CONTRIBUTION No 56

PAR / BY

Jidah A. NADER

Canada

114 UNE REGLE D'EVALUATION DE LA G ~ & R O S ~ D'UN PLAN DE RETRAHE D'ENTREPRISE, FOND^ SUR LA D U U ~

JIDAH S. NADER, PH. D., ASSURANCES I3 FINANUS, MONlRIk (CANADA)

RESUME

La durk a Ct6 largement utilisk dam SClaboration de stratbgies d'investissement associ&s aux paefeuilles de retraite. Toutefois, son utilisation dans la gestion &s engagements en matihe de retraite est rest& extr&mnent limit&. Cet dele expose une &gle f& sur la durQ, applicable l'6valuation de SCvolution de la g&.rosid d'un plan de retraite et de ses effets sur l'entreprise sponsor. La r&gle emploie une mesure de la dur& fond& sur le m d l e d'6valuatian bien connu & Gordon, pour Claborer une expression de forme f m k de la relation entre le d t marginal pour l'entreprise et l'hmornie dsultant du changement de ghQosit6 du plan de retraite. Ainsi, la regle tlimine la nhssit6 d'incorporer des projections actuarielles et des incidences fiscales dam le processus d'6valuation de l'tvolution de la gMosid du plan de retraite. Un exemple n&que illustre l'applicatian de la r&gle de durQ.

A DURATION RULE FOR EVALUATING PENSION 115 PLAN GENEROSITY

CONCORDIA UNIVERSlTY

FACULTY OF COMMERCE AND ADMINISIRATION

1455 DE MAISONNEW BLVD. WEST MON-IREAL, CANADA

H ~ G 1MB

ABSTRACT

Duration has been used widely in the development of investment strategies f a pension portfolios. However, its use in pension liability management has remained extremely limited. This papcr develops a duration rule for evaluating changes in pension plan generosity and their effects on the sponsoring firm. The rule employs a duration measure based on the well - known Gordon model of stock valuation, to develop a c l o d - form expression of the relationship between the f m ' s incremental costs and savings from changing pension plan generosity. In this manner, the rule eliminates the need to incorporate actuarial projections and tax effects into the process of evaluating pension plan generosity changes. A numerical example is provided to illustrate the application of the duration rule.

With greater emphasis being placed on integrating the benefit area into the pension management process, there is a growing need for procedures that can be used in evaluating a firm's pension benefit policy. A key element of benefit policy and, increasingly, of the collective bargaining process, is pension plan generosity 2. Evidence from the Iabor market indicates that a systematic increase in real genemity of the U.S. private pension system has taken place over the entire 30 - year period since World War 113. Firms' apparent motivation for granting such generosity increases consists of extracting current wage concessions from their workers. However, this wage - pension tradeoff has been examined primarily from an employee perspective? whereas procedures for evaluating its effects on f m have yet to be developed.

This paper develops a procedure in which a duration measure, based on Gordon's constant growth model, is employed to evaluate corporate policy towards pension plan generosity. Durand, who first developed this measure for common stocks, suggested that it might be useful for selecting growth stocks for pension portfolios, whenever the duration of the pension liabilities was too long to be matched by portfolios consisting entirely of bonds.5 Ever since, the measure's use in pension management has been confined to the investment area6 By extending its application to the pension benefit area, this paper demonsaates that the constant - growth duration measure may also have a role to play in pension liability management.

116 JIHAD S. NADER

Duration and Constant Growth : Basic Relationships

For a perpetual stream which grows at a constant rate g, and for which the appropiate discount rate is k (k > g), duratim is exp.essed as :

l b o other expressions occur in the development of the duration rule below. 'Ihe fmt is (1 + g) 1 (k - g), and it can readily be obtained from Equation (1) as follows :

The second expression is (1 + g) / (1 + k), and it can be obtained from Equation (2) as follows :

(D - l)(k - g) - 1 + g.

Dividing by 1 + k:

From Equation (I), we can write Equation (4) as :

1 ( D - - a 1 + k' which further simplifies to :

Derivation of the Duration Rule

For simplicity, we consider the case of a defined benefit pension plan which uses a sewice - related final salary famula.8 If the T i grants a non - retroaceive increase in the plan's generosity, then the firm's resulting incremental pension costs will have a present value :

A DURATION RULE FOR EVALUAnON PENSION PLAN GENEROSITY 117

where

b = initial pension benefit rate as a percentage of final salary, b* = increased benefit rate, W o = current wage demand, given benefit rate b, W*O = reduced wage &mand, given an haeased benefit rate b*,

T = years remaining until retirement,

g = salary growth rate assumption, and k = discount rate for valuing the pension liabilities.

