MATH 1090 Sec. 5
Definitions: An annuity is the fixed amount of payment.
(a) An annuity certain: the payments begin and end on fixed dates. (discuss this in this section)
• An ordinary annuity (or annuity immediate): a type of annuities that has the periodic payments made at the end of the period.
Note: We will discuss the ordinary annuities that have payment intervals that coincide with the the compounding period of the interest.
• An annuity due : a type of annuities that has the periodic payments made at the beginning of the period. (See Ex.4.)
Ex. rent payments, insurance premiums, etc.
The term of an annuity due is from the first payment to the end of one period after the last payment.
(b) A contingent annuity: the payments are not regular.
Formula:
• If $R is deposited at the end of each period for n periods in an annuity that earns interest at a rate of i per period, the future value of the annuity (i.e., ordinary annuity) will be
S = R ·
(1 + i ) n − 1 i
.
• If $R is deposited at the beginning of each period for n periods in an annuity that earns interest at a rate of i per period, the future value of the annuity (i.e., annuity due) will be
S due
= R ·
(1 + i ) n − 1 i
(1 + i ) .
Ex.1 (#6): Find the future value of an ordinary annuity of $300 paid quarterly for 5 years, if the interest rate is 12%, compounded quarterly.

Ex.2 (#16) If $4000 is deposited at the end of each half year in an account that earns 6.2% compounded semiannually, how long will it be before the account contains $120,000?
Definition: A sinking fund is such that a borrower makes periodic deposits that will grow to the amount of the entire principal in the future.
Ex.3 (#14) A sinking fund is established to discharge a debt of $80,000 in 10 years. If deposits are made at the end of each 6-month period and interest is paid at the rate of 8%, compounded semiannually, what is the amount of each deposit?
Ex.4 (#22) Find the future value of an annuity due of $1500 each month for 3 years if the interest rate is
12%, compounded monthly.