A $1,000,000 ordinary annuity with annual payments, at the end of each year, earns 6% compounded annually and lasts for 30 years. However, this annuity is a "decreasing" annuity, in that the annual payments "decrease" by 1% each and every year from previous year's payment, until the 30th and last payment. What are the first and last payments of this annuity? Thanks for any help.
The easiest way to approach this, is to sum up all the payments based on one $1 and then simply divide $1,000,000 by this PV, which will give the 1st. payment. Then the last payment would be: 0.99^(n -1). A good calculator, such as Wolfram/Alpha, can easily sum them up very rapidly, such as I have done here: ∑[0.99^n / 1.06^(n+1), n, 0, 29] =12.445866730.....
Then, the first payment=$1,000,000 / 12.445866730 =$80,347.96.
And the last payment will be=$80,347.96 x 0.99^(30-1) =$60,033.75.
Here is the W/A link: http://www.wolframalpha.com/input/?i=%E2%88%91%5B0.99%5En+%2F+1.06%5E(n%2B1),+n,+0,+29%5D