Lecture Notes on Compounding and Financial Calculations

Introduction to Compounding Periods

• Annual Rate of Return (R): Always quoted on an annual basis in finance, even if compounded more frequently.
• Compounding Frequency (m): Represents how often interest is compounded within a year.
  • Examples:
    • m=1: Annual compounding.
    • m=2: Semi-annual.
    • m=3: Quarterly.
    • m=12: Monthly.
    • m=365: Daily.
    • Continuous Compounding: Theoretical concept with constant compounding intervals.

Effective Annual Rate (EAR)

• Purpose: Compares different financial products with varying compounding frequencies by converting rates to an annual equivalent.
• Example:
  • Bank A: 12.5% compounded annually.
  • Bank B: 12% compounded monthly (equivalent to a 12.68% effective annual rate).
• Formula: (EAR = (1 + \frac{R}{m})^m - 1).

Annuities and Perpetuities

Perpetuities

• Definition: A constant stream of cash flows that last indefinitely.
• Present Value Formula: (PV = \frac{C}{R}), where C is the cash flow and R is the discount rate.
• Examples:
  • British Consol Bonds.
  • Preferred Stock.

Growing Perpetuities

• Definition: Cash flows grow at a constant rate (G).
• Present Value Formula: (PV = \frac{C}{R-G}).

Annuities

• Definition: A series of fixed payments made over a specified period.
• Present Value Formula: Derived from the difference between two perpetuities.
• Applications: Mortgages, car loans, student loans, bond coupon payments.
• Example Calculation: Determining how much loan to provide based on fixed car payments.

Future Value of Annuity

• Purpose: Calculates the accumulated value of regular payments at a future date.
• Example: Regular savings for a college fund.

Growing Annuities

• Definition: Annuities where payments grow at a constant rate.
• Example: Retirement plans with inflation-adjusted payments.

Present Value Analysis in Firm Valuation

• Concept: The value of a firm is the present value of its expected future cash flows.
• Challenges: Forecasting cash flow size, timing, and risk.

Using Excel for Financial Calculations

• Functions:
  • FV: Future Value calculation.
  • PV: Present Value calculation.
  • NPER: Number of periods.
  • PMT: Annuity payments.
  • RATE: Rate required to achieve a target future value.
• Example: Calculating future value of a college fund using Excel formulas.

Conclusion

• Understanding of discounting, compounding, and annuities is crucial for financial decision-making and product comparisons.
• Excel provides powerful tools to easily perform such calculations.