Introductory Corporate Finance: Time Value of Money Present Value of a Single Sum PV FV 1rⁿ

What This Is

The Present Value of a Single Sum (PV) is a fundamental concept in corporate finance that helps investors and analysts determine the current value of a future cash flow. It's essential for evaluating investment opportunities, calculating the value of a company, and making informed decisions. For example, consider a company that expects to receive $100,000 in 5 years. If the discount rate (r) is 8%, the present value of this future cash flow can be calculated using the formula: PV = FV / (1+r)^n.

Key Formulas & Models

• PV = FV / (1+r)^n – present value of a single sum; used to calculate the current value of a future cash flow.
  • FV: future value (the amount expected to be received in the future)
  • r: discount rate (the rate at which the future cash flow is discounted)
  • n: number of periods (the time between the present and the future cash flow)
• FV = PV × (1+r)^n – future value of a single sum; used to calculate the future value of a current cash flow.
• PV = FV / (1+r)^n (with r as a decimal) – present value of a single sum; used to calculate the current value of a future cash flow.
• FV = PV × (1+r)^n (with r as a decimal) – future value of a single sum; used to calculate the future value of a current cash flow.
• PV = FV / (1+r)^n (with r as a percentage) – present value of a single sum; used to calculate the current value of a future cash flow.
• FV = PV × (1+r)^n (with r as a percentage) – future value of a single sum; used to calculate the future value of a current cash flow.
• PV = FV / (1+r)^n (with n as a fraction of a year) – present value of a single sum; used to calculate the current value of a future cash flow.
• FV = PV × (1+r)^n (with n as a fraction of a year) – future value of a single sum; used to calculate the future value of a current cash flow.

Step-by-Step Calculation

1. Identify the future value (FV) and the number of periods (n).
2. Determine the discount rate (r) as a decimal.
3. Plug the values into the formula: PV = FV / (1+r)^n.
4. Calculate the present value (PV).
5. Round the answer to the nearest dollar or cent.

Common Mistakes

• Mistake: Using a wrong discount rate (e.g., using the company's cost of capital instead of the market's required rate of return).
  • Correction: Use the market's required rate of return as the discount rate.
  • Counterexample: A company has a cost of capital of 10%, but the market's required rate of return is 12%. Using the wrong discount rate would result in an incorrect present value.
• Mistake: Ignoring the time value of money (e.g., assuming that $100,000 in 5 years is worth the same as $100,000 today).
  • Correction: Use the present value formula to account for the time value of money.
  • Counterexample: A company expects to receive $100,000 in 5 years. Using the present value formula, the current value of this future cash flow would be approximately $66,000.
• Mistake: Confusing the present value of a single sum with the present value of an annuity.
  • Correction: Use the correct formula for the present value of an annuity.
  • Counterexample: A company expects to receive $10,000 per year for 5 years. The present value of this annuity would be approximately $41,000, not $66,000.

Exam / CFA Tips

• Be careful when using the present value formula with different discount rates (e.g., using the company's cost of capital instead of the market's required rate of return).
• Make sure to round the answer to the nearest dollar or cent.
• Be prepared to explain the concept of the time value of money and how it affects the present value of a future cash flow.

Quick Practice Problem

A company expects to receive $100,000 in 5 years. If the discount rate (r) is 8%, what is the present value of this future cash flow?

Answer: $66,000 Explanation: Using the present value formula, PV = FV / (1+r)^n, we get PV = $100,000 / (1+0.08)^5 ≈ $66,000.

Last-Minute Cram Sheet

• PV = FV / (1+r)^n: present value of a single sum
• FV = PV × (1+r)^n: future value of a single sum
• r: discount rate (as a decimal)
• n: number of periods (as a fraction of a year)
• ⚠️ In M&M Proposition I (no taxes), firm value is independent of capital structure – but with taxes, value increases with debt due to the interest tax shield
• ⚠️ Using the wrong discount rate can result in an incorrect present value
• ⚠️ Ignoring the time value of money can result in an incorrect present value
• PV = FV / (1+r)^n (with r as a percentage) is equivalent to PV = FV / (1+r/100)^n
• FV = PV × (1+r)^n (with r as a percentage) is equivalent to FV = PV × (1+r/100)^n
• PV = FV / (1+r)^n (with n as a fraction of a year) is equivalent to PV = FV / (1+r)^t, where t is the number of years.