A digital computer program for the computation of commutation tables

A DIGITAL COMPUTER PROGRAM FOR THE

COMPurATION OF C0~~1UTATION TABLES

by

Doretta Ann Binner

Submitted in Partiql FulfillmPnt of

the Requirements for

the

Degree

of

BACHELOR

OF

SCIENCE

from

the

MAS.SACHUSE'I'T.SErcrs

INSrITUTE OF TECHNOLOGY

May

1956

Signature redacted

Sigm=iture

of

Author ••• .rY. vr.~...-. ~r,r. r •• ~ .-•. ,.,.. ... •. • • • •.

Signature redacted

Signature

of

Thesis .Advisor •. •.-. wwu, ... ~ .~ . • n ~ ••• • • • -• .,, • • •

Signature redacted

Professor

L. F.

Hamilton

120 Bay State Road

Boston. )K.assachusetts

May 21.

1956

Secretary of the Faculty

~..assachusetts Institute of Technology

Cambridge

39.

Massachusetts

Dear Professor Hamilton:

In accordance with the requiremPnt~ for graduation

0

I herewith submit a thesis entitled

"A

Digital Computer

Program for The Computation

Of

Commutation Tables•.

Sincerely yours.

Signature redacted

TAB3LE ( CONTNTS Letter of Transmittal Table of Contents 2 Abstract

₃

IntrOduction

₄

Statement of Proble i

6

The IBM 650 Calculator ₉

Remarks Concerning the PrQgram 10

Remarks Concerning Processing 16

Appendix ₁₈

A Listing of Program with Annotations ₁₉

B Input Data - Mortality Rates qg 27

C Computed 1 _{ortality Tables}

29

D _{Computed Commutation Tables}

₃₃

E 402 Accounting Machine Plugging 41

F Constants ₄₂

G _{Computor Code}

43

The following thesis concerns the programming for digital computors solution of computations necessary to obtain Comutation Functions on both a traditional and continuous basis given the

mortality rates for any population sample at the prevailing interest rate. These computation were carried out on an IBM 650 computer to

at least sixteen digit accuracy, the limit of accuracy of available constants*

The field of insurance is a highly competitive one. An

insur-ance customer is quite obviously thrifty and conservative in nature, and hence, quite conscious of the cost involved. Since no immediate benefit is received for payments made, life insurance is a difficult

product to sell.. The actuarial department, which is responsible for computing premiums, reserves and nonforfeitures values on both exist-ing and proposed lines of insurance is, therefore, under pressure from the field personnel. In addition the insurance business is subject to very close supervision by the states in which insurance is written.

These agencies are established for much the sane reason as banking

control agengies, i.e., to protect the publics money. They set stringent and conservative limitations upon reserve funds and other related business procedures. Conflict can easily arise between these two forces.

These combines factors motivate the insurance companies to employ elaborate formulas in the calculation of premiums, reserves, nonforfeiture values and (when the company is a mutual one) dividends. The complexity

of the formulas is limited, however, by the cost of numerical solutions.

The insurance company is alert to the possibilities of high

speed digital computers, both as an economy in computation expense and as a possible means of solving more rapidly the problems vital

to efficient management, otherwise too intricate for hand solution.

Typical of insurance problems adaptable to digital computer solutions is the computation of Commutation Tables. Normally, insur-ance formulas make use of certain factors known as Commutation Functions. These factors are based upon motality rates for a population sample

comparable to the type of people to whom the policy in question will be sold. _{These functions also involve the prevailing interest rate} and are subsequently weighed with certain cost factors to yeild _a policy's cash value, dividend, premium and the like.

Since the same mortality figures and interest rate can easily be applied to many types of life insurance policies, each with different cost factors, it is desirable to tabulate those comutation functions

with considerable accuracy for all ages*.

* It is customary and genera3lDrmost _{feasible to consider all}

The Commutation Functions are conveniently defined in terms of certain functions of the interest rate. The customary symbols and the formulas which define them follow:

rate of interest i

rate of discount d = 1/ 1+ i

present value of $1

payable at the end

of one year =- V = 1 - d = 1 / +i

forceof interest=- 5 =

ln(1+

i)

An interest rate of 2.5% was employed in these calculations. The values of the corresponding functions accurate to sixteen places may be found in Appendix F.

