Annuities and perpetuities are key concepts in finance, helping us understand the value of regular payments over time. They're essential for everything from retirement planning to valuing investments, showing how money's value changes with time.

These tools let us calculate the worth of payment streams, whether they end after a set period or continue indefinitely. By grasping annuities and perpetuities, you'll be better equipped to make smart financial decisions and understand complex financial products.

Annuities: Ordinary vs Due

Defining Annuities

• An annuity is a series of equal payments or receipts that occur at evenly spaced intervals over a fixed period of time
  • Annuities are commonly used in financial planning, such as retirement income (pension payments) or loan repayments (mortgage payments)
• The two main types of annuities are ordinary annuities and annuities due, which differ in the timing of the payments or receipts

Ordinary Annuities

• Ordinary annuities have payments or receipts occurring at the end of each period
  • Example: A rental agreement where the tenant pays rent at the end of each month
• The future value and present value calculations for ordinary annuities assume that payments occur at the end of each period

Annuities Due

• Annuities due have payments or receipts occurring at the beginning of each period
  • Example: An insurance policy where premiums are paid at the beginning of each coverage period
• The future value and present value calculations for annuities due must account for the fact that payments occur at the beginning of each period
  • This typically results in higher future and present values compared to ordinary annuities, all else being equal

Importance of Distinguishing Between Annuity Types

• The distinction between ordinary annuities and annuities due is important because it affects the timing of cash flows and the calculation of their present and future values
• Using the wrong formula or assumption about payment timing can lead to inaccurate valuation and decision-making
  • For example, assuming an annuity due is an ordinary annuity would underestimate its true value

Future and Present Value of Annuities

Future Value of an Ordinary Annuity

• The future value of an ordinary annuity is the sum of all the payments at the end of the annuity term, assuming the payments are invested at a given interest rate
• The formula for the future value of an ordinary annuity is: $FVA = PMT × [(1 + r)^n - 1] / r$
  • $PMT$ is the periodic payment
  • $r$ is the interest rate per period
  • $n$ is the number of periods
• Example: If you invest $1,000 at the end of each year for 5 years at an annual interest rate of 5%, the future value of this ordinary annuity would be $5,525.63

Present Value of an Ordinary Annuity

• The present value of an ordinary annuity is the sum of all the payments discounted back to the present at a given discount rate
• The formula for the present value of an ordinary annuity is: $PVA = PMT × [1 - (1 + r)^{-n}] / r$
  • $PMT$ is the periodic payment
  • $r$ is the discount rate per period
  • $n$ is the number of periods
• Example: If you receive $1,000 at the end of each year for the next 5 years and the annual discount rate is 5%, the present value of this ordinary annuity would be $4,329.48

Applying Future and Present Value Concepts

• Understanding the future and present value of annuities is crucial for making informed financial decisions
  • Saving for retirement: Calculating how much you need to save each period to reach a target future value
  • Valuing investments: Determining the present value of expected future cash flows from an investment
• The choice between an ordinary annuity and an annuity due can have a significant impact on the outcome of these calculations

Future and Present Value of Annuities Due

Calculating the Future Value of an Annuity Due

• The future value of an annuity due is the sum of all the payments at the end of the annuity term, assuming the payments are invested at a given interest rate and the first payment occurs immediately
• The formula for the future value of an annuity due is: $FVAD = FVA × (1 + r)$
  • $FVA$ is the future value of an ordinary annuity
  • $r$ is the interest rate per period
• Example: Using the same example as the ordinary annuity, if you invest $1,000 at the beginning of each year for 5 years at an annual interest rate of 5%, the future value of this annuity due would be $5,801.91

Calculating the Present Value of an Annuity Due

• The present value of an annuity due is the sum of all the payments discounted back to the present at a given discount rate, with the first payment occurring immediately
• The formula for the present value of an annuity due is: $PVAD = PVA × (1 + r)$
  • $PVA$ is the present value of an ordinary annuity
  • $r$ is the discount rate per period
• Example: Using the same example as the ordinary annuity, if you receive $1,000 at the beginning of each year for the next 5 years and the annual discount rate is 5%, the present value of this annuity due would be $4,545.95

Comparing Annuities Due to Ordinary Annuities

• Annuities due have higher future and present values compared to ordinary annuities, all else being equal
  • This is because payments in an annuity due occur earlier, allowing for more time to earn interest or less discounting
• When deciding between an ordinary annuity and an annuity due, consider factors such as:
  • The timing of your cash flows and when you need the money
  • The interest rate environment and expected returns on investments
  • Any tax implications of receiving payments at different times

Perpetuities and Present Value

Understanding Perpetuities

• A perpetuity is an annuity that continues forever, with no end date
  • Perpetuities are a theoretical concept, as no investment truly lasts forever
• The present value of a perpetuity is the sum of the infinite series of discounted payments
• The formula for the present value of a perpetuity is: $PV = PMT / r$
  • $PMT$ is the periodic payment
  • $r$ is the discount rate per period
• Example: If a perpetuity pays $1,000 per year and the annual discount rate is 5%, the present value of this perpetuity would be $20,000

Applications of Perpetuities

• Perpetuities are often used to value securities with no maturity date, such as preferred stock or consols
  • Preferred stock: Pays a fixed dividend indefinitely, making it similar to a perpetuity
  • Consols: Government bonds with no maturity date, paying a fixed coupon forever
• The concept of a perpetuity can also be used to value long-term leases or other contracts with no specified end date

Growing Perpetuities

• A growing perpetuity is a perpetuity with payments that grow at a constant rate over time
• The formula for the present value of a growing perpetuity is: $PV = PMT / (r - g)$
  • $PMT$ is the initial periodic payment
  • $r$ is the discount rate per period
  • $g$ is the growth rate of the payments
• Example: If a growing perpetuity pays $1,000 in the first year and the payments grow at an annual rate of 2%, with a discount rate of 7%, the present value would be $20,000
• Growing perpetuities are useful for valuing investments with cash flows that are expected to increase over time, such as dividend-paying stocks with consistent growth