of the retirement benefit.
Solution
Actuarial present value
Example 8.3.6 For an offset type pension plan, you are given:
(i) The income benefit rate equals the excess of:
a. 2% of the three year final average salary times the number of years of service, over
b. 50% of the social insurance income benefit rate
(ii) The social insurance income benefit rate equals 2% of career average salary times the number of years of service.
(iii)
(iv) Salary increases occur only on anniversaries. (v)
Calculate
Solution
x x+10 x+37 x+38 x+39 x+40
Three year final average salary
Total future salary
Social insurance income benefit
i.e.
Then and
Substituting,
