annuity and perpetuity

Question

Sunrise Industries wishes to accumulate funds to provide a retirement annuity for its vice president of research, Asia Omar. Ms. Omar by contract will retire at the end of exactly 15 years. Upon retirement, she is entitled to receive an annual end-of-year payment of $50,000 for exactly 25 years. If she dies prior to the end of the 25-year period, the annual payments will pass to her heirs. Also sunrise want to assure a stability life for Asia Omar. Thus, they want to have a perpetuity which provide $15,000 each year. During the 15-year “accumulation period” Sunrise wishes to fund the annuity by making equal annual end-of-year deposits into an ACCOUNT earning 9% interest. Once the 25-year “distribution period” begins Sunrise plans to move the accumulated monies into an ACCOUNT earning a guaranteed 12% per year. After the 40th year, the interest rate will be 15%. Note that the first deposit will be made at the end of year 1, and that the first distribution payment Will be received at the end of year 15.

How large must Sunrise’s equal annual end-of-year deposits into the account be over the 15 year accumulation period to fund fully Ms. Asia Omar retirement annuity and perpetuity?

Answer 1

Assuming the $15,000 perpetuity begins after the 25-year retirement annuity ends:

First find how much money Sunrise needs at retirement in year 15.

The $50,000 payments last for 25 years, and the first payment is at the end of year 15. Since the first payment occurs immediately at retirement, this is an annuity due.

PV of retirement payments:

$50,000 × [(1 - 1/1.12^25) / .12] × 1.12

= $439,215.79

Next find the amount needed for the perpetuity.

After year 40, the account earns 15%, so the amount needed to provide $15,000 per year forever is:

$15,000 / .15 = $100,000

Discount that amount back to year 15 at 12%:

$100,000 / 1.12^24 = $6,588.21

So the total amount needed at year 15 is:

$439,215.79 + $6,588.21 = $445,804.00

Now find the annual deposit needed to accumulate this amount over 15 years at 9%.

Future value factor:

[(1.09^15 - 1) / .09] = 29.36092

Annual deposit:

$445,804 / 29.36092 = $15,183.59

So Sunrise must deposit approximately $15,184 per year for 15 years.