onsider an inheritance that pays the beneficiary an annuity of $500 that begins immediately (an annuity due) and then annually in year 1 through year 26 (for a total of 27 years) with one exception - the payment in year 21, and only 21, is not $500 but $0. Using an interest rate of 9%, determine the present value of this cash flow stream.

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INSTRUCTIONS: Place your answer in dollars and cents without using a dollar sign or a comma. For example, if your answer is one hundred thousand five hundred and seventy then place your answer as 100570. Work all analysis using at least four decimal places of accuracy.

To solve this problem, we need to calculate the present value of the annuity with one irregular payment. The annuity is an annuity due, meaning payments are made at the beginning of each year, except for year 21 when the payment is zero. We are given:

- Payment: $500 (except in year 21, where the payment is $0)
- Total payments: 27 years
- Interest rate: 9% (0.09)
- The payment stream starts immediately, so it’s an annuity due.

### Step-by-Step Process:

1. **Annuity Due Present Value Formula:**
   The present value of an annuity due can be calculated using the formula:

   \[
   PV = P \times \left( 1 + r \right) \times \left( 1 - (1 + r)^{-n} \right) / r
   \]
   where:
   - $ P $ is the payment per period,
   - $ r $ is the interest rate per period,
   - $ n $ is the number of periods.

   However, we will need to account for the fact that there is no payment in year 21. So we will first calculate the present value of a normal annuity due without that exception and then subtract the present value of the missing payment.

2. **Calculate the Present Value of a 27-Year Annuity Due:**
   Using the formula above for an annuity due:
   
   - $ P = 500 $
   - $ r = 0.09 $
   - $ n = 27 $

   Plugging these into the formula:

   \[
   PV_{\text{annuity due}} = 500 \times (1 + 0.09) \times \left( 1 - (1 + 0.09)^{-27} \right) / 0.09
   \]

3. **Present Value of the Missed Payment in Year 21:**
   The missed payment in year 21 will need to be discounted back to the present. Since this is an annuity due, the payment in year 21 is equivalent to year 20 in terms of discounting (because payments are made at the start of the year).

   The formula for the present value of a single payment is:

   \[
   PV_{\text{missed payment}} = P \times (1 + r)^{-20}
   \]
   where:
   - $ P = 500 $,
   - $ r = 0.09 $,
   - The payment is at year 21, so we discount it back 20 years.

   Plugging in the values:

   \[
   PV_{\text{missed payment}} = 500 \times (1 + 0.09)^{-20}
   \]

4. **Final Present Value:**
   The final present value will be the present value of the full annuity due minus the present value of the missed payment.

   \[
   PV_{\text{total}} = PV_{\text{annuity due}} - PV_{\text{missed payment}}
   \]

Let's calculate the values step by step.The present value of the cash flow stream is approximately \$5375.27.

Would you like more details on any part of the calculation? Here are five related questions to explore further:

1. How does an annuity due differ from an ordinary annuity?
2. What is the impact of the interest rate on the present value of an annuity?
3. How would the present value change if the payment missed occurred in a different year?
4. What formula is used to calculate the present value of a single future payment?
5. How can an annuity due be used in real-life financial planning?

**Tip:** When calculating the present value of an irregular cash flow, always adjust for missing or additional payments by discounting them back to the present separately.

Consider an inheritance that pays the beneficiary an annuity of $500 that begins immediately (an annuity due) and then annually in year 1 through year 26 (for a total of 27 years) with one exception - the payment in year 21, and only 21, is not $500 but $0. Using an interest rate of 9%, determine the present value of this cash flow stream.

This problem involves calculating the present value of an annuity due with payments of $500 for 27 years, except for a $0 payment in year 21. Learn how to use time value of money formulas to account for irregular cash flows at an interest rate of 9%.

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