You have just read an advertisement stating, “Pay us $100 per year for 10 years, and we will pay you $100 per year thereafter in perpetuity.” If this is a fair deal, what is the rate of interest?

To solve this problem, we need to determine the interest rate that makes the present value of the payments equal.

### Step 1: Understand the cash flows
- You pay $100 per year for 10 years.
- Starting from year 11, you receive $100 per year in perpetuity.

### Step 2: Present Value of Payments
The present value (PV) of the payments you make for the first 10 years can be calculated as the sum of a 10-year annuity:

\[
PV_{\text{payments}} = 100 \times \frac{1 - (1 + r)^{-10}}{r}
\]

where $r$ is the interest rate.

### Step 3: Present Value of Receipts
The present value of receiving $100 per year in perpetuity starting from year 11 can be calculated as:

\[
PV_{\text{receipts}} = \frac{100}{r} \times \frac{1}{(1 + r)^{10}}
\]

This is because the present value of a perpetuity starting at year 11 is the value of a perpetuity at year 10, discounted back 10 years to the present.

### Step 4: Set Up the Equation
For the deal to be fair, the present value of the payments must equal the present value of the receipts:

\[
100 \times \frac{1 - (1 + r)^{-10}}{r} = \frac{100}{r} \times \frac{1}{(1 + r)^{10}}
\]

### Step 5: Simplify and Solve for $r$

We can cancel out the 100s:

\[
\frac{1 - (1 + r)^{-10}}{r} = \frac{1}{r(1 + r)^{10}}
\]

This equation needs to be solved numerically since it is a transcendental equation.

### Step 6: Numerical Solution

This equation can be solved using numerical methods like Newton's method or by using financial calculators or software to find the interest rate $r$.

The solution to this equation gives us the interest rate, which is approximately $ r \approx 7.18\% $.

### Final Answer

The interest rate that makes this a fair deal is approximately **7.18%**.

Would you like more details on the steps or have any questions?

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Here are 5 related questions:
1. How do we calculate the present value of a perpetuity?
2. What methods can be used to solve transcendental equations numerically?
3. How does the interest rate impact the value of annuities?
4. What is the difference between an ordinary annuity and a perpetuity?
5. How would the solution change if payments were made semi-annually instead of annually?

**Tip:** Understanding the time value of money is crucial when analyzing investments and financial deals.

Discover how to calculate the interest rate that makes a financial deal fair by equating the present value of an annuity and a perpetuity. This guide walks through detailed steps, including key formulas and numerical methods, making it perfect for finance students and enthusiasts.

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