What it Takes to Become a Millionaire

I will tackle this problem from a mathematical standpoint.  I want to know how much money I need to put away each year in order to have $1,000,000 after n years.  I also want to know how long it will take to have $1,000,000 if I know the amount I can put away each year.


Formula:  actuarial accumulated value of an annuity paid at the end of the year

where n = time in years and i = annual interest rate.

Example 1:

Investor A just turned 40 years old and wants to be a millionaire by age 50.  He wants to know how much money to invest at the end of each year assuming his investments will earn an interest rate of 7%.

Solution 1:

In order to solve this equation we will need to introduce a new variable, A, where A is the amount paid (invested) at the end of each year.  We will also set the equation equal to the final accumulated value of $1,000,000.

$1,000,000 = A   * 

So i = .07 and n = 10 for this example.

Solving the equation for A we get A = $72,377.50.

In order for this investor to be a millionaire by his 50th birthday he will need to invest $72,377.50 at the end of each year earning 7%.

Example 2:  

Investor B at age 24 determines he will have $12,000 to invest at the end of each year.  Assuming a return of 7% per year he wants to know how many years it will take to become a millionaire.

Solution 2:

We know A = $12,000 so
$1,000,000 = $12,000    * 

Substituting for i = .07 and solving for n gives us:

n = 26.07

So it will take investor B a little over 26 years if he invests $12,000 at the end of each year that earns 7% interest.  He will be 50 years old.

Conclusion:

Investor A needs to put away a lot more money each year than Investor B.  This is due to Investor B having an additional 16 years of compounding before he reaches 50.

I wanted to give a fairly simple formula that anyone could use to come up with similar solutions.  Things can get a lot more complicated with varying interest rates, increasing payments and monthly contributions.