6.2 Annuities and Perpetuities

Time value of money concepts extend to annuities and perpetuities, which are series of equal payments. These financial tools are crucial for valuing streams of cash flows, like loan payments or rental income, over specific time periods or indefinitely.

Understanding annuities and perpetuities helps in financial decision-making. We'll explore different types, their present and future values, and how to calculate payments. This knowledge is essential for evaluating investments, loans, and long-term financial planning.

Annuity Types

Types of Annuities and Perpetuities

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• Annuity: Series of equal periodic payments or receipts over a fixed time period
  • Can be either an asset or a liability depending on whether the cash flows are inflows (asset) or outflows (liability)
  • Examples include car payments, mortgage payments, or rental income
• Ordinary Annuity: Annuity where the cash flows occur at the end of each period
  • Most common type of annuity
  • Cash flow timing aligns with the end of the compounding periods (annually, semi-annually, quarterly, monthly)
• Annuity Due: Annuity where the cash flows occur at the beginning of each period
  • Less common than ordinary annuities
  • Examples include rent payments or insurance premiums paid in advance
• Perpetuity: Annuity with an infinite time horizon where the periodic payments or receipts continue forever
  • Often used to value preferred stock which has no maturity date
• Growing Perpetuity: Perpetuity where the periodic payment or receipt grows at a constant rate each period
  • Example is a perpetual preferred stock with a fixed dividend growth rate

Annuity Valuation

Present Value of Annuities and Perpetuities

• Present Value of an Annuity: The value today of a series of equal periodic future cash flows discounted at the appropriate discount rate
  • Calculated using the formula: $PV = C \times \frac{1 - (1 + r)^{-n}}{r}$ where $C$ is the periodic cash flow, $r$ is the discount rate per period, and $n$ is the number of periods
  • Key inputs are the cash flow amount, discount rate, and time horizon
  • Allows for the comparison of an annuity cash flow stream to a lump sum amount
• Present Value of a Perpetuity: The value today of a series of equal periodic future cash flows that continue forever
  • Calculated using the formula: $PV = \frac{C}{r}$ where $C$ is the periodic cash flow and $r$ is the discount rate per period
  • Mathematically, the $\frac{1 - (1 + r)^{-n}}{r}$ term approaches $\frac{1}{r}$ as $n$ approaches infinity
  • Used in valuing preferred stock or ground leases with no maturity date

Future Value of an Annuity

• Future Value of an Annuity: The sum of a series of equal periodic cash flows accumulated to a future point in time at a given rate of return
  • Calculated using the formula: $FV = C \times \frac{(1 + r)^n - 1}{r}$ where $C$ is the periodic cash flow, $r$ is the rate of return per period, and $n$ is the number of periods
  • Represents the cumulative value of an annuity at the end of the annuity term
  • Can be used to solve for the required periodic contributions needed to accumulate to a target future amount

Solving for Annuity Payment

• Annuity Payment: The periodic cash flow of an annuity
  • Can be calculated by rearranging the present value of an annuity formula to solve for $C$: $C = PV \times \frac{r}{1 - (1 + r)^{-n}}$ where $PV$ is the present value, $r$ is the discount rate per period, and $n$ is the number of periods
  • Allows for structuring an annuity to achieve a target present or future value
  • Often used in calculating required loan payments such as mortgages or car loans