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Derek will deposit $5,097.00 per year for 9.00 years into an account that earns 11.00%. Assuming the first deposit is made 7.00 years from today, how much will be in the account 34.00 years from today?

5 answers

To find out how much Derek will have in his account 34 years from today, we need to calculate the future value of his deposits. He will deposit $5,097 annually for 9 years, starting 7 years from today. This means that the first deposit will be made at year 7, and the last deposit will be made at year 15.

Step 1: Calculate the number of years until the last deposit is made

Since the last deposit is made at year 15, and we want to find the future value at year 34, we need to calculate how many years each deposit will earn interest after it's made.

  1. Year of first deposit (7): This deposit will earn interest for \(34 - 7 = 27\) years.
  2. Year of second deposit (8): This deposit will earn interest for \(34 - 8 = 26\) years.
  3. Year of third deposit (9): This deposit will earn interest for \(34 - 9 = 25\) years.
  4. Year of fourth deposit (10): This deposit will earn interest for \(34 - 10 = 24\) years.
  5. Year of fifth deposit (11): This deposit will earn interest for \(34 - 11 = 23\) years.
  6. Year of sixth deposit (12): This deposit will earn interest for \(34 - 12 = 22\) years.
  7. Year of seventh deposit (13): This deposit will earn interest for \(34 - 13 = 21\) years.
  8. Year of eighth deposit (14): This deposit will earn interest for \(34 - 14 = 20\) years.
  9. Year of ninth deposit (15): This deposit will earn interest for \(34 - 15 = 19\) years.

Step 2: Calculate the future value of each deposit

The formula for the future value (FV) of a single deposit is:

\[ FV = P \times (1 + r)^n \]

Where:

  • \(P\) is the amount of the deposit ($5,097),
  • \(r\) is the interest rate (0.11),
  • \(n\) is the number of years until the account is evaluated.

Now we can calculate the future value of each deposit:

  1. Year 7 deposit (27 years of interest): \[ FV_1 = 5097 \times (1 + 0.11)^{27} \]
  2. Year 8 deposit (26 years of interest): \[ FV_2 = 5097 \times (1 + 0.11)^{26} \]
  3. Year 9 deposit (25 years of interest): \[ FV_3 = 5097 \times (1 + 0.11)^{25} \]
  4. Year 10 deposit (24 years of interest): \[ FV_4 = 5097 \times (1 + 0.11)^{24} \]
  5. Year 11 deposit (23 years of interest): \[ FV_5 = 5097 \times (1 + 0.11)^{23} \]
  6. Year 12 deposit (22 years of interest): \[ FV_6 = 5097 \times (1 + 0.11)^{22} \]
  7. Year 13 deposit (21 years of interest): \[ FV_7 = 5097 \times (1 + 0.11)^{21} \]
  8. Year 14 deposit (20 years of interest): \[ FV_8 = 5097 \times (1 + 0.11)^{20} \]
  9. Year 15 deposit (19 years of interest): \[ FV_9 = 5097 \times (1 + 0.11)^{19} \]

Step 3: Calculate the total future value

Finally, sum all the future values of the deposits to find the total amount in the account at year 34.

Calculating each term:

  1. \( FV_1 = 5097 \times (1.11)^{27} \approx 5097 \times 11.755609 \approx 59,847.34 \)
  2. \( FV_2 = 5097 \times (1.11)^{26} \approx 5097 \times 10.592385 \approx 54,007.40 \)
  3. \( FV_3 = 5097 \times (1.11)^{25} \approx 5097 \times 9.553977 \approx 48,923.13 \)
  4. \( FV_4 = 5097 \times (1.11)^{24} \approx 5097 \times 8.590821 \approx 43,694.20 \)
  5. \( FV_5 = 5097 \times (1.11)^{23} \approx 5097 \times 7.743052 \approx 39,415.80 \)
  6. \( FV_6 = 5097 \times (1.11)^{22} \approx 5097 \times 6.878094 \approx 35,209.62 \)
  7. \( FV_7 = 5097 \times (1.11)^{21} \approx 5097 \times 6.097895 \approx 31,214.30 \)
  8. \( FV_8 = 5097 \times (1.11)^{20} \approx 5097 \times 5.388682 \approx 27,571.67 \)
  9. \( FV_9 = 5097 \times (1.11)^{19} \approx 5097 \times 4.744512 \approx 24,224.73 \)

Total Future Value

Now summing them up:

\[ \text{Total FV} = 59,847.34 + 54,007.40 + 48,923.13 + 43,694.20 + 39,415.80 + 35,209.62 + 31,214.30 + 27,571.67 + 24,224.73 \approx 364,107.69 \]

So the total amount in the account 34 years from today will be approximately $364,107.69.

