WORKING PAPER SERIES

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1 ollege o Business Administration University o hode Island William A. Orme WOKING PE SEIES encouraging creative research Growth Opportunities, Stockholders' laim/liability on Pension Plans and oporate Pension Policies Yul W. Lee 2/2 No. 2 This working paper series is intended to acilitate discussion and encourage the exchange o ideas. Inclusion here does not preclude publication elsewhere. It is the original work o the authors and subject to copyright regulations. Oice o the Dean ollege o Business Administration Ballentine Hall 7 Lippitt oad Kingston, I

2 Growth Opportunities, Stockholders' laim/liability on Pension Plans and orporate Pension Policies Yul W. Lee* July 23 Abstract In this study, I derive the value o stockholders claim on a pension surplus and stockholders liability or a pension deicit in the post-eisa regulatory environment. Based on that valuation, I develop a model o corporate pension policies in which sponsoring irms weigh contributions to their pension plan against the exercise o growth options in the allocation o limited inancial resources. The model shows how corporate pension unding and asset allocation policies are shaped by the sponsoring irms characteristics, such as growth options, the marginal corporate tax rate, and regulatory variables such as the deicit reduction contribution rate, the variable-rate insurance premium, and the maximum possible raction o operating assets that can be seized by the Pension Beneit Guaranty orporation. I discuss the policy implications o the theoretical results. JEL lassiication: G; G23; G33 Keywords: Growth Option, Future Pension Beneit, Pension Funding, Pension isk *Lee is at the ollege o Business Administration, University o hode Island, Kingston, I 288; ylee@uri.edu; phone ; ax An earlier version o this paper was presented at the Fourth Financial isks International Forum held in Paris, France, arch, 2, and the Netsapr Symposium held at Amsterdam, Netherlands, January 22-24, 23.

3 Growth Opportunities, Stockholders' laim/liability on Pension Plans and orporate Pension Policies Abstract In this study, I derive the value o stockholders claim on a pension surplus and stockholders liability or a pension deicit in the post-eisa regulatory environment. Based on that valuation, I develop a model o corporate pension policies in which sponsoring irms weigh contributions to their pension plan against the exercise o growth options in the allocation o limited inancial resources. The model shows how corporate pension unding and asset allocation policies are shaped by the sponsoring irms characteristics, such as growth options, the marginal corporate tax rate, and regulatory variables such as the deicit reduction contribution rate, the variable-rate insurance premium, and the maximum possible raction o operating assets that can be seized by the Pension Beneit Guaranty orporation. I discuss the policy implications o the theoretical results. JEL lassiication: G; G23; G33 Keywords: Growth Option, Future Pension Beneit, Pension Funding, Pension isk

4 I. Introduction The recent inancial market crisis has brought a new attention to the unding adequacy o corporation-sponsored deined beneit pension plans. According to Financier 29, the aggregate deicit in pension plans sponsored by S&P 5 reached $49 billion in the midst o the 28 market crash, a shock reversal rom the $6 billion surplus seen a year earlier. ercer 23 reports that the aggregate pension deicit o S&P 5 companies increased by $73 billion to a record year-end high o $557 billion as o December 22 with a unded ration o 74%. uch o the pension deicit might be ultimately unded by tax payers' money via the Pension Beneit Guarantee orporation PBG. The PBG estimates that its inancial risk or potential termination o underunded pension plans sponsored by inancially weak corporations has increased to $295 billion in 22, an amount that has continued to worsen since the economic downturn in 28. According to a recent report 23 by the U.S. Government Accountability Oice 23, the single-employer pension program continues to be designated as high risk since July 23. Policy makers and ongress, thereore, are responsible or developing public policies and legislating laws that would encourage sponsoring corporations to practice sound pension policies. To meet such responsibilities, it is important to understand how sponsoring corporations shape their pension unding and asset allocation policies. In this research, I examine corporate deined beneit DB pension policies in a new perspective. In particular, I view DB pension plans as an alternative investment project competing with growth opportunities or unding. This new perspective provides insight into how sponsoring irms make pension unding and asset allocation decisions. As long as I understand, this is the irst study to examine a relation among sponsoring irm's growth opportunities, stockholders' claim/liability on pension plans, and corporate pension 2

5 policies. In this research, I develop a model o pension unding and asset allocation policies in the post-eisa regulatory environment. In this model, a representative sponsoring irm has two alternative investment projects competing or limited inancial resources: pension plans and growth options. As a result, the sponsoring irm weighs the risk-adjusted marginal expected return on pension unding against the risk-adjusted atertax marginal expected return on growth options. The risk-adjusted marginal expected return on pension unding consists o the tax saving and the risk-adjusted marginal expected return on the pension asset return discounted by the marginal pension surplus or deicit actor. Discounting o the pension surplus comes rom the act that overunded pension assets are not entirely considered the sponsoring irm s assets due to excise taxes and other restrictions on the conversion o overunded pension assets into operating assets. Burrow, Scholes, and enell 983, and Thomas suggest that sponsoring irms can withdraw excess pension assets rom their pension plans without paying excise and corporate taxes by making small reductions each year in amounts contributed to the plan. Firms sponsoring an overunded plan are exempt rom the minimum unding contributions Fs that are required or those with an underunded plan. As a result, irms sponsoring an overunded plan can save the Fs every year until the pension surplus is depleted. The annual Fs in a given year are determined by the pension beneits accrued in the previous year. The value o stockholders claim on excess pension assets, thereore, equals the present value o a stream o uture pension beneits that can be covered by excess pension assets. Discounting o the pension deicit results rom the act that irms sponsoring an underunded plan are only required to "slowly" remedy the pension deicit. Sponsoring irms are obligated to make deicit reduction contributions Ds to their plan and pay an 3

6 insurance premium to the Pension Beneit Guarantee orporation PBG every year until their plan becomes ully unded. The value o stockholders' liability or underunded pension plans, thereore, equals the present value o a sequence o Ds and insurance premiums paid every year until the pension deicit is remedied. This study shows that the pension asset allocation is determined by the marginal pension surplus discount actor PSDF or overunded plans, and the expected cost o severe underunding associated with plan termination or modestly underunded plans. The reason is that PSDF constitutes the reward or stock market investment, and the expected cost o severe underunding comprises the downside risk o stock market investment. It is shown that the optimal pension asset allocation is a mix o risky and riskless assets with a suiciently small PSDF or overunded plans, and with a suiciently large expected cost o severe underunding modestly underunded plans. For a severely underunded plan, however, the optimal pension asset allocation turns out to be the entire pension und invested in the risky asset. The reason is that the irm with a severely underunded plan has been already incurring the cost o severe underunding, which is a sunk cost. I suggest that an exponentially rising expected cost o severe underunding, which relects better the reality, would discourage the sponsoring irm o a severely underunded plan to make such an extreme allocation. I discuss policy implications o the theoretical results. In particular, I evaluate eectiveness o deicit reduction contributions and the variable-rate insurance premium which does not vary with pension asset risk. I ind that even the deicit reduction contribution rule tightened by the Pension Protection Act P o 26 would not induce sponsoring irms to make more voluntary contributions to their underunded pension plan. However, the variable-rate insurance premium would encourage sponsoring irms o an underunded plan to 4

