Joseph is 28 and would like to retire at 55. He has $25,000 to invest today and would like to have $1,000,000 in his retirement account when he retires. Joseph has found an investment that will pay him a return of 6% until retirement. How much will Joseph have to invest each month in order to reach his retirement?

Use the mortgage formula:

Future value = FV = 1000000
payment per period = P (to be found)
rate of interest (per period) = i (0.06/12=0.005 per month)
R = 1+i = integrated rate = 1.005
n = number of periods = (55-28)*12=324
then
FV=P(1+R+R²+R³+....+Rn-1)
=P(R^n-1)/(R-1)
1000000=P(1.005^324-1)/(.005)
=>
P=1000000*.005/(1.005^324-1)
=1239.85

Note: because of rounding, the FV of the above answer will be missing almost $3, while 1239.86 will yield an FV of $5.2 in excess.

What about the initial $25,000 that he starts with ?

i = .06/12 = .005

25000(1.005)^324 + R(1.005^324 - 1)/.005 = 1,000,000
R(806.5468...) = 874,181.64
R = $1083.86

Oops! Good catch! Thanks.

Elcia, please go with Reiny's answer.

To find out how much Joseph will have to invest each month to reach his retirement goal of $1,000,000, we can use the formula for the future value of an ordinary annuity:

Future Value = Payment × [(1 + interest rate)^number of periods - 1] / interest rate

Given:
- Joseph's current age: 28
- Joseph's desired retirement age: 55 (27 years from now)
- Joseph's initial investment: $25,000
- Joseph's target retirement amount: $1,000,000
- Interest rate: 6%

Step 1: Calculate the number of periods (months) Joseph will be investing:
Since Joseph plans to retire at age 55 and he is currently 28, the number of periods (months) Joseph will be investing is: (55 - 28) × 12 = 27 × 12 = 324 months

Step 2: Calculate the future value of Joseph's initial investment:
To calculate the future value of Joseph's initial investment of $25,000, we use the compound interest formula:
Future Value = Initial Investment × (1 + interest rate)^number of periods
Future Value = $25,000 × (1 + 0.06)^324 ≈ $301,660.10

Step 3: Calculate the additional monthly investment Joseph needs to reach his retirement goal:
To calculate the monthly investment needed, we rearrange the future value formula:
Payment = (Future Value × interest rate) / [(1 + interest rate)^number of periods - 1]
Payment = ($1,000,000 - $301,660.10) × 0.06 / [(1 + 0.06)^324 - 1]
Payment ≈ $698.34

Therefore, Joseph will need to invest approximately $698.34 per month to reach his retirement goal of $1,000,000 by age 55.