Lecture on Discounted Cash Flows: Annuities and Perpetuities

Introduction

• This lecture covers the arithmetic of discounted cash flows focusing on annuities and perpetuities.
• Recommended to download lecture notes from opentuition.com for better understanding.

Annuities

• Definition: An annuity is an equal cash flow each year for a set number of years.
• Example: A machine costs £45,000 and generates £8,000 annually for 8 years at a 15% discount rate.
  • Present Value (PV) calculation can be done individually for each year using discount factors.
  • Annuity tables simplify the process by providing the total discount factor for the number of years required.
  • For 8 years at 15%, the annuity discount factor is 4.487.
  • Net Present Value (NPV) is calculated as -£9,104 (negative NPV means rejection of investment).
• Annuity Tables: Used for equal cash flows each year; not applicable if cash flows vary.
• Caution: Annuity tables provide factors from year 1 to n; adjust for different start years.
  • Example: First cash flow in year 4 for 10 years, calculate using 13-year total minus 3-year total.

Perpetuities

• Definition: An ultimate annuity with equal cash flows each year indefinitely.
• Example: A machine costing £100,000 generates £12,000 in perpetuity at a 10% rate.
  • Use formula: Present Value = Cash Flow / r (where r is the interest rate).
  • For 10%, PV = £12,000 / 0.10 = £120,000.
  • NPV = £20,000 (positive NPV means acceptance of investment).
• Logic Behind Formula: Investing a lump sum at the bank to generate the same cash flow annually.

Applying Perpetuities to Delayed Cash Flows

• Example: £18,000 first received in year 5 in perpetuity at 5% interest.
  • Two Calculation Methods:
    1. Subtract total factor for the first 4 years from the perpetuity factor.
    2. Adjust using present value discount factor for the delay.
  • Both methods aim to find the present value by adjusting the perpetuity factor.

Conclusion

• Mastery of annuities and perpetuities is essential for discounted cash flow calculations.
• Use present value tables or annuity tables appropriately for individual flows or equal annual flows.
• Perpetuities require understanding and application of the formula for cash flows lasting indefinitely.