Annuities

Date: 12/31/2001 at 03:48:27
From: Russ Dodge
Subject: Annuities

If I have $100,000 at 5% interest and withdraw from it for 20 years, 
leaving a zero balance, how much monthly income will I have? 
$660 or $669?

Date: 12/31/2001 at 07:46:41
From: Doctor Jerry
Subject: Re: Annuities

Hi Russ,

I don't know the compounding period. I'll assume that the money is 
compounded each month.

The notation a_0 means a sub 0.

initial amount: a_0 (dollars)
interest rate: r (like 7%)
number of months withdrawals are made: N
monthly income collected at the end of each month, in the amount x
you start the process on the first day of some month

let p_0 be the amount remaining in the bank at time t=0 (months)
let p_j be the amount remaining in the bank at time t=j (months)

Okay,

   p_0 = a_0
   p_1 = a_0+a_0*r/(12*100)-x = a_0(1+r/1200)-x    

(amount deposited one  month ago, plus interest accrued on this 
amount, minus monthly income)

For convenience, let  1+r/1200 = w.  So,

   p_1 = a_0*w-x

   p_2 = p_1+p_1*r/(12*100)-x=p_1*w-x=a_0*w^2-x*w-x
   p_3 = p_2+p_2*r/(12*100)-x=p_2*w-x=a_0*w^3-x*w^2-x*w-x

So,

   p_j = a_0*w^j-x[w^{j-1}+w^{j-2}+...w+1]

Now,  w^{j-1}+w^{j-2}+...w+1=(1-w^j)/(1-w), finite geometric sum. So,

   p_j = a_0*w^j -x*(1-w^j)/(1-w).

Well, a_0 = 100000, r = 5, j = 12*20 = 240, and we want p_{240} = 0.  
So, w = 1+5/1200 and

0 = 100000*(1+5/1200)^{240} - x(1-(1+5/1200)^{240})/(1-(1+5/1200)).

If you solve this for x, you will find that x = 659.956

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/