Introductory Corporate Finance: Capital Budgeting - Equivalent Annual, Annuity EAA for Unequal Lives

What This Is

Equivalent Annual Annuity (EAA) for Unequal Lives is a method used to calculate the present value of unequal cash flows over multiple periods. This concept is crucial in corporate finance as it helps investors and analysts evaluate the value of projects with varying cash inflows and outflows. For instance, consider a company investing in a new project with initial costs of $100,000, followed by annual cash inflows of $20,000 for the first 5 years, $30,000 for the next 3 years, and $40,000 for the final 2 years. Using EAA, we can calculate the present value of these cash flows and determine the project's net present value.

Key Formulas & Models

• EAA = ?[CFt / (1 + r)^t] – equivalent annual annuity; calculates the present value of unequal cash flows.
• CFt: cash flow at time t
• r: discount rate (e.g., WACC)
• t: time period (e.g., year)
• PV = EAA / r – present value; calculates the present value of the EAA.
• IRR = r – internal rate of return; the discount rate that makes the EAA equal to zero.
• NPV = ?[CFt / (1 + IRR)^t] – net present value; calculates the present value of the cash flows using the IRR.
• WACC = wd × rd(1?T) + wps × rps + we × re – weighted average cost of capital; used as the discount rate.
• wd: weight of debt
• rd: cost of debt
• T: tax rate
• wps: weight of preferred stock
• rps: cost of preferred stock
• we: weight of equity
• re: cost of equity
• DOL = Q(P?V) / (Q(P?V)?F) – degree of operating leverage; measures EBIT sensitivity to sales.
• Q: sales
• P: price per unit
• V: variable costs per unit
• F: fixed costs

Step-by-Step Calculation

1. Determine the cash flows for each period (CFt).
2. Choose a discount rate (r) based on the project's risk and the company's WACC.
3. Calculate the EAA using the formula EAA = ?[CFt / (1 + r)^t].
4. Calculate the present value (PV) using the formula PV = EAA / r.
5. Calculate the internal rate of return (IRR) using the formula IRR = r.
6. Calculate the net present value (NPV) using the formula NPV = ?[CFt / (1 + IRR)^t].

Common Mistakes

• Mistake: Using the wrong discount rate (e.g., using the cost of debt instead of WACC).
• Correction: Use the WACC as the discount rate, as it reflects the company's overall cost of capital.
• Counterexample: A company with a WACC of 10% and a cost of debt of 5% should use 10% as the discount rate, not 5%.
• Mistake: Ignoring the time value of money (TVM) when calculating the present value.
• Correction: Use the TVM formula to calculate the present value, as it takes into account the time value of money.
• Counterexample: A company with a cash flow of $100,000 in 5 years should use the TVM formula to calculate the present value, not just use the cash flow as is.
• Mistake: Confusing the EAA with the NPV.
• Correction: The EAA is a measure of the present value of unequal cash flows, while the NPV is a measure of the present value of the cash flows using the IRR.
• Counterexample: A company with an EAA of $100,000 and an NPV of $120,000 should not confuse the two measures.

Exam / CFA Tips

• Tip: Be careful when using the WACC as the discount rate, as it may not reflect the project's specific risk.
• Tip: Use the TVM formula to calculate the present value, as it takes into account the time value of money.
• Tip: Be careful when confusing the EAA with the NPV, as they are two different measures.

Quick Practice Problem

A company has EBIT of $10M, interest $2M, tax 25% – compute DFL.

Answer: DFL = EBIT / (1 - T) = $10M / (1 - 0.25) = $13.33M

Explanation: The degree of financial leverage (DFL) is calculated by dividing the EBIT by the after-tax EBIT.

Last-Minute Cram Sheet

• EAA: equivalent annual annuity; calculates the present value of unequal cash flows.
• PV: present value; calculates the present value of the EAA.
• IRR: internal rate of return; the discount rate that makes the EAA equal to zero.
• NPV: net present value; calculates the present value of the cash flows using the IRR.
• WACC: weighted average cost of capital; used as the discount rate.
• DOL: degree of operating leverage; measures EBIT sensitivity to sales.
• CFt: cash flow at time t
• r: discount rate (e.g., WACC)
• t: time period (e.g., year)
• T: tax rate
• 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