Calculating Present Value for Future Savings

How much money should be deposited today in an account that earns 3.5 % compounded monthly so that it will accumulate to $ 14 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 3.5% interest compounded monthly, we can use the formula for the future value of a lump sum compounded at a certain interest rate:

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

Where:

• ( FV ) = future value ($14,000)
• ( PV ) = present value (the amount we want to find)
• ( r ) = annual interest rate (3.5% or 0.035)
• ( 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 = 14,000 )
• ( r = 0.035 )
• ( n = 12 )
• ( t = 4 )

Calculating ( PV ):

1. Calculate ( r/n ): [ r/n = 0.035/12 \approx 0.00291667 ]

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

3. Calculate ( (1 + r/n)^{nt} ): [ (1 + 0.00291667)^{48} \approx (1.00291667)^{48} \approx 1.148882 ]

4. Now, substitute back into the formula for ( PV ): [ PV = \frac{14,000}{1.148882} \approx 12,186.56 ]

Thus, the amount of money that should be deposited today is approximately $12,186.56.

Final Answer: The amount of money that should be deposited is $12,186.56.