June 1, 2012

Optimal corporate pension policy for de�ned

bene�t plans in the presence of PBGC insurance

Katarzyna Romaniuk �

Abstract

This paper develops a general continuous-time framework for de�ning the optimal corporate pension policyfor de�ned bene�t (DB) plans in the presence of PBGC insurance. Interactions between the �rm�s optimalinvestment and �nancing policies and the optimal portfolio strategy for DB plans are studied. In normal,non-distressed times, the optimal pension portfolio rule from the equityholders�perspective is acceptableto both the participants and PBGC. However, this is no longer the case when the �rm approaches �nancialdistress. The PBGC put then emerges in the consolidated balance sheet, and the �rm�s optimal portfoliobehavior becomes more aggressive. We recommend that the PBGC obtain a control right on pensiondecisions made by sponsoring companies near distress.

JEL classi�cation : C61; D92; G11; G22; G23; G31; G32.

Keywords : optimal corporate pension policy; de�ned bene�t pension plans; optimal portfolio; PBGCinsurance; �nancial distress.

�Universidad de Santiago de Chile, Department of Economics, and Université de Paris 1 Panthéon-Sorbonne,PRISM. Universidad de Santiago de Chile, FAE, Department of Economics, Lib. B. O�Higgins 3363, EstacionCentral, Santiago, Chile. E-mail: katarzyna.romaniuk@usach.cl, romaniuk@univ-paris1.fr.

"All of a company�s major business and �nancing decisions should be informed by a well-

structured analysis that views capital allocation, capital structure, and pension plan decisions as

parts of an integrated strategy." (Merton, 2006)

There exist two views - traditional and corporate - of pension policy (Bodie et al., 1985).

The traditional view emphasizes that the de�ned bene�t (DB) pension plan is to be managed

separately from the �rm and in the best interests of the plan�s participants. The corporate view

considers the �rm�s extended balance sheet, which incorporates the pension assets and liabilities,

and aims to manage the pension plan in the best interests of the �rm�s equityholders.

A widespread view is that, when using a non-consolidated balance sheet, one works with

an economically untrue representation of a �rm. Suboptimal decisions are frequently made. In

particular, pro�table investment projects may be rejected because the weighted average cost of

capital (WACC) is typically overstated (Merton, 2006). If pension liabilities are not considered,

estimates of leverage are downward biased (Shivdasani and Stefanescu, 2010).

The integrated approach is supported by evidence, beginning with Friedman (1983) and Bodie

et al. (1985). When valuing the �rm, the market takes into account the pension surplus or

de�cit (Old�eld, 1977; Feldstein and Seligman, 1981; Feldstein and Morck, 1983; Bulow et al.,

1987). The market assessment of the �rm�s risk incorporates the pension risk (Jin et al., 2006).

Rauh (2006) emphasizes that mandatory contributions to DB pension plans can limit the �rm�s

capital expenditures, especially for �rms facing �nancing constraints. Bergstresser et al. (2006)

show that a procedure of earnings manipulation via the de�nition of projected returns on pension

assets exists, and that this manipulation in turn impacts the pension portfolio policy. Following

Shivdasani and Stefanescu (2010), �rms do consider their pension assets and liabilities when

determining their leverage ratios.

The corporate pension policy view develops several types of arguments, which give (often con-

tradictory) advice as to the form of the pension plan�s policy. Beginning with tax considerations,

1

Black (1980) and Tepper (1981) advocate for a fully funded pension plan invested entirely in bonds

to exploit the favorable tax treatment accorded the pension fund. The Pension Bene�t Guaranty

Corporation (PBGC) insurance e¤ect advances the moral hazard issue, linked to the existence

of the DB plan guarantee; �rms in �nancial di¢ culty choose to underfund their pension plans

and to invest in stocks (Sharpe, 1976; Treynor, 1977).1 With regard to contribution considera-

tions, Black (1989) suggests that companies seeing to avoid paying high contributions when their

�nancial condition is weak invest in cash.

Though not related to the corporate pension policy view, asset-liability management (ALM)

mechanisms appear as one of the major arguments when de�ning the pension plan�s portfolio

strategy. Liability hedging principles can call for an investment in stocks, under the projected

bene�t obligation (PBO) view of liabilities, as stocks provide a hedge against salary in�ation

(Black, 1989; Lucas and Zeldes, 2006).2 However, when considering the accumulated bene�t

obligation (ABO), a heavy investment in bonds using a duration-matching strategy is advised

(Bodie, 1990; Merton, 2006).

The cited arguments focus mainly on the form of the pension plan�s policy. Let us now turn to

the topic of the interactions between the fundamental decisions made by the �rm on the one hand

and those made by the pension plan on the other hand. Whereas the pension plan�s decisions

essentially concern the funding level and portfolio form, the latter will be our focus due to the

law-constrained character of the choice of funding (Rauh, 2009). For the �rm, investment and

�nancing decisions are considered.

Let us focus �rst on pension assets. Black (1980) emphasizes that stocks in the pension fund

have a leverage-like character. Increasing the bond investment in the pension portfolio thus reduces

the �rm�s gearing. Merton (2006) acknowledges this principle by stating that an increase in asset

1 The evidence is mixed as to the existence of the PBGC insurance e¤ect in practice. This e¤ect is documentedby Bodie et al. (1985), whereas Gallo and Lockwood (1995), Coronado and Liang (2005) and Rauh (2009) proposecontradicting results.

2 Rauh (2009) �nds empirical evidence of the application of this strategy.

2

risk - an increase in the pension equity investment - must be o¤set by less risk-taking in the capital

structure - a decrease in leverage - if the equity risk is to remain unchanged. Furthermore, by

increasing the pension bond investment, more risk can be taken in operating activities (Merton,

2006).3

Considering pension liabilities, Shivdasani and Stefanescu (2010), focusing on the tax-deductible

character of pension contributions, show that managers partially substitute pension-related de-

ductions for interest deductions when de�ning their capital structure. They provide evidence that

an increase in the pension liability to total assets ratio is associated with a (less-than-proportional)

decrease in the debt to total assets ratio.

The existing literature o¤ers both theoretical and empirical elements of response concerning

the form of the corporate pension policy adopted or to adopt by the �rm. However, it lacks

a general, uni�ed framework in which the fundamental characteristics of the optimal corporate

pension policy would be determined. Building such a setting constitutes our objective. We aim to

propose a framework that is as general as possible, where the characteristics of and interactions

between the �rm�s optimal investment and �nancing policies and the DB pension plan�s optimal

portfolio strategy are analyzed. The modeling foundations are based on the seminal continuous-

time optimal portfolio problem designed by Merton (1971). We opt for an integrated balance sheet

perspective, where both the �rm�s conventional assets and debt and the pension plan�s assets and

liabilities are considered.

We prove that, when moving from the non-consolidated to the consolidated perspective, the

�rm (DB plan) additionally hedges against the DB plan (�rm) variables. The existing literature

has adopted a perspective less large than ours and has fundamentally regarded the pension port-

folio policy from the corporate perspective solely as a hedging tool, used to optimize the �rm�s

investment or �nancing decisions (Merton, 2006). In our more general setting, the optimal policies

3 Friedman (1983) and Gallo and Lockwood (1995) document that �rms tend to o¤set their business risk throughthe pension plan�s portfolio policy.

