Elizabeth has a choice between two equivalent investments.
Option A: Invest $1,000 and receive $100 at the end of every two years forever.
Option B: Invest $1,000 and receive $170 at the end of every three years forever.
Define i_A as the effective annual interest rate for investment A and i_B as the effective annual interest rate for investment B. 
calculate i_A/i_B

We are given two investment options, and we need to find the ratio of the effective annual interest rates, $ \frac{i_A}{i_B} $, for these investments. 

### Option A: 
- Invest $1,000 and receive $100 every 2 years forever.

This means the cash flow is $100 every 2 years, and we want to determine the effective annual interest rate $ i_A $.

The present value of the perpetuity in Option A can be written as:
\[
PV = \frac{100}{(1+i_A)^2} + \frac{100}{(1+i_A)^4} + \frac{100}{(1+i_A)^6} + \dots
\]
This is a perpetuity with payments every 2 years. The present value formula for such a perpetuity with a payment of $100 every 2 years is:
\[
PV = \frac{100}{(1+i_A)^2 - 1}
\]
Since the initial investment is $1,000, we equate this to the present value of the perpetuity:
\[
1000 = \frac{100}{(1+i_A)^2 - 1}
\]

### Option B: 
- Invest $1,000 and receive $170 every 3 years forever.

Similarly, the present value of the perpetuity in Option B can be written as:
\[
PV = \frac{170}{(1+i_B)^3} + \frac{170}{(1+i_B)^6} + \frac{170}{(1+i_B)^9} + \dots
\]
The present value formula for this perpetuity is:
\[
PV = \frac{170}{(1+i_B)^3 - 1}
\]
Again, the initial investment is $1,000, so we equate this to the present value:
\[
1000 = \frac{170}{(1+i_B)^3 - 1}
\]

### Step 1: Solve for $ i_A $
We now solve for $ i_A $ from the equation:
\[
1000 = \frac{100}{(1+i_A)^2 - 1}
\]

### Step 2: Solve for $ i_B $
We also solve for $ i_B $ from the equation:
\[
1000 = \frac{170}{(1+i_B)^3 - 1}
\]

### Step 3: Calculate $ \frac{i_A}{i_B} $

Let's solve these equations to find the effective interest rates and their ratio. I'll do that now.The solutions for the effective interest rates $ i_A $ and $ i_B $ are as follows:

- $ i_A = \frac{-1 + \sqrt{110}}{10} $
- $ i_B = \frac{-1 + \sqrt[3]{1170}}{10} $

The ratio $ \frac{i_A}{i_B} $ is:

\[
\frac{i_A}{i_B} = \frac{-1 + \sqrt{110}}{-1 + \sqrt[3]{1170}}
\]

Let's compute the numerical value of this ratio next.The ratio of the effective annual interest rates $ \frac{i_A}{i_B} $ is approximately 0.908. 

This means that the interest rate for investment A is about 90.8% of the interest rate for investment B.

Would you like any further clarification or additional details?

Here are 5 related questions to explore:
1. How do perpetuities work in financial mathematics?
2. What are the implications of choosing between different cash flow frequencies?
3. How does the present value of a perpetuity depend on the interest rate?
4. How can the effective annual interest rate be compared across different investments?
5. What are other financial instruments that use similar perpetuity formulas?

**Tip:** When comparing investments, consider not only the interest rates but also the timing and frequency of cash flows, as these can affect the overall value over time.

Elizabeth has a choice between two equivalent investments. Option A: Invest $1,000 and receive $100 at the end of every two years forever. Option B: Invest $1,000 and receive $170 at the end of every three years forever. Define i_A as the effective annual interest rate for investment A and i_B as the effective annual interest rate for investment B. Calculate i_A/i_B.

Learn how to calculate the ratio of effective annual interest rates for two perpetuity investments with different cash flow frequencies. Investment A involves $100 every 2 years, and Investment B involves $170 every 3 years. This solution explores perpetuity formulas and present value concepts, ideal for undergraduate finance students.

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