I plan to buy a pension plan that will pay me $80,000 at the end of
each year, starting at my 66th birthday and ending at my 85th
birthday. (If I am not lucky to live that long, my beneficiaries will
receive the payments over the same period.) I am now 30 years old and
according to the pension plan I must pay equal annual premiums(at
year-end) starting on my 31st and ending on my 65th birthday. If the
pension fund expects to earn a constant 6% rate of return on my
premium payments, what is the annual premium I have to pay for this
pension plan over the next 35 years?

Holla-ga,

The approach I would take to such a question is to draw a timeline
with arrows showing the payments and disbursements. Unfortunately,
this is a text based forum, so I'm going to settle for the second best
method - the table :

Birthday          Net cashflow
  30th                   0
  31st                  -x
   .                     .
   .                     .
  65th                  -x
  66th                80,000
   .                     .
   .                     .
  85th                80,000

Here x is the annual premium.

The best way of solving this problem is to break it down into parts we
understand (standard compound discounting and annuities) as follows.

Step 1) Find out the annuity value of the pension as of the 65th
birthday. At your 65th birthday, you will receive 80,000 at the end of
each year for the next 20 year. This is exactly the definition of an
annuity, so we can use the annuity formula :

    C (       1   )
P = - (1 - -------)
    r (    (1+r)^n)

The value of payments as of the 65th birthday, P65, is

      80000 (        1    )
P65 = ----- (1 - ---------) = $1,101,186.49
       .06  (    (1.06)^20)

(We use 6% throughout this question because this is the rate at which
money in the account grows over time, and if we think backwards, the
rate at which it discounts to the present).

We now need to discount this value back to the present (35 years
previous to the 65th birthday). To do this we just use the Present
Value formula :

PV = FV / (1+r)^n

So P30 = 1101186.49 / (1.06^35) = $143,270.11

We now know the present value of the disbursements. We know that at a
6% growth rate, the present value of the premium payments must also be
$143,270.11.

As of the 30th birthday, the payments form a 35 year annuity, so we
can reuse the annuity formula :

    x (       1   )
P = - (1 - -------)
    r (    (1+r)^n)

In this case we know P, r and n - we are solving for x, the premium.

             x  (        1    )
143270.11 = --- (1 - ---------) = x * 14.49
            .06 (    (1.06)^35)

Hence x = 143270.11 / 14.49 = $9881.89

So an annual premium of $9881.89 would just cover the value of the
80,000 payments upon retirement.

Hope this helps and good luck

calebu2-ga

Useful resources :
http://www2.bc.edu/~balduzzp/sylmf801f01.html - Investments (Time
value of money) course taught at Boston College.