3

of the �rm and pension plan are driven by speculative and hedging motives, and in each optimal

policy, the hedging activity considers, apart from the state variables, both the variables related to

the pension plan and those related to the �rm.

A speci�c situation occurs when the �rm approaches �nancial distress. The PBGC insurance

put then emerges in the consolidated balance sheet. We �nd that all of the terms of the pension

plan�s optimal portfolio policy become divided by the delta of the call symmetric to the PBGC

put. Consequently, both risk-shifting and risk-management incentives should become stronger, as

predicted by Rauh (2009).

In normal, non-distressed times, the optimal pension portfolio rule from the equityholders�per-

spective appears acceptable to the participants and PBGC. When moving from the participants�

to the equityholders�perspective, the di¤erence is that the hedging mechanisms become extended

to the �rm variables. However, the situation changes when the sponsoring �rm approaches dis-

tress. The equityholders�optimal portfolio rule becomes more aggressive, potentially putting both

the participants and the PBGC at risk. The PBGC should thus play a more active role than it

currently plays when a sponsoring �rm�s �nancial condition deteriorates because the company is

then induced to gamble with the pension assets at the PBGC�s expense. The paper�s recommen-

dation is to grant the PBGC a control right on pension decisions as soon as the sponsoring �rm

approaches distress.

The paper is organized as follows. The �rst section builds and solves the model. The sec-

ond section discusses the main results. The third section presents the principles to be followed

by equityholders, participants and PBGC in non-distress and distress times. The last section

concludes.

1 The model

We consider a �rm that has created a DB pension plan for its employees.

The entire reasoning is in terms of market value.

4

1.1 The balance sheets

One �rst studies the �rm�s and DB plan�s simpli�ed balance sheets (BS) separately. One obtains,

respectively, the following:

BS (1)AF DF

XF

BS (2)AP LP

XP

where AF and AP , respectively represent the �rm�s and pension plan�s assets, DF is the �rm�s

debt, LP are the pension plan�s liabilities, and XF � AF �DF and XP � AP �LP are the �rm�s

equity and DB plan�s net position, respectively.

The consolidated balance sheet incorporates both the �rm�s and DB plan�s items. It is written

as follows:

BS (3)

AF DF

AP LP

XF+P

where XF+P � XF +XP is the consolidated balance sheet equity.4

Let us introduce the PBGC insurance. The insured sponsoring company has the ability to

pass the burden of its DB plan de�cit, if such a de�cit occurs, to the PBGC. The insured �rm

buys a European put option, written on the DB pension plan�s assets and with the pension

liabilities as the strike (Hsieh et al., 1994). This put PP value at maturity T is written as

PP (T ) =Max(LP (T )�AP (T ); 0). One is thus induced to consider the following balance sheet:

4 Merton (2006) proposes a similar balance sheet form.

5

BS (4)

AF DF

AP LP

PP X

with X � XF + XP + PP being the consolidated balance sheet equity if the PBGC put

emerges.5 , 6

However, it is important to emphasize that the put PP can only be exercised by the sponsoring

�rm under speci�c conditions. A distress termination can only occur when "PBGC has determined

that the plan sponsor and each of its corporate a¢ liates have satis�ed at least one of the following

�nancial distress tests (...): a petition has been �led seeking liquidation in bankruptcy; a petition

has been �led seeking reorganization in bankruptcy, and the bankruptcy court (or an appropriate

state court) has determined that the company will not be able to reorganize with the plan intact

and approves the plan termination; it has been demonstrated that the sponsor or a¢ liate cannot

continue in business unless the plan is terminated (...)."7 The company must thus be in very

serious �nancial trouble, in or near bankruptcy, for the put to be exercisable. It then appears that

the balance sheet above holds only for a particular case. It is only when the company�s �nancial

health deteriorates that the put PP emerges in the balance sheet.

In the consolidated setting, in the presence of PBGC insurance, one thus obtains two types

of balance sheets: the non-distress BS (3) and distress BS (4), with the PBGC put appearing

only in the distress con�guration. A unique balance sheet for both non-distress and distress

con�gurations, with the put present in the two regimes, would not be realistic; it would imply

that the put could be exercised by a �nancially healthy sponsoring �rm. Inversely, a situation

can occur when the �rm approaches distress, but the DB plan is solvent. The distress balance

sheet BS (4) then holds, but the put is out of the money. Consequently, the option will not be

5 Godwin and Key (1999) provide some empirical support for the market�s valuation of the PBGC put option.

6 The call-put parity implies that AP + PP � LP = CP .7 http://www.pbgc.gov/res/factsheets/page/termination.html

6

exercised.

With regard to the main characteristics of the PBGC put, one must �rst mention that the value

of this option depends on the �rm�s assets and debt (and not only on the pension plan�s assets

and liabilities). This characteristic is taken into account in our modeling via the introduction of

the two non-distress and distress balance sheets. The put is exercisable only in the case of the

sponsoring �rm being in or near bankruptcy. The put value thus depends on the �rm�s �nancial

health, which is driven by the value of the �rm�s assets and debt.

Second, the PBGC put is in practice American. The option exercise time is stochastic and cor-

responds to the �rm�s bankruptcy (see, e.g., Marcus, 1985; Pennacchi and Lewis, 1994). However,

we choose the European-style modeling, implying a nonrandom bankruptcy time, for simpli�ca-

tion purposes, and in line with Hsieh et al. (1994). The reason for this choice is that introducing

the American-type put feature in our modeling would lead to complexities and compromise the

determination of an e¤ective response to our paper�s main question: what are the characteristics

of and interactions between the �rm and DB plan�s optimal policies? The unique objective of

the cited papers, which opted for the American-type modeling, was to price the PBGC insurance,

leading to a setting that was less large than that in which we are working.

1.2 The optimization programs

Let us de�ne the optimization programs. BS (4) constitutes the benchmark balance sheet.

The �rm manager chooses policies that are optimal from the equityholders�point of view.8 ,

9 He has the following program at date t:

MaxEt [U(Z(T ))] (1)

where Et stands for the expectation, conditional on the information available in t, U is the

8 The compatibility, or not, of the interests of equityholders, participants and PBGC is discussed in the thirdsection.

9 Agency issues between the manager and equityholders are thus ignored.

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utility function, assumed to be increasing and concave in Z and respecting the Inada conditions:

limZ!1 UZ = 0 and limZ!0 UZ =1, with UZ being the derivative of U with respect to Z, and

Z � ��AF �DF

�+ �

�AP � LP

�+ PP (2)

with �; �; = f0; 1g. T is the �rm�s horizon.10

The de�nition of Z allows the analysis of several relevant cases:

(i) � = 1 and � = = 0: non-consolidated �rm-only balance sheet - BS (1);

(ii) � = 1 and � = = 0: non-consolidated DB plan-only balance sheet - BS (2);

(iii) � = � = 1 and = 0: consolidated balance sheet without the PBGC put, as materialized

by BS (3);

(iv) � = � = = 1: consolidated balance sheet with PBGC put, as represented by BS (4).

