Question

Burt Sbeez is saving up for his retirement. Today is his 40th birthday. Burt first started saving when he was just 25 years old. On his 25th birthday, Burt made the first contribution to his retirement account when he deposited $3,000. Each year on his birthday, Burt has contributed another $3,000 to the account. The 16th (and last) of these contributions is made today. The account has paid interest at the rate of 4.2% APR, compounded monthly. Burt wants to close the account today and immediately move the money to a stock fund which is expected to earn a return of 9%/ year. Burt’s plan is to continue making contributions to this account each year on his birthday. His next contribution will be one year from now (at age 41) and his final planned contribution will be on his 67th birthday. If Burt wants to accumulate $4,500,000 in his account by the instant that he reaches age 67, how much must he contribute each year until age 67 (27 contributions in all) to achieve his goal?

Answer 1

Amount of contribution =	3000
no. of contributions made in retirement account (n) =	16
Interest rate is compounded monthly =	4.20%
no. of compounding in year (m) =	12
Effective annual interest formula (i)= (1+(r/m))^m-1
(1+(4.2%/12))^12-1
0.0428180072	or 0.04282
Contribution made at 25 th birthday means start of year and till 40. So future value of annuity formula become applicable.
Future value of annuity formula = P *{ (1+r)^n - 1 } / r
3000*((1+0.04282)^16 -1)/0.04282
66971.98255
So, at age of 40 years, amount accumulated and transferred to fund shall be $66971.98
Future value of $66971.98 received at age of 67 year shall be as follows
n = 27 years, interest rate = 9%
Future value = P*(1+i)^n
66971.98*(1+9%)^27
686133.4618
HIs goal is to accumulate at 67 is	4500000
out of which deposit of $69839.72 will produce=	686133.46
Balance future value required =	3813866.54
Contribution is made at end of year. So future value of annuity fomrula shall become applicable.
Future value of annuity formula = P *{ (1+r)^n - 1 } / r
3813866.54= P *(1+9%)^27 -1)/9%
3813866.54 = P *102.723134
P =37127.629
So, he should contribute $37127.63 at end of each year with previous amount to accumulate $4,500,000 at his retirement.