Erin O'Reilly was recently employed by the human

resources department of a moderate$-$sized engineer$-$

ing firm. Management is considering the adoption

of a defined$-$benefit pension plan in which the firm

will pay $75$ percent of an individual's last annual

salary if the employee has worked for the firm

for $25$ years. The amount of the pension is to be

reduced by $3$ percent for every year less than $25,$

so that an individual who has been employed for

$15$ years will receive a pension of $45$ percent of the

last year's salary $[75$ percent $-(103\%)].$ Pension

payments will start at age $65,$ provided the indi$-$

vidual has retired. There is no provision for early

retirement. Continuing to work after age $65$ may

increase the individual's pension if the person has

worked for less than $25$ years or if the salary were

to increase.

One of the first tasks given O'Reilly is to esti$-$

mate the amount that the firm must set aside today

to fund pensions. While management plans to hire

actuaries to make the final determination, the man$-$

agers believe the exercise may highlight some prob$-$

lems that they will want to be able to discuss with

the actuaries. O'Reilly was instructed to select two

representative employees and estimate their annual

pensions and the annual contributions necessary to

fund the pensions.

O'Reilly decided to select Arnold Berg and Van$-$

essa Barber. Berg is $58$ years old, has been with the

firm for $27$ years, and is earning $$34,000.$ Barber

is $47,$ has been with the firm for $3$ years, and earns

$$42,000$ annually. O'Reilly believes that Berg will

be with the firm until he retires; he is a competent

worker whose salary will not increase by more than

$4$ percent annually, and it is anticipated he will re$-$

tire at age $65.$ Barber is a more valuable employee,

and O'Reilly expects Barber's salary to rise at least

$7$ percent annually in order to retain her until retire$-$

ment at age $65.$

To determine the amount that must be invested

annually to fund each pension, O'Reilly needs $($in

addition to an estimate of the amount of the pen$-$

sion$)$ an estimate of how long the pension will be

distributed $($i$.$e$.,$ life expectancy$)$ and how much the

invested funds will earn. Since the firm must pay an

interest rate of $8$ percent to borrow money, she de$-$

cides that the invested funds should be able to earn

at least that amount.

While O'Reilly believes she is able to perform

the assignment, she has come to you for assistance

to help answer the following questions.

If each individual retires at age $65,$ how much

will his or her estimated pension be$?$

Life expectancy for both employees is $15$ years

at age $65.$ If the firm buys an annuity from an

insurance company to fund each pension and

the insurance company asserts it is able to

earn $9$ percent on the funds invested in the an$-$

nuity, what is the cost or the amount required

to purchase the annuity contracts?

If the firm can earn & percent on the money

it must invest annually to fund the pension,

how much will the firm have to invest annu$-$

ally to have the funds necessary to purchase

the annuities?

What would be the impact of each of the fol$-$

lowing on the amount that the firm must in$-$

vest annually to fund the pension?

a$)$ Life expectancy is increased to $20$ years.

b$)$ The rate of interest on the annuity contract

with the insurance company is reduced to

$7$ percent.

c$)$ Barber retires at age $62$ instead of $65.$A