Answer:
We can use the formula for the future value of an annuity to solve this problem. An annuity is a series of equal payments made at regular intervals. In this case, we are making a single lump-sum deposit now to grow into the desired future value.
The formula for the future value of a lump-sum deposit is:
FV = PV x (1 + r/n)^(n*t)
Where:
FV = future value
PV = present value (the amount to be deposited now)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
In this case, we want to find the present value (PV) required to grow into the future value (FV) of $800,000 in 25 years with an annual interest rate of 5% compounded monthly, so:
FV = $800,000
r = 5% = 0.05 (annual interest rate)
n = 12 (monthly compounding)
t = 25 (number of years)
Substituting these values into the formula, we get:
$800,000 = PV x (1 + 0.05/12)^(12*25)
Simplifying the exponent on the right-hand side, we get:
$800,000 = PV x (1.004167)^300
Dividing both sides by (1.004167)^300, we get:
PV = $800,000 / (1.004167)^300
Using a calculator or spreadsheet, we can evaluate this expression to find:
PV ≈ $267,227.92
Therefore, you would need to deposit approximately $267,227.92 now into the account to reach your retirement goal of $800,000 in 25 years with an annual interest rate of 5% compounded monthly.