The optimization program is thenMaxEt�U(XF (T ))

�,MaxEt

�U(XP (T ))

�,MaxEt

�U(XF+P (T ))

�and MaxEt [U(X(T ))] in cases (i), (ii), (iii) and (iv), respectively.11

One is tempted to consider environments where 0 � �; � � 1 and not only �; � = f0; 1g, aiming

to include cases where the relative weighting of the equityholders�and participants�interests would

vary. The resulting cases could be di¤erent from the two extreme con�gurations essentially dealt

with in this paper, i.e., that either the participants�interests or the equityholders�interests are

taken into account. These two extreme con�gurations are represented by case (ii) on the one hand

and cases (iii) and (iv) on the other hand.

However, the consideration of environments where 0 < �; � < 1 is mathematically unac-

ceptable. The entire logic of the paper�s modeling is built on and justi�ed by the balance sheet

structure. With 0 < �; � < 1, the function Z has no sense, �nancially. Intuition suggests, however,

10 For BS (1), BS (3) and BS (4), one directly concludes that T is the �rm�s horizon. In particular, for thedistress (4) con�guration, T is the bankruptcy date, if bankruptcy eventually occurs. Regarding BS (2), T is thepension plan�s horizon, where T normally coincides with the �rm�s horizon, as the pension plan�s normal functioningis conditional on the �rm being in activity. The pension plan�s horizon could turn out to be shorter than the �rm�shorizon in the case of a voluntary termination of the pension plan.

11 Foundations for the programs (i) and (ii) formulations can be found in Romaniuk (2011a) and Romaniuk(2007), respectively.

8

that cases in between the non-consolidated and consolidated settings would lead to behavioral pat-

terns in between the behavioral patterns characterizing the two extreme con�gurations described

in the paper.

With regard to the general form of the optimization programs, a standard assumption in the

corporate �nance literature is the �rm�s risk neutrality. The objective function of maximizing

utility can thus appear to be nonstandard. However, the paper�s objective requires the building

of optimization programs of comparable structure for cases (i) to (iv) to be able to compare the

results obtained in the four cases.

All of the optimization programs focus on the same variable type: the di¤erence between assets

and liabilities. The di¤erence between the �rm�s and the pension plan�s optimization programs

lies in the attitude toward risk. One would naturally opt for risk aversion when considering the

pension plan, and for risk neutrality when considering the �rm.

It appears di¢ cult to analyze the optimal portfolio policy under the assumption of risk neu-

trality. One is thus induced to de�ne risk aversion as the general principle. This general principle

appears justi�ed when one understands that the setting of the maximization of the expected

utility of wealth (as implied by risk aversion) is in a sense more general than the setting of the

maximization of expected wealth (as implied by risk neutrality), as one reduces the �rst to the

second by assuming a speci�c form (a¢ ne) of the utility function. Consequently, one chooses the

assumption of risk aversion as the unifying assumption.

Let us focus on the decisions to be made optimally, which will enable the de�nition of our

problem control variables. The major decisions a �rm makes are those related to investment and

�nancing, whereas the main task faced in managing the DB plan is the de�nition of a portfolio

strategy.

We shall consider that the variables representative of the �rm�s investment and �nancing deci-

sions are the assets to equity ratio xAF � AF

XF and the debt to equity ratio xDF � DF

XF , respectively,

with xAF � xDF = 1 (Romaniuk, 2011a). As emphasized by the author, the investment decision

9

is de�ned more by the AF level than by the ratio AF

XF itself. One considers that the �rm�s total

asset value is representative of the chosen level of investment activity. When expanding, the total

asset value increases; when downsizing, the latter decreases. When new investment projects are

undertaken, new assets are acquired; when eliminating existing projects, some assets are sold.12

Expanding (downsizing) also increases (reduces) the expected discounted value of future cash

�ows.13

The DB plan�s portfolio policy is de�ned by the proportions of the plan�s assets AP to invest

in the stock market index S and in the riskless asset �. We have:

AP = XSS +X�� (3)

where XS and X� the number of assets S and �, respectively. The portfolio strategy is then

characterized by the plan�s asset proportions xS � XSSAP to attribute to S and x� � X��

AP to give

to �, with xS + x� = 1.

The optimization program is thus the following:

MaxxAF ;xSEt [U(Z(T ))] (4)

In cases (i), (ii), (iii) and (iv), one obtains the optimization programsMaxxAFEt�U(XF (T ))

�,

MaxxSEt�U(XP (T ))

�, Max

xAF

;xSEt�U(XF+P (T ))

�and Max

xAF

;xSEt [U(X(T ))], respectively.

1.3 The variables dynamics

The relevant variables obey the following dynamics:

dAF (t)

AF (t)= �AF (t; Y (t))dt+ �AF (t; Y (t))dBA

F

(t) (5)

12 For Biais et al. (2010), downsizing indeed leads to the liquidation of a fraction of the �rm�s assets. ForDeMarzo and Fishman (2007), investment is a decision to expand or contract the �rm, the size of the latter beingmeasured by its (physical) scale.

13 Following DeMarzo and Fishman (2007), when rescaling the �rm, all of the cash �ows associated with the �rmare also rescaled.

10

dDF (t)

DF (t)= �DF (t; Y (t))dt+ �DF (t; Y (t))dBD

F

(t) (6)

dS(t)

S(t)= �S(t; Y (t))dt+ �S(t; Y (t))dB

S(t) (7)

d�(t)

�(t)= r(t; Y (t))dt (8)

dLP (t)

LP (t)= �LP (t; Y (t))dt+ �LP (t; Y (t))dB

LP (t) (9)

dc(t)

c(t)= �c(t; Y (t))dt+ �c(t; Y (t))dB

c(t) (10)

dw(t)

w(t)= �w(t; Y (t))dt+ �w(t; Y (t))dB

w(t) (11)

where c represents the contribution �ow from the �rm to the DB plan and w is the with-

drawal �ow from the DB plan, taking the form of pension bene�ts paid to the retired employ-

ees. �i(t; Y (t)), the bounded function of time t and the vector of K state variables Y , denotes

the expectation of the instantaneous variation rate of i, �i(t; Y (t)), the bounded function of

t and Y , is its standard deviation, Bi(t) stands for a standard Brownian motion, instanta-

neously correlated with Bj(t) with coe¢ cient �ij , where dBi(t)dBj(t) = �ijdt, �1 � �ij � 1

and i; j =�AF ; DF ; S; LP ; c; w

. r(t; Y (t)) is the instantaneously riskless interest rate, which is

assumed to depend on t and Y .

K stochastic state variables are present in the economy. The k-th variable Yk dynamics is

written as follows:

dYk(t)

Yk(t)= �Yk(t; Y (t))dt+ �Yk(t; Y (t))dB

Yk(t) (12)

11

where �Yk(t; Y (t)) and �Yk(t; Y (t)), bounded functions of t and Y , are the expectation and

standard deviation, respectively, of the instantaneous variation rate of Yk, and BYk(t) stands for

a standard Brownian motion, instantaneously correlated with BYl(t) with coe¢ cient �YkYl , where

dBYk(t)dBYl(t) = �YkYldt, �1 � �YkYl � 1 and k; l = f1; 2; :::;Kg.