Substituing the left - hand side (LHS) of Equation (6) into Equation (7), we have :

The firm's incremental savings from such a negotiated increase in pension plan generosity consist of the current wage cotlcessian extracted from the workers. pojected as a stream over the next T years. ?he present value of this stream is :

A closed - form expression of Equation (9) can be obtained by recognizing that the summation appearing in this equation is equivalent to (i) the present value of a $1 perpetuity which starts at t = 0 and $om at g, minus (ii) the present value of amher perpetuity which starts with $ (1 + g) + at time T + 1 and grows at g. This differem between (i) and (ii) is represented by the term appearing in square brackets, in h e following closed - fom expression of Equation (9) :

Pursuant to Equations (2) and (6). Equation 00) can be written in terms of duration as follows :

s o - t w o - w$)[l+ (D - l)(l - (l (11)

In a&r to adhere to the v h e - rnaximizarjon rule, the f m should only grant an hawse in pension plan generosity if C o S So. From Equaticns (8) and (ll), this requires that :

(12) Rearranging terms and simplifying, we have :

Relation (13) provides a greatly simplified rule for f m i a l evaluation of corporate pension benefit policy. Its simplicity lies in that it is based entirely on current wage and pension data : thus the need for projections does not arise. In particular, it eliminates the need to consider actuarial d e c r m t s (mortality, disability and withdrawal) which affect the stream of incremental costs and savings, since the effects of such decrements are common to the numerator and denominator of the LHS of Relation (13) ; thus, these effects cancel out. Similarly, tax effects need not be amsidered, since they also cancel out in the same way.

A Numerical nlustration

The following example illustrates the use of the duration rule. Consider a pension plan of the type described above, and let b = 2%. Assume the fm's valuation of its pension liability is based on g = 3% and k = 1Wo. From Equation (I), D = 1.10 / (0.10 - 0.03) = 15.71 years. Assume further that the workforce is equally divided between two age cohorts, W years and 50 years d d respectively. For the 20 - year - olds, Wo = $30,000 per worker ; for the 50 - year - olds, Wo = $50,000 per worker, Management can extract wage demand cancessions equal to l%$f WO from both coh ts , if the pension benefit rate is increayl to b* = 3%. That is, W 0 = 0.99 ($30JMO) = $29,700 for the 20 - year - olds, and W 0 = 0.99 ($50,000) = $49,500 for the 50 - year - olds. Retirement occurs at age 65 ; thus T = 45 f a the younger cohort and T = 15 for the older uht Using Relation (13), we have :

LHS - 0.97 and RHS - 6.40 for 20-year-olds; LHS - 0.97 and RHS - 1.83 for 50-year-olds.

Since LHS < RHS for both cohorts, the proposed increase in pension plan generosity is favorable to the fm. In the event that LHS < RHS f a one cohort and LHS > RHS for the other, the overall effect of increasing pension plan genemity would be favorable to the fmn, if the LHS -RHS difference for the first cohort ovwwhelms (in absolute magnitude) the LHS - RHS difference for the second cohort

Finally, the duration rule can readily be generalized to retroactive increases in pension plan generosity. Jf N denotes total years of service, from hiring date to retirement (N 2 T), then such generosity increases can be analyzed by substituting l/N in place of l/T in the RHS of Relation (13).

FOOTNOTES

1; See R. J. Amott and P. L. Bernstein, 'The Right Way to Manage your Pension Fund", Harvard Business Review, January/February 1988, pp. 95 - 102. 2. See D.M. McGill, Fundamentals of Private Pensions, 5th e d (Homewood, IL : Richard D. Irwin, Inc., 1984), p. 95. 3. See R. A. Ippolito, Pensions, Economics and Public Policy (Homewood, IL : Dow Jones - Irwin, 1986). p. 90. 4. See, for example, B . Schiller and R. Weiss, "Pensions and Wages : A Test for Equalizing Differences", Review of Economics and Statistics, November 1980, pp. 529

A DURATION R W WR EVALUAnON PENSION PLAN GENIXOSITY 119

-538 ; and R.S. Smith, "Compensating Differentials for Pensions and Uncledunding in the Public Sector", Review of Economics and Statistics, August 1981, pp. 463 - 467.

5. D. Durand, "Growth Stocks and the Petersburg Paradox", Journal of Finance, September 1957, pp. 348-363.

6. See, for example, J.A. Boquist, G.A. Racette and G.G. Schlarbaurn, "Duration and Risk Assessment for Bonds and Common Stocks", Joumal of Finance, December 1975, pp. 1360 - 1365 ; A.O. Wdiams and P.F. Heifer, '"g Security Price Risk Using Duration and Price Elasticity", Jwnal of Finance, May 1982, pp. 3!39411 ; and PA. Casabona, F.J. Fabozzi, and J.C. Francis, "How to Apply Duration to Equity Analysis", Joumal of Portfolio Managemet, Winter 1984, pp. 52 - 58.

7. Durand, "Growth Stocks", op. cit 8. For a detailed discussion of pensian plan design, see McGill. Fundamentals of Private Pensions, ap. cit

JIHAD S. NADER

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4 Duration Rule for Evaluating Dension Plan C e n e r o ~ i t y