On pension fund reserves

Citation preview

on m m t o n m m

mmrnm®

by Douis Garftm

A dissertation submitted in partial fulfillment of the require­ ments for the degree of Doctor of Philosophy, in the Department of Mathematics, in the Graduate Col­ lege of the State University of Iowa May* 1942

P roQ uest N u m b e r: 10311010

All rights reserved INFORMATION TO ALL USERS The q u a lity o f this re p ro d u c tio n is d e p e n d e n t u p o n th e q u a lity o f th e c o p y su b m itte d . In th e unlikely e v e n t th a t th e a u th o r did n o t send a c o m p le te m anuscript a n d th e re a re missing p a g e s , th e s e will b e n o te d . Also, if m a te ria l h a d to b e re m o v e d , a n o te will in d ic a te th e d e le tio n .

uest, P roQ uest 10311010 Published b y P roQ uest LLC (2017). C o p y rig h t o f th e Dissertation is h e ld by th e Author. All rights reserved. This work is p ro te c te d a g a in s t u n authorized c o p y in g u n d e r Title 17, U n ited States C o d e M icroform Edition © P roQ uest LLC. P roQ uest LLC. 789 East E isenhow er P arkw ay P.O . Box 1346 Ann Arbor, Ml 48106 - 1346

Acknowledgement

I wish to thank Professor Henry L« Rietz most sincerely for the encouragement and very real assistance he has given me in the pre­ paration of this dissertation*

O0WEOT8

Chapter 1 IX

Rage Introduction • • * • • * * * • • ♦ * « *

1

On the Mature of Pena ion Fund Heeerve® * 8

I. What la a Beoerv© Spstemf « • * * * *6 2« XXX If

V

To Whom do Beserve® Belong? * *

* * #8

Soma Criteria For -A Reserve Baal® *

» • 12

Interpretation and Application of the Criteria « * * * * « • • » • * * * • «

%&

1*

Fund® of the Saving© Bank Type # «

M

2#

Fund® of the Croup Annuity Type * » IS

$•

Fund® of the Classical Type * * * * 20

4#

Employer Contributions « * • * * *

2©

Applications to Three Typical Funds* «

SI

1*

Chicago Park Employee* Annuity and Benefit Fund * • * • « • • « • * *

*32

2*

Minnesota Tcachens* Retirement Fund 44

S*

Mew Tork State Teacher® Retirement

System » « * * • » * • » • • • * *

SI

Bibliography • » » • • * * * * • « « • * *

68

1

Chapter X INTRODUCTION The terms pension fund, pension system* and retirement system* among others * have been loosely used to mean any of a great variety of schemes for the payment of retirement or death benefits to various classes of people* They will be used here to mean schemes applying to staffs of employes; the employer may be any organization* public or private# Rather typical of pension fund history is the development of the New York City Police Pension Fund*

Es­

tablished In 1857* it was the first municipal pension system 1 in the United States# It was set up originally due to a desire on the part of the public to care for the fanillies of policemen injured or ki

in the performance of duty*

but developed slowly into a superannuation or service re­ tirement plan*

Th© assets of the fund were supplied from

public subscriptions and from certain city revenues which by law were payable directly to the fund rather than to the city treasury*

FeusIons and other benefits were paid as they

matured* and since no actuarial valuations were Made * the only indication of the progress of the fund was the amount of these matured payments*

Thus* in the early years of the

3-Buck* G. B.* "PENSION AND RFTIREMRNT SYSTEMS IN THE UNITED STATES OF AMERICA11* Tran®#* Eighth Int*nat11 Cong* of Actuaries* Vol* 1 (1927)* pp* 358-369*

Jhand, since no appropriation® from the city were required. Its condition appeared to be excellent* In 1871 provision was made for small contribu­ tions from the policemen, and in 1893 these were Increased to 2% of salary*

It was becoming increasingly evident that

the revenues were still not enough, however, to keep pace with the ever-growing obligations*

t!The city thereafter

definitely adopted the policy of making good deficiencies by taxation*”^ year to year prospective annuitants became more numerous.

The maturing claims were doubly accentuated

by ag© and increasing disability on the one hand, and the absence of adequate legal provisions for meeting increasing % liabilities on the other* Finally, in 1913 an actuarial investigation of the fund was made, revealing for the first time the true cost of the benefits provided*

This report

and a similar report on the Public School Teachersr Re­ tirement Fund of th© City of lew York published in the same year attracted so much public interest th© Commission on Pensions of the City of Hew York was established to make a comprehensive survey of th© pension funds of the City and of pension system® In other states and In foreign countries. A report was published In three parte In 1917 and 1918* Subsequently all of th© lew York City pension funds were ^Report on th© Police Pension Fund of the City of Hew York, 1913, p* 27.

