Question

you are saving for retirement .to live comfortably, you decide you will need to have $3 million by the timeyou are 65. today is your 20th birthday,and you decide, starting and continuingon every birthday up to including your 65th birthday, that you will put the same amount into a savings account . if the interest rate is 7%,how much you set aside rach year to make sure that you will have $3 million in the account on your 65th birthday?

Answer 1

To determine how much you need to set aside each year to reach $3 million in the savings account by your 65th birthday, we can use the concept of the future value of an annuity.

Given:

Target amount: $3,000,000

Interest rate: 7%

Period: From your 20th birthday to your 65th birthday (46 years)

Using the future value of an annuity formula:

FV = P * [(1 + r)^n - 1] / r

Where:

FV = Future value (target amount)

P = Annual contribution

r = Interest rate

n = Number of periods (years)

Plugging in the values:

$3,000,000 = P * [(1 + 0.07)^46 - 1] / 0.07

Now, let's solve for P:

P * [1.07^46 - 1] / 0.07 = $3,000,000

P * 50.6700721 = $3,000,000

P = $3,000,000 / 50.6700721

P ≈ $59,150.47

Therefore, you would need to set aside approximately $59,150.47 each year from your 20th birthday to your 65th birthday to accumulate $3 million in the savings account, assuming a 7% interest rate

Step-by-step explanation:

Certainly! Let's break down the calculation step by step:

We start with the formula for the future value of an annuity:

FV = P * [(1 + r)^n - 1] / r

FV represents the future value or target amount, P is the annual contribution, r is the interest rate, and n is the number of periods or years.

We plug in the given values:

FV = $3,000,000 (target amount)

r = 7% (interest rate)

n = 46 years (from your 20th birthday to your 65th birthday)

Substituting the values into the formula, we have:

$3,000,000 = P * [(1 + 0.07)^46 - 1] / 0.07

Here, we first calculate the value within the brackets:

(1 + 0.07)^46 - 1 ≈ 50.6700721

Then, we rearrange the equation:

P * 50.6700721 = $3,000,000

Finally, we solve for P by dividing both sides of the equation by 50.6700721:

P = $3,000,000 / 50.6700721 ≈ $59,150.47

So, to reach the target amount of $3 million by your 65th birthday, you would need to set aside approximately $59,150.47 each year.

This assumes a consistent annual contribution and a 7% interest rate