Actuarial Outpost Decreasing annuity problem
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#1
07-01-2018, 01:08 PM
 actsci123451 SOA Join Date: Feb 2018 Posts: 8
Decreasing annuity problem

Francois purchases a 10-year annuity-immediate with annual payments of 10X.

Jacques purchases a 10-year decreasing annuity-immediate which also makes annual payments. The payment at the end of year 1 is equal to 50. At the end of year 2, and at the end of each year through year 10, each subsequent payment is reduced over what was paid in the previous year by an amount equal to X.

At an annual effective interest rate of 7.072%, both annuities have the same present value.

Calculate X, where X < 5.

Can someone please explain why you can't use the decreasing annuity formula ( [n- a-angle-n] /i ) to solve this?
I did 10X a-angle-10 = X(Da)-angle-10 at 7.072%

And when in general can you use the decreasing annuity formula and when can you not?
#2
07-01-2018, 01:23 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 7,795

The payments under the stated decreasing annuity, are 50, 50-X, .........., 50 -9X, while to use the standard decreasing annuity formula X would have to be 5. Use the more general P,Q formula with P = 50 and Q = -5.
#3
07-01-2018, 03:18 PM
 actsci123451 SOA Join Date: Feb 2018 Posts: 8

Thanks for the reply. So you can only use the DA formula when the payments decrease by a constant amount?
But aren't the payments here decreasing by a constant amount, X, each year? Im still not understanding why I can't do X (Da)angle n.
#4
07-01-2018, 03:56 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 7,795

Quote:
 Originally Posted by actsci123451 Thanks for the reply. So you can only use the DA formula when the payments decrease by a constant amount? But aren't the payments here decreasing by a constant amount, X, each year? Im still not understanding why I can't do X (Da)angle n.
the Da formula without a leading coefficient is the pattern n, n-1, n-2,...,1. This can be scaled with a multiple. You are really better off never using the Da, and Ia formulas and just using the P,Q formula. Da angle n is P = n, Q=-1. The P,Q formula can be adapted to any arithmetic sequence and any number of payments..
#5
07-01-2018, 05:54 PM
 actsci123451 SOA Join Date: Feb 2018 Posts: 8

I see, so you can only use DA when in the form n, n-1, ...1. Thanks!