Calculating Present Value for Compound Interest Investment
To determine the present value required for an investment to grow to a future amount, we will use the present value formula for compound interest:
\( PV = \frac{FV}{{(1 + r/n)}^{(nt)}} \)
where:
- PV = present value (the amount to be invested now)
- FV = future value (the desired accumulated amount)
- r = annual interest rate (as a decimal)
- n = number of times the interest is compounded per year
- t = number of years the money is invested
Given information:
- FV = $110,000
- r = 2% or 0.02 (as a decimal)
- n = 4 (since the interest is compounded quarterly)
- t = 4 years
Plugging these values into the formula gives us the present value:
\( PV = \frac{110,000}{{(1 + 0.02/4)}^{(4*4)}} \)
\( PV = \frac{110,000}{{(1 + 0.005)}^{16}} \)
\( PV = \frac{110,000}{{1.005}^{16}} \)
Now we will calculate \(1.005^{16}\) and then divide 110,000 by this result:
\( 1.005^{16} = 1.082856 \) (rounded to six decimal places)
So,
\( PV = \frac{110,000}{1.082856} \)
\( PV ≈ 101,576.65 \)
Therefore, the amount to be invested now, or the present value needed, is approximately $101,576.65.
