Bella, who is currently 25 years old, wants to invest money into a retirement fund so as to have $2,000,000 saved up when she retires at age 65. If she can earn 12% per year in an equity fund, calculate the amount of money she would have to invest in equal annual amounts and alternatively, in equal monthly amounts starting at the end of the current year or month respectively.
To calculate the amount of money Bella would have to invest in equal annual or monthly amounts, we can use the formula for the future value of an ordinary annuity.
1. Let's start with the calculation for equal annual investments:
- Bella wants to retire at age 65, so she has 65 - 25 = 40 years until retirement.
- She wants to have $2,000,000 saved up, and she can earn 12% per year.
- Using the future value of an ordinary annuity formula, the equation becomes:
FV = P * ((1 + r)^n - 1) / r,
where FV is the future value, P is the annual investment, r is the annual interest rate, and n is the number of years.
- Plugging in the values, we get:
$2,000,000 = P * ((1 + 0.12)^40 - 1) / 0.12.
- Now we solve for P:
P = $2,000,000 * 0.12 / ((1 + 0.12)^40 - 1).
2. Now let's calculate the amount of money Bella would have to invest in equal monthly amounts:
- Since there are 12 months in a year, the number of payments over 40 years will be 40 * 12 = 480.
- The interest rate remains the same at 12% per year, but we need to adjust it for monthly compounding.
- The monthly interest rate is 12% / 12 = 1% = 0.01.
- The formula for the future value of an ordinary annuity now becomes:
FV = P * ((1 + r)^n - 1) / r,
where FV is the future value, P is the monthly investment, r is the monthly interest rate, and n is the number of months.
- Plugging in the values, we get:
$2,000,000 = P * ((1 + 0.01)^480 - 1) / 0.01.
- Solving for P:
P = $2,000,000 * 0.01 / ((1 + 0.01)^480 - 1).
By calculating the above equations, you can find the amount of money Bella would need to invest in equal annual or monthly amounts to have $2,000,000 saved up when she retires at age 65.