7 make more voluntary contributions. I suggest a simple structural change in the variable-rate insurance premium which would discourage sponsoring irms to underund their pension plan. The model developed in this study produce several testable empirical implications. They include a negative relation between sponsoring irms' growth options and pension unding, and a relation between the time distribution o uture pension beneits and pension unding among others. The remainder o the paper is organized as ollows. Section II reviews the literature. Section III derives the value o stockholders claim/liability on pension plans and develops a model o corporate pension unding and asset allocation decisions. Section IV and Section V discusses policy implications and empirical implications, respectively. Section VI sets orth the conclusions. II. elated Literature Firms sponsoring DBPPs promise to provide a ixed amount o retirement beneit to their employees based primarily on employee tenure, age, and salary. equired by the EISA o 974, irms are obligated to earmark a certain amount o assets to meet their pension obligations. These pension plan unds are generally invested in equities and ixed income securities. Firms are also required to make inancial contributions to their pension unds based on certain ormula speciied by the law and tax code. Despite a decline in their relative importance, due to a shit to deined contribution plans over the last 3 years, DBPPs still account or an important part o private sector retirement plans in the US. About 2% o all public irms in ompustat, mostly large irms, have DBPPs, covering about 44 million US 5

8 workers and employees, and aggregate DBPP assets in the US private sector amounted to $2.7 trillion as o the end o 22. Both academic researchers and practitioners have devoted considerable attention to the issue o optimal pension unding and asset allocation strategies or DBPP-sponsoring irms. Subsequent to the passage o the EISA in 974, several studies made important contributions to the corporate pension plan management. However, most o the previous studies ocus on one particular aspect o pension unding such as pension insurance and tax exemption o pension contributions. Sharpe 976, and Treynor 977 illustrate that in the absence o taxes and given the structure o a ixed insurance premium charged by the PBG, irms can maximize the put value and accordingly maximize shareholder wealth by employing a mini-max strategy in pension plan management, achieving a minimum level o pension unding while investing in a maximum level o risky assets. In contrast, Black 98 and Tepper 98 suggest a max-mini strategy rom a tax beneit perspective, achieving a maximum level o plan unding and a minimum level o investment in low-tax risky assets. By contrast, in this study, the entire beneits pension unding provides or stockholders are incorporated into the corporate pension unding decision. Those beneits include tax saving, pension put, and stockholders' claim on the pension asset return. auh 26 inds that pension sponsors decrease capital investment in response to a reduction in inancial resources caused by required pension contributions. auh's inding indirectly supports the premise o this study that pension unding competes with unding o growth opportunities or sponsoring irms' scarce inancial resources. ahu 26, however, The data are published by Investment ompany Institute. Total assets held in DBPPs represented 3.% o total employer-sponsored plan assets or $9.9 trillion in 22. For comparison, deined contribution pension plans held $5.4 trillion in assets, among which the largest portion o assets was held in 4k accounts $3.6 trillion in assets. 6

9 does not investigate how sponsoring irms allocate available inancial resources between voluntary pension pending and unding o other investment projects. ore recently, Love, Smith, and Wilcox 2, LSW hereater, contributes to the pension literature by investigating how government regulations can aect corporate pension decisions in a model in which sponsoring irms trade o the need to compensate employees or the inancial risk in their pension beneit against the cost advantage that may be gained by exploiting underpriced pension insurance. Both LSW and this study show that government pension policies are major determinants o corporate pension unding and asset allocation policies. However, there are several important dierences between LSW and this study. In LSW, sponsoring irms make the pension unding decision to minimize compensation costs. In this study, sponsoring irms view pension unding as an alternative investment project. Thereore, sponsoring irms weigh the marginal expected return on pension unding against that on unding growth opportunities. In LSW, employees receive only a partial pension beneit guarantee rom the PBG and are thus actively involved in the design o the pension beneit structure. This study takes the structure o pension beneits as a given, and implicitly assumes that employees receive a ull promised pension beneit guarantee rom the PBG. According to the PBG, in act, ew D plan participants are aected by the statutory maximum beneit coverage. In LSW, the corporate pension policies are driven by policy variables such as the minimum unding requirement, the maximum beneit guarantee level, the structure o the insurance premium, and inancial distress costs. In their model, thereore, sponsoring irms would make the same pension decisions and thus maintain the roughly same pension unding level and asset composition in a given regulatory environment. In this study, the corporate pension policies are determined by the marginal expected return on growth opportunities and 7

10 stockholders' claim/liability on pension plans. As a result, sponsoring irms would make dierent pension decisions depending on their growth opportunities, even i they operate in the same regulatory environment. In regard to the corporate pension asset allocation policy, several authors attempt to explain why irms invest pension unds so heavily in risky assets. ost notably, Sundaresan and Zapatero 997, and Lucas and Zaldes 26 develop a model in which irms can invest in equity to hedge against increases in uture pension beneits that are positively correlated with the stock market. Frank 22 inds that irms tax beneits are positively associated with the percentage o their pension assets invested in bonds. Bergstresser, Desai, and auh 26 demonstrate that irms may inlate short-term earnings by increasing the assumed rate o return on plan assets, which would, in turn, result in a greater share o plan assets invested in equities. auh 29 inds that irms with poorly unded pension plans and weak credit ratings allocate a greater share o pension und assets to saer securities, whereas irms with well-unded pension plans and strong credit ratings invest more heavily in equity. This study departs rom current literature in determinants o corporate pension asset allocation policies. In particular, this study inds that corporate pension asset allocation policies are determined by stockholders' claim on pension surplus and costs related to severe pension underunding. III. The odel The model describes the behavior o a representative irm promising a pension beneit in the post-eisa regulatory environment. Thereore, I take the existence o the DBPP and the PBG, the government pension insurer, as a given. I use the model to investigate two decisions the irm makes: how much ree cash low to use to und the pension plan in a given year and how to allocate the pension und among various risk classes o assets. 8

11 A. Setting In the model, the irm seeks to maximize the value o stockholder equity. The irm is expected to operate in multiple periods. At the end o each period, say, a time t, the irm is bestowed with a ree cash low X t generated rom the previous period s operation and other assets in place. The ree cash low and other assets in place constitute total assets in place A t. Each period, the irm receives short-lived growth options G t that will become worthless i not exercised immediately. 2 The irm has no outstanding debt. It ollows that the value o stockholder equity V t at the end o a representative time period t is equal to V t = A t G t where A t = X t other assets in place. The irm has two choices or the disposal o ree cash low X t : to exercise a subset o or all current growth options and/or temporarily park it in a risky inancial asset that I use to represent the aggregate stock market or 2 to make voluntary contributions to the pension plan. I use the terms a risky inancial asset and the stock market interchangeably throughout this paper. The exercise o current growth options turns growth opportunities into operating assets. Such growth options-turned operating assets are assumed to last one period and to generate a gross return G y with a mean G y and a range [, or an amount y o capital investment. 3 The mean return G y is assumed to decrease in the amount o investment y i.e., a diminishing return on investment. The operating assets in place are assumed to produce a gross one plus return o A, with a mean A and a range [,. The operating assets in place, a risky inancial asset, and a riskless asset are assumed to produce a gross one 2 The irm may expect a stream o growth options that will arrive in the uture. All uture growth options arriving ater period t, however, are assumed away or the sake o simplicity. 3 This assumption is made or the sake o simplicity. The assumption o short-lived growth options turned operating assets can be relaxed so that they last more than one period without changing the main results qualitatively. 9

