Question

Consider a person who begins contributing to a retirement plan at age 25 and contributes for 40 years until retirement at age 65. For the first ten years, she contributes $3,700 per year. She increases the contribution rate to $5,700 per year in years 11 through 20. This is followed by increases to $10,700 per year in years 21 through 30 and to $15,700 per year for the last ten years. This money earns a return of 10 percent.

First compute the value of the retirement plan when she turns age 65. (Round your answer to 2 decimal places.)

Compute the annual payment she would receive over the next 40 years if the wealth was converted to an annuity payment at 9 percent. (Round your answer to 2 decimal places.)

Answer 1

part A
1)
Value of $3,700 per year after 10 years(Future value of annuity)				=	Amt[{(1+r)^n}-1]/r
			Where
			amount	=	$3,700
			rate (.r)	=	10%
			time=t	=	10
	Future value of annuity(when she is 35 yrs old)			=	$3,700*[{(1+0.1)^10}-1]/0.1
				=	$3,700*[{(1.1)^10}-1]/0.1
				=	$3,700*[2.5937-1]/0.1
				=	$3,700*[1.5937]/0.1
				=	$58,966.90
	Future value of $58966.6 after 30 more years
				=	Amt(1+r)^n
				Where
				amount	=	$58,967
				rate (.r)	=	10% or 0.1
				time=t	=	30 years
Future value(when she is 65 years old) after 30 more years						=	$58966.9*(1+0.1)^30
						=	$58966.9*(1.1)^30
						=	$58966.9*17.4494
						=	$1,028,937.02
2)	Value of $5,700 per year(Future value of annuity)for 10 years				=
						Amt[{(1+r)^n}-1]/r
				Where
				amount	=	$5,700
				rate (.r)	=	10%
				time=t	=	10
		Future value of annuity(when she is 45 yrs old)			=	$5,700*[{(1+0.1)^10}-1]/0.1
					=	$5,700*[{(1.1)^10}-1]/0.1
					=	$5,700*[2.5937-1]/0.1
					=	$5,700*[1.5937]/0.1
					=	$90,840.90
	Future value of $90840.9 after 20 more years(when she will 65 yrsold)
				=	Amt(1+r)^n
				Where
				amount	=	$90,841
				rate (.r)	=	10% or 0.1
				time=t	=	20 years
Future value(when she is 65 years old) after 20 more years						=	$90840.9*(1+0.1)^20
						=	$90840.9*(1.1)^20
						=	$90840.9*6.7275
						=	$611,132.15
3)
	Value of $10,700 per year(Future value of annuity)for 10 years				=	Amt[{(1+r)^n}-1]/r
				Where
				amount	=	$10,700
				rate (.r)	=	10%
				time=t	=	10
		Future value of annuity(when she is 55 yrs old)			=	$10,700*[{(1+0.1)^10}-1]/0.1
					=	$10,700*[{(1.1)^10}-1]/0.1
					=	$10,700*[2.5937-1]/0.1
					=	$10,700*[1.5937]/0.1
					=	$170,525.90
	Future value of $170,525.9 after 10 more years (when she will be 65 yrs old
				=	Amt(1+r)^n
				Where
				amount	=	$170,526
				rate (.r)	=	10% or 0.1
				time=t	=	10 years
Future value(when she is 65 years old) after 10 more years						=	$170,525.9*(1+0.1)^10
						=	$170,525*(1.1)^10
						=	$170,525*2.5937
						=	$442,290.69
4)
	Value of $15,700 per year(Future value of annuity)for 10 years				=	Amt[{(1+r)^n}-1]/r
				Where
				amount	=	$15,700
				rate (.r)	=	10%
				time=t	=	10
		Future value of annuity(when she is 65 yrs old)			=	$15,700*[{(1+0.1)^10}-1]/0.1
					=	$15,700*[{(1.1)^10}-1]/0.1
					=	$15,700*[2.5937-1]/0.1
					=	$15,700*[1.5937]/0.1
					=	$250,210.90
	Total amount when she is 65 years old =amount in 1+amount in2 +amount in 3+amount in4
				=	1028937.02+$611,132.15+$442,290.69+$250,210.0
				=	$2,332,569.86
Part B
	annual payment she would receive for next 40 years(Present value of annuity)
					=	Amt[1-(1+r)^-n]/r			=
				here
		Present value of annuity			=	$2,332,569.86
				r	=	9%
				time=n=		40
				amt	=	?
		Therefore
				2,332,569.86	=	Amt[1-(1+0.09)^-40]/0.09
				2,332,569.86	=	Amt[(1-0.03184)/0.09]
				2,332,569.86	=	Amt*10.75736
			2,332,569.86/10.75736		=	amt
			$216,834.79		=	amt
		She will receive $216,834.79 annually for the next 40 years
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