Present and Future Value

Present and Future Value

• Present Value (PV) represents the current worth of a future sum of money, factoring in a specific discount rate.
• Future Value (FV) is the value of a present sum of money after it has increased over time due to interest.

Time Value of Money

• Money is worth more today than the same amount in the future.
• This is due to earning potential, inflation, and risk/uncertainty.
• Earning potential allows money to be invested and grow.
• Inflation erodes purchasing power.
• Future money is inherently less certain than current money.

Future Value Relationships

• Higher interest rates increase future value.
• More periods increase future value.
• A higher present value leads to a higher future value.

Present Value Relationships

• Higher interest rates decrease present value.
• More periods decrease present value.
• A higher future value results in a higher present value.

Simple vs. Compound Interest

• Simple interest earns interest only on the principal amount.
• The formula for simple interest is: Simple Interest = PV * r * t
• Compound interest earns interest on both the principal and accumulated interest.
• The formula for future value with compound interest is: Future Value = P (1 + r/n)^(nt)
• Compound interest grows money faster than simple interest.

Annuity Value and Time

• Increasing the length of time decreases the present value of an annuity because of increased discounting periods.
• Increasing the time period increases the future value of an annuity, allowing for more interest accumulation.

Annuity Value and Interest Rate

• Increasing the interest rate increases the future value of an annuity because of the higher rate of growth.
• Increasing the interest rate decreases the present value of an annuity because of more discounting.

Perpetuity Value and Payment

• A higher payment every period increases the present value of a perpetuity, given the same discount rate.
• The formula for PV of a perpetuity is PV = C / r, where C is the payment per period and r is the discount rate.

Annuity Value and Interest Rate

• If the interest rate increases, the present value of annuities decreases, given the same periods and payment.
• Increasing interest rates decrease present value.

Annuity Future Value and Interest Rate

• If the interest rate increases, the future value of annuities increases, given the same periods and payment.
• Higher interest rates provide more growth, so future value increases

Perpetuity Value and Interest Rate

• A higher interest rate decreases the present value of perpetuities, given the same payments.
• Future payments are worth less today when discounting occurs

Compounding Frequency and EAR

• More frequent compounding increases the Effective Annual Rate (EAR).
• Interest will be earned more often providing a greater overall EAR

Key Takeaways

• More time decreases PV and increases FV.
• Higher interest rates decrease PV and increase FV.
• More frequent compounding increases EAR.

Investment Evaluation Methods

• Net Present Value (NPV) accounts for the time value of money and directly measures added value.
• IRR accounts for the time value of money and is commonly used. But it can be misleading with unconventional cash flows and assumes reinvestment at IRR.
• Payback Period offers simplicity and is useful for liquidity analysis, but ignores the time value of money and cash flows after the payback period.
• Average Accounting Return (AAR) is based on accounting numbers and is simple to calculate, but it ignores the time value of money and uses book values instead of cash flows.
• Profitability Index (PI) considers the time value of money and is useful for comparing projects with different initial costs, but can be misleading for mutually exclusive projects.

Decision Rules for Project Acceptance

• Accept a project if NPV > 0, indicating positive added value.
• Accept a project if IRR > required return.
• Accept a project if the payback period is less than a preset time limit.
• Accept a project if AAR > required return.
• Accept a project if PI > 1, indicating a profitable project.

Nonconventional Cash Flows

• Nonconventional cash flows switch signs more than once.
• This can lead to multiple or no IRR solutions.
• Decision-making can become difficult in this situation.
• Use NPV instead of IRR for better accuracy.

Mutually Exclusive Projects

• Mutually exclusive projects involve choosing one project which means rejecting another.
• IRR and PI may rank projects differently than NPV.
• Picking the wrong project could negatively impact the overall value.
• Use NPV to decide since it directly measures value added.

NPV vs IRR

• Always use NPV.
• NPV directly measures added value.
• IRR can be misleading due to nonconventional cash flows or mutually exclusive projects.

Key Takeaways

• NPV is the best method.
• IRR is useful, but can be misleading.
• The payback period & AAR are simple, but ignore TVM.
• PI is useful, but not for mutually exclusive projects.

Incremental Cash Flows

• Incremental Cash Flows change as a direct result of taking on a project.
• Only include cash flows directly related to the project.
• Formula: Incremental Cash Flow = Project Cash Flows - Cash Flows Without Project

Stand-Alone Principle

• The stand-alone principle states that a project should be evaluated based on its own cash flows, separate from the company's other operations.
• This helps make it easier to analyze the project's profitability.

Cash Flows in Project Cash Flow

• Sunk Costs should not to be included, because they are irrecoverable.
• Opportunity costs should be included, representing the value of the next best alternative.
• Side Effects/Erosion should be included, as they represent a real cost affecting the business.
• Changes in Net Working Capital (NWC) should be included and are recovered at the end of the project.
• Financing Costs should not be included, focus on project cash flows, not how it is financed.
• Tax Effects should be included, because taxes directly affect cash flow.

Financial Statements

• A firm’s income statement shows its actual financial performance.
• A pro forma income statement is a projected income statement specifically for a project.
• Key Difference: Pro forma focuses only on the project.

Operating Cash Flow Approaches

• Operating Cash Flow (OCF) measures cash generated from operations.
• Bottom-Up Approach: OCF = Net Income + Depreciation (used when there's no interest expense)
• Top-Down Approach: OCF = Sales - Costs - Taxes (this approach ignores non-cash expenses like depreciation)

Computing Project Cash Flows

• Total Project Cash Flow is computed as follows:
Operating Cash Flow (OCF) - Net Capital Spending (NCS) - Changes in Net Working Capital (ΔNWC)
• This accounts for cash generated, investments made, and working capital needs.

After Tax Salvage Value

• After-Tax Salvage Value (ATSV) is the amount received from selling an asset, after taxes on capital gains.
• The Formula for After-Tax Salvage Value is: ATSV = Salvage Value - (Tax Rate) * (Salvage Value - Book Value)

Depreciation Methods

• Straight-Line Depreciation evenly depreciates an asset over time, is simple & predictable.
• MACRS (Modified Accelerated Cost Recovery System) asset depreciates more in early years.
• This process provides more tax savings earlier.
• MACRS is commonly used because it allows for faster tax benefits.

Scenario vs. Sensitivity Analysis

• Scenario Analysis examines multiple possible outcomes and view of risk (best case, worst case, base case).
• It provides a broad view of risk.
• Analysis relies on estimated probabilities.
• Sensitivity Analysis examines how one variable affects project outcomes and help identify key risk factors & is more detailed.
• Analysis changes one variable at a time, which may oversimplify risks.

Comparative Summary

• Key Differences: Scenario Analysis changes multiple factors at once (big-picture), while Sensitivity Analysis focuses on one factor at a time (detailed view).

Key Takeaways

• Only include relevant cash flows.
• Exclude sunk costs, financing costs.
• Use pro forma statements to project financials.
• The Operating Cash Flow equation is Net Income + Depreciation (for bottom-up approach).
• After-tax salvage value adjusts for capital gains taxes.
• The MACRS depreciation saves more taxes earlier than straight-line depreciation.
• Finally, scenario analysis looks at multiple outcomes while sensitivity analysis focuses on one variable.