Actuarial notation - AbsoluteAstronomy.com

Actuarial notation is a shorthand method to allow actuaries

Actuary

An actuary is a business professional who deals with the financial impact of risk and uncertainty. Actuaries provide expert assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms ....

to record mathematical formulas that deal with interest rates

Interest

Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money, or money earned by deposited funds....

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, a standard linear system has yet to emerge.

Interest rates

is the annual effective interest rate

Effective interest rate

The effective interest rate, effective annual interest rate, annual equivalent rate or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears.It is used to compare the...

, which is the "true" rate of interest over a year. Thus if the annual interest rate is 12% then

is the nominal interest rate

Nominal interest rate

In finance and economics nominal interest rate or nominal rate of interest refers to the rate of interest before adjustment for inflation ; or, for interest rates "as stated" without adjustment for the full effect of compounding...

convertible

times a year, and is numerically equal to

times the effective rate of interest over one

^th of a year. For example,

is the nominal rate of interest convertible semiannually. For example, if the effective annual rate of interest is 12%, then

represents the effective interest rate every six months. Since

, we have

and hence

. The ^"(n)" appearing in the symbol

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

Bond (finance)

In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...

(see also fixed income securities) and similar monetary financial liability instruments, whereas home mortgages frequently convert interest monthly. Following the above example again where

, we have

since

.

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

Debit

Debit and credit are the two aspects of every financial transaction. Their use and implication is the fundamental concept in the double-entry bookkeeping system, in which every debit transaction must have a corresponding credit transaction and vice versa.Debits and credits are a system of notation...

, 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.

is the present value

Present value

Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...

of 1 at interest rate i or discount factor over a year, (since it is multiplied by an amount of money one year in the future to obtain the value of this aforementioned amount today); that is, it can be obtained from

.
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:

. If you need 25 in 5 years the amount of money you need now is:

.

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

Present value

of that amount.

is the annual effective discount rate. From

, 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

years before, is

, which is analogous to the formula

for present value evaluated

years later.

, the nominal rate of discount convertible

times a year, is analogous to

. Discount is converted on an

^th-ly basis.

, the force of interest, is the limiting value of the nominal rate of interest when

increases without bound:

δ = Lim _m→∞

In this case, interest is convertible continuously.

The general relationship between

and

is:

And their numerical value is compared as follows:

Life tables

A life table

Life table

In actuarial science, a life table is a table which shows, for each age, what the probability is that a person of that age will die before his or her next birthday...

(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.

is the number of people alive, relative to an original cohort, at age

. As age increases the number of people alive decreases.

is starting point: the number of people alive at age 0. This is known as the radix of the table.

is the limiting age of the mortality tables.

is zero for all

shows the number of people who die between age

and age

. You can calculate

using the formula

is the probability of death between the ages of

and age

is the probability of a life age

surviving to age

Since the only possible alternatives from one year (age

) to the next (age

) are living or dying, the relationship between these two probabilities is:

These symbols also extend to multiple years, by adding the number of years at the bottom left of the basic symbol.

shows the number of people who die between age

and age

is the probability of death between the ages of

and age

is the probability of a life age

surviving to age

An important information that can be obtained from the life table is the life expectancy

Life expectancy

Life expectancy is the expected number of years of life remaining at a given age. It is denoted by ex, which means the average number of subsequent years of life for someone now aged x, according to a particular mortality experience...

is the curtate expectation of life for the people alive at age

. This is the expected number of complete years remaining to live (you may think of it as the number of birthdays they will celebrate).

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,

is a linear interpolation

Linear interpolation

Linear interpolation is a method of curve fitting using linear polynomials. Lerp is an abbreviation for linear interpolation, which can also be used as a verb .-Linear interpolation between two known points:...

between

and

. i.e.

Annuities

The basic symbol for the present value of an annuity

Annuity (finance theory)

The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...

. 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.

If the payment period of an annuity is independent of any life event, this is known as an annuity-certain. Otherwise, in particular if payments end on the beneficiary

Beneficiary

A beneficiary in the broadest sense is a natural person or other legal entity who receives money or other benefits from a benefactor. For example: The beneficiary of a life insurance policy, is the person who receives the payment of the amount of insurance after the death of the insured...

's death, it is called a life annuity

Life annuity

A life annuity is a financial contract in the form of an insurance product according to which a seller — typically a financial institution such as a life insurance company — makes a series of future payments to a buyer in exchange for the immediate payment of a lump sum or a series...

(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

years (in other words: the value one period before the first of n payments). This value is obtained from:

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:

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 payment of is made every one

^th of a year for a total number 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 value of these annuities are compared as follows:

because cash flows at later time has a smaller present value compared with the cash flows of the same size but at earlier time.

The subscript which represents the rate of interest may be replaced by or , and is often omitted if the rate is clearly known under 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 not longer be valid, and particular formulas can be developed for particular movements of the rate.

Life annuities

Life 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

Actuarial present value

In actuarial science, the actuarial present value of a payment or series of payments which are random variables is the expected value of the present value of the payments, or equivalently, the present value of their expected values....

of the annuities.

The age of the annuitant is put at the bottom-right, without the "angle".

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 the 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 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:

, where

is the age of the annuitant,

is the number of years of guaranteed payments, and

is the number of payments per year, and

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
The Actuarial Present Value of life contingent payments can be treated as the mathematical expectation of the present value random variable, or calculated through the current payment form.

Life insurance

The basic symbol for life insurance

Life insurance

Life insurance is a contract between an insurance policy holder and an insurer, where the insurer promises to pay a designated beneficiary a sum of money upon the death of the insured person. Depending on the contract, other events such as terminal illness or critical illness may also trigger...

. The following notation can then be added:

Notation to the top-right indicates the timing of death payment. A lack of notation means payments are made at the end of the year of death. A figure in parenthesis (for example ) 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 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 nothing indicates payment at the end of the period indicated.

For example:

indicates a life insurance benefit of 1 payable at the end of the year of death.

indicates a life insurance benefit of 1 payable at the end of the month of death.

indicates a life insurance benefit of 1 payable at the (mathematical) instant of death.

Force of mortality

Among actuaries, force of mortality refers to what economists and other social scientists call the hazard rate and is construed as an instantaneous rate of mortality at a certain age measured on an annualized basis.

In a life table

Life table

In actuarial science, a life table is a table which shows, for each age, what the probability is that a person of that age will die before his or her next birthday...

, we consider the probability of a person dying from age (x), that is, a person age x, to (x+1), a person age x + 1, called q_x. In the continuous case, we could also consider the conditional probability

Conditional probability

In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

of a person, who attained age (x), dying from age (x) to age (x+Δx) as:

where F_X(x) is the distribution function

Distribution function

In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...

of the continuous age-at-death random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

, X. As Δx tends to zero, so does this probability in the continuous case. The approximate force of mortality is this probability divided by Δx. If we let Δx tend to zero, we get the function for force of mortality, denoted as μ(x):

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Interest rates

Life tables

Annuities

Life annuities

Life insurance

Force of mortality

See also

External links