To determine the amount of money that needs to be deposited now, we can break the problem into smaller parts, identify relevant concepts, and then analyze the calculation step by step.

### 1. ### Break Down the Problem
1. Identify the cash flow: $4000 per year for 5 years starting from 3 years from now (i.e., from Year 3 to Year 7).
2. Determine the present value of this cash flow at Year 2.
3. Calculate the amount that needs to be deposited today (Year 0) to achieve this present value.

### 2. ### Relevant Concepts
- The present value (PV) formula is given by:
$$
PV = \frac{C}{(1 + r)^n}
$$
where $ C $ is the cash flow, $ r $ is the interest rate, and $ n $ is the number of periods.

### 3. ### Analysis and Detail
1. **Calculate the present value at Year 2:**
- The cash flows occur at Year 3 to Year 7: $4000 in Years 3, 4, 5, 6, and 7.
- The total present value at Year 2 starts with calculating each cash flow's present value:

$$
PV = \sum_{i=1}^{5} \frac{C}{(1 + r)^{t_i}}
$$
Where $ C = 4000 $ and $ t_i $ represents the years from Year 2:
- For Year 3: $ t_1 = 1 $ (PV = $ \frac{4000}{(1 + r)^1} $)
- For Year 4: $ t_2 = 2 $ (PV = $ \frac{4000}{(1 + r)^2} $)
- For Year 5: $ t_3 = 3 $ (PV = $ \frac{4000}{(1 + r)^3} $)
- For Year 6: $ t_4 = 4 $ (PV = $ \frac{4000}{(1 + r)^4} $)
- For Year 7: $ t_5 = 5 $ (PV = $ \frac{4000}{(1 + r)^5} $)

Thus,
$$
PV = 4000\left(\frac{1}{(1 + r)^1} + \frac{1}{(1 + r)^2} + \frac{1}{(1 + r)^3} + \frac{1}{(1 + r)^4} + \frac{1}{(1 + r)^5}\right)
$$

2. **Calculate the deposit today:**
- Now, we also need to calculate the present value of the amount that needs to be deposited today (Year 0) to get the present value calculated at Year 2. The formula is:
$$
Deposit = \frac{PV}{(1 + r)^2}
$$

### 4. ### Verify and Summarize
- We need a value for $ r $ (the interest rate) to get numerical answers. If you provide that, I can plug it into the formulas above and simplify the results.

### Final Answer
Using the formulas above, the amount to be deposited today can be found once we know the interest rate $ r $. Calculating the present value of the series of cash flows and then discounting will yield the final amount.

Question

To determine the amount of money that needs to be deposited now, we can break the problem into smaller parts, identify relevant concepts, and then analyze the calculation step by step.

### 1. ### Break Down the Problem
1. Identify the cash flow: $4000 per year for 5 years starting from 3 years from now (i.e., from Year 3 to Year 7).
2. Determine the present value of this cash flow at Year 2.
3. Calculate the amount that needs to be deposited today (Year 0) to achieve this present value.

### 2. ### Relevant Concepts
- The present value (PV) formula is given by:
  $$
  PV = \frac{C}{(1 + r)^n}
  $$
  where $ C $ is the cash flow, $ r $ is the interest rate, and $ n $ is the number of periods.

### 3. ### Analysis and Detail
1. **Calculate the present value at Year 2:** 
   - The cash flows occur at Year 3 to Year 7: $4000 in Years 3, 4, 5, 6, and 7.
   - The total present value at Year 2 starts with calculating each cash flow's present value:

$$
PV = \sum_{i=1}^{5} \frac{C}{(1 + r)^{t_i}}
$$
Where $ C = 4000 $ and $ t_i $ represents the years from Year 2:
- For Year 3: $ t_1 = 1 $ (PV = $ \frac{4000}{(1 + r)^1} $)
- For Year 4: $ t_2 = 2 $ (PV = $ \frac{4000}{(1 + r)^2} $)
- For Year 5: $ t_3 = 3 $ (PV = $ \frac{4000}{(1 + r)^3} $)
- For Year 6: $ t_4 = 4 $ (PV = $ \frac{4000}{(1 + r)^4} $)
- For Year 7: $ t_5 = 5 $ (PV = $ \frac{4000}{(1 + r)^5} $)

Thus,
$$
PV = 4000\left(\frac{1}{(1 + r)^1} + \frac{1}{(1 + r)^2} + \frac{1}{(1 + r)^3} + \frac{1}{(1 + r)^4} + \frac{1}{(1 + r)^5}\right)
$$

2. **Calculate the deposit today:** 
   - Now, we also need to calculate the present value of the amount that needs to be deposited today (Year 0) to get the present value calculated at Year 2. The formula is:
$$
Deposit = \frac{PV}{(1 + r)^2}
$$

### 4. ### Verify and Summarize
- We need a value for $ r $ (the interest rate) to get numerical answers. If you provide that, I can plug it into the formulas above and simplify the results.

### Final Answer
Using the formulas above, the amount to be deposited today can be found once we know the interest rate $ r $. Calculating the present value of the series of cash flows and then discounting will yield the final amount.

Knowee AI · Accepted Answer