Question

You are saving for retirement. To live​ comfortably, you decide you will need to save $ 4 million by the time you are 65. Today is your 25 th ​birthday, and you​ decide, starting today and continuing on every birthday up to and including your 65 th ​birthday, that you will put the same amount into a savings account. If the interest rate is 3 %​, how much must you set aside each year to make sure that you will have $ 4 million in the account on your 65 th ​birthday?

Answer 1

Answer: $50,846.3701

Explanation:

Need to save $4 million to live comfortably,

Interest rate, r = 3%

N = 40 years

$Present\ value=\frac{FV_{N} }{(1+i)^{N}}$

$Present\ value=\frac{4,000,000 }{(1+0.03)^{40}}$

$Present\ value=\frac{4,000,000 }{3.262}$

                             = 1,226,241.57

$Present\ value\ of\ annuity= C\times\frac{1}{i}\times(1-\frac{1}{(1+i)^{N}}) + C$

$1,226,241.57= C\times\frac{1}{0.03}\times(1-\frac{1}{(1.03)^{40}})+C$

$1,226,241.57= C\times\frac{1}{0.03}\times(1-\frac{1}{(1.03)^{40}})+C$

$1,226,241.57=C[\frac{1}{0.03}\times(1-0.3065)+1]$

$1,226,241.57=24.1166\times C$

$C=\frac{1,226,241.57}{24.1166}$

         = $50,846.3701

Hence, $50,846.3701 will be the annual payment to have $4 million in the account on 65th birthday.