Chapter 6: Discounted Cash Flow Valuation

Learning Objectives

• Determine future and present value of investments with multiple cash flows.
• Explain calculation of loan payments and interest rates.
• Describe loan amortization process.
• Discuss accurate and common misquotations of interest rates.

Chapter Outline

• Future and Present Values of Cash Flows
• Annuities and Perpetuities Valuation
• Comparing Interest Rates and Effects of Compounding
• Loan Types and Amortization

Future Value with Multiple Cash Flows

• Example: Deposit of $100 at 8% interest.
  • After year one: $100 * 1.08 = $108
  • After second year: $208 * 1.08 = $224.64
  • Total future value after year two = $224.64

Alternate Solutions to Future Value

1. Future value of each cash flow can be calculated separately then summed:
   • First deposit: $100 * (1 + 0.08)² = $116.64
   • Second deposit (only deposited for 1 year): $100 * (1 + 0.08) = $108
   • Total Future Value = $116.64 + $108 = $224.64
2. Compound accumulated balance year-on-year.

Saving Up Revisited

• Current account: $7,000, with annual deposits of $4,000 at 8%.

1. End of Year 1: $7,000 * 1.08 + $4,000 = $11,560
2. End of Year 2: $11,560 * 1.08 + $4,000 = $16,484.80
3. End of Year 3: $16,484.80 * 1.08 + $4,000 = $21,803.58
   • End of Year 4: $21,803.58 * 1.08 = $23,547.87

Present Value with Multiple Cash Flows

• Example: If needing $1,000 in one year and $2,000 in two years with a 9% interest rate:
  • PV of $1,000 in 1 year: $1,000 / 1.09 = $917.43
  • PV of $2,000 in 2 years: $2,000 / (1.09)² = $1,683.36
  • Total PV = $917.43 + $1,683.36 = $2,600.79

Present Value Calculation Methods

1. Discount cash flows individually and sum:
   • For $1,000 over 5 years: Calculate each PV and add.
2. Discount back one cash flow at a time and add.

Example: Investment Cash Flows

• $200, $400, $600, $800 over four years with 12% interest:
  • PV = $200/(1.12) + $400/(1.12)² + $600/(1.12)³ + $800/(1.12)⁴ = $1,432.93

Annuity Cash Flows

• An annuity consists of equal cash flows over equal periods.
  • Example: $500 paid annually for 3 years at 10%:
  • PV = $500 * [PVIFA(10%, 3)] = $1,243.43

Present Value Annuity Formula

• PV of annuity = C * [ (1 - (1 + r)⁻ⁱ) / r ]
  • Where C = cash flow, r = interest rate, and i = number of periods.

Annuity Tables

• Tables provide present value factors.
  • PV factor for 3 periods at 10% = 2.4869.

Financial Calculator for Annuities

• Use PMT key to find annuity present values.
  • Input cash flow & rate; do not enter future value.

Effective Annual Rates (EAR)

• Differences between quoted rates and effective rates.
  • Example: 10% compounded semiannually = 5% per period, with an EAR of 10.25%:
  • Formula: EAR = (1 + (quoted rate / m))^m - 1

Loan Types and Amortization

• Pure Discount Loans: Lump sum repaid in the future (e.g., T-bills).
• Interest-Only Loans: Pay interest until principal is returned.
• Amortized Loans: Payments reduce principal over time.

Example: Amortized Loan Schedule

• 5-year loan of $5,000 at 9% with equal payments:
  • Each annual payment: $1,285.46 based on calculated annuity present value factor.

Partial Amortization

• Loan payments for a fraction of time, balloon payment after:
  • Example: $100,000 mortgage with 5-year term and balloon payment after 60 months:
  • Monthly payment calculation leads to a large payment due at loan expiration.

Summary of Key Concepts

• PV = Present value; FV = Future value; r = interest rate; t = time periods; C = cash flow.
• The future value of repeated cash flows calculated using annuity future value factor.
• Present value and future value relationship calculations can shift methods.
• Calculation of the present value of perpetuities, annuities, and varying cash flows addressed dynamically in financial scenarios.