AnnuitiesDate: 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/ |
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