Calculating Present Value for Investment

How much money should be deposited today in an account that earns 2.5 % compounded monthly so that it will accumulate to $ 10 comma 000 in 4 ​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 as​ needed.)

To find out how much money should be deposited today (the present value, (PV)) in an account that earns 2.5% interest compounded monthly, we can use the formula for the future value of a lump sum compounded interest:

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

Where:

• (FV) = future value ($10,000)
• (PV) = present value (the amount we want to find)
• (r) = annual interest rate (2.5% or 0.025)
• (n) = number of times interest is compounded per year (monthly, so (n = 12))
• (t) = number of years (4 years)

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

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

Now, we can plug in the values:

• (FV = 10,000)
• (r = 0.025)
• (n = 12)
• (t = 4)

Calculating (PV):

1. Calculate (r/n): [ r/n = 0.025/12 \approx 0.00208333 ]

2. Calculate (nt): [ nt = 12 \times 4 = 48 ]

3. Calculate ((1 + r/n)^{nt}): [ (1 + 0.00208333)^{48} \approx (1.00208333)^{48} \approx 1.104486 ]

4. Now, substitute back into the (PV) formula: [ PV = \frac{10,000}{1.104486} \approx 9059.64 ]

Thus, the amount of money that should be deposited today is approximately $9059.64.

So, the final answer is:

[ \text{The amount of money that should be deposited is } \text{$9059.64}. ]