Question

Suppose you deposit $20,000 at the end of each of the next 30 years into a retirement account. Immediately after your last deposit, you take the entire accumulated value in your account and purchase a 20-year annuity, which will pay you X at the beginning of each year for 20 years. The price of this 20-year annuity is equal to the present value (at the time of purchase) of the 20 annual cash flows. The effective annual interest rate through the entire 50-year period is 10%. Find X.

Answer 1

Solution:
	X= $351,298.29 if nothing is rounded off intermediate calculation
	or $351,298
Working Notes:
	At end of 30 years we will have future value of annuity $20,000
	At end of 30 years purchase price of Annuity of next 20 years will the 30 years future value of annuity we get.
	And this amount will receive X beginning of each for then next 20 years .
	Hence, the Amount paid to purchase this annuity will be equal to present value of annuity due of $X , as payment will be received at beginning of years .
First of all	Amount at end of 30 years
	Future value of annuity = P x ((1+i)^n - 1)/I
	P= payments paid per year =$20,000
	I=interest rate = 10%=0.10
	n= no. Of years= 30 Year
	Future value of annuity at t=30 = P x ((1+i)^n - 1)/i
	Future value of annuity at t=30 = 20,000 x ((1+0.10)^30 - 1)/0.10
	=$3,289,880.454377729
	=$3,289,880
Now	The Value of present value of annuity due purchased at end of 30 years for next 20 years value is $3,289,880.454377729
	present value of annuity due= Px(1+i)[ 1-1 /(1 + i)^n]/ i
	P= yearly payment = $X
	let i= interest rate per period = 10% per year
	n= no. Of period = 20 x 1 =200
	PV of annuity due=$3,289,880.454377729
	present value of annuity due = Px[ 1-1 /(1 + i)^n)]/ i
	=P x(1+i)[ 1-1 /(1 + i)^n)]/ i
	$3,289,880.454377729 =X (1+0.10)(1-1/(1+0.10)^20)/0.10
	$3,289,880.454377729/9.364920092 = X
	X= $351,298.29
	X= $351,298
Please feel free to ask if anything about above solution in comment section of the question.