MORE GENERAL ANNUITIES
CONTINUOUS ANNUITY • A special case of annuities payable more frequently than interest is convertible is one in which the frequency of payment become infinite. • Although difficult to visualize in practice a continuous annuity is of considerable theoretical and analytical significance. • It is also useful as an approximation to annuities payable with great frequency, such as daily.
Continuous Annuity
Present value Accumulated value
Example: There is $40,000 in a fund which is accumulating at 4% per annum convertible continuously. If money is withdrawn continuously at the rate of $2400 per annum, how long will the fund last?
PAYMENTS VARYING IN ARITHMETIC PROGRESSION • Annuities also have varying payments instead of considered all annuities have had level payments. • For varying annuity, the payment period and interest conversion period will be assumed equal and coincide. • Any type of varying annuity can be evaluated by taking the present value or the accumulated value of each payment separately and summing the results. • On occasion, this may be the only feasible approach. • However, there are two types of commonly-encountered varying annuities for which relatively simple expressions can be developed, and will be considered.
• “Varying annuity” and “variable annuity” are two different things. • A “variable annuity” :• A type of life annuity in which payments vary according to the investment experience of an underlying investment account, usually one invested in common stocks. • Considered to have variations which are specified in advance and do not depend on some external event. • In arithmetic progression, there are two special cases which often appear and have special notation which are increasing annuity and decreasing annuity.
PAYMENTS VARYING IN ARITHMETIC PROGRESSION Increasing Annuity
Decreasing Annuity
Present value
Present value
Accumulated value
Accumulated value
PAYMENTS VARYING IN GEOMETRIC PROGRESSION • The annuity payments themselves follow a compound rate increase or decrease. • This annuity can readily be evaluated by directly expressing the annuity value as a series with each payment multiplied by its associated present or accumulated value. • Since the payments and the present or accumulated values are both geometric progressions, the terms in the series for the annuity value constitute a new geometric progression.
Example: A series of payments is made at the beginning of each year for 20 years with the first payment being $100. Each subsequent payment through the tenth year increases by 5% from the previous payment. After the tenth payment, each payment decreases by 5% from the previous payment. Calculate the present value of these payments at the time the first payment is made using an annual effective rate of 7%. Answer to the nearest dollar.