Definition
A perpetual annuity (or perpetuity) is a financial instrument where the recipient receives or pays a constant annual amount indefinitely. Even though the term “annuity” suggests annual payments, it can also apply to other regular intervals. The primary feature of a perpetuity is its infinite duration.
The present value (PV) of a perpetual annuity is calculated with the following formula:
\[ P = \frac{a}{i} \]
Where:
- \( P \) = Present value of the perpetuity
- \( a \) = Annual sum (or periodic payment)
- \( i \) = Interest rate (expressed as a decimal)
Examples
-
Stock Dividends: If a company promises to pay a fixed dividend of $5 annually forever, and the interest rate is 5%, the present value of this perpetuity is:
\[ P = \frac{5}{0.05} = $100 \]
-
Scholarships: An endowment fund that pays $10,000 annually forever to support a scholarship, with an interest rate of 4%, has a present value of:
\[ P = \frac{10000}{0.04} = $250,000 \]
Frequently Asked Questions (FAQs)
1. What is a perpetual annuity?
A perpetual annuity is a series of identical payments made at regular intervals that continue indefinitely.
2. How do you calculate the present value of a perpetuity?
The present value (PV) of a perpetuity is calculated by dividing the annual payment amount by the interest rate: \[ P = \frac{a}{i} \].
3. What is an example of a perpetuity in real life?
A common example is a perpetually paying dividend stock, where shareholders receive an annual dividend indefinitely.
4. Can the interest rate change in a perpetuity?
For the given PV formula, it is assumed that the interest rate remains constant. However, in the real world, fluctuating interest rates can affect the valuation.
5. What’s the difference between an annuity and a perpetuity?
An annuity is a series of payments made for a finite period, while a perpetuity continues indefinitely.
- Annuity: A series of payments made at equal intervals. Unlike perpetuities, annuities last for a specific period.
- Present Value (PV): The current worth of a future sum of money, given a specific rate of return.
- Discount Rate: The interest rate used to discount future cash flows to their present values.
Online References
Suggested Books for Further Studies
- “Principles of Corporate Finance” by Richard A. Brealey, Stewart C. Myers, and Franklin Allen - A comprehensive guide to fundamental finance principles, including annuities and perpetuities.
- “Financial Management: Theory & Practice” by Eugene F. Brigham and Michael C. Ehrhardt - Detailed insights into financial management concepts, inclusive of various financial instruments.
Accounting Basics: “Perpetual Annuity” Fundamentals Quiz
### What is a perpetual annuity?
- [x] An instrument that pays or receives a constant amount annually indefinitely.
- [ ] An annuity that lasts for a specific number of years.
- [ ] A sum paid periodically for less than a year.
- [ ] A temporary financial arrangement.
> **Explanation:** A perpetual annuity, or perpetuity, is characterized by the indefinite continuation of annual (or periodic) payments.
### How do you calculate the present value of a perpetuity that pays $100 annually with an interest rate of 5%?
- [x] $2000
- [ ] $20
- [ ] $500
- [ ] $1000
> **Explanation:** The present value (P) is calculated as: \\[ P = \frac{100}{0.05} = $2000 \\].
### If the annual sum of a perpetuity is $50 and the interest rate is 2%, what is the present value?
- [x] $2500
- [ ] $25
- [ ] $1000
- [ ] $2580
> **Explanation:** Using the formula for PV: \\[ P = \frac{50}{0.02} = $2500 \\].
### Which of the following correctly distinguishes a perpetuity from a regular annuity?
- [x] A perpetuity makes payments indefinitely, whereas a regular annuity makes payments for a fixed period.
- [ ] A perpetuity has variable payments while a regular annuity does not.
- [ ] Both make payments indefinitely.
- [ ] Both make payments for a fixed period.
> **Explanation:** A perpetuity continues indefinitely, while a regular annuity makes payments for a predetermined period.
### What would cause the present value of a perpetuity to increase?
- [x] A decrease in the interest rate.
- [ ] An increase in the interest rate.
- [ ] A decrease in the annual payment.
- [ ] An unchanged interest rate.
> **Explanation:** Since PV is inversely proportional to the interest rate, a lower interest rate results in a higher present value.
### What is an example of a perpetuity?
- [x] A stock paying a fixed dividend indefinitely.
- [ ] Mortgage payments.
- [ ] Rental payments under a lease.
- [ ] A bond with a maturity date.
> **Explanation:** A perpetuity involves payments that continue indefinitely, such as eternal dividend payments from a stock.
### You have a perpetuity paying $1,000 annually, with an interest rate of 8%. What is the present value?
- [x] \$12,500
- [ ] \$1,000
- [ ] \$8,000
- [ ] \$80,000
> **Explanation:** The present value is obtained by \\[ P = \frac{1000}{0.08} = \$12,500 \\].
### Which financial concept is assumed constant in the perpetuity formula?
- [x] Interest rate.
- [ ] Annual sum.
- [ ] Inflation rate.
- [ ] Real estate value.
> **Explanation:** The formula for calculating the present value of a perpetuity presumes a constant interest rate.
### What happens to the present value of a perpetuity if the annual sum stays the same and the interest rate increases?
- [ ] The present value increases.
- [x] The present value decreases.
- [ ] The present value fluctuates randomly.
- [ ] The present value remains unchanged.
> **Explanation:** A higher interest rate reduces the present value of perpetual payments, as the denominator in the formula increases.
### In what scenario would the concept of perpetuity be practically applicable?
- [x] Setting up an endowment fund paying out regular scholarships.
- [ ] Leasing an office space.
- [ ] Taking a fixed-term loan.
- [ ] Issuing a debenture with a maturity period.
> **Explanation:** Perpetuity is well-suited for schemes like endowment funds where the payment continues indefinitely to support a specific purpose.
Thank you for exploring the concept of perpetual annuities and challenging yourself with our quiz. This reinforces your understanding and application in real-world financial scenarios. Keep up the excellent learning!
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