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Chapter 5: Problem 31

The following exercises explore applications of annuities. Calculate the annual payouts \(C\) to be given perpetually on annuities having present value \(\$ 100,000\) assuming respective interest rates of \(r=0.03, r=0.05\), and \(r=0.07\).

Short Answer

Expert verified
Payments are $3,000 for r=0.03, $5,000 for r=0.05, and $7,000 for r=0.07.

Step by step solution

01

Understanding the Problem

We need to determine the constant annual payout \( C \) from an annuity that lasts indefinitely (a perpetuity), given different interest rates. The present value of the annuity is provided as \( P = \$100,000 \).
02

Perpetuity Formula Introduction

For a perpetuity, the annual payment \( C \) can be calculated using the formula: \[ C = P \times r \], where \( P \) is the present value of the annuity and \( r \) is the interest rate.
03

Calculate Payout for Interest Rate 0.03

Plug the interest rate \( r = 0.03 \) into the formula: \( C = 100,000 \times 0.03 \). Thus, \( C = 3,000 \).
04

Calculate Payout for Interest Rate 0.05

Plug the interest rate \( r = 0.05 \) into the formula: \( C = 100,000 \times 0.05 \). Thus, \( C = 5,000 \).
05

Calculate Payout for Interest Rate 0.07

Plug the interest rate \( r = 0.07 \) into the formula: \( C = 100,000 \times 0.07 \). Thus, \( C = 7,000 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perpetuity
A perpetuity is a type of financial product that pays out a fixed amount of money forever. It's like an annuity that never ends. When discussing perpetuities, it is important to understand that they provide a regular stream of infinite payments. The key characteristic of a perpetuity is its never-ending payments. This is why it is often used in theoretical finance to model situations where cash flows are expected to last indefinitely. Given its endless nature, the value of a perpetuity is calculated using a specific formula, which helps determine the amount you receive every period. The formula to calculate the annual payment of a perpetuity is: \[ C = P \times r \] Where:
  • \( C \) is the annual cash payment.
  • \( P \) is the present value of the perpetuity.
  • \( r \) is the interest rate.
Perpetuities are fundamental in finance because they offer a simple way to understand the concept of the time value of money over an infinite horizon. Despite their theoretical nature, they form the basis for many practical financial valuation techniques.
Interest Rates
Interest rates are a critical factor in determining the amount you can earn from a financial investment like a perpetuity. They represent the cost of borrowing money or the gain from lending money. When it comes to perpetuity calculations, the interest rate helps determine how much you will be paid each year. A higher interest rate will lead to a higher annual payment, while a lower interest rate will result in a smaller payment. The effect of interest rates is straightforwardly seen in the perpetuity formula: \[ C = P \times r \]This means interest rate \( r \) directly affects the payout \( C \). Understanding interest rates is crucial for making informed financial decisions, as they influence the returns you can expect from your investments and affect your borrowing costs. In the context of perpetuities, they help in understanding how rate changes can impact the income from the perpetuity over time.
Present Value
The present value is a financial concept that represents the current worth of a sum of money that will be received in the future. In the context of perpetuities, the present value is the amount of money required today that will fund the perpetuity, allowing for endless payments in the future. The idea behind present value is that money available today is worth more than the same amount in the future due to its potential earning capacity. This is central to the concept of the time value of money. In perpetuity calculations, the present value \( P \) is what you initially invest or deposit to get the perpetual payments. With this amount, the formula to calculate the annual payment is:\[ C = P \times r \]
  • \( P \) is the present value (in our case, \$100,000), which ensures the stable cash flow.
  • The formula shows how the present value must adjust with changing interest rates to maintain the same financial outcome.
The notion of present value is essential for comparing investments, evaluating investment opportunities, and making decisions based on money's future purchasing power.

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