To determine the lump sum that needs to be invested to grow to [tex]$71,000 in 15 years at an annual interest rate of 12%, compounded monthly, we will follow these steps:
### Step-by-Step Solution:
1. Identify the given values:
- Future Value (FV) = $[/tex]71,000
- Annual Interest Rate (r) = 12% = 0.12
- Number of compounding periods per year (m) = 12 (monthly compounding)
- Number of years (t) = 15
2. Calculate the monthly interest rate:
- Monthly interest rate (i) = Annual Interest Rate / Number of compounding periods per year
[tex]\[
i = \frac{0.12}{12} = 0.01
\][/tex]
3. Compute the total number of compounding periods:
- Total Number of Periods (n) = Number of compounding periods per year * Number of years
[tex]\[
n = 12 \times 15 = 180
\][/tex]
4. Use the formula for compound interest to solve for the present value (PV):
The formula for compound interest is:
[tex]\[
FV = PV \times (1 + i)^n
\][/tex]
Rearrange the formula to solve for the present value:
[tex]\[
PV = \frac{FV}{(1 + i)^n}
\][/tex]
Substitute the given values into the formula:
[tex]\[
PV = \frac{71,000}{(1 + 0.01)^{180}}
\][/tex]
5. Perform the calculation:
[tex]\[
PV = \frac{71,000}{(1.01)^{180}}
\][/tex]
[tex]\[
PV = \frac{71,000}{6.0}
\][/tex]
6. Result:
[tex]\[
PV \approx 11841.62
\][/tex]
So, the lump sum that must be invested at 12% compounded monthly for the investment to grow to [tex]$71,000 in 15 years is $[/tex]11841.62.
