Final answer:
To have the same amount at retirement as investing $6,000 per year, the lump-sum investment needed today, assuming a 7.5% annual interest rate over 38 years, would be approximately $101,823.
Step-by-step explanation:
To calculate the lump-sum investment needed today to achieve the same retirement savings as investing $6,000 per year at a 7.5% annual return rate for 38 years, we can use the present value of an annuity formula. However, the scenario you've provided is akin to investing $6,000 each year at the end of the period (ordinary annuity), but for a lump sum, we're looking to find the present value of these investments made at the beginning of the period (annuity due). To convert an ordinary annuity to an annuity due, we multiply the present value of the ordinary annuity by (1+r), where r is the interest rate.
Firstly, let's calculate the future value of the ordinary annuity:
The future value of an annuity (ordinary) formula is: FV = PMT * [((1 + r)^n - 1) / r]
Where:
PMT = $6,000
r = 7.5% or 0.075 annual
n = 38 years
By calculating the future value and then finding the present value of that future value, we determine the lump sum needed today.
The future value of the annuity:
FV = $6,000 * [((1 + 0.075)^38 - 1) / 0.075]
FV = $6,000 * [(1.075^38 - 1) / 0.075]
FV ≈ $6,000 * 209.764
FV ≈ $1,258,584
Now let's find the present value of the future value calculated:
The present value formula is: PV = FV / (1 + r)^n
Where:
FV = $1,258,584
r = 7.5% or 0.075
n = 38 years
Calculating our present value:
PV = $1,258,584 / (1 + 0.075)^38
PV ≈ $1,258,584 / 13.294
PV ≈ $94,719
To adjust for the annuity due (lump-sum investment made today), we multiply by (1 + 0.075), giving:
The adjusted present value for an annuity due (lump sum needed today):
Adjusted PV = $94,719 * (1 + 0.075)
Adjusted PV ≈ $94,719 * 1.075
Adjusted PV ≈ $101,823
Therefore, the lump-sum investment needed today to have the same amount at retirement as saving $6,000 per year for 38 years at a 7.5% return would be approximately $101,823.