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

To solve the problem using a financial calculator, we can set this up using the following variables:

  1. N (Number of periods): This is the total number of deposits Derek will make. Since he will make 9 annual deposits, \(N = 9\).

  2. I/Y (Interest rate): This is the annual interest rate. Since the account earns 11%, you will enter \(I/Y = 11\).

  3. PV (Present Value): Since Derek is making deposits into the account (which is a cash outflow), you will typically enter this as \(PV = 0\) because you are accounting for the future value of his deposits rather than having an initial amount in the account.

  4. PMT (Payment): This is the amount of the annual deposit. Derek will deposit $5,097 each year, so you will enter \(PMT = 5097\).

  5. FV (Future Value): This is the amount we want to calculate after 34 years. However, the future value calculation will involve finding out how much each deposit grows over time and summing those future values.

    To calculate FV using a financial calculator for multiple cash flows, you can either calculate the FV of each individual deposit separately (as previously calculated) or you can set \(FV = 0\) for the calculations, often the calculator will allow you to find the accumulated future value just by entering the PMT and the compound interest.

Steps on the Calculator:

  • Set N = 9 (the number of deposits)
  • Set I/Y = 11 (the annual interest rate)
  • Set PV = 0 (initial value in the account before the first deposit)
  • Set PMT = 5097 (the annual deposit amount)
  • Calculate FV, and make sure to account for the fact that the first deposit occurs in year 7, which means the effective cash flow starts at that point, followed by the subsequent years until the last contribution at year 15.

Important Note:

When you calculate the FV, if your calculator does not allow for separate cash flow timing, you may end up with an inaccurate future value because you need to account for how long each deposit is invested.

A manual calculation is required to ensure the timing of the cash flows is appropriately considered, but using a finance calculator in this way will give you an approximation.

Final Format for Inputs:

  • N = 9
  • I/Y = 11
  • PV = 0
  • PMT = 5097
  • FV = (Calculate using the FV function based on inputs, or sum of individual FV if calculating each deposit)

If your calculator handles cash flow timing (like the Cash Flow function in some advanced calculators), you can enter each cash flow along with the corresponding years, otherwise, you will need to sum the future values of each deposit as mentioned earlier.

okay, I solve FV and got FV= 72,193.76549. Next step. what do I enter on a finance calculator for N, i/y, PV, PMT, and FV to solve for FV at year 34?

Great, you've calculated the future value of the 9 annual deposits made starting 7 years from today, and you've correctly identified that the total FV at the end of those deposits is approximately $72,193.77.

Now you need to find out the value of this future amount at year 34. Since the deposits will finish at year 15, you'll need to allow this future value to grow from year 15 to year 34.

Steps to Calculate FV at Year 34

  1. Determine the Growth Period:

    • The time from year 15 to year 34 is \(34 - 15 = 19\) years.

  2. Enter Values in the Financial Calculator:

    • We treat the \(72,193.77\) as the present value (PV) at year 15, and we want to find out the future value (FV) at year 34.

Inputs for the Financial Calculator:

  • N: The number of years until you want to find the future value at year 34, which is 19 years.
  • I/Y: The annual interest rate, which remains at 11%.
  • PV: The present value, which will be the future value previously calculated as \(-72,193.77\) (it’s entered as a negative because it’s an outflow when calculating FV).
  • PMT: Since we are just growing the existing future value and not making any additional payments, enter \(PMT = 0\).
  • FV: You will be calculating this value.

Final Input Values:

  • N = 19
  • I/Y = 11
  • PV = -72,193.77 (as it is an inflow when calculating future value)
  • PMT = 0
  • FV = ? (This is what you will calculate)

Calculation:

Using the calculator, after entering these values, compute FV to find out how much the account will be worth at year 34.

  1. Calculate FV using the calculator, and it will compound the present value at the given interest rate for the specified number of years.

This will give you the total amount in Derek's account at year 34, which should be significantly higher than the amount he contributed due to compound interest over those 19 years.