The National Center for Health Statistics (NCHS) prepares, at 10 year intervals, life tables for the death registration states based on a census of population and deaths in a three year period containing the census year. The series of tables began with tables for the period 1900-1902. The most recent set of tables is for the period 1989-1991. The actuarial functions published in this study are based on the most recent set of tables as published in "U. S. Decennial Life Tables for 1989-91", Vol. 1, No. 1, DHHS Publication No. PHS-98-1150-1 by the National Center for Health Statistics. A printed copy may be obtained from the Government Printing Office (phone number: 202-512-1800) for a price of $3.50. The publication number which must be given to the Government Printing Office when requesting the publication is 017-022-01395-9.

The actuarial functions in this study can be used to calculate present values of expected future payments contingent upon the death or survival of a designated individual. A series of periodic payments to continue during the life of an individual is called a life annuity. A payment to be made on the death of an individual is called a life insurance.

As with the 1989-91 decennial life tables, the actuarial functions included in here were calculated by sex and for both sexes combined for the total population, the white population, the population other than white and the black population. For each race sex combinations described, actuarial functions were tabulated using interest rates of 1%, 2%, 3%, 4%, 5%, 6%, 7%, 8%, 9%, 10%, 11%, and 12%. The values were calculated to more decimal places than are shown and were rounded individually by the computer. Therefore, there may be small discrepancies from the mathematical relationships that should hold among the various columns.

Actuarial textbooks1 should be consulted for more detailed information about actuarial functions than is presented in this study.

Let *i* denote the annual assumed rate of interest expressed in decimal form (e.g., *i* = .07 if the rate is 7 percent). Then $1 deposited now will accumulate at compound interest to (1+*i*)^{n} at the end of *n* years. Conversely, the present value today of a payment of $1 due at the end of *n* years is (1+*i*)^{-n} = *v*^{n} where *v* is used to denote the reciprocal of (1+*i*). Thus, the present value of a series of payments of $1 due at the end of each of the next 10 years is given by the equation

*v* + *v*^{2} + *v*^{3} + ... + *v*^{10} .

Now suppose that the series of payments is paid at the end of the year only if an individual now exact age *x* is alive at the time the payment is to be made. In the life table *l*_{x} is the number of survivors to age *x* out of *l*_{0} births. Therefore, using the mortality experience of a specific life table, the probability of an individual exact age *x* surviving to age *x* + *t* is

*l*_{x+t} ÷ l_{x} .

The formula for the present value of the series of payments of $1 due at the end of each of the next 10 years to an individual exact age x provided that individual is alive at the time each payment is to be made becomes

(*vl*_{x+1} ÷ *l*_{x}) + (*v*^{2}*l*_{x+2} ÷ *l*_{x}) + ... + (*v*^{10}*l*_{x+10} ÷ *l*_{x}) .

If the period of payments is extended from 10 years to the lifetime of the individual the present value or life annuity, denoted *a*_{x}, is as follows

*a*_{x} = (*vl*_{x+1} ÷ *l*_{x}) + (*v*^{2}*l*_{x+2} ÷ *l*_{x}) + (*v*^{3}*l*_{x+3} ÷ *l*_{x}) + ...

where the addition continues to the end of the life table2.

The preceding formula provides a means of calculating *a*_{x}, the present value at age *x* of a life annuity of $1 per year. However, it would be laborious to make this calculation for many different ages and interest rates. Thus a shortcut method of tabulating certain auxiliary values called commutation functions is used. If each term of the right hand side of the equation were multiplied by *v*^{x} ÷ *v*^{x} the equation becomes

*a*_{x} = ((*v*^{x+1}*l*_{x+1}) ÷ (*v*^{x}*l*_{x})) + ((*v*^{x+2}*l*_{x+2}) ÷ (*v*^{x}*l*_{x})) + ...

If we define the following commutation functions

*D*_{x} = *v*^{x}*l*_{x}

and

*N*_{x} = *D*_{x} + *D*_{x+1} + *D*_{x+2} + ... ,

the formula can be written as

*a*_{x} = *N*_{x+1} ÷ *D*_{x} .

The preceding formula is a special case of the general life annuity formula. The value at age *k* of a life annuity of *R* per annum under which the first payment is made at age *y* and the last payment at age* j*-1 (all payments being contingent upon the survival of the annuitant) is

*R*(*N*_{y} - *N*_{j}) ÷ *D*_{k} .

A life insurance is a contract to pay a designated amount of money on the death of a certain individual, called the insured. In the life table, the probability of an individual exact age *x* dying between ages *x* + *t* and *x* + *t* + 1 can be expressed as

(*l*_{x+t} - *l*_{x+t+1}) ÷ *l*_{x}

or

*d*_{x+t} ÷ *l*_{x}

where *d*_{x+t} = *l*_{x+t} -* l*_{x+t+1} denotes the number of deaths between ages *x* + *t* and *x* + *t* + 1 in the life table. Then the present value, denoted *A*_{x}, of the payment of $1 on the next birthday of the insured following his date of death is as follows

*A*_{x} = (*vd*_{x} ÷ *l*_{x}) + (*v*^{2}*d*_{x+1} ÷* l*_{x}) + (*v*^{3}*d*_{x+2} ÷ *l*_{x}) + ...

Multiplying each term of the right hand side of the equation by *v*^{x} ÷ *v*^{x} yields

*A*_{x} = ((*v*^{x+1}*d*_{x}) ÷ (*v*^{x}*l*_{x})) + ((*v*^{x+2}*d*_{x+1}) ÷ (*v*^{x}*l*_{x})) + ...

Introduction of two new commutation functions,

*C*_{x} = *v*^{x+1}*d*_{x}

and

*M*_{x} = *C*_{x} + *C*_{x+1} + *C*_{x+2} + ...

gives

*A*_{x} = *M*_{x} ÷ *D*_{x} .

In fact, there is a general life insurance formula,

*R*(*M*_{y} - *M*_{j}) ÷ *D*_{k} .

which gives the present value determined at age *k* for life insurance coverage commencing at age *y* and terminating at age *j* with *R* as the amount of insurance.

1Bowers, N.L. Jr., et.al., Actuarial Mathematics (2nd ed.), Itasca,IL Society of Actuaries, 1997

2In this study, the end of the life table differed with each race-sex combination of the population. The ending age was chosen to be consistent with the values published in "U.S. Decennial Life Tables for 1989-91". Actuarial functions, however, are not shown beyond age 109.