Unit 6: Investment Decisions - Subjective Questions

Capital budgeting is the process of identifying, evaluating, and selecting long-term investments whose benefits are expected to extend beyond one accounting period.

Discounted cash flow techniques are preferred because they:

• Recognize the time value of money.
• Consider cash flows over the project's entire economic life.
• Focus on cash flows rather than accounting profit.
• Incorporate the project's risk through the discount rate or adjusted cash flows.
• Provide an objective basis for comparing investment alternatives.
• Relate investment decisions to the objective of maximizing shareholders' wealth.

The main discounted techniques are Net Present Value, Internal Rate of Return, Profitability Index, and the Discounted Payback Period. Their common principle is to calculate present values using:

where is the cash flow in period and is the relevant discount rate.

Net Present Value (NPV) is the difference between the present value of a project's future cash inflows and the present value of its cash outflows.

It is calculated as:

where:

• = net cash flow in period
• = required rate of return or cost of capital
• = project life
• = initial investment

Decision rules:

• Accept an independent project if .
• Reject it if .
• The firm is indifferent if .
• For mutually exclusive projects, normally select the project with the highest positive NPV.

A positive NPV indicates that the investment is expected to earn more than the required return and increase shareholders' wealth.

The Year 4 cash flow includes both the operating inflow and salvage value:

The project's NPV is:

Present values are approximately:

• Year 1:
• Year 2:
• Year 3:
• Year 4:

Therefore:

Recommendation: The project should be accepted because its NPV is positive. It is expected to increase the value of the firm by approximately $24,875.36.

The Internal Rate of Return (IRR) is the discount rate at which the present value of a project's cash inflows equals the present value of its cash outflows. At this rate, the project's NPV is zero.

It is determined by solving:

Decision rules:

• Accept the project if is greater than the required rate of return.
• Reject the project if is lower than the required rate.
• The firm is indifferent if equals the required return.

IRR may be estimated through trial and error, interpolation, a financial calculator, or spreadsheet software. It is popular because it expresses profitability as a percentage, but it can produce misleading results for non-conventional cash flows or mutually exclusive projects.

At a discount rate of 15%:

This gives approximately:

At a discount rate of 20%:

This gives approximately:

Using linear interpolation:

Thus, the project's estimated IRR is approximately 18.12%. It should be accepted if the required rate of return is below this rate.

The Discounted Payback Period (DPP) is the time required for the cumulative present value of a project's cash inflows to recover its initial investment.

Each cash inflow is first discounted as follows:

Differences from the traditional payback period:

• Traditional payback uses undiscounted cash flows, whereas DPP recognizes the time value of money.
• DPP is generally longer than the traditional payback period.
• DPP gives a more realistic measure of investment recovery.
• Both methods ignore cash flows occurring after the payback cutoff.
• Neither method directly measures the total wealth created by the project.

A project is accepted if its discounted payback period is shorter than the maximum period specified by management.

The discounted cash inflows are:

• Year 1:
• Year 2:
• Year 3:
• Year 4:

Cumulative discounted inflow after Year 3 is:

The amount still unrecovered is:

The fraction of Year 4 required is:

Therefore:

The project recovers its discounted investment in approximately 3.75 years.

Cash-flow estimation should follow these principles:

• Use incremental cash flows, meaning only changes caused by accepting the project.
• Exclude sunk costs, because they have already been incurred and cannot be recovered.
• Include opportunity costs, such as income sacrificed by using an existing asset.
• Include side effects, including positive synergies and negative effects such as product cannibalization.
• Estimate cash flows on an after-tax basis.
• Include changes in net working capital.
• Treat financing costs separately because they are normally reflected in the discount rate.
• Include terminal cash flows, such as salvage value and working-capital recovery.
• Use cash flows rather than accounting profits.
• Maintain consistency between inflation assumptions in cash flows and the discount rate.

These principles ensure that the project's true contribution to the firm's value is measured.

The initial cash outlay includes all incremental cash flows occurring when the project begins. A general expression is:

where:

• = purchase cost of the new asset
• = installation, transportation, and setup costs
• = initial increase in net working capital
• = proceeds from selling an old asset
• = tax payable on the sale of the old asset, or minus the tax saving when it is sold at a tax loss

Other relevant costs may include site preparation, testing, employee training, and opportunity costs of resources used by the project. Sunk costs and financing charges are excluded. The initial outlay usually occurs at time zero and is therefore not discounted.

Operating cash flow after tax can be derived from revenue and operating costs. Let be revenue, be cash operating costs, be depreciation, and be the tax rate.

Earnings before interest and tax are:

Taxes are:

Because depreciation is a non-cash expense, it is added back:

Rearranging gives:

Thus, the main calculation methods are:

• Income approach:
• Tax-shield approach:
• Top-down approach:

The term is the depreciation tax shield. Interest expense is normally excluded because financing effects are reflected in the discount rate.

Terminal cash flow is the incremental cash flow occurring at the end of a project's economic life. Its major components are:

• The final year's operating cash flow.
• After-tax proceeds from disposing of project assets.
• Recovery of net working capital.
• Closure, removal, or environmental restoration costs.

The after-tax salvage value is generally:

where is salvage value, is tax book value, and is the tax rate.

• If , the gain normally creates a tax liability.
• If , the loss may create a tax saving, subject to applicable tax rules.

Net working capital recovered at termination is normally a cash inflow and is generally not taxable because it represents the release of funds previously invested rather than income.

Similarities:

• Both recognize the time value of money.
• Both use cash flows over the project's life.
• Both generally give the same accept-or-reject decision for conventional independent projects.

