Actuarial notation
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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.
Example notationInterest ratesFailed to parse (Missing texvc executable; please see math/README to configure.): \,i\! is the annual Effective interest rate, which is the "true" rate of interest over a year. Thus if the annual interest rate is 12% then Failed to parse (Missing texvc executable; please see math/README to configure.): \,i = 0.12\! . Failed to parse (Missing texvc executable; please see math/README to configure.): \,i^{(m)}\! is the Nominal interest rate convertible Failed to parse (Missing texvc executable; please see math/README to configure.): m times a year, and is numerically equal to Failed to parse (Missing texvc executable; please see math/README to configure.): m times the effective rate of interest over one Failed to parse (Missing texvc executable; please see math/README to configure.): m th of a year. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): \,i^{(2)}\!
is the nominal rate of interest convertible semiannually. If the effective annual rate of interest is 12%, then Failed to parse (Missing texvc executable; please see math/README to configure.): \,i^{(2)}/2 = 0.06\!
which means that the effective interest rate every six months is Failed to parse (Missing texvc executable; please see math/README to configure.): \,i = 0.06\!
is not an "exponent." It merely represents the number of interest conversions, or compounding times, per year. Semi-annual compounding, (or converting interest every six months), is frequently used in valuing bonds, (see also fixed income securities), and similar monetary financial liability instruments. Whereas home mortgages frequently convert interest monthly, in which case the effective monthly rate, convertible twelve times per year, is Failed to parse (Missing texvc executable; please see math/README to configure.): \,i^{(12)}/12\! , or 1% effective per month, again following the example above where the annual interest rate is 12% and Failed to parse (Missing texvc executable; please see math/README to configure.): \,i = 0.12\! . Effective and nominal rates of interest are not the same because interest paid in earlier measurement periods "earn" interest on interest in later measurement periods; which is called "compound" interest. That is, nominal rates of interest credit interest to an investor, (alternatively charge, or debit, interest to a debtor), more frequently than do effective rates. The result is more frequent compounding of interest income to the investor, (or interest expense to the debtor), when nominal rates are used. Failed to parse (Missing texvc executable; please see math/README to configure.): \,v\!
is the discount factor over a year, which can be obtained from Failed to parse (Missing texvc executable; please see math/README to configure.): \,v = {(1+i)}^{-1}\!
. A discount factor 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: Failed to parse (Missing texvc executable; please see math/README to configure.): \,1 \times v\! . If you need 25 in 5 years the amount of money you need now is: Failed to parse (Missing texvc executable; please see math/README to configure.): \,25 \times v^5\! . Alternatively, the discount factor is the factor that should be multiplied with the amount one year from now so as to discount to the present value of that amount. Failed to parse (Missing texvc executable; please see math/README to configure.): \,d\! is the annual effective rate of discount. From Failed to parse (Missing texvc executable; please see math/README to configure.): \,(1-d) = v = {(1+i)}^{-1}\! , the rate of discount is computed by reference to a balance of money at the end of a measurement period, (but paid or accrued at the beginning of a measurement period), which is in contrast to a rate of interest which is calculated by reference to a balance of money at the beginning of a measurement period, (but paid or accrued at the end of a measurement period). The rate of interest - the present value of 1 now, evaluated Failed to parse (Missing texvc executable; please see math/README to configure.): \,n
years before, is Failed to parse (Missing texvc executable; please see math/README to configure.): \,{(1-d)}^{n}\!
, which is analogous to the formula Failed to parse (Missing texvc executable; please see math/README to configure.): \,{(1+i)}^{n}\! for present value evaluated Failed to parse (Missing texvc executable; please see math/README to configure.): \,n years later. Failed to parse (Missing texvc executable; please see math/README to configure.): \,d^{(m)}\! , the nominal rate of discount convertible Failed to parse (Missing texvc executable; please see math/README to configure.): \,m\!
times a year, is analogous to Failed to parse (Missing texvc executable; please see math/README to configure.): \,i^{(m)}\!
. Discount is converted on an Failed to parse (Missing texvc executable; please see math/README to configure.): m th-ly basis.
increases without bound:
In this case, interest is convertible continuously.
and Failed to parse (Missing texvc executable; please see math/README to configure.): \,d\! is:
Failed to parse (Missing texvc executable; please see math/README to configure.): \,(1+i) = (1+\frac{i^{(m)}}{m})^{m} = e^{\delta} = (1-\frac{d^{(m)}}{m})^{-m} = (1-d)^{-1}\!