12 plus return o A with a mean o A and a range o [,, with a mean o and a range o [,, and >, respectively. 4 The corporate capital gain tax rate is assumed to be the same as the corporate income tax rate. The irm is assumed to sponsor a DBPP and to be enrolled in the PBG s pension beneit insurance program. As a result, the irm possesses plan assets P t and pension liabilities L t. The pension plan has two possible unding statuses: overunded and underunded. I the market value o plan assets exceeds the present value o pension liabilities, the pension plan is considered overunded. Otherwise, it is considered underunded. B. Stockholders laim/liability on the Pension Plan Stockholders claim on overunded pension assets and their share o ununded pension liabilities are complex and dier with irm characteristics and regulatory variables. In this section, I derive the value o stockholders claim on pension surplus and their liability or pension deicit. B.Stockholders laim on Pension Surplus The stockholders o a irm sponsoring an overunded plan may be entitled to only a raction o the excess pension assets. Existing law requires that the irm must pay an excise tax o up to 5% in addition to the normal 34% corporate tax or pay a lower excise tax and share the excess pension assets with the plan participants i the irm converts the excess pension assets into operating assets through a reversion. As a result, the reversion would potentially leave stockholders with only 6 cents or each dollar o reversion. 4 Without loss o generality, G y, A, and are assumed to be stationary with the same mean and is the same or all time periods or the sake o simplicity.

13 As suggested by Burrow, Scholes, and enell 983, and Thomas 989, sponsoring irms have an indirect way o moving excess pension assets back into operation without paying excise and corporate taxes. One o them is to reduce the amount contributed to the pension plan each year. 5 The EISA o 974 and several rounds o subsequent legislation i.e., the P o 987, the etirement Protection Act o 994, and the P o 26 require irms sponsoring an underunded plan to make mandatory contributions that amount to the greater o the Fs and the Ds. The Fs consist o new pension beneits accrued in the current year the normal cost and a raction o the unding shortall currently %. The Ds comprise the irst year s contribution o 8 3% o any underunding and installment payments o the rest o the underunding over a period o three to ive years. A irm sponsoring an overunded plan, however, is exempt rom those mandatory contributions. As a result, the irm can gradually convert excess pension assets into operating assets by applying the pension surplus toward the required minimum level o unding contributions until the pension surplus is depleted. As a result, the value o stockholders net claim on a pension surplus N PS is equal to the discounted value o a stream o new pension beneits to be accrued in the uture: T accr PS PBi N t = it i= t φpb T accr Tt λ over P t L t 2 where PB i accr = expected new pension beneits to be accrued in time i, = the corporate cost o capital; 5 Burrow, Scholes, and enell 983 discuss two other methods: increasing the raction o total employee compensation in the orm o promised pensions and increasing early retirement beneits. Unlike the method considered in the present study, their two methods may not be easily implemented since the compensation package must be negotiated with employees.

14 T = inal period in which new pension beneits are covered by remaining excess pension assets, ϕ < ϕ < = raction o the inal period s pension beneits that can be covered by remaining excess pension assets, λ over = discount actor or a pension surplus. Notice rom expression 2 that uture pension beneits are discounted back at the corporate cost o capital since the pension plan itsel is not allowed to raise the external und against its pension assets. Only the sponsoring irm can raise the external und to cover pension beneits in the capital market against the sponsoring irm's entire assets i.e., both operating and pension assets. The inal period T is determined by the relation P T L T <. Let PB paid t denote the pension beneits paid out rom pension assets to retired employees in time t, and denote the long-term average annual gross expected return on pension assets. 6 The expected value o pension assets at time T, P T, can be expressed as: T T P T = j=t,j P t i=t j=i,j T PB i paid = Tt P t T Ti paid i=t PB i 3 The value o pension liabilities at time T consists o the uture value o a stream o new pension beneits to be accrued rom time t to T minus the uture value o a stream o pension beneits paid out rom time t to T plus the uture value o pension liabilities outstanding as o time t. electing the actual practice, pension liabilities are assumed to grow at the gross riskless rate so that The value o pension liabilities at time T can be expressed as: 6 As the pension surplus becomes smaller in size over time, the sponsoring irm may rebalance pension assets and change the asset composition which would, in turn, change the return on pension assets. 2

15 L T = Tt T L t Ti accr T i=t PB i Ti paid PB i i=t 4 The relation between the pension surplus P t L t at time t, a stream o new pension beneits to be accrued, and a stream o pension beneits paid out rom time t to T can be expressed as: P t L t = T Ti i=t PB Tt i accr φpb T accr Tt T i=t Ti Ti Tt PB paid i Tt Tt L t 5 Notice rom Expressions 2 and 5 that λ over = i = = ; and λ over < i > =. In a more plausible case where > >, λ over can be greater or less than one, depending on,, and,. The reason that λ over can be greater than one is that pension assets are assumed to grow in value aster than pension liabilities so that the expected pension surplus increases over time. B2. Stockholder Liability or Pension Deicit As discussed beore, irms o an underunded pension plan are obligated to make the greater o Fs and Ds to the pension plan every year until it becomes ully unded. To relect the structure o tax-deductible Ds imposed on a irm sponsoring an underunded plan, I assume that the irm is required to contribute a constant raction, k, < k <, o the pension deicit to the pension und every year until the pension deicit is remedied. In exchange or providing insurance coverage, the PBG requires irms sponsoring an underunded plan to pay a variable-rate premium currently $9 per $, o underunding in addition to a ixed premium o $35 per participant. Let m denote the variable-rate premium as the percentage o ununded pension liabilities. Notice that similar to pension beneits, the pension deicit can be inanced only against the sponsoring irm's entire assets. The value o stockholders net liability or a pension deicit NL PD can then be expressed as 3

16 NL t PD = τk m = i t ˆ it it it k Pt Lt it τk m [ P t L t ] k k λ under P t L t 6 where λ under is the discount actor or pension deicit. Notice rom expression 6 that i = = and m =, λ under = τ, which means that sponsoring irms are liable only or a raction o the pension deicit. The EISA o 974 and several rounds o later legislation also state that i irms sponsoring an underunded plan ail to make the minimum required contributions and the unpaid amounts total more than $ million, the PBG can perect and enorce a statutory lien on the irm s property, up to 3% o its net worth. The net worth is widely interpreted as the air market value o equity o the sponsoring irm, excluding pension assets and liabilities. As a result, the augmented pension assets consist o the pension plan assets plus a raction o the irm s stockholder equity without pension assets and liabilities. It ollows that the value o stockholders net claim on the augmented pension assets N A is: N t A = max{φv t λ under P t L t, } 7 where φ = raction o the intrinsic value o the irm s equity that can be seized by the PBG in the event o the irm s ailure to make required contributions, V t = A t G t. Taken together, the value o stockholders equity o the irm at the end o time t can be expressed as 4

17 V t = - φa t G t max{φv t λ PS P t L t, } 8 where λ PS = λ over i P t > L t, λ under i P t < L t.. Pension Funding Decision At the beginning o every period, the sponsoring irm repeats the same pension unding and asset allocation decisions. The pension unding decision involves allocation o the ree cash low X t between the exercise o current growth options and contribution to the pension und. The pension asset allocation decision entails allotment o the pension und between risky and riskless inancial assets. In this section, I analyze the irm s pension unding decision, given pension asset allocation. I consider two alternative circumstances: with internal inancing and with external inancing.. With Internal Financing This subsection analyzes the irm s pension unding decision with internal inancing. Let α, < α <, denote the raction o the ree cash low X t that the irm allocates to the exercise o current growth options and/or investment in the stock market. The remainder α is allocated to the pension und. This study ocuses on the pension unding decision in a given period and not a dynamic cash low allocation decision between current and uture growth options. The sponsoring irm s operating assets in place are assumed to generate a steady stream o cash low suicient to und growth options and the pension plan every period. I the irm chooses to contribute part or all o the ree cash low to the pension und, 5

