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Financial Management: The Time Value of Money - Annuities and Other Topics

The Time Value of Money - Annuities and Other Topics

Ordinary Annuities

  • An annuity is a series of equal payments made at the end of equidistant points in time over a finite period.

  • Solving for Payment (PMT) in an Ordinary Annuity: Calculate the periodic payment needed to reach a future goal. This involves rearranging the FV formula or using a financial calculator/Excel.

  • Solving for Interest Rate (i) in an Ordinary Annuity: Determine the rate of return required for an investment to grow to a specific future amount given periodic payments. Typically solved using a financial calculator or Excel.

  • Solving for Number of Periods (n) in an Ordinary Annuity: Calculate the time it takes for an annuity to reach a target future value. Easier to solve using a financial calculator or Excel.

  • The Present Value (PV) of an Ordinary Annuity: Measures the current value of a stream of future equal cash flows.

Amortized Loans

  • Definition: A loan repaid in equal periodic payments, which constitute an ordinary annuity. Examples include home mortgages and auto loans.

  • Calculation: Loan payments (PMT) are determined by treating the loan amount as the present value (PV) of an ordinary annuity.

  • Loan Amortization Schedule: Details how each payment is allocated between interest and principal over the loan's life. Early payments consist mostly of interest.

  • Monthly Payments: For loans with monthly payments, the annual interest rate (i) is divided by 12, and the number of periods (n) is multiplied by 12.

  • Outstanding Loan Balance: Equals the present value of all remaining future payments.

Annuities Due

  • Definition: An annuity where all cash flows occur at the beginning of each period.

  • Future Value (FV) Computation: The future value of an annuity due is compounded for one additional period compared to an ordinary annuity.

  • Present Value (PV) Computation: The present value of an annuity due is discounted back for one less period, meaning each cash flow is received earlier.

Perpetuities

  • Definition: An annuity that continues indefinitely (forever), having no specified maturity.

  • Two types:

    • Level Perpetuity: Payments are constant over time.

    • Growing Perpetuity: Cash flows increase at a constant rate (g) each period.

  • Calculating the Present Value of a Level Perpetuity: This involves dividing the constant payment by the interest (discount) rate.

  • Calculating the Present Value of a Growing Perpetuity: This involves considering the payment at the end of year 1, the interest rate per period (i > g), and the constant growth rate of payments.

Complex Cash Flow Streams

  • Description: Cash flow patterns that combine single payments and various types of annuities (ordinary, due, perpetuities).

  • Solution Strategy:

    • Break down the complex stream into simpler components (individual cash flows or standard annuities).

    • Calculate the present (or future) value of each individual component.

    • Sum the present (or future) values of all components to determine the total value, ensuring all values are brought to the same point in time (e.g., today for PV).

    • Utilizing a timeline is critical for visualizing and solving complex problems accurately.