SIbid., p. 31

s

reorganised on an actuarial basic* Directly paralleling this trend of development as it has, the pension movement in the United States is young in comparison with European systems, but even in Europe it is only a relatively short time since actuaries began to take any significant part in pension fund operations* On© reason for the lack of actuarial participation in the early pension systems was that many of th© promoters of these systems had no knowledge of actuarial science or of th© work of actuaries.

Another significant reason, however, is that

the actuaries themselves did not want to undertake the work on such schemes on the ground that th© rules were so complicated with details that reliable estimates of costs 4 were impossible* It was not until 1908 that the first comprehensive exposition of an actuarial theory for pension funds was published by H* W, Manly*®

Within the next few

years papers were published by King and M* Lauchlan further a extending the subject* Bine© that time a great deal of work has been don© on pension fund theory and the general public is becoming more and mor© educated regarding the costs that pension systems entail* '^SietB, FUNDS, R. A,

H. I.

L. ,OH THK STATUS OF CERTAIN CURRENTPF.NSI0H A.,vol. 3 (1914), pp. 33-34.

5Manly, H. W..OH THE VALUATION OF STAFFPENSION FUNDS, J. I. A., vol. 36 (1908), p. 809. ®Klng, 0., OH STAFF PENSION FUNDS, J. I. A., vol. 39 (1905), p. 189. H*Lauchlan, J. J., ON A METHOD OF CALCULATING THi-i PRESENT VALUES OF PROSPECTIVE PKHSIONS BASED OH SALARY, ETC., T. F. A., vol. 6 (1911), pp. 8-36.

4

Although, there are many questions still unanswered regarding th© fundamental principles upon which pension systems ought to be cons true ted f it Is quite generally agreed by pension fund authorities that, except possibly for a system on a nationwide scale and involving a large pro­ portion of the population (such as th© Social Security system for example), the only way that the soundness of a retirement system can be assured is by the establishment of some sort of a reserve system*

The classical studies of

Manly, Ming, and M*Lauchlan presuppose a reserve basis; th© 1916 Report of the Illinois Pension Laws Commission says that n th© reserve plan is regarded by the Commission as the only plan by which the financial soundness of a pension fund can. be a®stared and th© proper payments for public service b© mad© at the time they are rendered, instead of being left for payment at some later time, with no certainty of their being actually paid at that timefr; and the Report on th© Pension Funds of th© City of Mew York (Part III, 1918) says, ^Benefits allowable on account of future service should be provided on a reserve basis, the amounts necessary to purchase, at the time of retirement, an employ©*s full retirement allowance, being fttecumulated by both the city and th© employe during the active service period of each employ© •** With the possible exception of large scale systems as noted above, opinion on this point remains relatively unchanged.

5

There has been much lees agreement, however, as to the methods to be used in calculating the amounts of re­ serves*

In th© classical methods developed by Manly and King

much us© is made of rates of withdrawal from the system and of salary scales*

Due to the great number of factors which

affect these rates and their intimate dependence upon variable economic conditions there has been a great deal of objection in th© past two or three decades to their use In the way In which the classical methods employ them*

Indeed,

many retirement systems have been established according to plans which do not require their us© at all*

On the other

hand, there ©re still many sound pension systems whose operations are all based, essentially, on calculations by the classical procedures* Th© present study is an effort to isolate those principles of reserve calculation upon which there seems to be more or less universal a^eement and to work out a reserve method based upon them which will eliminate as much as possible the features to which exception has been taken*

6

Chapter IX OH THE HATUHS OF PENSION FUHD RESERVES 1*

What Is a He serve System? A reserve system for a pension fund may he defined

as an arrangement for accumulating in advance funds for the payment of future benefit s.

The nature of th© benefits and

th© sources from which they are to be supported are im­ portant not only in determining methods by which such funds may be accumulated, but in deciding whether or not a reserve system Is even desirable. Xt has come to be quite generally accepted that **a contributory system of annuities is th© only one which society can permanently s u p p o r t . T h a t Is, pension systems In which th© benefits are paid for partly by th© employe and partly by the employer are considered the desirable type, especially in the case of funds for public employes. Furthermore, It Is usually agreed that contributions made by an employe belong to him and should not be forfeited under any circumstances; hence they are simply to be held In trust for him by the fund.

For such contributions, then,

there will be no question about the amount of reserves s the amount that must be on hand at any time on account of the benefits which they are to furnish Is exactly the accumulated amount of the contributions which the employe has mad©. 1WhIttall, W. J, H., ROTES OH SOME EE CERT DEFELOP ivR PR OF PENSION PROBLEMS IK AMERICA, J* I. A., vol. 51 (1910), pp .106 ff.