Differences:

• NPV expresses the result as an absolute monetary value; IRR expresses it as a percentage return.
• NPV assumes intermediate cash flows can be reinvested at the cost of capital; the traditional IRR interpretation assumes reinvestment at the IRR.
• NPV can be calculated for all cash-flow patterns; IRR may produce multiple or no meaningful rates for non-conventional cash flows.
• NPV directly measures the expected addition to shareholders' wealth.
• IRR may rank mutually exclusive projects incorrectly when their scale or timing differs.

When NPV and IRR conflict for mutually exclusive investments, NPV should generally be preferred because it is consistent with wealth maximization.

NPV and IRR can produce conflicting rankings because of:

• Scale differences: A small project may have a high percentage return but create less total wealth than a larger project.
• Timing differences: One project may generate early cash flows while another generates larger later cash flows.
• Project-life differences: Projects may operate for unequal periods.
• Reinvestment assumptions: NPV effectively uses the cost of capital, while IRR traditionally assumes reinvestment at the project's IRR.
• Non-conventional cash flows: Multiple changes in cash-flow signs may produce multiple IRRs.

The conflict should normally be resolved by selecting the project with the higher positive NPV at the firm's appropriate cost of capital. Incremental cash-flow analysis may also be used: subtract the cash flows of one project from those of the other and determine whether the incremental investment earns more than the required return.

Incremental IRR is the IRR calculated on the differences between the cash flows of two mutually exclusive projects. It determines whether the additional investment in the larger or more expensive project earns an adequate return.

The procedure is:

1. Arrange the projects in order of initial investment.
2. Subtract the smaller project's cash flows from the larger project's cash flows.
3. Calculate the IRR of these incremental cash flows.
4. Compare the incremental IRR with the required rate of return.

Decision rule:

• If incremental , choose the larger project.
• If incremental , choose the smaller project.

Incremental IRR can help explain a ranking conflict, but the final decision should be checked against incremental NPV. If cash flows are non-conventional or the projects have unequal lives, the NPV method remains more reliable.

Risk analysis in capital budgeting examines the uncertainty and variability associated with a project's future cash flows and investment value. A project is riskier when actual outcomes can differ substantially from expected outcomes.

Principal methods include:

• Sensitivity analysis: Changes one variable at a time to determine its effect on NPV or IRR.
• Scenario analysis: Changes several related variables simultaneously under best-case, base-case, and worst-case scenarios.
• Break-even analysis: Identifies the sales volume or value of another variable at which NPV equals zero.
• Probability analysis: Assigns probabilities to possible cash flows and calculates expected values and dispersion.
• Decision-tree analysis: Evaluates sequential decisions and contingent outcomes.
• Simulation: Generates a distribution of possible project outcomes using repeated random trials.
• Risk-adjusted discount rate: Applies a higher discount rate to riskier projects.
• Certainty equivalent approach: Converts risky cash flows into risk-free equivalent amounts.

The selected method should reflect the project's complexity and the availability of reliable data.

Sensitivity analysis measures how a project's NPV or IRR changes when one key input changes while all other inputs remain constant.

Procedure:

1. Calculate the base-case NPV.
2. Identify critical variables such as sales volume, selling price, operating cost, project life, or salvage value.
3. Change one variable by a specified percentage.
4. Recalculate NPV for each change.
5. Compare the results and identify the variables to which NPV is most sensitive.

Advantages:

• Simple to understand and apply.
• Identifies critical value drivers.
• Helps management focus on uncertain assumptions.
• Supports break-even and contingency analysis.

Limitations:

• Changes only one variable at a time.
• Ignores relationships among variables.
• Does not assign probabilities to outcomes.
• Results depend on the chosen range of variation.
• It identifies sensitivity but does not directly measure the probability of loss.

The annual net cash flow is:

The five-year annuity factor at 10% is approximately . Therefore:

For NPV to equal zero, the required annual cash flow is:

The break-even annual revenue is therefore:

The allowable revenue decline is:

As a percentage:

Thus, the base-case NPV is approximately $65,356, and annual revenue can decline by about 12.32% before NPV becomes zero.

The Certainty Equivalent Approach adjusts risky expected cash flows into amounts that decision-makers consider equivalent to certain cash flows. Each expected cash flow is multiplied by a certainty equivalent coefficient:

where:

• = expected risky cash flow in period
• = certainty equivalent coefficient, generally between and
• = certainty equivalent cash flow

The adjusted cash flows are discounted at the risk-free rate:

A lower indicates greater risk. The project is accepted if the resulting NPV is positive. The method separates the adjustment for risk from the adjustment for time value, but estimating reliable certainty equivalent coefficients can be difficult and subjective.

Certainty Equivalent Approach:

• Adjusts the numerator by reducing expected cash flows.
• Discounts adjusted cash flows at the risk-free rate.
• Uses certainty equivalent coefficients that may differ by year.
• Separates risk from the time value of money.

Its formula is:

Risk-Adjusted Discount Rate Approach:

• Leaves expected cash flows unchanged.
• Adjusts the denominator by adding a risk premium to the discount rate.
• Usually applies one risk-adjusted rate across the project life.

Its formula is:

The certainty equivalent method is theoretically more precise because risk can be adjusted separately for each period. The risk-adjusted discount rate method is simpler but may overstate risk for distant cash flows when a constant premium is compounded over time.

First, convert the risky expected cash flows into certainty equivalent cash flows:

• Year 1:
• Year 2:
• Year 3:

Discount these adjusted cash flows at the risk-free rate:

The present values are approximately:

• Year 1:
• Year 2:
• Year 3:

Therefore:

The risk-adjusted NPV is approximately $28,723.24. Since it is positive, the project should be accepted.