3.3 Annuities and Perpetuities

Annuities and perpetuities are crucial financial tools for managing regular payments over time. They come in different forms, each with unique calculations for future and present values. Understanding these concepts is key for making informed decisions about investments, loans, and long-term financial planning.

Ordinary annuities, annuities due, and perpetuities each have specific formulas for calculating their values. These formulas help determine the worth of payment streams, whether they're finite or ongoing indefinitely. Mastering these calculations is essential for evaluating various financial products and opportunities.

Annuities

Ordinary annuities vs annuities due

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• Annuities involve series of equal payments made at regular intervals (monthly, quarterly, annually)
• Ordinary annuities have payments occurring at the end of each period
  • Rent payments made at the end of each month
  • Interest payments on bonds paid semi-annually
• Annuities due have payments occurring at the beginning of each period
  • Lease payments made at the beginning of each month
  • Insurance premiums paid at the start of the coverage period

Future and present value of ordinary annuities

• Future Value of an Ordinary Annuity (FVOA) calculates the sum of all payments at a future date
  • $FVOA = PMT \times \frac{(1+r)^n-1}{r}$
    • $PMT$ represents periodic payment amount
    • $r$ denotes interest rate per period
    • $n$ indicates number of periods
  • Assumes payments grow at the given interest rate
• Present Value of an Ordinary Annuity (PVOA) determines the value of all future payments discounted to the present
  • $PVOA = PMT \times \frac{1-\frac{1}{(1+r)^n}}{r}$
  • Discounts each future payment back to the present using the interest rate
  • Useful for valuing annuities, loans, or leases

Future and present value of annuities due

• Future Value of an Annuity Due (FVAD) calculates the sum of all payments at a future date
  • Each payment grows for one additional period compared to an ordinary annuity
  • $FVAD = FVOA \times (1+r)$
  • Multiplies FVOA by $(1+r)$ to account for the extra period of growth
• Present Value of an Annuity Due (PVAD) determines the value of all future payments discounted to the present
  • Each payment is discounted one period less compared to an ordinary annuity
  • $PVAD = PVOA \times (1+r)$
  • Multiplies PVOA by $(1+r)$ to account for payments occurring at the beginning of each period

Perpetuities

Perpetuities and present value calculation

• Perpetuities are annuities that continue forever with no end date
  • Preferred stock dividends expected to be paid indefinitely
  • Endowment funds that provide ongoing income to organizations
• The Present Value of a Perpetuity (PVP) calculates the value of all future perpetual payments discounted to the present
  • $PVP = \frac{PMT}{r}$
    • Assumes payments occur at the end of each period (ordinary perpetuity)
  • Divides the periodic payment by the interest rate
  • For a perpetuity due (payments at the beginning of each period), the formula is adjusted:
    • $PVP_{due} = \frac{PMT}{r} \times (1+r)$
    • Multiplies the ordinary perpetuity formula by $(1+r)$