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The problem in the image states:

The following two investment options are viewed under an annual effective interest rate of \(i\):
1. Investment A is a 20-year zero-coupon bond which matures at par value 320.
2. Investment B is a perpetuity-immediate paying an annual payment starting with 8 and having each successive payment increase by 8% from the previous payment.

If the elasticity of each investment is 8, then find the value of \(i\).

Give your answer as a decimal rounded to two places (i.e., XX.XX). If your answer was 3%, 2.94%, you would answer 2.94.

To solve this problem, we'll start by analyzing the details of the investments and their elasticity:

1. **Investment A (Zero-coupon bond)**:
   - Par value at maturity = 320
   - Duration = 20 years
   - Let \( P \) be the present value of the bond.
   - \( P = \frac{320}{(1+i)^{20}} \)

2. **Investment B (Perpetuity-immediate)**:
   - Starting payment = 8
   - Each payment increases by 8% annually.
   - The payment at time \( n \) is \( 8(1.08)^{n-1} \).
   - The present value of a perpetuity-immediate with payments increasing by a geometric progression is given by:
     \[
     PV = \frac{8}{i - 0.08}
     \]

The elasticity \( \epsilon \) of each investment is given by:
   \[
   \epsilon = - \frac{i}{PV} \cdot \frac{dPV}{di}
   \]

Given that the elasticity for both investments is 8, we will use this to find the value of \(i\).

Let's first calculate the present value for each investment.

### Investment A:
\[ P_A = \frac{320}{(1+i)^{20}} \]

### Investment B:
\[ P_B = \frac{8}{i - 0.08} \]

Using the elasticity condition \( \epsilon = 8 \), we need to solve for \(i\).

Let's start with the bond (Investment A):
\[ 8 = -\frac{i}{\frac{320}{(1+i)^{20}}} \cdot \frac{d}{di}\left(\frac{320}{(1+i)^{20}}\right) \]

Next, for the perpetuity (Investment B):
\[ 8 = -\frac{i}{\frac{8}{i - 0.08}} \cdot \frac{d}{di}\left(\frac{8}{i - 0.08}\right) \]

Let's solve these equations to find the value of \(i\).The solutions for the effective annual interest rate \(i\) from both investments are:

1. **Investment A (20-year zero-coupon bond)**:
   - \(i = \frac{2}{3} \approx 0.6667\)

2. **Investment B (Perpetuity-immediate)**:
   - \(i \approx 0.0914\)

Since both investments should yield the same interest rate \(i\), we need to determine which one makes sense in the given context. Given the problem structure and the realistic interest rates, the plausible value here is \(i \approx 0.0914\).

Therefore, the value of \(i\) to two decimal places is:

\[ i = 0.0914 \approx 0.09 \]

Would you like further details on the steps, or do you have any additional questions?

Here are 8 related questions to consider:
1. What is the general formula for the present value of a zero-coupon bond?
2. How do you derive the present value of a perpetuity with increasing payments?
3. What is the definition of elasticity in financial terms?
4. How does the interest rate affect the present value of future payments?
5. Can you explain the concept of perpetuity-immediate in more detail?
6. What are the key differences between a zero-coupon bond and a perpetuity?
7. How would the solutions change if the bond had semi-annual compounding?
8. What other factors could influence the choice between these two investments?

**Tip:** Always double-check your calculations and assumptions, especially in finance, where small errors can lead to significant differences.

Learn how to determine the effective annual interest rate (i) from two investments: a 20-year zero-coupon bond and a perpetuity-immediate with increasing payments. Detailed step-by-step solution included. Suitable for college-level students studying finance and mathematics.

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