The pension plan�s liabilities LP are stochastic. The view of liabilities adopted in this paper

is thus the PBO view, in line with Black (1989) and Lucas and Zeldes (2006). When valuing

liabilities, one discounts with the expected return rate of securities of equivalent risk. However,

one can move directly to the ABO view simply by assuming that �LP = 0. The discount rate is

then the riskless interest rate.

1.4 The optimal policies

Table 1 presents the optimal investment and portfolio policies in cases (i) to (iv).

Appendix A develops the proof.

Appendix B proposes a technical description of the structure of the optimal policies.

12

Table 1: Optimal policies in cases (i), (ii), (iii) and (iv)

�rm�s optimal investment policy xAF DB plan�s optimal portfolio policy xS

cases (i) and (ii) xiAF = � JXF

JXFXFXF

�AF��DF

�2AF

+�2DF�2�AFDF

xiiS = �JXP

JXPXPXPXP

AP

�S�r�2S

� �AFDF��2DF

�2AF

+�2DF�2�AFDF

+LP

AP

�SLP�2S

+ cXF

�AF c��DF c

�2AF

+�2DF�2�AFDF

� cAP

�Sc�2S

�Xk

JXF YkYk

JXFXFXF

�AF Yk��DF Yk

�2AF

+�2DF�2�AFDF

+ wAP

�Sw�2S

�Xk

JXP YkYk

JXPXPXPXP

AP

�SYk�2S

case (iii) xiiiAF = � JXF+P

JXF+PXF+PXF+PXF+P

XF

�AF��DF

�2AF

+�2DF�2�AFDF

xiiiS = � JXF+P

JXF+PXF+PXF+PXF+P

AP

�S�r�2S

� �AFDF��2DF

�2AF

+�2DF�2�AFDF

�XF

AP

xAF (�SAF��SDF )+�SDF

�2S

�AP

XF xS�AF S��DFS

�2AF

+�2DF�2�AFDF

+LP

AP

�SLP�2S

+ LP

XF

�AFLP��DFLP

�2AF

+�2DF�2�AFDF

+ wAP

�Sw�2S

+ wXF

�AFw��DFw

�2AF

+�2DF�2�AFDF

�Xk

JXF+P YkYk

JXF+PXF+PXF+PXF+P

AP

�SYk�2S

�Xk

JXF+P YkYk

JXF+PXF+PXF+PXF+P

XF

�AF Yk��DF Yk

�2AF

+�2DF�2�AFDF

case (iv) xivAF = � JXJXXX

XXF

�AF��DF

�2AF

+�2DF�2�AFDF

xivS = � JXJXXX

X

AP�1+PP

AP

� �S�r�2S

� �AFDF��2DF

�2AF

+�2DF�2�AFDF

� XF

AP�1+PP

AP

� xAF (�SAF��SDF )+�SDF

�2S

�AP (1+PP

AP )XF xS

�AF S��DFS

�2AF

+�2DF�2�AFDF

�LP (�1+PP

LP )AP

�1+PP

AP

� �SLP�2S

�LP (�1+PP

LP )XF

�AFLP��DFLP

�2AF

+�2DF�2�AFDF

+ w

AP�1+PP

AP

� �Sw�2S

+ wXF

�AFw��DFw

�2AF

+�2DF�2�AFDF

� 1

AP�1+PP

AP

�PKk=1 P

PYkYk

�SYk�2S

� 1XF

PKk=1 P

PYkYk

�AF Yk��DF Yk

�2AF

+�2DF�2�AFDF

�Xk

JXYkYk

JXXXX

AP�1+PP

AP

� �SYk�2S

�Xk

JXYkYk

JXXXXXF

�AF Yk��DF Yk

�2AF

+�2DF�2�AFDF

The subscripts on J (the indirect utility function) and on PP denote partial derivatives. �op

stands for the covariance between any variables o and p.

13

In the three cases (i), (iii) and (iv), one obtains the �rm�s optimal �nancing policy xDF by

recalling that xDF = xAF � 1. As demonstrated by Romaniuk (2011a), all of the portfolio terms

in xDF , with the exception of the second term, are identical to the xAF terms. The second term,

� �AFDF��2DF

�2AF

+�2DF�2�AFDF

in xAF , becomes � �2AF

��AFDF

�2AF

+�2DF�2�AFDF

in xDF .

2 Main results

2.1 The consolidated non-distress environment

2.1.1 Main result: transition from the non-consolidated to the consolidated envi-ronment

When moving from the non-consolidated to the consolidated perspective, new hedge funds emerge in

the optimal policies: the �rm (DB plan) additionally hedges against the DB plan (�rm) variables.

2.1.2 The mechanisms driving the consolidated optimal policies

In the existing literature, the pension portfolio policy from the corporate perspective has been

fundamentally regarded solely as a hedging tool, used to optimize the �rm�s investment or �nancing

decisions. The general advice appears to be low equity investment in the pension fund, enabling

more risk-taking in operating activities or in the capital structure (Merton, 2006). According to

this approach, the pension portfolio rule, as de�ned by Eq. xiiiS in Table 1, would reduce to the

hedging term for the �rm variables.14 , 15

The view proposed in this paper appears to be more general than that adopted in the existing

literature. The optimal pension portfolio policy incorporates the element described in this liter-

ature. However, it also includes other elements: �rst the speculative fund, representative of the

risk-reward arbitrage, then hedging terms for the stochastic variables characteristic of the pension

plan�s (and not only �rm�s) activity, and eventually the state-variable hedge fund. All the terms

14 One refers to the AF - DF hedging term, i.e., �XF

APxAF (�SAF ��SDF )+�SDF

�2S

.

The latter term will constitute the sole element of the optimal portfolio policy if the optimization program takesthe form MinxS�

2XF+P , under the assumption of non-stochastic L

P and w variables.

15 The equations referenced in sections 2 and 3 are shown in Table 1.

14

encountered in Eq. xiiiS together form the optimal pension portfolio policy under a fairly general

understanding.

In addition, we show a mechanism not yet derived in the literature: how the �rm�s conventional

decisions evolve when moving from the separate to the consolidated perspective. Here, the main

modi�cation is that these rules now incorporate hedging terms for the pension plan variables.

To conclude, in our general setting, the �rm and pension plan�s optimal policies are driven by

speculative and hedging motives, and in each optimal policy, the hedging activity considers, apart

from the state variables, both the variables related to the pension plan and those related to the

�rm.

2.2 The consolidated distress environment

2.2.1 Main result: the pension portfolio distortions induced by the PBGC put

When the PBGC put emerges in the consolidated balance sheet, all of the terms of the DB plan�s

optimal portfolio policy become divided by the delta of the call symmetric to the PBGC put. The

optimal portfolio behavior becomes more aggressive.

2.2.2 The direction of the PBGC put e¤ect

In the simpli�ed Black-Scholes (1973) world, characterized by a call on a geometric Brownian

motion underlying and with constant strike, the delta of the call can be easily derived, and it is

directly concluded that it lies between 0 and 1. Dividing a term by the Black-Scholes delta thus

leads to an increase in this term�s absolute value.