18 the irm receives a one-time tax credit on the amount contributed. At the beginning o every period, the irm repeats the same pension unding decision to maximize the market value o stockholder equity: ax V t = E t [ V t ] α = OA φe t [ ~ X t ~ A A t X t G y dy operating assets in place α growth options turned operating assets τ{ ~ X t ~ A A t X t α G y dy }] φτ αx t corporate tax paid tax saving rom pension contributions E t [ φv t PP λ PS { ~ P t αx t L t } φv t { ~ P t αx t L t } > ] stockholders claim on augmented pension assets where OA = the discount rate or the sponsoring irm's operating assets, PP = the discount rate or the pension plan, = the discount rate or the sponsoring irm's non-pension assets that consists o operating assets and tax beneit rom pension contributions, ~ = gross one plus return on pension assets, i.e., ~ ω; ~ ω ~, ω < ω < = raction o plan assets invested in the risky inancial asset. The sponsoring irm weighs the discounted risk-adjusted marginal expected atertax return on current growth options against the discounted risk-adjusted marginal expected return on a pension contribution. The marginal expected ater-tax return on current growth options is a decreasing unction o α due to the diminishing return on investment. The marginal expected return on a pension contribution consists o the discounted marginal 9 6

19 corporate tax rate and the discounted expected tax-ree return on plan assets. Notice rom expression 9 that the return made by plan assets is not subject to the corporate tax, while the return earned by the exercise o current growth options is taxable. The exemption o the return on pension assets rom the corporate tax constitutes another beneit o a pension contribution in addition to its tax deductibility. Let α, < α <, denote the allocation ratio at which - αx t is equal to the maximum possible tax-deductible contribution a irm can make to an overunded pension plan. 7 With no tax beneit available, the irm would rather use the ree cash low to exercise current growth options and/or invest in the stock market. Thereore, a irm sponsoring an overunded plan would allocate at least αx t to exercise current growth options and/or invest in the stock market. Proposition Suppose that the objective unction 9 is concave in α. Then the optimal α, where α α, or an overunded plan and α, or an underunded plan, is determined at which OA [ G α τ G α Proba φ{e a G α τe a G α }] = τ PP [Probb PSDF E b ProbcPDDF* E c ] where PSDF arginal Pension Surplus Discount Factor Tα t, PDDF arginal Pension Deicit Discount Factor τk m, k a P t L t < φv t, 7 The DBPP-related laws set the maximum tax-deductible contributions beyond which the irm o an overunded plan is not allowed to receive a tax credit. Furthermore, the irm may be subject to an excise tax on contributions above the maximum. The etirement Protection Act o 994 and the P o 26 state that the maximum tax-deductible contributions may not exceed the greatest o the Fs, 2 the amount necessary to ully und the plan s current liability, and 3 the normal cost, plus the plan s past service cost with amortization periods reduced to years. 7

20 b P t L t >, c φv t P t L t <, 8 or α = α or or an overunded plan and α = or or an underunded plan. Proo: See Appendix. For ease o understanding the results o Proposition, I take a simpler case, where stockholders claim on the augmented pension assets is always positive. In such a case, the optimal α* is determined at which OA [ α τ α ] G G = τ PP [ProbbPSDF E b Probc PDDF* E c ] Note rom expression that the optimal α* is determined at which the risk-adjusted marginal expected ater-tax return on growth options equals the discounted marginal corporate tax rate plus a weighted average o the risk-adjusted expected tax-ree returns on pension assets discounted by the marginal discount actors or overunded and underunded plans, with the weights being the probabilities o the pension plan becoming overunded and underunded due to pension contributions and pension asset return. Expression indicates that an additional dollar o voluntary contribution to the pension plan would enable the sponsoring irm to save the marginal corporate tax, and increase stockholders' claim on excess pension assets or reduce the burden o making mandatory pension contributions by the 8 Since the irm can temporarily park the ree cash low in the stock market, G α >. Otherwise, α is determined at which OA [ τ Proba φ{e a G α τe a G α ] = [τ ] PP [ProbbPSDF E b Probc PDDF * E c ] and α is divided between current growth options and the risky inancial asset. The division o α is determined at which G α τ G α = τ. Since this study ocuses on the pension unding decision, I assume that G α > or simplicity. 8

21 expected tax-ree return on plan assets discounted by the marginal discount actors. The marginal pension surplus discount actor PSDF measures the present value o one dollar o the pension beneit to be accrued in the arthest uture which is covered by an additional dollar o voluntary contribution. The marginal pension deicit discount actor PDDF measures the ater-tax ractional D and variable-rate insurance premium to be reduced by an addition dollar o voluntary contribution. In the case where stockholders claim on the augmented pension assets is negative, the sponsoring irm would surrender up to a raction currently 3% o stockholder equity. In such an event, stockholders o the sponsoring irm and the PBG would share the marginal expected ater-tax return on current growth options. I the pension deicit is so large that even the augmented pension assets ails to cover it, it would be in the best interest o stockholders or the sponsoring irm to walk away rom the pension plan i.e., ile a distress termination, laying all pension obligations on the PBG. 9 For both overunded and underunded pension plans, several common actors determine voluntary pension contributions that, in turn, determine pension unding. Those actors include the irm-speciic characteristics such as the expected return on current growth options, the marginal corporate tax rate, and the time distribution o uture pension beneits. For both overunded and underunded plans, a greater marginal expected return on current growth options leads to lower pension unding. The eect o the marginal corporate tax rate, 9 I a sponsoring irm issues a notice o intent to terminate its DBPP, the PBG examines the irm s inancial condition to determine whether the sponsoring irm satisies one o several inancial distress tests. The inancial distress tests include whether it has been demonstrated that the sponsor or ailiate cannot continue in business unless the plan is terminated. I the PBG approves o the application or distress termination, the sponsoring irm and each controlled group member are jointly and severally liable to the PBG under EISA Section 462b or the total amount o ununded beneit liabilities determined in accordance with 29 F 422 Subpart D. I the PBG determines that a plan does not qualiy or distress termination, the plan will remain ongoing under close monitoring by the PBG. Bereskin 29 notes that sponsoring irms may be permitted to terminate their plans only when iling or bankruptcy protection since the passage o the Single-Employer Pension Plan Amendment Act o

22 however, is not the same or overunded and underunded plans. In the case o an underunded plan, unlike or an overunded pension plan, an increase in the marginal corporate tax rate would not necessarily result in more voluntary pension contributions. While the sponsoring irm receives an immediate tax credit rom a pension contribution, an additional dollar contribution to an underunded plan would reduce stockholders net pension liability only by one minus the marginal corporate tax rate. The reason is that the tax deductibility o pension contributions has been already actored into the stockholders net pension liability. orollary α α, decreases in PSDF, i.e., Tα t. Proo: See Appendix. Pension unding is also partially determined by the time distribution o uture pension beneits through PSDF. It should be noticed that PSDF depends on how ar in the uture the last one dollar o excess pension assets is used to cover the accrued pension beneits. As a result, i the pension surplus is too large in size or ew pension beneits are expected to be accrued in the near uture, the marginal expected return on pension contribution would be small relative to that on growth options. In such a case, the irm would rather invest the entire ree cash low in current growth options and/or the stock market. In the other extreme case, i a large amount o pension beneits is expected to be accrued in the imminent uture, the marginal expected return on pension contribution would surpass that on growth options. In such an event, the sponsoring irm would contribute the entire ree cash low to the pension und. orollary 2 α* decreases in k or m. Proo: See Appendix. 2

