Annuities are…

  • Money you borrowed: mortgages, etc. ⇒ pay regularly to pay off large amount
  • Money you lent: social security credit, etc. ⇒ give large money, get regular payments later

Amortization is the same thing but from a different perspective ⇒ You’re spreading a one-time cost into multiple years. Formulae are same if the recurring payments are of the same amount.

Types of Annuities

  • Ordinary Annuity: payment occurs at the end of each period
  • Simple Ordinary Annuity:

Formulae for Constant Payment

  • : Term; time from first payment to last payment (in years)
  • : Present value of annuity of terms.
  • : Total payment accrued after periods (not a present value!)
  • for formula simplification
  1. Preset value of annuity (=) formula. (see below for more)
    • (i.e. the principal loan amount, mortgage loan amount, etc.)
  1. Payment accrued after all periods (= =Future value of annuity after all payments made) (Demonstration, use geometric sum formula)
  1. Simplified formula of PV using 1. and 2.
  1. Remaining balance after periods (=)

Portion of the -th payment that goes to… Principal

  • Total Principal ⇒


  1. Every period, interest is fully paid.
  2. Interest is applied to the remaining balance (not to be confused with )
  1. Total Interest ⇒

⇒ Thus the total payment:

  • Consider -periodic compounding (= times every year)
  • Cost of Loan: ignoring the time value of money, sum of all interest payments
  • As montly payment increases, number of payment periods decays exponentially.

Formulae for Variable Interest Rates

Variables are same as above, except:

  • : payment after period
  • : interest rate during period
  1. Present value of annuity ()