Ralph wants to quit his job and move to Hawaii on December 25, 2015. Once there, he anticipates that he will need to make annual withdrawals of 15000 dollars (starting on December 25, 2016) to supplement his income from working as a cabana boy, and he wants the money to last 10 years (i.e. he'll make 10 withdrawals total). His plan is to make annual deposits, starting on December 25, 2000 and ending on December 25, 2015, into an account paying 7.9 percent interest rate compounded annually. How large should each deposit be for Ralph to realize his goal? Answer correct to 3 decimal places. 

To determine the size of Ralph's annual deposits, we need to break the problem into two parts:

1. **Withdrawal Phase (Dec 25, 2016 - Dec 25, 2025):**
   Ralph plans to withdraw $15,000 annually for 10 years from an account earning 7.9% interest, compounded annually.

   We need to calculate the **present value** of these withdrawals as of December 25, 2015. This present value will be the amount Ralph needs to have in his account by December 25, 2015 to satisfy his withdrawal needs. This present value can be computed using the formula for the present value of an ordinary annuity:

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

   Where:
   - $ W = 15,000 $ (annual withdrawal),
   - $ r = 0.079 $ (annual interest rate),
   - $ n = 10 $ (number of withdrawals).

2. **Deposit Phase (Dec 25, 2000 - Dec 25, 2015):**
   Ralph plans to make equal annual deposits into an account earning 7.9% interest from December 25, 2000 to December 25, 2015 (16 deposits in total). The future value (FV) of these deposits as of December 25, 2015 must be equal to the present value calculated in step 1.

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

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

   Where:
   - $ D $ is the annual deposit,
   - $ r = 0.079 $ (interest rate),
   - $ t = 16 $ (number of deposits).

We first calculate the present value of the withdrawals, and then use that value to calculate the annual deposit $ D $.

### Step 1: Calculate the Present Value of the Withdrawals

\[
PV = 15,000 \times \frac{1 - (1 + 0.079)^{-10}}{0.079}
\]

I'll calculate this first.The present value of the 10 annual withdrawals, starting from December 25, 2016, is approximately \$101,106.79. This is the amount Ralph needs to have in his account by December 25, 2015.

### Step 2: Calculate the Annual Deposits

Now, we use the future value of an annuity formula to determine the size of the annual deposits Ralph should make over 16 years. The future value should equal \$101,106.79, and the formula is:

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

We rearrange this to solve for $ D $:

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

Where:
- $ FV = 101106.79 $ (future value of withdrawals),
- $ r = 0.079 $ (interest rate),
- $ t = 16 $ (number of deposits).

Let's calculate $ D $.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, we already determined the present value of Ralph's withdrawals to be approximately \$101,106.79. To continue calculating the annual deposits manually, you can use the formula I provided:

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

Where:
- $ FV = 101106.79 $,
- $ r = 0.079 $,
- $ t = 16 $.

If you'd like me to walk you through solving this step by step, feel free to ask!

Would you like more details on any specific part of this? Here are some related questions:

1. How does compounding interest affect deposit calculations?
2. What are the key differences between present and future value in annuities?
3. How can you adjust this model for different interest rates?
4. How do changing withdrawal periods impact the required savings?
5. What happens if withdrawals start immediately after the last deposit?

**Tip**: Always double-check your assumptions (like interest rates or number of periods) when planning financial decisions!

Ralph wants to quit his job and move to Hawaii on December 25, 2015. Once there, he anticipates that he will need to make annual withdrawals of 15000 dollars (starting on December 25, 2016) to supplement his income from working as a cabana boy, and he wants the money to last 10 years (i.e. he'll make 10 withdrawals total). His plan is to make annual deposits, starting on December 25, 2000 and ending on December 25, 2015, into an account paying 7.9 percent interest rate compounded annually. How large should each deposit be for Ralph to realize his goal? Answer correct to 3 decimal places.

Learn how to calculate the annual deposits Ralph needs to make in an account with 7.9% interest to achieve his goal of withdrawing $15,000 annually for 10 years, starting from December 25, 2016. This step-by-step solution covers the use of present and future value formulas for annuities.

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