Derek will deposit $5,000.00 per year for 15.00 years into an account that earns 7.00%. The first deposit is made next year. Derek has $15,000.00 in the account today. How much will be in the account 20.00 years from today?

Answer 1

To find the total amount in the account 20 years from today, we need to consider both the future value of Derek's deposits and the future value of the existing balance.

Step 1: Future Value of Existing Balance

Derek currently has $15,000 in the account. This amount will earn interest for 20 years. We can calculate the future value using the formula for compound interest:

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

Where:

• \(FV\) = future value
• \(PV\) = present value ($15,000)
• \(r\) = annual interest rate (7% or 0.07)
• \(n\) = number of years (20)

Substituting the values into the formula:

\[ FV = 15000(1 + 0.07)^{20} \]

Calculating this:

\[ FV = 15000(1.07)^{20} \] \[ FV = 15000 \times 3.8697 \quad (\text{approximately, using a calculator}) \] \[ FV \approx 58045.50 \]

Step 2: Future Value of Annual Deposits

Derek will be making annual deposits of $5,000 for 15 years. Since the first deposit will be made next year, these deposits will not earn interest for the entirety of the 20 years. Instead, they will earn interest for a decreasing number of years (from 15 years down to 1 year).

The future value of a series of deposits can be calculated using the future value of an annuity formula:

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

Where:

• \(P\) = annual deposit ($5,000)
• \(r\) = annual interest rate (0.07)
• \(n\) = number of deposits (15)

Substituting the values:

\[ FV = 5000 \times \frac{(1 + 0.07)^{15} - 1}{0.07} \]

Calculating this step-by-step:

1. Calculate \((1 + 0.07)^{15}\):

\[ (1.07)^{15} \approx 2.7591 \quad (\text{approximately}) \]

2. Subtract 1 from this result:

\[ 2.7591 - 1 = 1.7591 \]

3. Divide by the interest rate:

\[ \frac{1.7591}{0.07} \approx 25.13 \quad (\text{approximately}) \]

4. Multiply by the deposit amount:

\[ FV \approx 5000 \times 25.13 \approx 125656.52 \]

Step 3: Total Amount in the Account

Now we add the future value of the existing balance and the future value of the deposits:

\[ Total , FV = FV_{\text{existing}} + FV_{\text{deposits}} \] \[ Total , FV \approx 58045.50 + 125656.52 \approx 183702.02 \]

Final Result

The total amount in Derek's account 20 years from today will be approximately $183,702.02.

Answer 2

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

Answer 3

To solve this problem using a finance calculator, you can break it down into two separate calculations: one for the future value of the existing amount and another for the future value of the annuity (the deposits). Below is how you would set up each calculation using the common finance calculator notation:

Step 1: Future Value of Existing Balance

You will calculate the future value (FV) of the current amount of $15,000 after 20 years at an interest rate of 7%.

• N (number of periods): 20
• I/Y (interest rate): 7 (this means 7%)
• PV (present value): -15,000 (negative because this is your current amount, cash outflow)
• PMT (payment): 0 (no additional payments are made)
• FV (future value): ? (this will be the answer from the calculation)

Step 2: Future Value of Annual Deposits

Next, you will calculate the future value (FV) of the $5,000 annual deposits made for 15 years, which will have 15 deposit periods.

• N (number of periods): 15
• I/Y (interest rate): 7 (7%)
• PV (present value): 0 (no current amount for this deposit series)
• PMT (payment): -5,000 (negative because these are cash outflows)
• FV (future value): ? (this will be the answer from the calculation)

Summary of Inputs for the Calculator

1. For Future Value of Existing Balance:

   • N = 20
   • I/Y = 7
   • PV = -15,000
   • PMT = 0
   • FV = ? (Calculate to find FV)

2. For Future Value of Annual Deposits:

   • N = 15
   • I/Y = 7
   • PV = 0
   • PMT = -5,000
   • FV = ? (Calculate to find FV)

Finding Total Future Value

Finally, after calculating the future values from both calculations, you would add the two future values together to get the total value in the account at the end of 20 years.

Make sure to keep track of your signs (positive/negative) for cash inflows vs. outflows while calculating.