Chapter 5: Discounted Cash Flow Valuation

Discounted Cash Flow Valuation Notes

Key Concepts and Skills

  • I. Uneven Cash Flows

  • A. Present Value: The current value of future cash flows, discounted at a particular interest rate.

  • B. Future Value: The value of an investment after it has gained interest for a specific period.

  • II. Annuities and Perpetuities

  • A. Present value ordinary annuities: The present value of a series of equal payments made at regular intervals.

  • B. Future value ordinary annuities: The future value of a series of equal payments made at regular intervals.

  • C. Present value annuities due: Similar to ordinary annuities except payments start at the beginning of each period.

  • D. Perpetuities: A stream of equal cash flows that occur at regular intervals indefinitely.

  • III. Interest Rate Calculations

  • A. Annual Percentage Rate (APR): The nominal interest rate, quoted annually.

  • B. Effective Annual Rate (EAR): The interest rate accounting for compounding within the year.

  • IV. Amortized Loan With Fixed Payment

  • Understanding how fixed payments impact loan principal and interest over time.

  • V. Importance of Chapter 5

  • Essential concepts for all business students regarding cash management, loans, investments, and financial planning.

Multiple Cash Flows

  • Two Main Computational Methods:

  1. Use of TVM keys (N, I/Y, PV, PMT, FV) for solutions involving equal payments.

  2. Cash Flow Worksheet for solving problems involving unequal payments.

Present Value of Uneven Cash Flows

  • Use the principle of additivity to sum the present value of each cash flow.

  • Example: Calculate the PV of cash flows of $200, $400, $600, and $800 over four years at a 12% discount rate using the formula:

  • PV = CF / (1 + R)^T (where CF = cash flow, R = interest rate, T = time)

Calculation Steps:

  • Year 1: $200 / (1.12)^1 = $178.57

  • Year 2: $400 / (1.12)^2 = $318.88

  • Year 3: $600 / (1.12)^3 = $427.07

  • Year 4: $800 / (1.12)^4 = $508.41

  • Total PV = $1,432.93

Cash Flow Worksheet Method

  • Steps to enter cash flows into a financial calculator to find NPV.

  • Clear CF Worksheet.

  • Enter cash flows and their frequencies (F01).

  • Calculate NPV at the given interest rate.

Annuities and Perpetuities

  • Definitions:

  • Annuity: A series of equal payments at regular intervals.

  • Perpetuity: An infinite series of equal payments.

Formulas

  1. Annuities:

  • PV = PMT × [(1 - (1 + r)^-t) / r]

  • FV = PMT × [((1 + r)^t - 1) / r]

  1. Perpetuity:

  • PV = PMT / r

Interest Rate Calculations

  • APR vs EAR:

  • APR is the nominal rate; EAR accounts for compounding.

  • Comparison: Always use EAR to compare different investments.

Computing Interest Rates:

  • If monthly rate is 0.5%, then APR = 0.5% * 12 = 6%.

  • Compute EAR using formulas depending on compounding frequency.

Amortized Loans

  • Each payment reduces the principal and covers interest expenses.

  • Example Calculation: For a $5000 loan at 8% interest over 4 years.

  • Annual payment calculated to be approximately $1509.60.

Conclusion: Importance for Business Students

  • Understanding cash flow valuation is crucial for financial decision-making in personal and business finance.

  • Concepts are directly applicable to common financial situations like mortgages, student loans, and budgeting for large expenses.

Discounted Cash Flow Valuation Notes

Key Concepts and Skills

  • I. Uneven Cash Flows

  • A. Present Value: The current value of future cash flows, discounted at a particular interest rate.

  • B. Future Value: The value of an investment after it has gained interest for a specific period.

  • II. Annuities and Perpetuities

  • A. Present value ordinary annuities: The present value of a series of equal payments made at regular intervals.

  • B. Future value ordinary annuities: The future value of a series of equal payments made at regular intervals.

  • C. Present value annuities due: Similar to ordinary annuities except payments start at the beginning of each period.

  • D. Perpetuities: A stream of equal cash flows that occur at regular intervals indefinitely.

