Implementing the NPV Rule: Detailed Analysis and Calculation Steps

Introduction to Implementing the Net Present Value (NPV) Rule

• The primary objective of Lecture 9 is to explore the practical implementation of the NPV rule, specifically focusing on the transition from accounting figures to free cash flows and the application of adjusted discount rates.

• The process of calculating NPV involves three main phases:

  • Phase 1: Calculate the taxable operating profits that are directly attributable to the specific project. This is used to determine the corporation taxes associated with embarking on the project.

  • Phase 2: Calculate the before-tax incremental operating cash flows. From this figure, deduct the calculated taxes and incorporate net capital spending to arrive at the corresponding after-tax free cash flows.

  • Phase 3: Calculate the NPV of the project using Discounted Cash Flow (DCF) analysis. This involves discounting the after-tax free cash flows back to the present using the project’s required rate of return.

Converting Accounting Profits into Cash Flows

• Implementing DCF requires the ability to convert standard accounting profits into actual cash flows because companies manage data via accounting information systems, and tax liabilities are calculated based on profits, not raw cash movements.

• The fundamental reason for discounting cash flows rather than accounting profits is that the two figures are fundamentally different:

  • Depreciation: Depreciation is an accounting allocation and is not a cash flow. The actual cash flow event occurs when the asset is purchased. However, the firm is permitted to deduct capital allowances (which are tax-specific versions of depreciation) when calculating taxable profits.

  • Working Capital (WC): Since accounting records for sales and costs do not always happen simultaneously with the exchange of cash, changes in Working Capital reflect these timing differences. Changes in WC also reflect institutional investments in inventory (stock) or cash reserves necessary for operations.

Understanding Depreciation and Capital Allowances

• Depreciation is defined as the allocation of the cost of an investment over several accounting periods. It is strictly a non-cash expense used for internal and external reporting.

• Capital Allowance is the equivalent of depreciation but is strictly governed by tax law. It determines the actual tax relief a company receives on its capital investments.

• Common methods for calculating depreciation/allowances include:

  • Straight Line Method: The cost of the asset is divided equally over its useful life. For example, an asset costing $\text{}1,000,000$ with a life of $4$ years yields a depreciation of $\frac{\text{}1,000,000}{4} = \text{}250,000$ per year.

  • Reducing Balance Method: This is an accelerated write-down method where a fixed percentage is applied to the remaining book value each year.

• Reducing Balance Example ($1,000,000$ initial investment at a $25\%$ rate):

  • Year 1: Starting Value = $\text{}1,000,000$; Depreciation = $\text{}250,000$; Year-end Value = $\text{}750,000$.

  • Year 2: Starting Value = $\text{}750,000$; Depreciation = $\text{}187,500$; Year-end Value = $\text{}562,500$.

  • Year 3: Starting Value = $\text{}562,500$; Depreciation = $\text{}140,625$; Year-end Value = $\text{}421,875$.

  • Year 4: Starting Value = $\text{}421,875$; Depreciation = $\text{}105,469$; Year-end Value = $\text{}316,406$.

Practical Calculation of After-Tax Cash Flows

• To calculate the cash flow for a specific project year, follow the logic of reconciling profit and cash adjustments.

• Example Data:

  • Sales: $500,000$

  • Operating expenses (excluding depreciation): $280,000$

  • Accounting depreciation: $50,000$

  • Capital allowance: $42,000$

  • Change in working capital: $-10,000$

  • Corporation tax rate: $33\%$

• Step 1: Calculate Taxable Profits:

  • $\text{Taxable Profits} = 500,000 - 280,000 - 42,000 = 178,000$

• Step 2: Calculate Tax Liability:

  • $\text{Tax} = 0.33 \times 178,000 = 58,740$

• Step 3: Calculate After-Tax Cash Flow:

  • Sales less expenses (operating cash flow): $500,000 - 280,000 = 220,000$

  • Add decrease in working capital (note: a negative change of $-10,000$ implies a release of cash): $+10,000$

  • Less taxes: $-58,740$

  • Total After-tax cash flow: $\text{}171,260$

Incremental Cash Flow Principles

• Cash flows must be calculated after corporation tax because the discount rate used is typically an after-tax rate.

• All figures must be incremental, meaning they must represent the change in the firm's total cash flow that occurs as a direct result of accepting the project.

• Guidelines for Incremental Analysis:

  • Ignore Sunk Costs: Losses, revenues, or costs already incurred (past expenses) are irrelevant and must be ignored.

  • Consider Side-effects: Account for project interdependencies, such as erosion (cannibalization of existing sales) or synergy.

  • Consider Opportunity Costs: Factor in the value of assets or resources that could be used in their next best alternative application.

  • Ignore Arbitrarily Allocated Costs: Only include costs that change as a result of the project; ignore overheads allocated by the firm that would exist regardless of the project.

• Financing Costs: Interest payments and other financing costs are excluded from cash flow calculations to avoid double-counting, as these are captured in the discount rate.

• Inflation Consistency: Nominal cash flows should be discounted at a nominal discount rate.

Determining the Discount Rate (WACC)

• The appropriate discount rate for calculating NPV is the project’s Weighted Average Cost of Capital (WACC).

• The WACC formula is expressed as:

  • $WACC = r_D(1 - T_C)(\frac{D}{V}) + r_E(\frac{E}{V})$

• Variable Definitions:

  • $r_D$: Required return on debt (cost of debt capital).

  • $r_E$: Required return on equity (cost of equity capital).

  • $T_C$: Corporation tax rate.

  • $D$: Market value of the company’s debt.

  • $E$: Market value of the company’s equity.

  • $V$: Total market value of the company ($V = D + E$).

• This formula reflects the combined after-tax cost of all sources of finance, weighted by their proportion in the firm's capital structure.

Example WACC Calculation

• Consider a project with the following financial parameters:

  • Required return on debt ($r_D$): $0.12$ ($12\%$)

  • Required return on equity ($r_E$): $0.28$ ($28\%$)

  • Corporation tax rate ($T_C$): $0.33$ ($33\%$)

  • Debt-to-Value ratio ($\frac{D}{V}$): $0.4$ ($40\%$)

  • Equity-to-Value ratio ($\frac{E}{V}$): $0.6$ ($60\%$)

• WACC Formula Application:

  • $WACC = 0.12(1 - 0.33)(0.4) + 0.28(0.6)$

  • $WACC = (0.12 \times 0.67 \times 0.4) + 0.168$

  • $WACC = 0.03216 + 0.168 = 0.20016$

  • Rounded WACC = $20\%$

Summary Checklist for NPV Implementation

• Taxation: Find the extra tax paid specifically because the project was undertaken.

• Cash Flow: Generate the after-tax free cash flows, ensuring they are strictly incremental (excluding sunk costs and including opportunity costs).

• Discounting: Use the project's required rate of return, which is the weighted average of all after-tax costs of finance (WACC).