In our more complex setting, obtaining such a straightforward result appears di¢ cult. Nonethe-

less, one can intuitively believe that the general trend implied by the reasoning in the Black-Scholes

world holds in our more sophisticated environment. If the PBGC put emerges, the agent opti-

mizes with respect to the former (no-put) payo¤ augmented by the put. Intuition then leads us

to assume that his optimal behavior should be riskier: in the case of an unfavorable outcome, he

has the possibility of passing the burden of part of the pension liabilities to the insuring company

15

by exercising the put.

2.2.3 The forces of risk shifting and risk management

Rauh (2009) emphasizes that, for a sponsoring �rm approaching distress, two types of mechanisms

- risk shifting and risk management - determine together the chosen pension risk level. On the

one hand, risk shifting implies that the �rm is tempted to increase the pension risk degree, as

if the assets perform well, the �rm can manage to avoid bankruptcy and will then face lower

funding requirements, whereas in the case of bankruptcy, the PBGC will take responsibility for

the generated pension de�cit. On the other hand, risk-management incentives work in the opposite

direction: they tend to decrease the pension risk level. If bankruptcy is avoided, but assets have

performed poorly, large contributions may be necessary and can themselves �nancially threaten the

company or at least prevent the �rm from making pro�table investments. Rauh (2009) concludes

that on average, the second e¤ect dominates among U.S. �rms.

Let us build a link between the mechanisms described above and our modeling. One �rst

recalls that the terms of an optimal portfolio strategy, as de�ned by Eq. xivS for instance, struc-

turally incorporate two element types. The �rst term is the speculative fund, representative of the

risk-taking behavior; the remaining terms are hedging elements, which represent risk-management

mechanisms (Romaniuk, 2011b). One can consider that the speculative term is here, in a larger

sense, a representation of the risk-shifting mechanism, as in the case of bankruptcy, the conse-

quences of this risk-taking will be e¤ectively borne by the PBGC. One thus obtains a solution

de�ning the optimal portfolio policy, which includes both the risk-shifting term and the risk-

management terms.

When the probability of �nancial distress becomes signi�cant, the PBGC put appears in the

balance sheet, and the optimal portfolio rule takes the form xivS . When compared with the no-put

case, represented by Eq. xiiiS , the main di¤erence in moving to the put case is that all of the

terms of the optimal portfolio strategy become divided by the delta of the corresponding call. The

16

risk-shifting and risk-management mechanisms should thus both become more pronounced when

incorporating the put.

Whereas it seems that for U.S. �rms, the risk-management mechanism dominates, the theo-

retical setting we propose encompasses the dominance of risk shifting and the dominance of risk

management. Depending on whether the speculative proportion is higher (or lower) than the pro-

portion representing the sum of the hedging terms, risk shifting dominates (or is exceeded by) risk

management. When comparing Eqs. xiiiS and xivS , it appears that an important reason why the

speculative or hedging e¤ect would dominate in the distress case is that it was already dominant

in the no-distress con�guration.

3 The principles to be followed by equityholders, partici-pants and PBGC in times of non-distress and distress

3.1 The equityholders�perspective

The sponsoring company�s equityholders aim to optimize with respect to their wealth. They

prefer to work within a consolidated balance sheet setting, as it is more realistic than the non-

consolidated perspective. It would obviously be practical to the �rm�s equityholders to be able to

use the pension as an optimizing tool in their own interests.

Choosing the equityholders�perspective, the �rm�s optimal investment policy follows Eq. xivAF

(Eq. xiiiAF ), and the DB plan�s optimal portfolio strategy is de�ned by Eq. xivS (Eq. xiiiS ) if the

PBGC put emerges (is not present). For the �rm, any divergence from the behavior induced by

the given equations leads to a suboptimal outcome.

3.2 The participants�and PBGC�s perspective

If asked for their preferred portfolio strategy, the pension plan�s participants would opt for sound

ALM principles, as de�ned by program MaxEt�U(XP (T ))

�(Romaniuk, 2007). For the �rm to

respect the participants�interests as de�ned above, the company would choose the �rm�s optimal

investment and �nancing policies from the �rm�s (consolidated) perspective, but the DB plan�s

17

optimal portfolio policy - from the participants� (non-consolidated) perspective. A mix of the

solutions of the (iii) (or (iv)) and (ii) optimizations programs would be obtained. The �rm�s

optimal investment policy would be de�ned as proposed by Eq. xivAF or Eq. xiiiAF , depending on

whether the PBGC put is considered, and the DB plan�s optimal portfolio strategy would follow

Eq. xiiS .

However, the legislator would encounter di¢ culties when aiming to force the �rm to follow

the given pension portfolio policy. First, there is the di¢ culty of establishing the right incentives.

Second, it would not be evident to estimate this portfolio strategy in practice, with the objective

of �xing the benchmark to be observed by the �rm.

It could then appear more appropriate to consider the participants�interests in a larger sense

than that implied by program MaxEt�U(XP (T ))

�. The pension plan participants�overwhelming

goal is in fact to receive the pension promised.16 Essentially, they should not oppose the

consolidated balance sheet perspective, nor should they oppose the equityholders�consideration

of their pension as an optimizing tool, so long as the pension promised is not endangered.

From the participants�point of view, this pension can only be jeopardized when in a given

con�guration. In normal times, when the sponsoring company is healthy, this should not be the

case; the minimum funding requirements should guarantee a su¢ cient funding level and, even more

fundamentally, the company has a legal responsibility for the pension promised. The situation

can change if the company approaches �nancial distress. Allowing then the company to use the

pension as an optimizing tool in the equityholders�interests can turn out to be dangerous for the

participants, who may not receive the total amount of bene�ts promised,17 and for the PBGC,

who will have to �nance the pension plan de�cit in the case of the company�s bankruptcy.

In normal, non-distressed times, allowing the sponsoring company to follow the optimal pension

portfolio policy from its point of view does not appear to be a dangerous decision for either the

16 If a surplus-sharing principle is included in the pension formula, a more complex situation can emerge, as theparticipants will aim to generate a surplus.

17 The PBGC guarantee is subject to a ceiling.

18

participants or the PBGC. Indeed, we have seen that the major di¤erence when comparing the

optimal portfolio rule from the participants� and equityholders� perspectives, as de�ned by Eqs.

xiiS and xiiiS , respectively, is that moving from the �rst to the second perspective leads to hedging

against the �rm variables. Even a portfolio rule containing only the �rm-variable hedging term,

as implied by Merton�s (2006) analysis, does seem reasonable, especially when one recalls that

according to this approach, low equity investment is advised.

When the company�s �nancial health deteriorates, the �rm�s behavior becomes more aggressive.

We have seen that when the PBGC put emerges in the �rm�s consolidated balance sheet, the major

distortion observed is that all of the terms of the DB plan�s optimal portfolio policy become divided

by the delta of the symmetric call. The PBGC put induced distortions, while favorable to the �rm

in the context of �nancial distress, are obviously unfavorable to the participants and the PBGC.

