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Derek will deposit $4,735.00 per year for 10.00 years into an account that earns 10.00%. Assuming the first deposit is made 6.00 years from today, how much will be in the account 31.00 years from today?

3 answers

To solve this problem, we can break it down into two parts: first, we need to calculate the future value of Derek's deposits at the time they stop being deposited, and then we will find out how much that amount will grow over the remaining years until the specified time.

Step 1: Calculate the future value of the deposits

Derek deposits $4,735.00 per year for 10 years, starting 6 years from today.

1. Calculate the future value of an annuity:

The formula for the future value of an annuity is:

\[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \]

Where:

  • \( FV \) = Future Value of the annuity
  • \( PMT \) = Payment per period ($4,735.00)
  • \( r \) = Interest rate per period (10% or 0.10)
  • \( n \) = Number of periods (10 years)

Substituting in the values:

\[ FV = 4735 \times \frac{(1 + 0.10)^{10} - 1}{0.10} \]

Calculating \((1 + 0.10)^{10}\):

\[ (1 + 0.10)^{10} \approx 2.5937 \]

Now, substituting back into the FV formula:

\[ FV = 4735 \times \frac{2.5937 - 1}{0.10} \approx 4735 \times \frac{1.5937}{0.10} \approx 4735 \times 15.937 \approx 75,389.55 \]

So, at the end of Year 15 (which is when the last deposit is made for 10 years starting 6 years from today), the future value of the annuity is approximately $75,389.55.

Step 2: Calculate the future value at the end of an additional 16 years

After the last deposit is made at the end of year 15, we need to find out how much this amount will grow over the next 16 years (from Year 15 to Year 31).

We will use the future value formula as:

\[ FV = PV \times (1 + r)^t \]

Where:

  • \( FV \) = Future Value
  • \( PV \) = Present Value ($75,389.55)
  • \( r \) = Interest rate (0.10)
  • \( t \) = Number of years (16)

Substituting in the values:

\[ FV = 75,389.55 \times (1 + 0.10)^{16} \]

Calculating \((1 + 0.10)^{16}\):

\[ (1 + 0.10)^{16} \approx 4.505 \]

Now substituting back into the FV formula:

\[ FV = 75,389.55 \times 4.505 \approx 339,375.45 \]

Final Answer

Therefore, the total amount that will be in the account 31 years from today is approximately:

\[ \boxed{339,375.45} \]

what value do I enter on a finance calculator for N, i/y, PV, PMT, and FV?

To use a financial calculator for this problem, here's how you can fill out the parameters for \(N\), \(I/Y\), \(PV\), \(PMT\), and \(FV\):

First Part (First 10 deposits)

  1. N (Number of periods for the annuity deposit):

    • Since the deposits are made for 10 years, enter 10.

  2. I/Y (Interest rate per period):

    • The annual interest rate for the account is 10%, so enter 10.

  3. PV (Present Value):

    • Since we are calculating the future value of the deposits only, the present value is 0 for this part. Enter 0.

  4. PMT (Payment per period):

    • The amount Derek deposits each year is 4735. Enter -4735 (the negative sign indicates cash outflow).

  5. FV (Future Value):

    • You will calculate this value after entering the above. This is the future value of the annuity after 10 years.

Calculate the future value of the annuity after 10 years of deposits:

After inputting these values, use the calculator to compute FV. This will give you the amount after 10 years of deposits.

Second Part (Growing the future value)

Now you have the FV from the first part, which you'll use as the present value for the second part of your calculation. Now you want to find out what this amount grows to over the next 16 years.

  1. N (Number of periods after deposits stop):

    • Since you want to calculate the future value for 16 more years, this will be 16.

  2. I/Y (Interest rate per period):

    • Still 10.

  3. PV (Present Value):

    • Use the FV amount you got from the first step (let’s say it resulted in X). Enter X as a positive value because this is an investment you will be growing (the cash inflow in this context).

  4. PMT (Payment per period):

    • Since no additional payments are being made during this period, enter 0.

  5. FV (Future Value):

    • This will be the value you are solving for; after entering the above, calculate the FV using the calculator.

Summary of Calculator Inputs

  • First part:

    • N = 10
    • I/Y = 10
    • PV = 0
    • PMT = -4735
    • FV = (Calculate this)

  • Second part:

    • N = 16
    • I/Y = 10
    • PV = (Use FV from first part)
    • PMT = 0
    • FV = (Calculate this)

Once you've performed the calculations, the final FV will give you the total amount in the account after 31 years.