Question

Derek wants to withdraw $10,298.00 from his account 6.00 years from today and $13,278.00 from his account 10.00 years from today. He currently has $3,347.00 in the account. How much must he deposit each year for the next 10.0 years? Assume a 6.24% interest rate. His account must equal zero by year 10.0 but may be negative prior to that.

Answer 1

To determine how much Derek must deposit each year for the next 10 years to meet his future withdrawal needs while considering interest, we can break the problem down into steps.

### Step 1: Calculate the Future Value (FV) of current amount ($3,347) at the withdrawal times.

**1. Future Value Withdrawals**

Derek needs $10,298 in 6 years and $13,278 in 10 years. We will calculate the future value of the initial amount he currently has at the specified points in time:

**Future Value Formula:**
\[
FV = PV \times (1 + r)^n
\]
Where:
- \(PV\) = present value (initial investment)
- \(r\) = interest rate (0.0624)
- \(n\) = number of years

#### Calculate FV for Year 6
- PV: $3,347
- r: 6.24% = 0.0624
- n: 6

\[
FV(6) = 3347 \times (1 + 0.0624)^6 \approx 3347 \times 1.484812 \approx 4,963.13
\]

#### Calculate FV for Year 10
- PV: $3,347
- n: 10

\[
FV(10) = 3347 \times (1 + 0.0624)^{10} \approx 3347 \times 1.713735 \approx 5,741.44
\]

### Step 2: Calculate the future value at the time of each withdrawal.

**Total Future Value Available at Year 6 and Year 10**:
- After 6 years: $4,963.13
- After 10 years: $5,741.44

### Step 3: Determine total amounts needed at those times.

#### Year 6 needs:
- Needs: $10,298
- Available: $4,963.13
- Additional needed:

\[
10,298 - 4,963.13 = 5,334.87
\]

#### Year 10 needs:
- Needs: $13,278
- Available: $5,741.44
- Additional needed:

\[
13,278 - 5,741.44 = 7,536.56
\]

### Step 4: Calculate how much must be deposited each year.

To find out how much Derek needs to deposit each year, we can set up the following future value of a series formula for each required amount.

**Future Value of a Series Formula (Annuity)**
\[
FV = PMT \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- \(PMT\) = annual payment (deposit)
- \(r\) = interest rate (0.0624)
- \(n\) = number of years of deposits

**Deposits for Year 6**
We want the future value of the deposits after 6 years to equal $5,334.87.

\[
5,334.87 = PMT_1 \times \frac{(1 + 0.0624)^6 - 1}{0.0624}
\]
Calculating the annuity factor:

\[
(1 + 0.0624)^6 - 1 = 0.484812 \to \frac{0.484812}{0.0624} \approx 7.7731
\]

Now substituting in:

\[
5,334.87 = PMT_1 \times 7.7731
\]

Solve for \(PMT_1\):

\[
PMT_1 = \frac{5,334.87}{7.7731} \approx 685.06
\]

**Deposits for Year 10**
We want the future value of the deposits after 10 years to equal $7,536.56.

\[
7,536.56 = PMT_2 \times \frac{(1 + 0.0624)^{10} - 1}{0.0624}
\]
Calculating the annuity factor:

\[
(1 + 0.0624)^{10} - 1 = 0.713735 \to \frac{0.713735}{0.0624} \approx 11.4308
\]

Now substituting in:

\[
7,536.56 = PMT_2 \times 11.4308
\]

Solve for \(PMT_2\):

\[
PMT_2 = \frac{7,536.56}{11.4308} \approx 658.08
\]

### Step 5: Conclusion

Since Derek can deposit these amounts each year starting today, ideally, we want the total equal sum deposited to cover both needs over the 10 years.

Therefore, the total deposit he needs to make per year on average over 10 years is:

\[
Total PMT = PMT_1 + PMT_2 = 685.06 + 658.08 = 676.57
\]

In conclusion, Derek must deposit approximately **$676.57 each year for the next 10.0 years** to cover his expected withdrawals at the specified future times, assuming a 6.24% interest rate.