The equityholders know that, in the case of bankruptcy, the limited-liability property allows

them not to pay for the �rm�s generated de�cit. When approaching �nancial distress, it is in

their interest to gamble with the assets to try to (maybe) reverse the situation via a high-risk

strategy. As actions on the �rm�s conventional assets are very frequently constrained, such as by

debt covenants, these conventional assets are a less convenient tool for these risky strategies than

the pension assets, whose portfolio form remains unconstrained. The current law does not require

taking account of the liability characteristics when designing the pension portfolio form (Wilcox,

2006). Debt agreements do not restrict the latter, either, possibly because of the low priority

the PBGC faces in the event of default (Rauh, 2009). In addition, this low level of priority itself

induces the �rm to gamble with the assets backing the pension liability when the probability of

bankruptcy is high (Rauh, 2009).

The main conclusion of our analysis is that the moment for the PBGC to step in is not the

sponsoring company�s bankruptcy but when the �rm approaches �nancial distress.

19

3.3 The reform step implied by the model results

The PBGC should obtain the legal right to directly intervene at the sponsoring company level as

soon as a deterioration in the company�s �nancial health occurs, as measured, for instance, by a

given credit rating downgrade. More precisely, the PBGC should then gain immediate and direct

access to all of the pension plan data available to the company, this in the objective of quickly

making its own evaluation of the pension plan�s current �nancial status and prospects. The

company and PBGC should together build a viable and realistic funding and portfolio program

for the pension plan, whose respect would be internally checked by the PBGC. This internal

control could be lifted when the company manages to converge to its �nancial balance.

What we thus fundamentally propose is that the PBGC immediately gain a control right on what

happens in the pension plan as soon as the sponsoring company�s �nancial health is threatened.

Instead of adopting a passive approach, the PBGC should directly take preemptive steps to prevent

a possible deterioration in the pension plan�s �nancial status.

The Early Warning Program constitutes a control tool for the PBGC over troubled companies.

Under this program, the PBGC adopts a preemptive attitude, trying to avoid the future occurrence

of large losses.18 The underlying mechanism consists of a monitoring system for companies with

weak credit ratings or underfunded pension plans. Representing the program�s focus are corporate

transactions that could jeopardize pensions and lead to a pronounced increase in the risk of long-

run loss to the PBGC. Of particular interest are transactions that can importantly weaken the

�nancial support for a pension plan, as the breakup of a controlled group or a leveraged buyout.

If the transaction to occur is judged to pose a threat, the PBGC aims to negotiate additional

protections for the pension plan.

However, the PBGC�s current �nancial status shows that the Early Warning Program�s e¤ec-

tiveness is limited. We must agree with Rauh (2009) when he states that "the PBGC exerts little

18 http://www.pbgc.gov/prac/risk-mitigation.html

20

control over the corporation when it is a going concern, outside of the contribution requirements

and the collection of the insurance premiums." This state must be changed, in particular when

dealing with �nancially troubled sponsoring companies. One possibility of reform would be to

develop the existing Early Warning Program by extending the range of monitoring activities and

the PBGC�s control rights.

4 Conclusion

We developed a general continuous-time model for de�ning the optimal corporate pension policy

for DB plans in the presence of PBGC insurance. The pension plan�s optimal portfolio strategy

and the �rm�s optimal investment and �nancing policies were derived and analyzed. We suggested

that in normal, non-distressed times, adopting the equityholders�point of view when de�ning the

pension portfolio strategy does appear to be acceptable for the participants and PBGC. However,

cautiousness should be the rule as soon as the sponsoring company approaches �nancial distress,

at which point the aggressiveness of the equityholders� optimal behavior should increase. The

PBGC should thus elaborate more constraining control procedures on sponsoring companies facing

�nancial di¢ culties than those applied today.

The reasoning this paper adopts from Merton (1971) implies a quantitative focus, emphasizing

the expectations and (co)variances of the involved variables, where the motives of speculation and

hedging play a crucial role. This reasoning appears to be particularly well suited for studying

diversi�cation e¤ects between the �rm�s assets and debt on the one hand and the pension plan�s

assets and liabilities on the other hand. Among the main arguments driving the pension plan

policy�s de�nition as cited in the introduction, the ALM principles and the PBGC insurance e¤ect

are ideal candidates for analysis in the chosen setting. However, other arguments may be less easy

to address in a setting as general as ours. For instance, taxes, though implicitly considered via

the introduction of state variables, are not the focus of the analysis we propose. To analyze their

e¤ects adequately, one would need to produce a paper exclusively focused on the taxation issue,

21

due to the inherent complexity of the problem.

The PBGC�s current troubles call for raising the issue of the optimal insurance contract de-

sign. Today, there are obvious con�icts of interest between the PBGC and the sponsoring �rms,

especially when the latter approach �nancial distress. The consequences of these con�icts are

eventually entirely borne by the PBGC and potentially by taxpayers, if a �nancial rescue of the

PBGC becomes necessary. The interests of the sponsoring �rms must be aligned (or at least

become more compatible) with those of the PBGC via an adequate contract design to avoid the

further accumulation by the PBGC of pension de�cits as heavy as in recent years.

22

APPENDICES

APPENDIX A: SOLUTION TO THE OPTIMIZATION PROGRAM

a. The Z dynamics

Let us determine the dynamics of Z. One di¤erentiates Eq. (2) by taking account of the DB

plan�s asset composition, de�ned by Eq. (3), and of the �ows of contributions c and withdrawals

w, to obtain the following:

dZ = ��dAF � dc� dDF

�(13)

+��XSdS +X�d� + dc� dw � dLP

�+ dPP

The Z dynamics is written as follows:

dZ

Z=

�XF

Z

�xAF

dAF

AF� c

XF

dc

c� xDF

dDF

DF

�(14)

+�AP

Z

�xSdS

S+ x�

d�

�+

c

APdc

c� w

APdw

w� LP

APdLP

LP

�+ PP

Z

dPP

PP

Using xAF � xDF = 1 and xS + x� = 1, the Z dynamics becomes:

dZ

Z=

�XF

Z

�xAF

�dAF

AF� dD

F

DF

�+dDF

DF� c

XF

dc

c

�(15)

+�AP

Z

�xS

�dS

S� d��

�+d�

�+

c

APdc

c� w

APdw

w� LP

APdLP

LP

�+ PP

Z

dPP

PP

23

b. The PP dynamics

One needs to derive the put PP dynamics.

As PP (t; AP ; LP ; Y1; Y2; :::; YK), applying Ito�s lemma to the function PP yields:

dPP = PPt dt+Xm

PPmdm+1

2

Xm

Xn

PPmndmdn (16)

with m;n =�AP ; LP ; Y1; Y2; :::; YK

and PPm denoting the partial derivative of the put with

respect to the variable m, PPmn its second derivative with respect to m and n.

The Black-Scholes (1973) non-arbitrage riskless portfolio � takes the form:

� = �PP +Xm

PPmm (17)

The self-�nancing condition implies:

d� = �dPP +Xm

PPmdm (18)

Combining Eqs. (16) and (18) yields:

d� = �PPt dt�1

2

Xm

Xn

PPmndmdn (19)

The portfolio �, being riskless, earns the riskless rate r. The � dynamics follows:

d� = r�dt (20)

The combination of Eqs. (17), (19) and (20) implies:

�PPt dt�1

2

Xm

Xn

PPmndmdn = r(�PP +Xm

PPmm)dt (21)

Replacing Eq. (21) in Eq. (16), the put dynamics becomes:

24

dPP

PP=

"r +

Xm

PPmm

PP(�m � r)

#dt+

Xm

PPmm

PP�mdB

m (22)

where the variable m dynamics is considered under its general form: dmm = �mdt+ �mdBm.

