of each year for years (in other words: the value one period before the first of n payments). This value is obtained from: ( in the denominator matches with 'i' in immediate) represents the present value of an annuity-due, which is a series of unit payments at the beginning of each year for years (in other words: the value at the time of the first of n payments). This value is obtained from: ( in the denominator matches with 'd' in due) is the value at the time of the last payment, the value one period later. If the symbol is added to the top-right corner, it represents the present value of an annuity whose payments occur each one th of a year for a period of years, and each payment is one th of a unit. , is the limiting value of when increases without bound. The underlying annuity is known as a continuous annuity. The present values of these annuities may be compared as follows: To understand the relationships shown above, consider that cash flows paid at a later time have a smaller present value than cash flows of the same total amount that are paid at earlier times. The subscript which represents the rate of interest may be replaced by or , and is often omitted if the rate is clearly known from the context. When using these symbols, the rate of interest is not necessarily constant throughout the lifetime of the annuities. However, when the rate varies, the above formulas will no longer be valid; particular formulas can be developed for particular movements of the rate. Life annuities A life annuity is an annuity whose payments are contingent on the continuing life of the annuitant. The age of the annuitant is an important consideration in calculating the actuarial present value of an annuity. The age of the annuitant is placed at the bottom right of the symbol, without an "angle" mark. For example: indicates an annuity of 1 unit per year payable at the end of each year until death to someone currently age 65 indicates an annuity of 1 unit per year payable for 10 years with payments being made at the end of each year indicates an annuity of 1 unit per year for 10 years, or until death if earlier, to someone currently age 65 indicates an annuity of 1 unit per year