Overview
This lecture covers the fundamentals of simple annuities, including definitions, distinguishing features, key formulas, and step-by-step sample problems to compute future values, present values, and periodic payments.
Simple vs. General Annuities
- A simple annuity has equal payment intervals and interest periods.
- A general annuity has payment intervals different from the interest period.
Types of Annuities
- Ordinary annuity: payments made at the end of each interval.
- Annuity due: payments made at the beginning of each interval.
- Annuity certain: payments occur for a definite period.
- Contingent annuity: payments continue for an indefinite period.
Key Concepts & Formulas
- Future value (F) of a simple ordinary annuity: ( F = R \times \frac{(1 + i)^n - 1}{i} )
- Present value (P) of a simple annuity: ( P = R \times \frac{1 - (1 + i)^{-n}}{i} )
- Periodic payment (R) from future value: ( R = \frac{F \times i}{(1 + i)^n - 1} )
- Periodic payment (R) from present value: ( R = \frac{P \times i}{1 - (1 + i)^{-n}} )
- ( i = \frac{r}{m} ) where r = nominal interest rate, m = number of compounding periods per year.
- ( n = m \times t ) where t = years, n = total number of payments.
Problem-Solving Steps (Examples)
- Identify payment interval, compounding frequency, and whether it is a simple or general annuity.
- Substitute known values into the relevant formula.
- Use a calculator for exponents and fractions.
- For cash price of an item with down payment and installment, sum down payment and present value of installments.
Worked Examples
- Example: Monthly savings of 3,000 at 9% compounded monthly for 6 months produces a future value of 18,340.89.
- Example: 200 saved each month at 0.25% compounded monthly for 6 years grows to 14,507.02.
- Example: Retirement withdrawals of 36,000 quarterly for 20 years at 12% compounded quarterly require a deposit of 1,087,227.48.
- Example: Buying a car with 200,000 down payment, 16,200 monthly for 5 years, and 10.5% interest results in a total price of 953,702.20.
- Example: Borrowing 100,000 repaid in three annual equal payments at 8% requires 38,803.35 per year.
- Example: To amass 500,000 in 12 years at 1% compounded semiannually, deposit 19,660.31 every six months.
Key Terms & Definitions
- Simple Annuity — annuity with matching payment and interest periods.
- General Annuity — annuity with differing payment and interest periods.
- Ordinary Annuity — payment at the end of each interval.
- Annuity Due — payment at the beginning of each interval.
- Future Value (F) — total value of all payments at the end of the annuity.
- Present Value (P) — value today of all future payments.
- Periodic Payment (R) — regular payment amount every interval.
- Interest Rate per Period (i) — nominal rate divided by compounding periods.
- Term (t) — length of the annuity in years.
- Total Number of Payments (n) — number of periods times years.
Action Items / Next Steps
- Practice using provided formulas with a calculator.
- Identify the type of annuity before solving problems.
- Review sample problems and attempt similar exercises for mastery.