One can rewrite Eq. (22) as follows:

dPP

PP=

2664 r + PPAPAP

PP (�AP � r)

+PPLPLP

PP (�LP � r) +PK

k=1 PPYk

YkPP (�Yk � r)

3775 dt (23)

+PPAP

AP

PP�AP dBA

P

+PPLPLP

PP�LP dB

LP +KXk=1

PPYkYkPP

�YkdBYk

Let us develop the parameters of the AP dynamics.

The DB plan�s assets AP are invested in the assets S and �, in the proportions xS and x�,

respectively, with xS + x� = 1, implying:

dAP

AP= xS

�dS

S� d��

�+d�

�(24)

Incorporating the S and � dynamics, as de�ned by Eqs. (7) and (8), respectively, one obtains:

dAP

AP= [xS (�S � r) + r] dt+ xS�SdBS (25)

The put PP dynamics follows:

dPP

PP=

2664 r + PPAPAP

PP (xS (�S � r))

+PPLPLP

PP (�LP � r) +PK

k=1 PPYk

YkPP (�Yk � r)

3775 dt (26)

+PPAP

AP

PPxS�SdB

S

+PPLPLP

PP�LP dB

LP +KXk=1

PPYkYkPP

�YkdBYk

25

c. The Z dynamics (cont.)

One replaces the dynamics of AF , DF , c, S, �, w, LP and PP , as de�ned by Eqs. (5), (6),

(10), (7), (8), (11), (9) and (26), respectively, in Eq. (15), and factorizes the resulting Z dynamics

with respect to xS and LP

Z . One obtains the following:

dZ

Z=

2666666666666664

�XF

Z

�xAF (�AF � �DF ) + �DF � c

XF �c�

+AP

Z

�� + PPAP

�xS (�S � r)

+LP

Z

����LP + PPLP (�LP � r)

�+�AP

Z

�r + c

AP �c � wAP �w

�+ PP

Z

�r +

PKk=1 P

PYk

YkPP (�Yk � r)

�

3777777777777775dt (27)

+�XF

Z

�xAF

��AF dBA

F

� �DF dBDF�+ �DF dBD

F

� c

XF�cdB

c�

+AP

Z

�� + PPAP

�xS�SdB

S

+LP

Z

��� + PPLP

��LP dB

LP

+�AP

Z

� c

AP�cdB

c � w

AP�wdB

w�

+ PP

Z

KXk=1

PPYkYkPP

�YkdBYk

!

d. The optimization program

The �rm manager solves the optimization program (4) under the constraints of the Z and Yk

dynamics, as de�ned by Eqs. (27) and (12), respectively.

e. The �rst order conditions

Let the indirect utility function J be de�ned as:

J(Z(t); Y (t); t) � maxxAF ;xS

Et [U(Z(T ))] (28)

26

with J increasing, strictly concave in Z, once di¤erentiable with respect to t and twice di¤er-

entiable with respect to Z and Y .

The Hamilton-Jacobi-Bellman optimality condition states:

0 = maxxAF ;xS

DJ(Z(t); Y (t); t) (29)

where D the Dynkin operator, the Dynkin of J being de�ned by:

DJ = Jt + JZZ�Z +1

2JZZZ

2�2Z (30)

+Xk

JYkYk�Yk +1

2

Xk

Xl

JYkYlYkYl�YkYl

+Xk

JZYkZYk�ZYk

where the subscripts on J denote partial derivatives and �op stands for the covariance between

any variables o and p, while the variable Z dynamics is considered under its general form dZZ =

�Zdt+ �ZdBZ .

Replacing the parameters of the Z dynamics with their formulations as de�ned in Eq. (27)

and deriving DJ with respect to xAF and xS yields:

0 = JZZ�XF

Z(�AF � �DF ) (31)

+JZZZ2

26666666666666666664

��XF

Z

�2 �xAF

��2AF + �

2DF � 2�AFDF

�+��AFDF � �2DF

��+�XF

ZcZ (� � �) (�AF c � �DF c)

+�XF

ZAP

Z

�� + PPAP

�xS (�AFS � �DFS)

+�XF

ZLP

Z

��� + PPLP

�(�AFLP � �DFLP )

��XF

ZwZ � (�AFw � �DFw)

+�XF

Z PP

Z

PKk=1 P

PYk

YkPP (�AFYk � �DFYk)

37777777777777777775+Xk

JZYkZYk

��XF

Z(�AFYk � �DFYk)

�

27

0 = JZZ

�AP

Z

�� + PPAP

�(�S � r)

�(32)

+JZZZ2

26666666666666666664

�AP

Z

�� + PPAP

��2xS�

2S

+AP

Z

�� + PPAP

��XF

Z (xAF (�SAF � �SDF ) + �SDF )

+AP

Z

�� + PPAP

�cZ (� � �)�Sc

+AP

Z

�� + PPAP

�LP

Z

��� + PPLP

��SLP

�AP

Z

�� + PPAP

�wZ ��Sw

+AP

Z

�� + PPAP

� PP

Z

PKk=1 P

PYk

YkPP �SYk

37777777777777777775+Xk

JZYkZYkAP

Z

�� + PPAP

��SYk

f. The optimal policies

Using Eqs. (31) and (32), one determines the following optimal policies, respectively:

xAF = � JZJZZZ

Z

�XF

�AF � �DF

�2AF + �

2DF � 2�AFDF

(33)

��AFDF � �2DF

�2AF + �

2DF � 2�AFDF

�c (� � �)�XF

�AF c � �DF c

�2AF + �

2DF � 2�AFDF

�AP�� + PPAP

��XF

xS�AFS � �DFS

�2AF + �

2DF � 2�AFDF

�LP��� + PPLP

��XF

�AFLP � �DFLP

�2AF + �

2DF � 2�AFDF

+w�

�XF

�AFw � �DFw

�2AF + �

2DF � 2�AFDF

�

�XF

KXk=1

PPYkYk�AFYk � �DFYk

�2AF + �

2DF � 2�AFDF

�Xk

JZYkYkJZZZ

Z

�XF

�AFYk � �DFYk

�2AF + �

2DF � 2�AFDF

28

xS = � JZJZZZ

Z

AP�� + PP

AP

� �S � r�2S

(34)

� �XF

AP�� + PP

AP

� xAF (�SAF � �SDF ) + �SDF

�2S

� c (� � �)AP�� + PP

AP

� �Sc�2S

�LP��� + PPLP

�AP�� + PP

AP

� �SLP�2S

+w�

AP�� + PP

AP

� �Sw�2S

�

AP�� + PP

AP

� KXk=1

PPYkYk�SYk�2S

�Xk

JZYkYkJZZZ

Z

AP�� + PP

AP

� �SYk�2S

One obtains the optimal policies xAFand xS presented in Table 1 by recalling that

- in case (i), the �rm-only con�guration: � = 1, � = = 0 and Z = XF ;

- in case (ii), the DB pension plan-only con�guration: � = 1, � = = 0 and Z = XP ;

- in case (iii), the consolidated balance sheet without the PBGC put: � = � = 1, = 0 and

Z = XF+P ;

- in case (iv), the consolidated balance sheet with PBGC put: � = � = = 1 and Z = X.

