You are given an n-year annuity-due of 1 per year plus a final payment of X at time n + k - 1, for 0 < k < 1. The present value of all payments can be simplified to the following: Present Value = (1-(v^(n+k)))/d Find X. useful information d=i/(1+i) and v=1(1+i)

To find \( X \), we begin by analyzing the problem step by step. We are given an annuity-due of 1 per year for \( n \) years, plus a final payment of \( X \) at time \( n + k - 1 \), where \( 0 < k < 1 \). The present value of all payments can be simplified to:

\[
PV = \frac{1 - v^{n+k}}{d}
\]

We are tasked with finding \( X \), the final payment.

### Step 1: Define terms
- \( d = \frac{i}{1+i} \) is the discount rate.
- \( v = \frac{1}{1+i} \) is the discount factor.
- The annuity-due has payments of 1 per year at the start of each year for \( n \) years.

### Step 2: Present value of the annuity-due
The present value of an annuity-due with payments of 1 for \( n \) years can be written as:

\[
PV_{\text{annuity}} = \frac{1 - v^n}{d}
\]

### Step 3: Present value of the final payment \( X \)
The final payment \( X \) occurs at time \( n+k-1 \), and its present value is discounted back to time 0 using the discount factor \( v^{n+k-1} \). Thus, the present value of the final payment is:

\[
PV_{\text{final}} = X \cdot v^{n+k-1}
\]

### Step 4: Total present value
The total present value of all payments, including the annuity-due and the final payment, is the sum of their present values:

\[
PV = \frac{1 - v^n}{d} + X \cdot v^{n+k-1}
\]

### Step 5: Set up the equation
We are told that the present value of all payments is:

\[
PV = \frac{1 - v^{n+k}}{d}
\]

So, equating the two expressions for \( PV \), we get:

\[
\frac{1 - v^n}{d} + X \cdot v^{n+k-1} = \frac{1 - v^{n+k}}{d}
\]

### Step 6: Solve for \( X \)
First, multiply both sides of the equation by \( d \) to eliminate the denominators:

\[
1 - v^n + X \cdot d \cdot v^{n+k-1} = 1 - v^{n+k}
\]

Now, subtract \( 1 - v^n \) from both sides:

\[
X \cdot d \cdot v^{n+k-1} = v^n - v^{n+k}
\]

Finally, solve for \( X \):

\[
X = \frac{v^n - v^{n+k}}{d \cdot v^{n+k-1}}
\]

### Final Expression for \( X \):

\[
X = \frac{v^n - v^{n+k}}{d \cdot v^{n+k-1}}
\]

This is the expression for \( X \), which is the final payment of the annuity.

---

Would you like a more detailed explanation of any step? Here are five related questions to explore further:

1. How does the annuity-due differ from an ordinary annuity in terms of present value?
2. What happens to the value of \( X \) as \( k \) approaches 0 or 1?
3. How would you calculate \( X \) if you were given specific values for \( n \), \( k \), and \( i \)?
4. What is the significance of the discount factor \( v \) in annuity calculations?
5. How does changing the interest rate \( i \) affect the present value of the annuity?

**Tip:** When calculating present values, always ensure you're clear about the timing of payments—whether they occur at the start (annuity-due) or end (ordinary annuity) of each period.

You are given an n-year annuity-due of 1 per year plus a final payment of X at time n + k - 1, for 0 < k < 1. The present value of all payments can be simplified to the following: Present Value = (1-(v^(n+k)))/d. Find X. Useful information: d=i/(1+i) and v=1/(1+i).

Learn how to determine the final payment X for an n-year annuity-due with a payment structure that includes a final payment at time n + k - 1. This guide provides a detailed step-by-step solution, using concepts like present value, discount factors, and annuity valuation. Suitable for college-level finance or actuarial students.

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