Intro to Finance: Time Value of Money - Perpetuities PV PMT r

What This Is

A perpetuity is a type of investment that generates a constant cash flow forever. It's essential in finance to understand perpetuities because they help us value investments with long-term cash flows, such as bonds, dividend-paying stocks, and real estate investment trusts (REITs). For example, consider a $1,000 bond with a 5% coupon rate. If we assume the bond will pay the coupon rate forever, its present value (PV) can be calculated using the formula PV = PMT / r.

Key Formulas & Symbols

• PV = PMT / r where PV = present value, PMT = perpetual cash flow, r = periodic interest rate.
• PMT = PV × r where PMT = perpetual cash flow, PV = present value, r = periodic interest rate.
• r = PMT / PV where r = periodic interest rate, PMT = perpetual cash flow, PV = present value.
• PMT = FV × r where PMT = perpetual cash flow, FV = face value, r = periodic interest rate.
• FV = PMT / r where FV = face value, PMT = perpetual cash flow, r = periodic interest rate.
• PV = FV / (1 + r) where PV = present value, FV = face value, r = periodic interest rate.
• PMT = FV × r / (1 + r) where PMT = perpetual cash flow, FV = face value, r = periodic interest rate.

Step-by-Step Calculation

1. Determine the perpetual cash flow (PMT) from the investment.
2. Identify the periodic interest rate (r) associated with the investment.
3. Plug the values into the formula PV = PMT / r to calculate the present value (PV).
4. If necessary, convert the present value (PV) to a face value (FV) using the formula FV = PV × (1 + r).

Common Mistakes

• Mistake: Using a different discount rate for different cash flows in a perpetuity.
• Correction: Use a consistent discount rate for all cash flows in a perpetuity.
• Mistake: Forgetting to account for the time value of money when calculating the present value of a perpetuity.
• Correction: Always use the present value formula PV = PMT / r to calculate the present value of a perpetuity.
• Mistake: Assuming a perpetuity will last forever when the cash flows are actually finite.
• Correction: Always check the assumptions behind the perpetuity and consider the possibility of finite cash flows.

Exam / CFA Tips

• Tip: Be careful with the wording of the question. If it asks for the present value of a perpetuity, use the formula PV = PMT / r. If it asks for the face value, use the formula FV = PV × (1 + r).
• Tip: Make sure to identify the periodic interest rate (r) associated with the perpetuity. This is often the key to solving the problem.
• Tip: Be aware of the assumptions behind the perpetuity. If the cash flows are finite, you may need to use a different formula or approach.

Quick Practice Problem

A company is considering investing in a bond with a face value of $1,000 and a 5% coupon rate. If the discount rate is 8%, what is the present value of the bond?

Answer: $12,500. Explanation: Using the formula PV = PMT / r, we get PV = $50 / 0.08 = $625. However, this is the present value of the first coupon payment. To find the present value of the entire bond, we need to use the formula PV = FV / (1 + r) = $1,000 / (1 + 0.08) = $1,000 / 1.08 = $925. Then, we add the present value of the first coupon payment to get the total present value: $925 + $625 = $1,550. However, this is incorrect. We need to use the formula PV = PMT / r to find the present value of the bond. The correct answer is $12,500.

Last-Minute Cram Sheet

• The formula for the present value of a perpetuity is PV = PMT / r.
• The formula for the face value of a perpetuity is FV = PV × (1 + r).
• The periodic interest rate (r) is the key to solving perpetuity problems.
• Always check the assumptions behind the perpetuity.
• The perpetuity formula assumes the cash flows are constant and will last forever.
• The perpetuity formula assumes the discount rate is constant and will not change over time.
• The perpetuity formula assumes the cash flows are not affected by inflation.
• The perpetuity formula assumes the cash flows are not affected by taxes.
• The perpetuity formula assumes the cash flows are not affected by credit risk.