23 For an underunded pension plan, the pension unding is determined by PDDF which is, in turn, determined by the raction o the unding shortall, k, required to be made up every year, and the variable-rate premium, m. A higher k or m would increase PDDF, which, in turn, raises the marginal expected return on pension contributions, and, thus, would lead to a greater allocation o the ree cash low to the pension und. orollary 3 α* decreases in φ. Proo: See Appendix. An increase in φ, the maximum possible raction o stockholder equity that can be seized by the PBG, would raise the upper limit o the pension deicit or which stockholders o the sponsoring irm are liable. aking stockholders responsible or a greater share o ununded pension liabilities would induce the sponsoring irm to make more voluntary contributions to the pension plan. 2. With External Financing This subsection extends the analysis o the corporate pension unding decision to the case where the irm relies on external inancing. For simplicity, the irm is assumed to retain no ree cash low at the end o time t. The irm can borrow an external und F at a gross interest rate D F, with D = in the case o no deault risk. The interest rate D F increases in F due to a positive relation between F and deault risk. Since the PBG s claim receives a lower priority in the event o deault, debt claims are assumed to be senior to pension liabilities. With external inancing, objective unction 9 is modiied as ollows: Keating 993 argues that the PBG' arguments in avor o priority status in the event o a sponsoring irm's bankruptcy are weak. iting several reerences, Bereskin 29 notes that "the PBG's claims are typically relegated to that o other unsecured claims [in the event o sponsoring irms' bankruptcy]." Schroeder 25 reports that the PBG recovers only 7% o its claim o ununded pension liabilities in stockholder equity o the reorganized irm. 2

24 ~ V ] ax F, α V t = E t [ t = φ E t [ OA { ~ F ~ A A t G y dy α D FF τ{ ~ A A t F ~ α G y dy D F F}} τ αf ] ~ F ~ A At α G y dy DFF > ] E t [ φv t PP λ PS { where ~ P t αf L t } φv t { ~ P t αf L t } > ] D F = gross external inancing cost as a percentage o the amount o external inancing, with D F, F = amount o external inancing. Figure demonstrates the sponsoring irm s pension unding decision. The optimal amount o external inancing is determined or the amount at which the greater o the discounted marginal expected ater-tax return on current growth options and the discounted marginal expected return on a pension contribution is equal to the discounted marginal atertax cost o external inancing. Once the optimal amount o external inancing is determined, the optimal allocation o the external und between current growth options and the pension plan is determined in the same manner as with internal inancing. The only dierence is that with external inancing, the marginal expected returns on current growth options and pension contributions are conditional on the sponsoring irm remaining solvent. 22

25 D. Pension Asset Allocation Decision This section analyzes the pension asset allocation decision. For a given pension unding decision, the sponsoring irm s objective unction or pension asset allocation is reduced to: ax < ω < E t [ φv t PP λ PS { ~ P t αx t L t } φv t { ~ P t αx t L t } > ] 2 As discussed in Section III, the objective unction 2 indicates that the sponsoring irm is responsible or a pension deicit up to a raction φ o the sponsoring irm's stockholder equity, excluding pension assets and liabilities. It has been so ar assumed that there is no additional cost associated with a severe pension deicit on the part o the sponsoring irm. With a signiicant proportion o plan assets invested in the stock market, a market correction could cause plan assets to all signiicantly below pension liabilities in value. The P o 26 requires employers with at least one plan that is less than 8% unded to report annually additional inormation so that the PBG can closely monitor the situation. Thereore, the sponsoring irm would bear an additional cost o meeting reporting requirements and being monitored by the PBG. Furthermore, an increase in pension deicit would raise the probability o pension plan termination initiated by the PBG, which involves a plan termination-related cost on the part o the sponsoring irm. Upon plan termination, sponsoring irms must pay a termination premium o $,25 per participant in the year o termination and in each o ollowing two years established by the Deicit eduction Act o 25. In addition, sponsoring irms have to bear various types o direct and indirect costs associated with 23

26 plan termination, including the transaction costs o disposing o operating assets up to 3% o the equity value i the unpaid amount o Fs exceeds $ million. Bicksler and hen 985 also discuss the legal expenses arising rom lawsuits associated with plan termination and the costs resulting rom poor labor relations. In light o these observations, I assume that the sponsoring irm with a severe pension deicit incurs a percentage cost c proportional to the pension deicit exceeding a certain threshold. I then modiy stockholders net liability or a pension deicit as ollows: NL t PD λ under P t L t = τk m [ τk m c [ P t k P t k L t k ] i δl t < P t L t ; L t k ] i P t < δl t 3 where c is a constant with c > and δ is a constant with < δ <. In the ollowing analysis, the objective unction 2 is assumed to be concave in ω. Proposition 2. For an overunded pension plan i.e., P t αx t > L t, there exists a critical marginal pension discount actor, PSDF*, such that or PSDF < PSDF*, the optimal pension asset allocation is a mix o risky and riskless inancial assets i.e., < ω * < and, or PSDF > PSDF*, ω * =. 2. For a modestly underunded pension plan i.e., δl t < P t αx t < L t, there exists a critical c* such that, or c > c*, the optimal pension asset allocation is a mix o risky and riskless inancial assets i.e., < ω * < and, or c < c*, ω * =. Proo: See Appendix. 24

27 The intuition behind this result is ollowing. The sponsoring irm weighs beneits o allocating more pension assets toward the risky inancial asset against costs. For an overunded pension plan, a high PSDF means a relatively large amount o pension beneits to be accrued in the near uture, toward which the sponsoring irm can apply the pension surplus. With a suiciently high PSDF, the irm with an overunded pension plan would invest the entire pension und in the risky inancial asset since stockholders can claim most o the incremental excess pension assets resulting rom a stock market appreciation. Thereore, a suicient condition or a mix o risky and riskless assets is a suiciently low PSDF. It is a remote possibility, however, that a market correction causes an overunded pension plan to turn into severely underunded. Thereore, the cost o severe underunding is not relevant to the asset allocation decision or an overunded plan. For a modestly underunded plan, a greater allocation o pension assets toward the risky inancial asset would provide the sponsoring irm with the opportunity or curtailing Ds and saving the insurance premium by reducing the pension deicit through a capital gain. But it could also cause the irm to incur the expected cost o severe underunding. As the irm raises the percentage o stock market investment, the marginal expected gain i.e., PDDF remains constant, but the marginal expected cost o severe underunding increases since the probability o alling into a severely underunded status rises with the percentage o stock market investment. With a suiciently large cost o severe underunding, thereore, the irm would ind a mix o risky and riskless assets optimal. It is unlikely, however, that the underunded plan turns into overunded as a result o more stock market investment. As a result, or an underunded plan, PSDF is not a actor that determines the asset allocation decision. 25