7

The benefits which are furnished by contributions of the employer may b© divided into three broad groups 2 death benefits, disability benefits, and superannuation benefits.

Although th© results may sometimes be combined

into a single formula, th© calculation of costs is essentially a separate problem for each of these types of benefits, and for convenience will here be considered as such* The disability benefits of pension funds are of such a nature that ail risk of disability ceases when the employe enters upon his superannuation benefits.

For this

reason it is justifiable to regard th© disability benefit as a yearly renewable term insurance for th© value, in any year, of th© obligation that would be incurred should disability occur during that year.

By this method no re*

serves are set aside for any disability benefit until it has actually been entered upon.

Using exactly similar reasoning,

th© cost of benefits payable in case of th© death of an employe before retirement may also be handled on a yearly renewable term basis* Thus the question of reserves for future benefits reduces to a consideration of reserves for superannuation benefits which are to be furnished by the employer.

It is with a method for accumulating such

reserve funds that the remainder of the discussion will be chiefly concerned.

8

2*

To Whom do Reserves Belong? Suppose an employer sets up a reserve system for

accumulating funds with which to pay superannuation benefits that he has promised to his employes*

At certain intervals

he sets aside sums of money, Invests them, for that purpose* Bo these accumulations belong to the employe, then? they belong to the employer, or to both?

Or do

Various answers

have been given to these questions, reflecting different attitudes regarding th© nature of pension funds, Early pension schemes granted retirement allowances as gratuities pure and simple*

They were furnished by the

benevolence of the employer, and hence an employe had no claim on any retirement allowance'*

V

z - y

V

_ Tj_ .01 let. a + ...•+.oik,szil

^

where sx Is the salary earned in the (x+1 ) st year of age, 1. ©., he tween ages x and x+1.

The required interpretation

is that an annuity of k* per cent of the average salary earned over a period of z~y years multiplied by the number of years of service gives the same benefit as if each year of service is thought of as earning a retirement allowance of kf per cent of th© salary earned in that year*

With this In mind, consider an employe aged x who entered tli© service at age y, in a system which gives a

pension of k1 per cent of salary for each year of service, the earliest possible retirement age for this employe being z years*

The yearly pension earned by past service is

»01k* * (

*

**4qx~1 >* having the value at age z of

.01k9 * (sy+3y+i^,**+3.x-l)

5 and if payable certainly it has

the present value •Oik1 ' (sy+Sy+it

) ag

Hence,

taking into account the probabilities of death,- disability, and withdrawal, the reserve required is 9

.Qlk* » (sy+Sy+a+ * **4‘sx~l 5 at vs**" s-xpx where z„xPx

the probability that the employe will stay

in the service and satisfy the requirements for retirement at ag© 55. According to this formula the reserve at age z will be *0 1 k ? . (Sy+Sy+a* **m* az~l ^ az* an ©^o-anfc sufficient to provide the full retirement allowance.

The reserve at any

ag© x is th© value at that time (considering th© pertinent probabilities ) of th© superannuation benefit earned by past service. Also, the formula uses only data about salaries

25

actually earned#

Inasmuch as death, disability, or with­

drawal are purely future contingencies the probabilities employed cormofc be replaced by observed data, but they may be modified from time to time and the reserves are autom­ atically adjusted by such modifications*

Therefore, this

formulation of th© reserve requirement satisfies all four criteria without any qualifications. Since the rules of nearly any retirement system may "be broken down Into a combination of rules of the three types indicated, combinations of th© methods briefly illustrated in this chapter should suffice to provide re­ serve formulas for a rather general class of retirement systems* 4*

Employer Contributions Up to this point quite a bit has been said about the

amounts which ought to be held by retirement systems as reserves.

The next natural question which arises Is,

”What plan should be used to accumulate these required amounts?” A very simple answer is that the employer ought to contribute enough in any year to complete the reserve requirements for that year,

fore specifically, the reserve

necessary for all the active members of the fund can be calculated according to the theory which has been developed. Also, It Is possible to determine the accumulated balance of past contributions made by the employer for this purpose: consists of those contributions plus Interest earnings minus the amounts which have been allocated to the support of benefits already entered upon.

