Decreasing annuity question

DITmoonlight · #1 09-10-2008, 10:12 AM

I tried to solve this problem but i am only half way through it. I saw the solution but I still don't understand it. I would appreciate it if I get some help and see different answers so it makes sense to me. Thanks in advance.

Jane receives a 10-year increasing annuity-immediate paying 100 the first year and increasing by 100 each year thereafter. Mary receives a 10-year decreasing annuity-immediate paying X the first year and decreasing by X/10 each year thereafter. At an effective annual interest rate of 5%, both annuities have the same present value. Calculate X.

Answer is 864.

donny5k · #2 09-10-2008, 10:32 AM

Should just be 100 (Ia)_n = X (a_n - .1(a_n - nv^n)/i)
n=10, i=.05
Right side is the P,Q formula for arithmetic progression annuities with P the starting payment amount and Q the difference between payments. Left side is increasing annuity.

Edit:
Using BA II Plus, a_10 = 7.7217, (Ia)_10 = (a_10 *1.05 - 10v^10)/.05 = 39.3733,
so X = 3937.33/(7.7217 - .1(7.7217 - 10v^10)/.05) = 864.1

To clarify, left side is PV(Jane's annuity) and right side is PV(Mary's annuity).

atomic · #3 09-10-2008, 12:08 PM

Quote:

Originally Posted by DITmoonlight

I tried to solve this problem but i am only half way through it. I saw the solution but I still don't understand it. I would appreciate it if I get some help and see different answers so it makes sense to me. Thanks in advance.

Jane receives a 10-year increasing annuity-immediate paying 100 the first year and increasing by 100 each year thereafter. Mary receives a 10-year decreasing annuity-immediate paying X the first year and decreasing by X/10 each year thereafter. At an effective annual interest rate of 5%, both annuities have the same present value. Calculate X.

Answer is 864.

As usual, the solution is made clearer by writing out the way the cash flows look:

$PV_J = 100v + 200v^2 + \cdots + 1000v^{10}$

$PV_M = Xv + \left(X - \frac{X}{10}\right)v^2 + \cdots + \left(X - \frac{9X}{10}\right)v^{10}$ .

Now simplify:

$PV_J = 100(Ia)_{\overline{10|}.05}$

$PV_M = \frac{X}{10}(Da)_{\overline{10|}.05}$ .

Since

$(Ia)_{\overline{n|}} + (Da)_{\overline{n|}} = (n+1) a_{\overline{n|}}$ ,

we see that

$\frac{X}{10}\left(11a_{\overline{10|}} - (Ia)_{\overline{10|}}\right) = 100(Ia)_{\overline{10|}}$ .

We then recall that

$(Ia)_{\overline{n|}} = \frac{{\ddot a}_{\overline{n|}} - nv^n}{i}$ ,

and so

$a_{\overline{10|}} = 7.7217349$ ,

$(Ia)_{\overline{10|}} = 39.3737828$ ,

from which the solution readily follows.

I never thought I'd look back on this material and wish the exams I'm taking could be this easy.

DITmoonlight · #4 09-10-2008, 09:17 PM

Got it! thank you both for all your help!

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