Annuities and Perpetuities Overview

Key Concepts on Annuities

  • Annuity Definition: A financial product consisting of a series of equal payments made at regular intervals.

    • Ordinary Annuity: Payments made at the end of each period (e.g., receive $100 every year for 20 years).

    • Annuity Due: Payments made at the beginning of each period (e.g., receive $100 every year starting now).

  • Perpetuity Definition: A type of annuity that receives regular payments indefinitely (e.g., receive $200 every year for life).

Key Formulas

  • Present Value of an Ordinary Annuity (PVa):
    PVa = PMT × (1 - (1 + r)-n) / r

  • Future Value of an Ordinary Annuity (FVa):
    FVa = PMT × ((1 + r)n - 1) / r

  • Present Value of an Annuity Due (PVad):
    PVad = PMT × (1 - (1 + r)-n) / r × (1 + r)

  • Future Value of an Annuity Due (FVad):
    FVad = PMT × ((1 + r)n - 1) / r × (1 + r)

Example Calculations

  • Future Value Calculation:

    • Scenario: Saving $1,000 each year at 11% interest for 10 years.

    • FV = $1,000 × ((1 + 0.11)10 - 1) / 0.11 = $16,722.01.

  • Present Value Calculation:

    • Scenario: $12,000 annually for 20 years, starting next year with an opportunity cost of 6%.

    • PV = $12,000 × (1 - (1 + 0.06)-20) / 0.06 = $137,639.06.

Using Financial Calculators

  1. Entering Data for FV: For an ordinary annuity saving $1,000 for 10 years at 11% interest, enter:

    • PMT = 1000

    • n = 10

    • I/Y = 11

    • Calculate FV.

    • Output = $16,722.01.

  2. Present Value Calculation:
    Use similar steps for finding present value, adapting inputs as per scenario (e.g., PMT = -12000 for the present value scenario).

    • Output present value by entering data correctly and calculating.

Notes on Mortgage Loans

  • Mortgage Loan: The principal of a mortgage loan is repaid through equal periodic payments.

    • Amortization: Process of determining the equal payments that include both principal and interest components.

  • Example: For a $300,000 mortgage at 3% interest over 30 years:

    • Monthly payments (PMT) = $1,264.81.

Types of Perpetuities

  • Constant Perpetuity: Series of equal payments indefinitely.

    • Present Value Formula: PV = C / r

  • Growing Perpetuity: Payments increase at a constant rate.

    • Present Value Formula: PV = C / (r - g)

Growing Annuities and Perpetuities

  • Growing Annuity: Payments that increase over time.

    • PV g = PMT × [(1 - ((1 + g) / (1 + r))n) / (r - g)]

  • Example: First payment of $5,000 increasing at 4.5%.

    • PV calculations demonstrate differing outcomes between ordinary and growing computations.

Conclusion

These key concepts, formulas, and examples are essential for understanding annuities, perpetuities, and their associated calculations in financial contexts. They enable effective planning and assessments in finance and investment sectors.