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.