The difference between

it

26

what Is needed and what is available should b© the employerfs , contribution for th© year# Some arguments against this sort of procedure are to b© anticipated#

One objection may b© to the fact that

under this plan th© contributions in respect, of any parti­ cular employe Increase from year to year#'*'

The objection

***Xt can be shown algebraically that the contributions will tend to Increase* For example, consider a simple savings bank scheme with annual contributions of 4$ of a salary* Then, in th© notation previously defined, the accumulated con­ tributions at ag© x are sx;t ~ sx~l*t~l *0 4 sx-X Also, the reserve at age x - 1 is i mZ*Lt» < O • . «• V , S ■ «* S n S .. 1l X-l 25-X l^X-l x-lst-l %-lst-l xx-l where 1 ^ is the number living at age x according to th© multiple decrement service table# At age x g VX * _ia_ H .1 8X !t

■Sc = rSc-ljt-l + L

*04sx-ll -ia- . -3^=1 J Sc 1

s

. ax-I st-1 •(■3.+ i )

*04 z~x&x sx-l

1 1 1 *tX-A

r

Xx - 1

4 . .04

a *.xpi • ©x-i

3-x Thu© th© contribution on behalf of the employe at age x would have to be, if the experience of the fund followed the service table, #04 2-OLPx ®x-l Similarly, for th© next year It would be *0 4

z-(x+l)px+l‘8X

■

46*

Reserves for membership service * new entrants Let s^ be the salary actually earned by a member in i th© year of age x to x-tl; and let denote th© probability that a member aged x will live and remain In service for t years as given by a multiple decrement service table Involving rates of death In service and withdrawal from active member­ ship for any other reason* Case 1*

y t 25 The superammation pension will be 28$ of average

©alary, to b© entered upon at ag© y-t 55*

That is, the

pension will b© of annual rent ®«r *+•**

-*■+■**

34,

*28 - •00Q (V

*

sy+i+***+sjr+34)*

If

Is the value of ag© x of a superannuation annuity of payable under th© conditions imposed on such pensions, th©

1

required reserve at age y-f 55 will b© -

#

•L

8 ^ 5 4 )^

The full amount will be available at that time If

the reserve at age x (x> y) is taken to b© (1.0)

.OOB< V

V l + .. .

vy+35-X ay+3g .

At ages x>y+35, if th© member does not choose to r©tire after 55 years of service, the reserve formula should be modified to

Case 2*

\

25 < y ^35

x-y

Th® superannuation pension which may be entered upon (sv+ sYt 1+ ***+sS9\ at age 60 will be *28 ,_ P©** year* Thus the \ "6(i-y r,l ,r/ reserve required at ages x>y will be given by

(2*0)

60-xP3C V°°’X

{V V l +***+Sx-l) *60

except that if the member stay© in service beyond ag© 60, formula (1*1) should be used thereafter* Case 3*

35 ^ y $ 45* Assuming that the member

will retire at agey-f-25,

th© reserve necessary at that time

will be

Th© full reserve will be available

if the reserve at any .

earlier ag® x Is maintained at th©

amount

( 3 *°>

y+ 2 5 - x T

v y + 2 5 "X

< *0 1 1 £ >

(V

V

1 + * * *+ S x - l } “ y + B S

Reserves for member of this class who ar© older than y-f* 25 years should b e held according to formula (1*1)*

61

Case 4*

y > 45 The retirement pension will be 1*12^ of average

salary for each year of service, retirement assumed to be at ag© 70*

At age 70 the full reserve requirement will be

.03.12(y

ayf 1+ **-+ s 69 )(70-y) a^Q = ,001S{ W

l

+** •+s69) a|0

Reserve requirements at any earlier ag© x will be according

to the formula (4*0)

70-x^x v

(*0112) (®y+®y+j+ **

If the member stay® on in the service after age 70, the re­ serve at age x, 7 0 < x * y + 2 5 , should be (4,1)

(.0112) (a + s +...+« ,) a? j y y+1 x —a x and forages x>y-+85, formula (1*1) again applies; It will b© noticed that in all four cases unless salaries continue to increase appreciably, the reserves necessary at ages sibova the assumed retirement age may actually decrease from year to year*

This implies that con­

tributions in behalf of members older than these ages will usually be very small, and there may even be a refund avail­ able to the system in some cases* Reserves for membership aervice» present teachers For members who enter the system at ag© y with m years of prior service, th© four cases will become Case 1*

y i 25+ m

Case 2*

25 + m < y £ 35+ m

Case 3*

35+

Case 4*

y > 45 + m

y = 45+ m

If th© salaries earned during the prior service period are used In determining average salary on which to base

benefits, the formulation of reserve values will be exactly the same as for new entrants except that y is replaced by y-ra wherever it appears# On th© other hand, if only salaries earned during membership service are to fo© used in calculating benefits (since there i&ay be no record of salaries paid before the system was established), the formulas corresponding to (1*0), (1*1), (2*0), (5*0), (4*0), and (4*1) respectively, to b© used similarly, will be (1*01)

y+Sb-m-x^x

^ ***+sx-X ^®y+55-m