Tutorial 6 - Pension Mathematics
Lecturer: Trần Minh Hoàng
Actuarial Mathematics 2
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 1 / 11
Question
A group of employees is subjected to four decrements: death (d), withdrawal (w),
disability (i), and retirement (r). Probabilities of decrement follow the Standard
Service Table (see next slide).
a) Calculate the probability that a 45-year old employee terminates employment
by death within five years.
b) Calculate the probability that a 40-year old employee stays in service in the
next 2 years but withdraws in the next 3 years.
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 2 / 11
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 3 / 11
Question
You are given the salary rate function
s̄x = 1.05x .
The salary rate for a person exact age 30 is $50,000. Determine the salary earned
by this person between his 59th and 60th birthday, without using an
approximation.
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 4 / 11
Question
At ABC Company, a pension plan pays career average salary upon retirement at
age 65. Salaries increase 4% each year. An employee born on July 1, 1992 joins
the company on July 1, 2022 and retires at age 65. Calculate the replacement
ratio for this employee.
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 5 / 11
Question
You are given the salary scale:
1.04y
if y ≤ 50
sy =
(1.04 )(1.03y −50 ) if y > 50.
50
For someone joining the company at age 25, a defined benefit plan provides a life
annuity starting at age 65 with annual payments equal to 50% of career average
salary. Find the replacement ratio.
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 6 / 11
Question
In a defined contribution plan:
The employee will purchase a monthly annuity-immediate at age 65. You are
given that
(12)
a65 = 14.
The salary scale is sx = 1.03x , for integer x.
The employee joins the plan at exact age 40.
Funds invested in the plan earn 8% effective.
Retirement age is 65.
Contributions to the plan are made annually at the end of the year, based on
the salary paid for that year.
Calculate the percentage of salary that must be contributed to achieve a 50%
replacement ratio.
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 7 / 11
Question
In a defined contribution plan:
Salary increases continuously with the salary rate s̄x = 1.04x .
An employee joins the plan at exact age 35 and retires at age 65.
Funds invested in the plan earn 6% effective.
Contributions of 15% of the salary are made continuously, based on the
salary rate at that time.
(12)
ä65 = 13.5.
As an alternative, the employer can offer a defined benefit plan providing a
monthly annuity-due that is a percentage of the final 3-year average salary.
Determine the percentage which equates the benefits under the defined benefit
plan to the benefits under the defined contribution plan.
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 8 / 11
Question
A new participant, age 45, in a pension plan has a choice of two retirement
income benefit rates:
Under option 1, the retirement income benefit rate is 1.4% of the 5-year final
average salary times the number of years of service.
Under option 2, the retirement income benefit rate is 2.0% of the career
average salary times the number of years of service.
You are given:
s45+k = 1.05k for k = 0, 1, 2....
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.
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 9 / 11
Question
A company plans to implement either a defined benefit plan or a defined
contribution plan.
Upon normal retirement at exact age 65, the defined benefit plan would provide
an annual benefit, payable monthly, of 25% of 5-year final average salary.
The defined contribution plan would assign 6% of salary to each participant’s
account at the end of each calendar year. These accounts would accumulate at
5% interest each year. Upon normal retirement at exact age 65, this account
would be converted to an annual benefit, payable monthly, by dividing the
accumulated account by 9.239.
You are given:
A participant enters on January 1 at exact age x < 60.
The participant’s salary is 25,000 on January 1.
Salaries increase by 5% on January 1 each year.
a) Find expressions in terms of x for the annual benefit at normal retirement for
the defined benefit and defined contribution plans for this participant.
b) For what age at entry, x, is the benefit under a) equal to that under b)?
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 10 / 11
Question
In a defined benefit pension plan, benefits accrue at 1.5% per year.
Members who withdraw before retirement receive an annual life annuity-due
deferred to age 65 of the accrued percentage of their 3-year final average salary.
Members who die before retirement receive a lump sum payment of the salary in
the final year of death at the end of the year of death.
You are given:
All withdrawals are assumed to occur in the middle of the year.
4% of members withdraw between their 55th and 56th birthdays.
Mortality after retirement or withdrawal follows the Standard Ultimate Life
Table, with uniform distribution of deaths between integral ages.
i = 0.05
Susan is age 55. She joined the pension plan at age 40. She has earned $88,000 at
age 52, $90,000 at age 53, $92,000 at age 54, and will earn $94,600 at age 55.
a) Calculate the EPV of Susan’s withdrawal benefits over the next year.
b) Calculate the EPV of Susan’s death benefits over the next year.
Lecturer: Trần Minh Hoàng Tutorial 6 - Pension Mathematics Actuarial Mathematics 2 11 / 11