Failed to parse (Missing texvc executable; please see math/README to configure.): \, i > i^{(2)} > i^{(3)} > \cdots > \delta > \cdots > d^{(3)} > d^{(2)} > d
Life tablesA life table (or a mortality table) is a mathematical construction that shows the number of people alive (based on the assumptions used to build the table) at a given age, or other probabilities associated with such a construct. Failed to parse (Missing texvc executable; please see math/README to configure.): \,l_x\! is the number of people alive, relative to an original cohort, at age Failed to parse (Missing texvc executable; please see math/README to configure.): x . As age increases the number of people alive decreases. Failed to parse (Missing texvc executable; please see math/README to configure.): \,l_0\! is starting point: the number of people alive at age 0. This is known as the radix of the table. Failed to parse (Missing texvc executable; please see math/README to configure.): \omega\! is the limiting age of the mortality tables. Failed to parse (Missing texvc executable; please see math/README to configure.): \,l_n\! is zero for all Failed to parse (Missing texvc executable; please see math/README to configure.): \,n \geq \omega\! . Failed to parse (Missing texvc executable; please see math/README to configure.): \,d_x\! shows the number of people who die between age Failed to parse (Missing texvc executable; please see math/README to configure.): x and age Failed to parse (Missing texvc executable; please see math/README to configure.): x + 1 . You can calculate Failed to parse (Missing texvc executable; please see math/README to configure.): \,d_x\!
using the formula Failed to parse (Missing texvc executable; please see math/README to configure.): \,d_x = l_x - l_{x+1}\!
is the probability of death between the ages of Failed to parse (Missing texvc executable; please see math/README to configure.): x and age Failed to parse (Missing texvc executable; please see math/README to configure.): x + 1 . Failed to parse (Missing texvc executable; please see math/README to configure.): \,q_x = d_x / l_x\!
is the probability of a life age Failed to parse (Missing texvc executable; please see math/README to configure.): x surviving to age Failed to parse (Missing texvc executable; please see math/README to configure.): x + 1 .
Failed to parse (Missing texvc executable; please see math/README to configure.): \,p_x = l_{x+1} / l_x\!
Failed to parse (Missing texvc executable; please see math/README to configure.): \,p_x+q_x=1\!
Failed to parse (Missing texvc executable; please see math/README to configure.): \,_nd_x = d_x + d_{x+1} + \cdots + d_{x+n-1} = l_x - l_{x+n}\! shows the number of people who die between age Failed to parse (Missing texvc executable; please see math/README to configure.): x and age Failed to parse (Missing texvc executable; please see math/README to configure.): x + n . Failed to parse (Missing texvc executable; please see math/README to configure.): \,_nq_x\! is the probability of death between the ages of Failed to parse (Missing texvc executable; please see math/README to configure.): x and age Failed to parse (Missing texvc executable; please see math/README to configure.): x + n . Failed to parse (Missing texvc executable; please see math/README to configure.): \,_nq_x = _nd_x / l_x\!
is the probability of a life age Failed to parse (Missing texvc executable; please see math/README to configure.): x surviving to age Failed to parse (Missing texvc executable; please see math/README to configure.): x + n .
Failed to parse (Missing texvc executable; please see math/README to configure.): \,_np_x = l_{x+n} / l_x\!
Failed to parse (Missing texvc executable; please see math/README to configure.): \,e_x\! is the curtate expectation of life for the people alive at age Failed to parse (Missing texvc executable; please see math/README to configure.): x . This is the expected number of complete years remaining to live (you may think of it as the number of birthdays they will celebrate).
Failed to parse (Missing texvc executable; please see math/README to configure.): \,e_x = \sum_{t=1}^{\infty} \ _tp_x\!
is a linear interpolation between Failed to parse (Missing texvc executable; please see math/README to configure.): \,l_x\! and Failed to parse (Missing texvc executable; please see math/README to configure.): \,l_{x+1}\! . i.e.
Failed to parse (Missing texvc executable; please see math/README to configure.): \,l_{x+t} = (1 - t)l_x + tl_{x+1} \!
