Calculating Present Value for Investment

How much money should be deposited today in an account that earns 8 % compounded semiannually so that it will accumulate to $ 9000 in three​ years? LOADING... Click the icon to view some finance formulas. Question content area bottom Part 1 The amount of money that should be deposited is ​$    enter your response here. ​(Round up to the nearest​ cent.)

To find out how much money should be deposited today (the present value, (PV)) in an account that earns 8% interest compounded semiannually, we can use the formula for compound interest:

[ FV = PV \times (1 + r/n)^{nt} ]

Where:

• (FV) = future value ($9000)
• (PV) = present value (the amount we want to find)
• (r) = annual interest rate (0.08 for 8%)
• (n) = number of times interest is compounded per year (2 for semiannual)
• (t) = number of years (3)

We need to rearrange the formula to solve for (PV):

[ PV = \frac{FV}{(1 + r/n)^{nt}} ]

Now, substituting the values into the formula:

• (FV = 9000)
• (r = 0.08)
• (n = 2)
• (t = 3)

Calculating (PV):

1. Calculate (r/n): [ r/n = 0.08/2 = 0.04 ]

2. Calculate (nt): [ nt = 2 \times 3 = 6 ]

3. Calculate ((1 + r/n)^{nt}): [ (1 + 0.04)^{6} = (1.04)^{6} ]

   Now calculate ( (1.04)^6 ): [ (1.04)^6 \approx 1.265319 ]

4. Now substitute back into the (PV) formula: [ PV = \frac{9000}{1.265319} \approx 7117.69 ]

Thus, the amount of money that should be deposited today is approximately $7117.69.

Final Answer: The amount of money that should be deposited is $7117.69 (rounded to the nearest cent).