Actuarial notation

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Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with
interest rates An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
and
life tables In actuarial science and demography Demography () is the statistics, statistical study of populations, especially human beings. Demographic analysis examines and measures the dimensions and Population dynamics, dynamics of populations; ...
. Traditional notation uses a halo system where symbols are placed as
superscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the Baseline (typog ...
or
subscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the Baseline (typog ...
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, a standard linear system has yet to emerge.

# Example notation

## Interest rates

$\,i$ is the annual
effective interest rate The effective interest rate (EIR), effective annual interest rate, annual equivalent rate (AER) or simply effective rate is the percentage of interest rate, interest on a loan or financial product if compound interest accumulates over a year duri ...
, which is the "true" rate of interest over ''a year''. Thus if the annual interest rate is 12% then $\,i = 0.12$. $\,i^$ (pronounced "i ''upper'' m") is the
nominal interest rate In finance and economics, the nominal interest rate or nominal rate of interest is the rate of interest stated on a loan or investment, without any adjustments or fees. Examples of adjustments or fees # An adjustment for inflation(in contrast with ...
convertible $m$ times a year, and is numerically equal to $m$ times the effective rate of interest over one $m$''th'' of a year. For example, $\,i^$ is the nominal rate of interest convertible semiannually. If the effective annual rate of interest is 12%, then $\,i^/2$ represents the effective interest rate every six months. Since $\,\left(1.0583\right)^=1.12$, we have $\,i^/2=0.0583$ and hence $\,i^=0.1166$. The "(m)" appearing in the symbol $\,i^$ is not an "
exponent Exponentiation is a mathematics, mathematical operation (mathematics), operation, written as , involving two numbers, the ''Base (exponentiation), base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". W ...
." 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 A mortgage loan or simply mortgage (), in civil law (legal system), civil law jurisdicions known also as a hypothec loan, is a loan used either by purchasers of real property to raise funds to buy real estate, or by existing property owners ...
frequently convert interest monthly. Following the above example again where $\,i=0.12$, we have $\,i^=0.1139$ since $\,\left\left(1+\frac\right\right)^=1.12$. Effective and nominal rates of interest are not the same because interest paid in earlier measurement periods "earns" interest in later measurement periods; this is called compound interest. That is, nominal rates of interest credit interest to an investor, (alternatively charge, or
debit Debits and credits in double-entry bookkeeping are entries made in Account (accountancy), account ledgers to record changes in Value (economics), value resulting from business transactions. A debit entry in an account represents a transfer of v ...
, 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. The symbol $\,v$ represents the
present value In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
of 1 to be paid one year from now: :$\,v = ^\approx 1-i+i^2$ This present value factor, or discount factor, is used to determine 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 should invest now is: $\,1 \times v$. If you need 25 in 5 years the amount of money you should invest now is: $\,25 \times v^5$. $\,d$ is the
annual effective discount rate The annual effective discount rate expresses the amount of interest paid or earned as a ''percentage'' of the balance at the ''end'' of the annual period. It is related to but slightly smaller than the effective rate of interest, which expresses t ...
: :$d = \frac\approx i-i^2$ The value of $\,d$ can also be calculated from the following relationships: $\,\left(1-d\right) = v = ^$ The rate of discount equals the amount of interest earned during a one-year period, divided by the balance of money at the end of that period. By contrast, an annual effective rate of interest is calculated by dividing the amount of interest earned during a one-year period by the balance of money at the beginning of the year. The present value (today) of a payment of 1 that is to be made $\,n$ years in the future is $\,^$. This is analogous to the formula $\,^$ for the future (or accumulated) value $\,n$ years in the future of an amount of 1 invested today. $\,d^$, the nominal rate of discount convertible $\,m$ times a year, is analogous to $\,i^$. Discount is converted on an $m$''th''-ly basis. $\,\delta$, the force of interest, is the limiting value of the nominal rate of interest when $m$ increases without bound: :$\,\delta = \lim_i^$ In this case, interest is convertible continuously. The general relationship between $\,i$, $\,\delta$ and $\,d$ is: :$\,\left(1+i\right) = \left\left(1+\frac\right\right)^ = e^ = \left\left(1-\frac\right\right)^ = \left(1-d\right)^$ Their numerical value can be compared as follows: :$\, i > i^ > i^ > \cdots > \delta > \cdots > d^ > d^ > d$