AnnuitiesThe basic symbol for the present value of an annuity is Failed to parse (Missing texvc executable; please see math/README to configure.): \,a\! . The following notation can then be added:
If the payment period of an annuity is contingent of a life event, this is known as an annuity-certain. Otherwise, it is called a life annuity. Failed to parse (Missing texvc executable; please see math/README to configure.): a_{\overline{n|}i} (read a-angle-n-i)represents the present value of an annuity-immediate, which is a series of unit payment at the end of each year for Failed to parse (Missing texvc executable; please see math/README to configure.): n years. This value is obtained from:
Failed to parse (Missing texvc executable; please see math/README to configure.): \,a_{\overline{n|}i} = v + v^2 + \cdots + v^n = \frac{1-v^n}{i}
represents the present value of an annuity-due, which is a series of unit payment at the beginning of each year for Failed to parse (Missing texvc executable; please see math/README to configure.): n years. This value is obtained from:
Failed to parse (Missing texvc executable; please see math/README to configure.): \ddot{a}_{\overline{n|}i} = 1 + v + \cdots + v^{n-1} = \frac{1-v^n}{d}
is added to the top-right corner, it represents the present value of an annuity whose payment of is made every one Failed to parse (Missing texvc executable; please see math/README to configure.): m th of a year for a total number of Failed to parse (Missing texvc executable; please see math/README to configure.): n years, and each payment is one Failed to parse (Missing texvc executable; please see math/README to configure.): m th of a unit.
Failed to parse (Missing texvc executable; please see math/README to configure.): a_{\overline{n|}i}^{(m)} = \frac{1-v^n}{i^{(m)}}
, Failed to parse (Missing texvc executable; please see math/README to configure.): \ddot{a}_{\overline{n|}i}^{(m)} = \frac{1-v^n}{d^{(m)}}
is the limiting value of Failed to parse (Missing texvc executable; please see math/README to configure.): \,a_{\overline{n|}i}^{(m)}
when Failed to parse (Missing texvc executable; please see math/README to configure.): m
increases without bound. The underlying annuity is known as a continuous annuity.
Failed to parse (Missing texvc executable; please see math/README to configure.): \overline{a}_{\overline{n|}i}= \frac{1-v^n}{\delta}
Failed to parse (Missing texvc executable; please see math/README to configure.): 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}
which represents the rate of interest may be replaced by Failed to parse (Missing texvc executable; please see math/README to configure.): d or Failed to parse (Missing texvc executable; please see math/README to configure.): \delta , and is often omitted if the rate is clearly known under the context.
Life annuitiesLife annuities are those contingent on the death of the annuitant. The age of the annuitant is important information when we want to calculate the actuarial present value of the annuities.
For example: Failed to parse (Missing texvc executable; please see math/README to configure.): \,a_{65}\! indicates an annuity of 1 unit per year payable at the end of each year until death to someone currently age 65 Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): {\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: Failed to parse (Missing texvc executable; please see math/README to configure.): a_{x:\overline{n|}i}^{(m)} , where Failed to parse (Missing texvc executable; please see math/README to configure.): x is the age of the annuitant, Failed to parse (Missing texvc executable; please see math/README to configure.): n is the number of years of guaranteed payments, and Failed to parse (Missing texvc executable; please see math/README to configure.): m is the number of payments per year, and Failed to parse (Missing texvc executable; please see math/README to configure.): i is the interest rate. In the interest of simplicity the notation is limited and cannot show:
Life insuranceThe basic symbol for Life Insurance is Failed to parse (Missing texvc executable; please see math/README to configure.): \,A\! . The following notation can then be added:
) means the benefit is payable at the end of the period indicated (12 for monthly; 4 for quarterly; 2 for semi-annually; 365 for daily).
For example: Failed to parse (Missing texvc executable; please see math/README to configure.): \,A_x\! indicates a life insurance benefit of 1 payable at the end of the year of death. Failed to parse (Missing texvc executable; please see math/README to configure.): \,A_x^{(12)}\! indicates a life insurance benefit of 1 payable at the end of the month of death. Failed to parse (Missing texvc executable; please see math/README to configure.): \,\overline{A}_x\! indicates a life insurance benefit of 1 payable at the (mathematical) instant of death. See also
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