Question

Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 75.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 75.0 when he fully retires, he will begin to make annual withdrawals of $135,088.00 from his retirement account until he turns 86.00. He he will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 10.00% interest rate.

Answer 1

First step is calculation of present value of $135,088 annual withdrawals on 75th birthday.

a	Present value of annuity=	P* [ [1- (1+r)-n ]/r ]
P=	Periodic payment	135,088
r=	Rate of interest per period
	Annual interest	10.00%
	Number of interest payments per year	1
	Interest rate per period	0.1/1=
	Interest rate per period	10.000%
n=	number of periods:
	Number of years	11
	Periods per year	1
	number of periods	11
	Present value of annuity=	135088* [ (1- (1+0.1)^-11)/0.1 ]
	Present value of annuity=	877,404.80

Balance in investment account at the age of 75 required is $877,404.80

Second step is calculation of present value of  this fund balance at the age of 65:

Present value of money:	=	FV/ (1+r) ^N
Future value	FV=	$       877,404.80
Rate of interest	r=	10%
Number of years	N=	10
Present value	=	877404.8/ (1+0.1)^10
	=	$       338,277.53

Amount in fund required at the age of 65 is  $       338,277.53

Third step is calculation of annual payments to make fund balance  $       338,277.53

Payment required	=		FV*r /[(1+r)^n -1]
Future value	FV		338,277.53
Rate per period	r
Annual interest			10%
Number of interest payments per year			1
Interest rate per period			0.1/1=
Interest rate per period			10.000%
Number of periods	n
Number of years			40
Periods per year			1
number of periods			40
Period payment	=		338277.53*0.1/ [(1+0.1)^40 -1]
	=		764.31

Annual payment required is $764.31