## Life tables

A
life table In actuarial science and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, what the probability is that a person of that age will die before their next birthday ("probability of death ...
(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. In addition to the number of lives remaining at each age, a mortality table typically provides various probabilities associated with the development of these values. $\,l_x$ is the number of people alive, relative to an original cohort, at age $x$. As age increases the number of people alive decreases. $\,l_0$ is the starting point for $\,l_x$: the number of people alive at age 0. This is known as the radix of the table. Some mortality tables begin at an age greater than 0, in which case the radix is the number of people assumed to be alive at the youngest age in the table. $\omega$ is the limiting age of the mortality tables. $\,l_n$ is zero for all $\,n \geq \omega$. $\,d_x$ is the number of people who die between age $x$ and age $x + 1$. $\,d_x$ may be calculated using the formula $\,d_x = l_x - l_$ $\,q_x$ is the probability of death between the ages of $x$ and age $x + 1$. :$\,q_x = d_x / l_x$ $\,p_x$ is the probability that a life age $x$ will survive to age $x + 1$. :$\,p_x = l_ / l_x$ Since the only possible alternatives from one age ($x$) to the next ($x+1$) are living and dying, the relationship between these two probabilities is: :$\,p_x+q_x=1$ These symbols may also be extended to multiple years, by inserting the number of years at the bottom left of the basic symbol. $\,_nd_x = d_x + d_ + \cdots + d_ = l_x - l_$ shows the number of people who die between age $x$ and age $x + n$. $\,_nq_x$ is the probability of death between the ages of $x$ and age $x + n$. :$\,_nq_x = _nd_x / l_x$ $\,_np_x$ is the probability that a life age $x$ will survive to age $x + n$. :$\,_np_x = l_ / l_x$ Another statistic that can be obtained from a life table is
life expectancy Life expectancy is a statistical measure of the average time an organism is expected to live, based on the year of its birth, current age, and other demographic factors like sex. The most commonly used measure is life expectancy at birth ...
. $\,e_x$ is the curtate expectation of life for a person alive at age $x$. This is the expected number of complete years remaining to live (you may think of it as the expected number of birthdays that the person will celebrate). :$\,e_x = \sum_^ \ _tp_x$ A life table generally shows the number of people alive at integral ages. If we need information regarding a fraction of a year, we must make assumptions with respect to the table, if not already implied by a mathematical formula underlying the table. A common assumption is that of a Uniform Distribution of Deaths (UDD) at each year of age. Under this assumption, $\,l_$ is a
linear interpolation In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known poin ...
between $\,l_x$ and $\,l_$. i.e. :$\,l_ = \left(1 - t\right)l_x + tl_$

## Annuities

The basic symbol for the present value of an
annuity In investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose ...
is $\,a$. 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-certain. Otherwise, in particular if payments end upon the
beneficiary A beneficiary (also, in trust law, ''cestui que use'') in the broadest sense is a natural person or other legal entity who receives money or other employee benefit, benefits from a benefactor (law), benefactor. For example, the beneficiary of a lif ...
's death, it is called a
life annuity A life annuity is an annuity, or series of payments at fixed intervals, paid while the purchaser (or annuitant) is alive. The majority of life annuities are insurance products sold or issued by life insurance companies however substantial case l ...
. $a_$ (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 $n$ years (in other words: the value one period before the first of ''n'' payments). This value is obtained from: :$\,a_ = v + v^2 + \cdots + v^n = \frac$ ($i$ in the denominator matches with 'i' in immediate) $\ddot_$ represents the present value of an annuity-due, which is a series of unit payments at the ''beginning'' of each year for $n$ years (in other words: the value at the time of the first of ''n'' payments). This value is obtained from: :$\ddot_ = 1 + v + \cdots + v^ = \frac$ ($d$ in the denominator matches with 'd' in due) $\,s_$ is the value at the time of the last payment, $\ddot_$ the value one period later. If the symbol $\,\left(m\right)$ is added to the top-right corner, it represents the present value of an annuity whose payments occur each one $m$th of a year for a period of $n$ years, and each payment is one $m$th of a unit. :$a_^ = \frac$, $\ddot_^ = \frac$ $\overline_$ is the limiting value of $\,a_^$ when $m$ increases without bound. The underlying annuity is known as a continuous annuity. :$\overline_= \frac$ The present values of these annuities may be compared as follows: :$a_ < a_^ < \overline_ < \ddot_^< \ddot_$ 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 $i$ which represents the rate of interest may be replaced by $d$ or $\delta$, 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 The actuarial present value (APV) is the expected value of the present value of a contingent cash flow stream (i.e. a series of payments which may or may not be made). Actuarial present values are typically calculated for the benefit-payment or seri ...
of an annuity. * The age of the annuitant is placed at the bottom right of the symbol, without an "angle" mark. For example: $\,a_$ indicates an annuity of 1 unit per year payable at the end of each year until death to someone currently age 65 $a_$ indicates an annuity of 1 unit per year payable for 10 years with payments being made at the end of each year $a_$ indicates an annuity of 1 unit per year for 10 years, or until death if earlier, to someone currently age 65 $a_$ indicates an annuity of 1 unit per year until the earlier death of member or death of spouse, to someone currently age 65 and spouse age 64 $a_$ indicates an annuity of 1 unit per year until the later death of member or death of spouse, to someone currently age 65 and spouse age 64. $a_^$ 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 $_$ 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: $a_^$, where $x$ is the age of the annuitant, $n$ is the number of years of payments (or until death if earlier), $m$ is the number of payments per year, and $i$ is the interest rate. In the interest of simplicity the notation is limited and does not, for example, show whether the annuity is payable to a man or a woman (a fact that would typically be determined from the context, including whether the life table is based on male or female mortality rates). The Actuarial Present Value of life contingent payments can be treated as the mathematical expectation of a present value random variable, or calculated through the current payment form.