28 In contrast, the irm with a severely underunded plan, has been already incurring the cost o severe underunding, which is a sunk cost. As a result, beneits o stock market investment would outweigh its costs. In such a circumstance, the irm with a severely underunded plan may ind it optimal to invest the entire pension assets in the risky inancial asset. We have assumed a linear relationship between the pension deicit and the cost o severe underunding. It is more plausible, however, that the cost o severe underunding increases at an increasing rate with the size o pension deicit. With such an exponential relationship, even the irm with a severely underunded plan would preer to take a more conservative approach in asset allocation. The ollowing comparative statics result is obtained. orollary ω * T Pt increases in PSDF t, i.e., Lt t. Proo: See Appendix. The intuition behind this result is the same as discussed above or Proposition 2. The eect o k and m on the optimal pension asset allocation, ω *, is vague or modestly and severely underunded pension plans. It can be shown, however, that ω *, < ω * <, decreases in m under mild conditions. The reason is that a higher m would increase stockholders' liability or pension deicit. As a result, the irm would reduce allocation o pension assets to the risky inancial assets in response to a higher insurance premium. orollary 2 ω *, < ω * <, decreases in φ. Proo: See Appendix. 26

29 An increase in φ would cause the sponsoring irm to lose more o its operating assets to the PBG i a stock market correction makes the pension plan severely underunded. It ollows that a higher φ would induce the sponsoring irm to invest less in the risky inancial asset. IV. Policy Implications Previous sections show that or overunded and modestly underunded plans, pension unding and asset allocation decisions are inluenced by PSDF, PDDF, which is, in turn, determined by policy variables such as the D rate k, the variable-rate insurance premium m, and the raction o the irm s operating assets that can be seized by the PBG φ. This section evaluates the eectiveness o these policy variables in inluencing corporate pension unding and asset allocation policies. The model implicitly assumes that sponsoring irms are required to ully und their underunded pension plans within a certain period. Prior to the P o 26, however, sponsoring irms were generally required to und only 9% o pension liabilities and could delay unding ununded pension liabilities through accounting maneuvering or as many as 3 years Brown 28. Such a partial unding requirement can be easily incorporated into expression 6, the valuation o stockholders share o ununded pension liabilities. It can be shown that the partial unding requirement has the same eect on pension unding and asset allocation decisions as does the D rule. The P o 26 gradually phased in a % unding target and tightened the D rule such that sponsoring irms are required to remedy their pension deicits within our to seven years. I examine whether the tightened D rule can induce sponsoring irms to alter the unding and asset allocation strategies with the ollowing plausible numerical example. For the numerical example, I set Probc the probability o the plan being modestly 27

30 underunded =. k =.5, m =.9, =.5, =.2, =.2, and τ =.35, relecting evidence that sponsoring irms made an average annual return o 2% on their pension assets in the period o For those parameter values, PDDF is estimated to be.62. I we apply the estimate o.62 or PDDF to expression, the right-hand side o expression equals.448. This means that sponsoring irms o a modestly underunded plan make a meager 4.48% o return on each one dollar contributed to the pension plan. Such a relatively low return may explain why an overwhelming majority o sponsoring irms made no voluntary contributions to their pension und at all in the period o In light o the estimate above, we expect a majority o sponsoring irms to continue to make no voluntary contributions with the current D rule. The new D rule mandated in the P o 26 would certainly shorten the period in which underunded pension plans should be ully unded. However, the numerical example above suggests that the new D rule would not prevent sponsoring irms rom underunding their pension plan in the irst place. The reason is that the reward or voluntary contributions to the pension plan are relatively low. The PBG charges sponsoring irms a ixed premium o $35 per participant as o 2, plus.9% o any ununded vested beneits. The variable-rate premium, which is set by ongress, does not vary with pension asset risk, and, as a result, is widely believed to be below the economically air level. However, it would be practically impossible or the insurance premium to be set commensurate with pension asset risk due to a high measurement and monitoring cost. This study shows that a higher variable-rate premium m would induce sponsoring irms o an underunded plan to und their plan more. The current structure o the insurance premium, thereore, is eective in inducing sponsoring irms to improve their pension status, even i the premium is not designed to relect pension asset risk. 28

31 Furthermore, this study shows that k and m are interchangeable and have the same eect on the corporate pension decisions. The D rule has been already tightened and it may not be practically possible to tighten it urther. This study suggests that policy makers can adjust the variable-rate premium to obtain the same eect on the corporate pension unding policy as altering the D rule. This study also suggests that sponsoring irms o a severely underunded plan would choose to invest the entire pension und in the stock market regardless o k or m. The pension put i.e., the ability o sponsoring irms to walk away rom their pension plan by surrendering up to 3% o the equity value makes such an incentive stronger. As discussed in the previous section, an exponentially increasing cost o severe underunding may dampen such an incentive. However, such a cost structure alone may not be suicient enough to prevent sponsoring irms rom taking an excessive pension asset risk. When sponsoring irms are bullish on the stock market, they may ind the expected return on the stock market outweigh the expected cost o plan termination when making a pension asset allocation decision. A lat variable-rate premium rate is currently imposed on ununded pension liabilities regardless o the size o the pension deicit. This study suggests that i the variablerate premium is structured such that sponsoring irms o larger ununded pension liabilities are more heavily penalized, they would have less incentive to underund the pension plan. For instance, the variable-rate pension premium rate can be designed to be progressive so that a higher premium rate is charged to the sponsoring irms with a larger pension deicit. The results discussed in Section III suggest that the maximum possible raction o sponsoring irms operating assets that can be seized by the PBG, φ, is an eective policy tool that inluences sponsoring irms pension unding and asset allocation policies. A higher φ would encourage sponsoring irms to make more voluntary contributions to their pension 29

32 plan and allocate more o pension unds toward sae assets. However, a change o φ requires approval rom ongress, which is a long process. A possible remedy or such a protracted process is that ongress provides the PBG with the authority to change φ within a certain range without ongressional approval. V. Empirical Implications In this section, I discuss empirical implications o the theoretical results discussed in section III. Implication : Young irms with ample growth options have lower pension unding ratio than mature irms with limited growth options. Young irms may have more growth opportunities than mature irms since mature irms may have already exhausted most o their growth options. As a result, young irms with ample growth opportunities may preer to use inancial resources or exercise o growth options rather than pension unding. Implication 2: Sponsoring irms with a higher PSDF maintain a relatively higher pension unding ratio. Proposition shows that PSDF is a determinant o pension unding and a higher PSDF would lead to more pension unding. PSDF is determined by the size o pension surplus and the time distribution o pension beneits to be accrued in the uture. Speciically, PSDF is negatively related to the size o pension surplus and positively related to the amount o pension beneit to be accrued in the imminent uture. As a result, PSDF may be estimated as ollows: PSDF i,t = i PS i,t- Pension Surplus o irm i < PB i,t ; 3

33 PSDF i,t = i PB i,t < PS i,t- < PB i,t PB i,t ; PSDF i,t = 2 i PB i,t PB i,t < PS i,t- < PB i,t PB i,t 2 PB i,t2 ; PSDF i,t = min 3, PB 2 i, t * PBi, t * PS i, t PB i, t 2 i PB i,t PB i,t 2 PB i,t2 < PS i,t- Implication 3: For overunded pension plans, pension unding is positively related to the marginal corporate tax rate. For underunded pension plans, the tax eect is weak. For an overunded plan, a higher marginal corporate tax rate would enable the sponsoring irm to save more tax rom pension contributions. For an underunded plan, however, an increase in the marginal corporate tax rate has two opposite eects. It would allow the sponsoring irm not only to save more tax but also to reduce stockholders' claim on the pension asset return by the marginal corporate tax rate. The corporate tax eect, thereore, is weak or underunded plans. Implication 4:The irm with a low credit rating or a high debt ratio maintains lower pension unding relative to pension liabilities. With the pension plan inanced by an external und, the cost o external inancing is another actor or the pension unding decision. Other things being equal, the irm with a lower credit rating or a larger debt outstanding would incur a higher cost o external inancing. With a high cost o external inancing, pension unding is more likely to produce a negative return. arroll and Niehaus 998 report evidence on the impact o ununded pension liabilities and excess pension assets on corporate debt ratings. This study suggests that such causality may run the other way around. That is, the pension plan is underunded simply because o a high 3

