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.