Introductory Corporate Finance: Time Value of Money - Annuities Ordinary vs. Annuity, Due FV and PV Formulas

What This Is

An annuity is a series of equal cash flows received or paid at regular intervals. In corporate finance, annuities are crucial for calculating present value (PV) and future value (FV) of cash flows, which is essential for investment decisions, financing choices, and valuation. For instance, consider a company that plans to invest $100,000 in a project with expected annual cash flows of $20,000 for 5 years. Using annuity formulas, we can calculate the PV of these cash flows and determine the project's net present value (NPV).

Key Formulas & Models

• PV = PMT × [(1 - (1 + r)^(-n)) / r] – present value of an annuity; calculates the PV of a series of equal cash flows.
  • PMT: periodic payment (e.g., annual cash flow)
  • r: periodic interest rate (e.g., annual interest rate divided by the number of periods)
  • n: number of periods (e.g., years)
• FV = PMT × [(1 + r)^n - 1] / r – future value of an annuity; calculates the FV of a series of equal cash flows.
  • PMT: periodic payment (e.g., annual cash flow)
  • r: periodic interest rate (e.g., annual interest rate divided by the number of periods)
  • n: number of periods (e.g., years)
• PV of an annuity due = PMT × [(1 + r)^n - 1] / r × (1 + r) – present value of an annuity due; calculates the PV of a series of equal cash flows received at the beginning of each period.
  • PMT: periodic payment (e.g., annual cash flow)
  • r: periodic interest rate (e.g., annual interest rate divided by the number of periods)
  • n: number of periods (e.g., years)
• FV of an annuity due = PMT × [(1 + r)^n - 1] / r × (1 + r) – future value of an annuity due; calculates the FV of a series of equal cash flows received at the beginning of each period.
  • PMT: periodic payment (e.g., annual cash flow)
  • r: periodic interest rate (e.g., annual interest rate divided by the number of periods)
  • n: number of periods (e.g., years)
• Ordinary Annuity = PMT × [(1 - (1 + r)^(-n)) / r] – ordinary annuity; calculates the PV of a series of equal cash flows received at the end of each period.
  • PMT: periodic payment (e.g., annual cash flow)
  • r: periodic interest rate (e.g., annual interest rate divided by the number of periods)
  • n: number of periods (e.g., years)

Step-by-Step Calculation

1. Determine the periodic payment (PMT) and the number of periods (n).
2. Choose the interest rate (r) and the compounding frequency.
3. Select the type of annuity (ordinary or annuity due).
4. Apply the corresponding formula to calculate the present value (PV) or future value (FV).
5. Verify the calculation by checking the units and the sign of the result.
6. Interpret the result in the context of the problem.

Common Mistakes

• Mistake: Using the wrong formula for the type of annuity (e.g., using the ordinary annuity formula for an annuity due).
  • Correction: Make sure to use the correct formula based on the type of annuity.
• Mistake: Ignoring the compounding frequency when calculating the interest rate (r).
  • Correction: Adjust the interest rate (r) to reflect the compounding frequency.
• Mistake: Assuming the periodic payment (PMT) is constant when it's not.
  • Correction: Use a more complex formula or a financial calculator to handle non-constant periodic payments.
• Mistake: Failing to verify the calculation by checking the units and the sign of the result.
  • Correction: Double-check the calculation to ensure it's correct.

Exam / CFA Tips

• Tip: Be careful when distinguishing between ordinary and annuity due annuities.
• Tip: Use a financial calculator or a spreadsheet to handle complex annuity calculations.
• Tip: Make sure to adjust the interest rate (r) to reflect the compounding frequency.
• Tip: Verify the calculation by checking the units and the sign of the result.

Quick Practice Problem

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

Answer: DFL = 1 - (1 - (1 + r)^(-n)) / r = 1 - (1 - (1 + 0.1)^(-5)) / 0.1 = 0.64

Explanation: The DFL measures the sensitivity of EBIT to sales.

Last-Minute Cram Sheet

• 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 PV of an annuity due is equal to the PV of an ordinary annuity multiplied by (1 + r).
• The FV of an annuity due is equal to the FV of an ordinary annuity multiplied by (1 + r).
• The ordinary annuity formula is PV = PMT × [(1 - (1 + r)^(-n)) / r].
• The annuity due formula is PV = PMT × [(1 + r)^n - 1] / r × (1 + r).
• The FV of an annuity is equal to the PV of an annuity multiplied by (1 + r)^n.
• The PV of an annuity is equal to the FV of an annuity divided by (1 + r)^n.
• The interest rate (r) must be adjusted to reflect the compounding frequency.
• The periodic payment (PMT) must be constant for the annuity formulas to apply.
• The annuity formulas can be used to calculate the PV and FV of a series of equal cash flows.