34 borrowing cost which results rom a low credit rating. Such an external inancing constraint would eventually lead to pension plan underunding. Implication 5: The irm with a higher PSDF takes more pension asset risk. Proposition 2 and orollary show that the irm with a higher PSDF has an incentive to take more pension asset risk since the irm can apply the excess pension assets increased by a capital gain toward the required pension contributions. VI. Summary and onclusion In this study, pension unding is viewed as an alternative investment project and the sponsoring irm weighs pension unding against unding o growth opportunities. I derive the value o stockholders claim on overunded pension assets and liability or ununded pension liabilities. Based on that valuation, I develop a model o corporate pension unding and asset allocation policies in the post-eisa regulatory environment. This model shows how corporate pension policies are shaped by the sponsoring irm characteristics, such as growth opportunities and marginal corporate tax rate, and policy variables such as minimum unding requirements, the variable-rate insurance premium, and the maximum possible raction o a irm s operating assets that can be seized by the PBG. I also discuss the policy and empirical implications o the theoretical results. As an extension o this study, I am currently testing the empirical implications and ind that the preliminary test results strongly support the empirical predictions. Similar theoretical results are also provided by ooper and oss

35 Appendix Proo o Proposition : Objective unction 9 can be simpliied as: X t V t = OA [ A A t X t α G y dy τ{ G y dy }] τ αx t α X t OA ProbaφE c V t PP Probbλ over {E b P t αx t L t } PP Probc λ under {E c P t αx t L t } A T where λ over {E b P t αx t L t } = E b PB φpb T TP i= t t L t t. i i t For ϕ <, dierentiating A with respect to α and dividing by X t yields dv t /Xt = dα Probaφ[ OA G α τ G α τ ] [Probb PSDF E b Probc PDDF * E c ] A2 where PSDF Tα t ; PDDF τk m. k since dλ overp t L t dα = T PB d i i=t it dp t L t x dp tl t, and dα 33 T PB d i i=t it dp t L t dϕpb Tt/ dp t L t = TP t L t t, and dp tl t = E dα b X t. = dpb t... For ϕ =, expression A is not dierentiable with respect to α. In that case, objective unction 9 reaches a local maximum at a discontinuity point where Probaφ[ OA G α τ G α intersects with Probaφ τ PP [Probb PSDF E b Probc PDDF * E c ].

36 I expression A2 > or α = α or an overunded plan and or α = an underunded plan, and expression A2 < or α =, objective unction 9 reaches a local maximum at α α, or an overunded plan and at α, or an underunded plan where expression A2 = since objective unction 9 is concave in α by assumption. I expression A2 > α, < α <, α =. I expression A2 < α, < α <, α = α or an overunded plan and α = or an underunded plan. Q.E.D. Proo o orollary o Proposition : Notice rom A2 that or α = α, where < α <, Probaφ[ OA G α τ G α τ ] A3 PP [Probb PSDF E b Probc PDDF * E c ] = Dierentiating A3 with respect to PSDF Tα t yields: A3 = Tα t PP Probb E b <. It ollows that dα d Tα t = A3 Tα t A3 α < since A3 α < by assumption. Q.E.D. Proo o orollary 2 o Proposition : Dierentiating A3 with respect to k produces: A3 k = PP Probc { τ k ˆ τ k m k ˆ ˆ 2 } E c < since k k m = m > min m 2, m >. It ollows that 34

37 dα A3 = k dk A3 α < since A3 α < by assumption. Dierentiating A3 with respect to m produces: A3 = m PP τ Probc E k c < since > by assumption. It ollows that A3 dα = m dm A3 α < since A3 α < by assumption. Q.E.D. Proo o orollary 3 o Proposition : Dierentiating A3 with respect to φ produces: A3 = Proba[ φ OA G α τ G α τ ] < since or α = α, Probaφ[ OA G α τ G α τ ] PP [Probb PSDF E b Probc PDDF * E c ] = and thus it must be true that OA G α τ G α > τ. It ollows that A3 dα = φ dφ A3 α < since A3 α < by assumption. Q.E.D. Proo o Proposition 2: Deine t P t αx t. Objective unction 3 can be rewritten as: ax ω E t [ PP φv t PP λ PS { L t } > ] ~ P t αx t L t } φv t = E t [ PP φv t PP λ over { ω E t [ PP φv t PP λ under { ω E t [ PP φv t PP λ under { ω ~ t L t L ~ t t L ~ t t { } ~ > x] ~ } y < ~ < x ] } z < ~ < y ] P t αx t 35

38 = x - [ PPφ Vt PPλover{ ω t Lt ] df, Vt [ PPφ Vt PPλunder{ ω t Lt }] df, Vt x y [ PPφ Vt PPλunder{ ω } t Lt }] df, Vt y z A4 where x = ω Lt t, y = ω δlt t, and z = max[, ω Lt φvt t ]. Dierentiating expression A4 with respect to ω by applying the Leibniz integral rule yields: A4 = t [ T i ω PP dg x τ k m df, Vt k ˆ x y τ k m c df, Vt k ˆ ] A5 T PBi since λ over P t L t = i t and dλ overp t L t = dω i= t y z T PB d i i=t it dp t L t x dp t L t dω and T PB d i i=t it dp t L t = PB t... dϕpb T Tt / dp t L t = Tt, and dp t L t dω = t, and λ under P t L t = τk m[ P t k 36

39 L t ] and dλ underp t L t k dω = dτkm Pt k dp t x dp t dω and dp t =. dω t. For an overunded plan i.e.,., t P t αx t > L t, A5 ω<ωx= = t PP For ω =, T i dg > or ω < ω x = since >. A5 = ω t [ T i T i PP dg dg x x τ k m df, Vt k ˆ A6 y τ k m c df, Vt k ˆ ] y z where x = L t t, y = δl t t, and z = max[, Lt φv t t ]. Notice rom A6 that A6 < or T i = and A6 decreases in T i. Thereore, there exists a critical PSDF* such that or PSDF < PSDF*, A6 <. Since A5 > or ω close to zero and A6 < or ω = PSDF < PSDF*, and objective unction 3 is concave in ω by assumption, < ω < or PSDF < PSDF*,, and ω = or PSDF > PSDF*,. 2. For an underunded plan i.e., t P t αx t < L t, as ω approaches zero, 37

40 38 lim ω A5 = t PP [, ˆ t V df k m k τ > since lim ω x =, lim ω y = and >. For ω =, A5 ω = t PP [ x i T dg, ˆ t x y V df k m k τ, ˆ t y z V df k c m k τ ] A7 where x = t t L, y = t t δl, and z = max[, t t t L φv ]. Notice that, ˆ t y z V df k c m k τ < since y < < and, thus, A7 decreases in c. Thereore, there exists a critical c* such that or c > c*, A7 <. Since A5 > or ω close to zero and A5 < or ω = and c > c*, and objective unction 3 is concave in ω by assumption, < ω < or c > c*, and ω = or c < c*. Q.E.D. Proo o orollary o Proposition 2: Notice rom A5 that or ω = ω *, < ω * <, t PP [ x i T dg, ˆ t x y V df k m k τ, ˆ t y z V df k c m k τ ] = A8