The mortality experience of an insurance company over a period

of ten years, when approximately smoothed, is generally expressed as

a series of percentar-es of people of a given age that may be expected

to die between the X* and (X + 1) birthdays. These percentages

for the ages X= 0 to X = 100 are denoted by

q,.

The values to be found in Appendix C and D are based on the mortality rates, q%'s, of

A) "The Commissioners 1941 Standard Ordinary Mortality Table", and

B) "1941 Intermediate Mortality Table," These can be found listed in Appendix B.

The number of people living, 1%,and the number of people

dy-ing, dx, at age X are determined from these qts by assuming a given

number of people at age zero may be expected to comply to the ortality

7

rates. These factor. may be erpress as

where 1 is te avuter of people consi:dered at g-e 0.

The three frctrrs for echage. (q,,.1 , da)

constitute the Mortality Functicns. For a listing of these see

Appendix C. From these we obtain the following Commutation Functions.

which represents the present value of a dollar to be collected.X years

from now from each person alive at age X assuming collection is made at the beginning of the year.

which reprsent the present value of a dollar to be puid X years

from now to each person dying at age X assuming payment is made at

theend of the yeer.

hich reducer to

9,S

s.x

when it i' assumed that everyone is dead at age 100 and hence.

D 0 for X 99. Similcrly,

ezo

reduces to

'e.

/W~x =f- CS

These "traditicnsl" untion bFer.e7 on priiwni collection at

8

correspond to actual occurence. These values may be adjusted based on the assumptions that 1) premiums are received continuously up to

the moment of death, 2) claims are paid at the moment death occurs

and

3)

deaths are unifromly distributed at moment-ly intervals over the year of death. The "continuous" functions are then defined ass

C,,

Ac,

D

=

D-These formulas together with the derivation of the continous from the traditional functions are taken from the "Mortality Tables and Method of Computation of Nonforfeiture Benefits Applicable to

Individual Life Insurance Policies Issued at 1948 Premium Rates by Metropolitan Life Insurance Company" pages

3

and

4.

THE

im

650 CALCULATOR

In the organization of the computor program the following properties of the IBM 650 calculator are significant:

1) Words contain ten decimal digits

2) It has rapid access (2.7 sec.) to 2000 storage registers

for either instructions or numeric factors

3)

Instructions are in one over one address form

4) Instructions may be altered by program

5)

Output is in eighty digit units (punch cards)

The program of 286 instructions is stored in bands 0200, 0250,

0300, 0500, 0600, 0650, and 0700. In addition the library subroutine

for multiplication of two twenty digit numbers to yield a twenty digit product is stored in band 1750. Bands 1800, 1900, and 1950 are used for storage of constants. The q& values for X

=

0 to X

=

99

are loaded with the program into bands 1100 and 1150.

All output instructions are addressed to 1977. The output

lay-outs are as follows

COL.

11-20

5-6

21-40

41-6o

61-70

Zeros Age lx dq gx ones "D twos Cx Mx threes * fours "

The program is entered at location 0202, the first instruction in block 10.

Block 10 computes alternately the l; nd dx values for ages

0 to

99

punching out acard for each age as it is computed (the "zero"

cards referred to -bove). The most significent ten digits of the la's

and dA's are stored consecutively in r(1000-1099) and r(1200-1299) respectively. The least significant 10 digits are correspondingly

stored in r(1100-1199) and r(1300-13)9).

Calculation of the D's and Cats in Block 20 is likewise,

altern-ated. Each D ais _{it is computed is stored in the location previously storing}

11

BLOCK 10

eOMTE&SOREPUNCH 1 A xjl R CoMPUT E d PNJVC H

SET

At,

1+5k

Com P/TE & STORE

cop"PrEl

STORE

V

.v -V 'Oeener

A sreRE

va ji -- cJ

BLOC

K

20

919-149' d2 l FE A REDVE

FIG.

I

-

I

I

12

Ii~J

[~

4

1

SEr AGE A65 BY I COMvPUTE

S,.. -+ Fx ...P sx

(A S $x

BLOCK 30

-0fIii

x

,Mu

FIG.

