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

What This Is

The Future Value of a Single Sum (FV) formula is a fundamental concept in corporate finance, used to calculate the future value of a present value (PV) investment. It's essential for evaluating investment opportunities, estimating future cash flows, and making informed decisions. For instance, if you invest $10,000 today at a 5% annual interest rate for 5 years, the future value would be $11,518.19.

Key Formulas & Models

• FV = PV × (1+r)^n – future value of a single sum; used to calculate future cash flows.
  • PV: present value (initial investment)
  • r: annual interest rate (as a decimal)
  • n: number of years
• PV = FV / (1+r)^n – present value of a single sum; used to calculate initial investment.
• FV = PV × (1+r)^n × (1+g)^n – future value of a growing annuity; used to calculate future cash flows with growth.
  • g: growth rate (as a decimal)
• PV = FV / (1+r)^n × (1+g)^n – present value of a growing annuity; used to calculate initial investment with growth.
• FV = PMT × (((1+r)^n - 1) / r) – future value of an annuity; used to calculate future cash flows from a series of payments.
  • PMT: periodic payment
• PV = FV / (((1+r)^n - 1) / r) – present value of an annuity; used to calculate initial investment from a series of payments.
• FV = PV × (1+r)^n × (1+i)^n – future value of a single sum with inflation; used to calculate future cash flows with inflation.
  • i: inflation rate (as a decimal)

Step-by-Step Calculation

1. Identify the present value (PV) and the desired future value (FV) of the investment.
2. Determine the annual interest rate (r) and the number of years (n).
3. Apply the FV formula: FV = PV × (1+r)^n.
4. If the investment grows at a rate (g), apply the FV formula for a growing annuity: FV = PV × (1+r)^n × (1+g)^n.
5. If the investment is a series of payments, apply the FV formula for an annuity: FV = PMT × (((1+r)^n - 1) / r).
6. If the investment is affected by inflation, apply the FV formula with inflation: FV = PV × (1+r)^n × (1+i)^n.

Common Mistakes

• Mistake: Using the wrong interest rate (e.g., using the company's cost of debt instead of the opportunity cost of capital).
  • Correction: Use the opportunity cost of capital (e.g., WACC) as the discount rate.
• Mistake: Ignoring compounding frequency (e.g., monthly or quarterly compounding).
  • Correction: Adjust the interest rate and number of periods accordingly.
• Mistake: Confusing the time value of money with the risk-free rate.
  • Correction: Understand that the time value of money is the return on investment, not the risk-free rate.
• Mistake: Failing to account for inflation.
  • Correction: Adjust the interest rate and future value calculation to account for inflation.

Exam / CFA Tips

• Be prepared to apply the FV formula in different scenarios, such as calculating future cash flows, estimating initial investment, and evaluating investment opportunities.
• Understand the differences between the FV formula and other time value of money formulas, such as the PV formula.
• Be aware of common mistakes, such as using the wrong interest rate or ignoring compounding frequency.

Quick Practice Problem

A company has EBIT of $10M, interest $2M, and tax 25%. Compute the debt-free leverage (DFL) ratio.

Answer: DFL = (EBIT - Interest) / EBIT = ($10M - $2M) / $10M = 0.80

Explanation: The DFL ratio measures the company's ability to service its debt without relying on interest payments.

Last-Minute Cram Sheet

• FV = PV × (1+r)^n
• PV = FV / (1+r)^n
• FV = PV × (1+r)^n × (1+g)^n
• PV = FV / (1+r)^n × (1+g)^n
• FV = PMT × (((1+r)^n - 1) / r)
• PV = FV / (((1+r)^n - 1) / r)
• FV = PV × (1+r)^n × (1+i)^n
• Opportunity cost of capital (WACC) is the discount rate for investment decisions.
• Compounding frequency affects the interest rate and number of periods.
• Inflation affects the interest rate and future value calculation.
• ⚠️ 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.
• ⚠️ The risk-free rate is not the same as the opportunity cost of capital.
• ⚠️ The time value of money is the return on investment, not the risk-free rate.