  • III. Interest Rate Calculations

  • A. Annual Percentage Rate (APR): The nominal interest rate, quoted annually.

  • B. Effective Annual Rate (EAR): The interest rate accounting for compounding within the year.

  • IV. Amortized Loan With Fixed Payment

  • Understanding how fixed payments impact loan principal and interest over time.

  • V. Importance of Chapter 5

  • Essential concepts for all business students regarding cash management, loans, investments, and financial planning.

Multiple Cash Flows

  • Two Main Computational Methods:

  1. Use of TVM keys (N, I/Y, PV, PMT, FV) for solutions involving equal payments.

  2. Cash Flow Worksheet for solving problems involving unequal payments.

Present Value of Uneven Cash Flows

  • Use the principle of additivity to sum the present value of each cash flow.

  • Example: Calculate the PV of cash flows of $200, $400, $600, and $800 over four years at a 12% discount rate using the formula:

  • PV = CF / (1 + R)^T (where CF = cash flow, R = interest rate, T = time)

Calculation Steps:

  • Year 1: $200 / (1.12)^1 = $178.57

  • Year 2: $400 / (1.12)^2 = $318.88

  • Year 3: $600 / (1.12)^3 = $427.07

  • Year 4: $800 / (1.12)^4 = $508.41

  • Total PV = $1,432.93

Cash Flow Worksheet Method

  • Steps to enter cash flows into a financial calculator to find NPV.

  • Clear CF Worksheet.

  • Enter cash flows and their frequencies (F01).

  • Calculate NPV at the given interest rate.

Annuities and Perpetuities

  • Definitions:

  • Annuity: A series of equal payments at regular intervals.

  • Perpetuity: An infinite series of equal payments.

Formulas

  1. Annuities:

  • PV = PMT × [(1 - (1 + r)^-t) / r]

  • FV = PMT × [((1 + r)^t - 1) / r]

  1. Perpetuity:

  • PV = PMT / r

Interest Rate Calculations

  • APR vs EAR:

  • APR is the nominal rate; EAR accounts for compounding.

  • Comparison: Always use EAR to compare different investments.

Computing Interest Rates:

  • If monthly rate is 0.5%, then APR = 0.5% * 12 = 6%.

  • Compute EAR using formulas depending on compounding frequency.

Amortized Loans

  • Each payment reduces the principal and covers interest expenses.

  • Example Calculation: For a $5000 loan at 8% interest over 4 years.

  • Annual payment calculated to be approximately $1509.60.

Conclusion: Importance for Business Students

  • Understanding cash flow valuation is crucial for financial decision-making in personal and business finance.

  • Concepts are directly applicable to common financial situations like mortgages, student loans, and budgeting for large expenses.

Example Problems with Step-by-Step Solutions

  1. Present Value of Cash Flows

  • Calculate the present value of cash flows of $300, $500, and $700 over three years at a 10% discount rate.

  • Step 1: Year 1: $300 / (1.10)^1 = $272.73

  • Step 2: Year 2: $500 / (1.10)^2 = $413.22

  • Step 3: Year 3: $700 / (1.10)^3 = $526.36

  • Total PV = $272.73 + $413.22 + $526.36 = $1212.31

  1. Future Value of Annuities

  • What is the future value of an annuity with an annual payment of $200 over 5 years at an interest rate of 5%?

  • Step 1: FV = PMT × [((1 + r)^t - 1) / r]

  • Step 2: FV = 200 × [((1 + 0.05)^5 - 1) / 0.05]

  • Step 3: FV = 200 × [((1.27628) - 1) / 0.05]

  • Step 4: FV = 200 × 5.6525 = $1130.50

  1. NPV Calculation using Cash Flow Worksheet

  • Suppose there are cash flows of $100, $150, and $200 for years 1 to 3 at a 12% discount rate.

  • Step 1: Clear CF Worksheet.

  • Step 2: Enter cash flows as follows: CF0 = 0, CF1 = 100, CF2 = 150, CF3 = 200.

  • Step 3: Enter the frequency of each cash flow, then compute the NPV at 12%.

  • Total NPV = $276.94