29

APPENDIX B: DESCRIPTION OF THE OPTIMAL POLICIES STRUCTURE

The equations referenced in this Appendix are shown in Table 1.

a. Case (i): The �rm�s policies in a non-consolidated perspective

The optimal xiAF and xiDF are composed of four terms. The preference-dependent specu-

lative fund follows the objective of building the �nancially most interesting risk-reward couple��AF � �DF ;�2AF + �

2DF � 2�AFDF

�, while taking into account the relative risk tolerance degree,

as represented by the coe¢ cient � JXF

JXFXFXF . One then encounters the preference-independent

portfolio variance minimizing hedge against the �rm´s debt DF (the �rm´s assets AF ) in xiAF

(xiDF ). The preference-independent contribution hedge term follows, covering the stochastic evo-

lutions of the contribution process. The last term de�nes a preference-dependent state-variable

hedge fund, which constitutes a cover against the variations of the K state variables in�uencing

the evolution of the economy.

b. Case (ii): The pension plan�s policy in a non-consolidated perspective

Five terms are present in the optimal xiiS . The usual speculative fund, preference dependent

via the coe¢ cient � JXP

JXPXPXP and optimizing the risk-reward couple (�S � r;�2S), is followed by

four hedge funds. The �rst three funds are preference independent and cover the variations in

the plan�s liabilities LP , the contributions to the fund c and the withdrawals from the fund w,

respectively. The last optimal xiiS term, preference dependent, is a hedge against the state variable

variations.

c. Case (iii): Consolidated optimal policies in normal times

Six terms emerge in the optimal xiiiAF : the preference-dependent speculative fund, building

on the risk-reward arbitrage between �AF � �DF and �2AF + �2DF � 2�AFDF , four preference-

independent hedge funds against the DF , S, LP and w variations and the preference-dependent

state-variable hedge fund.

30

When comparing to the optimal investment policy in the �rm-only (i) con�guration, as mate-

rialized by Eq. xiAF , several di¤erences are observed. First, three new hedge funds emerge in the

consolidated case, which cover the S, LP and w variations. The investment policy is now impacted

by the hedging demand for stochastic variables characteristic of the DB pension plan. Second,

the contribution c hedge fund disappears in the consolidated case. The contributions are paid by

the �rm and received by the DB pension plan, leading the corresponding hedge fund to cancel out

in the consolidated con�guration. Third, in the speculative fund and in the state-variable hedge

fund, risk tolerance is measured with respect to XF+P and no longer XF .

For the optimal xiiiDF , the modi�cations are similar to those registered for the optimal xiiiAF .

With regard to the optimal xiiiS , one observes �ve terms: the preference-dependent speculative

fund, representing the risk-reward arbitrage between �S� r and �2S , three preference-independent

hedge funds against the AF and DF , LP and w variations and the preference-dependent state-

variable hedge fund.

When comparing to the non-consolidated DB plan-only (ii) case, characterized by the optimal

portfolio policy de�ned by Eq. xiiS , one notices the following di¤erences. First, a new hedge fund

emerges against the AF and DF variations: the DB pension plan�s optimal portfolio policy is now

impacted by a hedging demand for stochastic variables characteristic of the �rm. Second, the

contribution c hedge fund cancels out. Third, in the speculative fund and in the state-variable

hedge fund, risk tolerance is now measured with respect to XF+P and not XP .

To summarize, when moving from the non-consolidated to the consolidated �rm�s optimal

investment and �nancing policies and DB plan�s optimal portfolio strategy, new hedge funds

emerge: the �rm (DB plan) additionally hedges against the DB plan (�rm) variables. More

precisely, in the optimal xiiiAF and xiiiDF , one additionally hedges against the DB plan�s S, LP and

w; in the optimal xiiiS , one additionally hedges against the �rm�s AF and DF .

The second major di¤erence is the elimination of the contribution c hedge fund in all three

optimal policies. The non-emergence of the contribution hedge fund occurs because the consol-

31

idated approach leads to the consideration of the pension assets (liabilities) as the �rm�s assets

(debt). Consequently, the e¤ect of cash �ows directed from the �rm to the pension plan is seen

as neutral.

d. Case (iv): Consolidated optimal policies in distress times

Eq. xivAF shows that seven terms characterize the optimal investment policy: the preference-

dependent speculative fund, representative of the risk-reward arbitrage between �AF � �DF and

�2AF + �2DF � 2�AFDF , �ve preference-independent hedge funds against the variations of DF , S,

LP , w and the put PP driven state variables and the preference-dependent state-variable hedge

fund.

When compared to the no-PBGC-put case, the optimal investment policy thus being formu-

lated as de�ned by Eq. xiiiAF , the di¤erences are as follows. First, the S hedge fund is now

multiplied by 1 + PPAP . Second, the LP hedge fund becomes multiplied by 1 � PPLP . Third, the

put PP driven state-variable hedge fund emerges. Fourth, in the speculative fund and in the

preference-dependent state-variable hedge fund, risk tolerance is measured with respect to X and

no longer XF+P .

For the optimal xivDF , the modi�cations are similar to those registered for the optimal xivAF .

With regard to xivS , the optimal portfolio policy incorporates six terms: the preference-

dependent speculative fund, building on the arbitrage between �S � r and �2S , four preference-

independent hedge funds against the variations of AF and DF , LP , w and the put PP driven

state variables and the preference-dependent state-variable hedge fund.

When compared to the optimal portfolio policy characteristic of the no-PBGC-put case, as

de�ned by Eq. xiiiS , several di¤erences are observed. First, all of the terms are divided by 1+PPAP .

Second, the LP hedge fund is multiplied by 1 � PPLP . Third, the put PP driven state-variable

hedge fund emerges. Fourth, in the speculative fund and in the preference-dependent state-variable

hedge fund, risk tolerance is now measured with respect to X and not XF+P .

32

To summarize, the PBGC put emergence leads to major modi�cations. First, and most impor-

tant, the terms of the DB plan�s optimal portfolio policy become all divided by 1+PPAP . Second,

in the �rm�s optimal investment and �nancing policies, the DB plan�s portfolio policy related

term - the S hedge fund - becomes multiplied by 1 + PPAP . Third, there are two di¤erences in all

three optimal policies: the LP hedge fund becomes multiplied by 1�PPLP , and the put PP driven

state-variable hedge fund emerges.

Let us focus on the most important modi�cation presented above, i.e., that the terms of the

DB plan�s optimal portfolio policy become all divided by 1 + PPAP when the PBGC put emerges.

It appears to be useful to study the meaning of the expression 1 + PPAP . Recalling the call-put

parity AP +PP �LP = CP , one derives the latter with respect to AP to obtain 1+PPAP = CPAP .

33

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