## Life insurance

The basic symbol for a
life insurance Life insurance (or life assurance, especially in the Commonwealth of Nations) is a contract between an insurance policy holder and an insurance , insurer or assurer, where the insurer promises to pay a designated beneficiary a sum of money upon ...
is $\,A$. The following notation can then be added: * Notation to the top-right indicates the timing of the payment of a death benefit. A lack of notation means payments are made at the end of the year of death. A figure in parenthesis (for example $A^$) 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). * Notation to the bottom-right indicates the age of the person when the life insurance begins. * Notation directly above the basic symbol indicates the "type" of life insurance, whether payable at the end of the period or immediately. A horizontal line indicates life insurance payable immediately, whilst no mark above the symbol indicates payment is to be made at the end of the period indicated. For example: $\,A_x$ indicates a life insurance benefit of 1 payable at the end of the year of death. $\,A_x^$ indicates a life insurance benefit of 1 payable at the end of the month of death. $\,\overline_x$ indicates a life insurance benefit of 1 payable at the (mathematical) instant of death.

The basic symbol for premium is $\,P$ or $\,\pi$. $\,P$ generally refers to net premiums per annum, $\,\pi$ to special premiums, as a unique premium.

# Force of mortality

Among actuaries, force of mortality refers to what
economists An economist is a professional and practitioner in the social sciences, social science discipline of economics. The individual may also study, develop, and apply theories and concepts from economics and write about economic policy. Within this ...
and other social scientists call the
hazard rate Survival analysis is a branch of statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, ...
and is construed as an instantaneous rate of mortality at a certain age measured on an annualized basis. In a life table, we consider the probability of a person dying between age (''x'') and age ''x'' + 1; this probability is called ''q''''x''. In the continuous case, we could also consider the
conditional probability In probability theory, conditional probability is a measure of the probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is ...
that a person who has attained age (''x'') will die between age (''x'') and age (''x'' + Δ''x'') as: : where ''F''''X''(''x'') is the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of the continuous age-at-death
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
, X. As Δ''x'' tends to zero, so does this probability in the continuous case. The approximate force of mortality is this probability divided by Δ''x''. If we let Δ''x'' tend to zero, we get the function for force of mortality, denoted as ''μ''(''x''): :$\mu\,\left(x\right)=\frac\left\{1-F_X\left(x\right)\right\}$

*
Actuarial present value The actuarial present value (APV) is the expected value of the present value of a contingent cash flow stream (i.e. a series of payments which may or may not be made). Actuarial present values are typically calculated for the benefit-payment or seri ...
* Actuarial science *
Annual percentage rate The term annual percentage rate of charge (APR), corresponding sometimes to a nominal APR and sometimes to an effective APR (EAPR), is the interest rate for a whole year (annualized), rather than just a monthly fee/rate, as applied on a loan ...
* Mathematics of finance