Discounted Cash Flow Valuation - Chapter 6

Learning Objectives

• Determining future and present values of investments with multiple cash flows.
• Explaining loan payment calculations and discovering interest rates on loans.
• Describing loan amortization processes.
• Understanding interest rate quotations and possible misquotations.

Chapter Outline

• Future and Present Values of Multiple Cash Flows
• Valuing Level Cash Flows: Annuities and Perpetuities
• Impact of Compounding on Rates
• Types of Loans and Loan Amortization

Future Value with Multiple Cash Flows

• Example Calculation:
  • Deposit of $100 today at 8% interest.
  • Deposit another $100 at the end of the first year.
  • Total Amount Calculation:
  • End of Year 1:
    100 imes 1.08 = 108 + 100 = 208
  • End of Year 2:
    208 imes 1.08 = 224.64
• Two methods to calculate future values:
  • Compound accumulated balance forward each year.
  • Calculate FV of each cash flow and sum them.

Saving Up Revisited

• If you deposit $4,000 annually for three years, starting with $7,000:
  • Year 1:
    7,000 imes 1.08 + 4,000 = 11,560
  • Year 2:
    11,560 imes 1.08 + 4,000 = 16,484.80
  • Year 3:
    16,484.80 imes 1.08 + 4,000 = 21,803.58
  • Year 4:
    21,803.58 imes 1.08 = 23,547.87

Present Value with Multiple Cash Flows

• Example Case: Need $1,000 in one year and $2,000 in two years at 9%.
  • Present Value (PV) Calculation:
  • 
    PV = rac{2,000}{(1.09)^2} + rac{1,000}{(1.09)} = 1,683.36 + 917.43 = 2,600.79
• Two ways to determine PV:
  • Discount each cash flow back individually and sum.
  • Discount back one period at a time.

PV of Annuity Cash Flows

• Example: Cash flow of $500 for three years at 10%.
• PV Formula for Annuities: PV = rac{C}{1 - (1 + r)^{-t}}
  • Example Calculation for PV factor:
  1. Present value factor = rac{1 - 0.751315}{0.10}
  2. PV of annuity = 500 imes 2.48685 = 1,243.43

Finding Loan Payments

• Example: Borrowing $100,000 at 18% over 5 years:
  • Monthly payments of just under $32,000.
• Calculator Method:
  • Compute using PMT key for annuity.

Amortized Loans

• Commonly lead to fixed payments leading to total loan payoff.
• Example Calculation: $5,000, 5-year loan at 9%:
  • Use:
    C = rac{PV}{ ext{Annuity Factor}}
  • Total payment: $1,285.46 yearly; first interest payment $450, principal payment $835.46.

Quotes for Interest Rates

• Understanding APR and EAR: APR is nominal while EAR accounts for compounding effects.
• Example Comparisons:
  • Bank Rates: Determine best option among given EARs.

Growing Annuities and Perpetuities

• Present and future value calculation methods highlighted for perpetuities as well as growing cash flows like lottery payouts.

Conclusion

• Key concepts include future/present values, annuities, payment structures, and interest rate interpretations crucial for effective financial management in investments and loans.