41 where x = ω Lt ]. t, y = ω δlt t, and z = max[, ω Lt φvt t Dierentiating expression A8 with respect to Tt yields A7 Tt = t PP dg x > or all x since >. It ollows that dω d Tt = A7 Tt A7 ω > since A8 ω < by assumption. Q.E.D. Proo o orollary 2 o Proposition 2: I φ V t V t, then A5 at ω = ω is equal to: { ~ P t αx t L t } > or all T i x dg τ k m df, Vt k ˆ x y A9 τ k m c df, Vt k ˆ ] = y Dierentiating A9 with respect to k yields: = τ k, 2 df Vt A9 k m x y τ m c, 2 df Vt k y > since = and τm k 2 < τmc k 2 and y <. It ollows that dω A8 = K dk A8 ω > since A8 ω < by assumption. 39

42 Dierentiating A8 with respect to m generates: τ A8 = k, df Vt k x y τ df, Vt k < y since = and x <. It ollows that A8 dω = m dm A8 ω < since A8 ω < by assumption. Q.E.D. Proo o orollary 2: Dierentiating A5 with respect to φ produces: A5 φ = τ k m c k ˆ ω t V t z df = z, Vt < since z <. It ollows that dω A5 = φ dφ A5 ω < since A5 ω < by assumption. Q.E.D. 4

43 eerences Bereskin, F. L., 29, Determinants o deined-beneit pension plan policies, working paper, University o ochester. Bicksler, J. L., and A. H. hen, 985, The integration o insurance and taxes in corporate pension strategy, Journal o Finance, 4, Black, F., 98, The tax consequences o long-run pension policy, Financial Analysts Journal, 36, Brown, J.., 28, Guaranteed trouble: The economic eects o the Pension Beneit Guaranty orporation, Journal o Economic Perspectives, 22, Burrow, J. I.,. S. Scholes, and P. enell, 983, Economic implications o EISA, in Z. Bodie and J. B. Shoven Eds., Financial Aspects o the United States Pension System, University o hicago Press, hicago, pp arroll, T. J., and G. Niehaus, 998, Pension plan unding and corporate debt ratings, Journal o isk and Insurance, 653, ooper,., and T.W. oss, 22, Pensions: theories o underunding, Labor Economics 8, Financier, 29, Pension Plan Deicits The Impact On US orporations, press release, June. Frank,., 22, The impact o taxes on corporate deined beneit plan asset allocation, Journal o Accounting esearch, 37, Investment ompany Institute, 22, The US retirement market, available at Jin, L.,.. erton, and Z. Bodie, 26, Do a irm s equity returns relect the risk o its pension plan? Journal o Financial Economics, 8, 26. Keating, D, 993, hapter 's new ten-ton monster: the PBG and bankruptcy," innesota Law eview 77, Love, D.A., P.A. Smith, and D.W. Wilcox, 2, The eect o regulation on optimal corporate pension risks, Journal o Financial Economics 3, Lucas, D., and S. Zeldes, 26, Valuing and hedging deined beneit pension obligations The role o stocks revisited. Working paper, Northwestern University. ercer, 23, S&P 5 plan sponsors inish 22 with highest yearn-end pension deicit ever, press release, January. 4

44 auh, J. D., 26, Investment and inancing constraints: evidence rom the unding o corporate pension plans, Journal o Finance, 6, auh, J. D., 29, isk shiting versus risk management: Investment policy in corporate pension plans, eview o Financial Studies, 22, Schroeder,., 25, Big stakes in ailing airlines raise questions or U.S. pension agency, WSJ.com, 3 November. Sharpe, W. F., 976, orporate pension unding policy, Journal o Financial Economics, 3 3, Shivdasani, A., and I. Steanescu, 2, How do pensions aect corporate capital structure decisions? eview o Financial Studies, 23, Sundaresan, S., and F. Zapatero, 997, Valuation, optimal asset allocation and retirement incentives o pension plans, eview o Financial Studies,, U.S. Government Accountability Oice, 23, eport to ongressional ommittees: Highrisk series: An update, February Tepper, I., 98, Taxation and corporate pension policy, Journal o Finance, 36, 3. Treynor, J. L., 977, The principles o corporate pension inance, Journal o Finance, 32,

45 Figure : Pension Funding Decision with External Financing For both overunded and underunded pension plans, the optimal amount o external inancing is determined at the amount where the greater o the marginal expected ater-tax return on current growth options and the marginal expected return on a pension contribution is equal to the marginal ater-tax cost o external inancing. Once the amount o external inancing is determined, the optimal allocation o the external und between current growth options and pension contributions is determined at the ratio where the marginal expected ater-tax return on current growth options is equal to the marginal return on pension contributions. For simplicity, the ollowing graphs illustrate the optimal external inancing and pension unding decisions with no possibility o bankruptcy or augmented pension assets alling below pension liabilities in value. I deine b P t L t > and c P t L t <. ase : An external und is raised. G, D α τ α OA G G τ PP ProbbPSDFE b Probc PDDF E c D F -τ D F --τ D FF α F* F* 43

46 ase 2: No external und is raised. G, D D F -τ D F --τ D FF α τ α OA G G τ PP ProbbPSDFE b Probc PDDF E c F * = F 44

47 Founded in 892, the University o hode Island is one o eight land, urban, and sea grant universities in the United States. The,2-acre rural campus is less than ten miles rom Narragansett Bay and highlights its traditions o natural resource, marine and urban related research. There are over 4, undergraduate and graduate students enrolled in seven degreegranting colleges representing 48 states and the District o olumbia. ore than 5 international students represent 59 dierent countries. Eighteen percent o the reshman class graduated in the top ten percent o their high school classes. The teaching and research aculty numbers over 6 and the University oers undergraduate programs and 86 advanced degree programs. UI students have received hodes, Fulbright, Truman, Goldwater, and Udall scholarships. There are over 8, active alumnae. The University o hode Island started to oer undergraduate business administration courses in 923. In 962, the BA program was introduced and the PhD program began in the mid 98s. The ollege o Business Administration is accredited by The AASB International - The Association to Advance ollegiate Schools o Business in 969. The ollege o Business enrolls over 4 undergraduate students and more than 3 graduate students. ission Our responsibility is to provide strong academic programs that instill excellence, conidence and strong leadership skills in our graduates. Our aim is to promote critical and independent thinking, 2 oster personal responsibility and 3 develop students whose perormance and commitment mark them as leaders contributing to the business community and society. The ollege will serve as a center or business scholarship, creative research and outreach activities to the citizens and institutions o the State o hode Island as well as the regional, national and international communities. The creation o this working paper series has been unded by an endowment established by William A. Orme, UI ollege o Business Administration, lass o 949 and ormer head o the General Electric Foundation. This working paper series is intended to permit aculty members to obtain eedback on research activities beore the research is submitted to academic and proessional journals and proessional associations or presentations. An award is presented annually or the most outstanding paper submitted. Ballentine Hall Quadrangle Univ. o hode Island Kingston, hode Island