math lessons - Actuarial notation

Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.

Traditional notation uses a halo system where symbols are placed as superscript or subscript before or after the main letter. Example notation using the halo system can be seen below.

Various proposals have been made to adopt a linear system where all the notation would be on a single line without the use of superscripts or subscripts. Such a method would be useful for computing where representation of the halo system can be extremely difficult. However, no standard linear system has yet to emerge.

Contents

1 Example Notation

1.1 Interest Rates
1.2 Mortality Tables
1.3 Annuities

2 See also

3 External links

Example Notation

Interest Rates

$\,i\!$ is the interest rate applying over the year. Thus if the annual interest rate is 8% then $\,i = 0.08\!$

$\,i^{(12)}\!$ is the amount of interest which would be earned over a year where the interest is payable monthly. This amount is not the same as $\,i\!$ because interest paid in the early months can earn interest on itself later in the year.

$\,v\!$ is the discount rate over a year. $\,v\!$ can be obtained from the formula: $\,v = 1/(1+i)\!$ . A discount rate is used to obtain the amount of money that must be invested now in order to have a given amount of money in the future. For example if you need 1 in one year then the amount of money you need now is: $\,1 * v\!$ . If you need 25 in 5 years the amount of money you need now is: $\,25 * v^5.\!$

Mortality Tables

$\,l_x\!$ is the starting point. This shows the number of people alive at age x. As age increases the number of lives decreases.

$\,d_x\!$ shows the number of people who die between age x and age x+1. You can calculate $\,d_x\!$ using the formula $\,d_x = l_x - l_{x+1}\!$

$\,q_x\!$ is the probability of death between the ages of x and age x + 1.

$\,p_x\!$ is the probability of a life age x surviving to age x + 1.

Annuities

The basic symbol for annuities is $\,a\!$ . The following notation can then be added:

Notation to the top-right indicates the frequency of payment. A lack of notation means 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 indicates when payments are made. Two dots indicates an annuity payable at the start of the year, a horizontal line indicates an annuity payable continuously, whilst nothing indicates an annuity payable at the end of the year.

For example:

$\,a_{65}\!$ indicates an annuity of 1 unit per year payable at the end of each year until death to someone currently age 65

$a_{\overline{10|}}$ indicates an annuity of 1 unit per year payable for 10 years with payments being made at the end of the year

$a_{65:\overline{10|}}$ indicates an annuity of 1 unit per year for 10 years, or until death if earlier, to someone currently age 65

$a_{65}^{(12)}$ indicates an annuity of 1 unit per year payable 12 times a year (1/12 unit per month) until death to someone currently age 65

${\ddot{a}}_{65}$ indicates an annuity of 1 unit per year payable at the start of each year until death to someone currently age 65

or in general:

$a_{x:\overline{n|}i}^{(m)}$ , where $x$ is the age of the annuitant, $n$ is the number of years of guaranteed payments, and $m$ is the number of payments per year, and $i$ is the interest rate.

In the interest of simplicity the notation is limited and cannot show:

Whether the annuity is payable to a man or a woman

External links

Categories: Applied mathematics | Mathematical notation

01-04-2007 01:18:14
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