Editing Actuarial notation (section)

===Annuities===
[[Image:Annuities actuarial notation.svg|thumb|upright=1.9|Illustration of the payment streams represented by actuarial notation for annuities.]]
The basic symbol for the present value of an [[annuity]] is <math>\,a</math>. The following notation can then be added:

* Notation to the top-right indicates the frequency of payment (i.e., the number of annuity payments that will be made during each year). A lack of such notation means that payments are made annually.
* Notation to the bottom-right indicates the age of the person when the annuity starts and the period for which an annuity is paid.
* Notation directly above the basic symbol indicates when payments are made. Two dots indicates an annuity whose payments are made at the beginning of each year (an "annuity-due"); a horizontal line above the symbol indicates an annuity payable continuously (a "continuous annuity"); no mark above the basic symbol indicates an annuity whose payments are made at the end of each year (an "annuity-immediate").

If the payments to be made under an annuity are independent of any life event, it is known as an [[Annuity (European financial arrangements)#Annuity certain|annuity-certain]]. Otherwise, in particular if payments end upon the [[beneficiary]]'s death, it is called a [[life annuity]].

<math>a_{\overline{n|}i}</math> (read ''a-angle-n at i'') represents the present value of an annuity-immediate, which is a series of unit payments at the ''end'' of each year for <math>n</math> years (in other words: the value one period before the first of ''n'' payments). This value is obtained from:

:<math>\,a_{\overline{n|}i} = v + v^2 + \cdots + v^n = \frac{1-v^n}{i}</math>
(<math>i</math> in the denominator matches with 'i' in immediate)

<math>\ddot{a}_{\overline{n|}i}</math> represents the present value of an annuity-due, which is a series of unit payments at the ''beginning'' of each year for <math>n</math> years (in other words: the value at the time of the first of ''n'' payments). This value is obtained from:

:<math>\ddot{a}_{\overline{n|}i} = 1 + v + \cdots + v^{n-1} = \frac{1-v^n}{d}</math>
(<math>d</math> in the denominator matches with 'd' in due)

<math>\,s_{\overline{n|}i}</math> is the value at the time of the last payment, <math>\ddot{s}_{\overline{n|}i}</math> the value one period later.

If the symbol <math>\,(m)</math> is added to the top-right corner, it represents the present value of an annuity whose payments occur each one <math>m</math>th of a year for a period of <math>n</math> years, and each payment is one <math>m</math>th of a unit.

:<math>a_{\overline{n|}i}^{(m)} = \frac{1-v^n}{i^{(m)}}</math>, <math>\ddot{a}_{\overline{n|}i}^{(m)} = \frac{1-v^n}{d^{(m)}}</math>

<math>\overline{a}_{\overline{n|}i}</math> is the limiting value of <math>\,a_{\overline{n|}i}^{(m)}</math> when <math>m</math> increases without bound. The underlying annuity is known as a [[continuous annuity]].

:<math>\overline{a}_{\overline{n|}i}= \frac{1-v^n}{\delta}</math>

The present values of these annuities may be compared as follows:

:<math>a_{\overline{n|}i} < a_{\overline{n|}i}^{(m)} < \overline{a}_{\overline{n|}i} < \ddot{a}_{\overline{n|}i}^{(m)}< \ddot{a}_{\overline{n|}i}</math>

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 <math>i</math> which represents the rate of interest may be replaced by <math>d</math> or <math>\delta</math>, 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.