Math Problem Statement
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?
Solution
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:
where 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:
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:
Step 5: Simplify and Solve for
We can cancel out the 100s:
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 .
The solution to this equation gives us the interest rate, which is approximately .
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?
Here are 5 related questions:
- How do we calculate the present value of a perpetuity?
- What methods can be used to solve transcendental equations numerically?
- How does the interest rate impact the value of annuities?
- What is the difference between an ordinary annuity and a perpetuity?
- 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.
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Math Problem Analysis
Mathematical Concepts
Time Value of Money
Annuities
Perpetuities
Present Value Calculation
Formulas
Present Value of an Annuity: PV = C × (1 - (1 + r)^-n) / r
Present Value of a Perpetuity: PV = C / r
Discounting a Perpetuity to a Future Value: PV = (C / r) × (1 / (1 + r)^n)
Theorems
Annuity Present Value Formula
Perpetuity Present Value Formula
Suitable Grade Level
Undergraduate Finance or Advanced High School (Grades 11-12)
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