2

CK

I

&

13

r -- --- -- -- - -- -COMPUT Ea&SonE

COPUTE 8 SrORE7

INCREAS A xx BY

-L

BLOCK 6

BLOCK

4-0

INCREAS5 AGE BY I 90?- AGE XX N

_S57

_ACE

coneluT E

BL OCK 50

(i..,

AS SPECIAL cAsA ser ACer' conpo-TE5 STORE C't *x A0 Q-BY

0

0

B LoCK 70

'

FIlG. 3

Io3 -STOP

(:CA.q

I

114

the l, from which it was obtained. Similarly the Ca's replace the

d's in stornpe. The aopropriate exit from the summtion subroutine

is then set up and the program enters Black 30.

Block 30 then computes each Mv st-rting at age

99

and working

backward to zero. As each M is computed it and the correspnding

D, is punched out to form the series of "one"cards. The D es remain in storage.

Block 40 set the sunrnation subroutine (Block 30) to compute the4Is and punch the "two" cards similarly to the "one's".

In Block 50r(1400-15 99) are employed as temporary storage

for the first term of eachD3, When these are all computed the second term and their sum is obtained for each age .- 'The DE's thus obtained

are substitted for the DAIs in storage.

The C's obtP1ned in one step from the Cs and replacethem

in storage in Block

60.

It then exits to the surmation subroutine

to obtain the "four" cards containing CA and My..

Control is returned to block 70 to set up subroutine to sum

D%'s. It provides a stop after the "three" cards containing D, and

At nc point in the program a ts rethMsi's, ornre

stored in other than the punch registers.

The machine time required for each run is approximately

6

minutes and 40 seconds. A further breakdown is available.

Block 10 Punch first hundred cards

Block 20 Calculations

Block 30--40 Punch second and third hundred cards Block

5..60

Calculations

Block 70 Puch fourth and fifth hundred cards

TOTAL

Read out was at maximum punch speed for the

533.

evident from the momentary stopage of the conse-alights the fact that 500 cards were produced in about 285 sec. over 100 per min. Punch time accounted for over 70% of' machine time.

60

see.

50 see.

115 sec.

65

110

400 sec.

This wae

as weil as

or slightly

the total

As is evident from figure

4,preparing

a problem for computor

solution extends beyond the actual programming.

Instructions are first punched on Single Instruction Load Cards having the following layout

_QL. CONTACTS

1-10

6900040003

21-22 24 23-26 Instruction Location 27-30 8000 31-40 Instruction

Columns 23-26 and 31-40 are key punched as are the alphabetic representation of the numeric instruction code in Col. 12-15.

Veri-fication of these is taken for granted in the flow chart. The constants in columns 1-10, 21-22, and 27-30 are then gang punched.

Computor solution has been covered in the previous section. The cards are then hand separated into 1) the zeros, ages

0-99, 2) the twos, ages 99-0 and fours, ages 99-0 and

₃₎

the ones,

ages 99-0 and three, ages 99-0. These three groups, when appropri-ately sorted, are then ready for tabulation.

The wiring diagram in Appendix E will serve to explain the

layout obtained in Appendix D. The layouts of Appendices B and C

are considerably more straight forward.

DOC-

4

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42

INPUT & OUTPUT

71 PCH Punch a card

TESTS

45

BRNZ Take alternate instruction if accumulator is other than zero

TPPER ACCUMULATOR

10 AU Add upper without reset

11 SU Subtract upper without reset

21 STU Store upper in memory

60

RAU Reset and add upper

MULTIPLY-DIVIDE

19 MULT Multiply

LCWER ACCUMULATOR

15 AL Add lower without reset

16 SL Subtract lower without reset

20 STL Store lower in memory

65RAL

Reset and add lower

DISTRIBUTOR

24 STD Store distributor

69 LD Load distributor

44

MISCELLANEOUS

01 STOP Stop

22 STDA Replace positions

5-8

of distributor with corresponding position of lower accumulator

and store result

47 BROV Take alternate instruction in case of accumulator overflow and overflow on shift and count, and

Insurance

Mortality Talbes and Method OComputationf Nonforfeiture Benefits Applicable to Individual Life Insurance Policies

Issued at 1948 Premium Rates by Metropolitan Life Insureance Company

Digital Computing Equipment

IBM type

65

magnetic drum data*pressing machine

manual of operation problem-planning aids

IB-M Electric junc ardrhed d accounting machine principles 9L oteration card punch, type 24 printing card punch, type 26

sorters types 82, 80 and

75

accounting machine, type 402

A Descriotion of e IBM Bunch-Card stem by F.M. Verzuh

(Oct. 1953)