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W h i t e P a P e r

L i a b i L i t y - D r i v e n

i n v e s t m e n t

March 2010

Issue #2

Benjamin Bruderresearch & Development Lyxor asset management, [email protected]

Guillaume Jamet research & Development Lyxor asset management, [email protected]

Guillaume Lasserre Quantitative management Lyxor asset management, [email protected]

Q u a n t R e s e a R c h b y Ly x o R 1

L i a b i L i t y- D r i v e n i n v e s t m e n tissue # 2

Foreword

Liability-Driven Investments as well as Asset and Liability Management refer to thosesituations in which investors must monitor the difference between their assets and theirliabilities. Conversely, Asset Management refers to managing assets with no reference toany liabilities whatsoever. Since it is unlikely that an investor has no liabilities at all, mostreal-world investment situations can be categorized as Liability-Driven Investment.

Unfortunately, unlike Asset Allocation, which offers quite a well-established framework,LDI and ALM cannot refer to any well-identified theoretical body. As such, most financialinstitutions are forced to make their own way through the interactions between asset alloca-tion and liability hedging within an ever-changing accounting and prudential environment.

Despite this absence of a theoretical body, a soft consensus has emerged that LDI andALM might not be of such practical importance. Long-term statistics, supported by decadesof growth in stock markets, have shown that historically, equities would always perform inthe long run, typically eight years. There was the Japanese case, of course, but that wasvery specific. Historically, a well-balanced equity portfolio would always outperform fixed-income liabilities. Liability hedging could therefore only appear as a costly, useless solution.Institutions should focus more on their long-run asset allocation and separate that matterfrom changes in their liabilities. Based on the literature on long-run investments, moststrategies have converged towards a balanced, constant-mix portfolio approach. The equityexposure was essentially country-driven, depending on the local financial culture.

Some years ago, due to the constant decrease in interest rates, many institutions realizedthat investing had become a harder task, since more “alpha” was needed to “cover” unhedgedliabilities. Another analysis would have been to acknowledge that because liabilities were nothedged, the necessary returns on the asset side were varied over time. By offering seeminglylow risk and steady yields, hedge funds as well as structured credit products appeared tobe the right solution to face this combination of decreasing interest rates and an unhedgedinstitutional gap.

Unfortunately, the dislocation of part of the hedge fund industry, the major crisis sufferedby securitization products as well as the equity markets’ widespread drawdown has shedcrude light on this consensus. First, the risk of long-term poor equity returns appearsto be real. Second, hedging issues can no longer be hidden by the alpha quest and needto be addressed thoroughly. Eventually, market acceleration illustrates both the necessityof addressing volatility as a specific risk and considering governance structures capable ofhandling dynamic investment strategies.

In the years to come, financial institutions as well as accounting and prudential playerswill have to cope with this new reality. We at Lyxor see this as a major trend and want tohelp in addressing it. This is the aim of this second issue of the Lyxor White Paper Series.

Nicolas Gaussel



Executive Summary

Defined benefits pension plans guaran-tee the pension level of their members.These pensions are generated from pastcontributions of the members and the spon-sor through a pension fund. From thesecontributions, the pension fund investmentpolicy should generate enough performanceto ensure pension payments without requir-ing any extra contribution. As such, theimportant matter is not whether the in-vestment policy performs in absolute terms,but the behavior of the strategy with re-spect to the liabilities.

This kind of problem is knownas liability-driven investment, or as-set/liability management in the financeindustry. To achieve this goal, the keyis to compare the asset value of the fund toits liabilities. From this perspective, we see

that, to some extent, there is only one con-sistent liability valuation method. Indeed,actuarial methods can produce good esti-mates for future payments amounts. Fromthese estimates, we show that future pay-ments can be exactly replicated by a port-folio, essentially canceling out the liabilityrisks. The present value of the liabilities isdefined without ambiguity as their buyoutprice, i.e. the current value of the liabil-ity hedging portfolio (LHP). This portfoliostrategy can be achieved using short-termfixed income investment, together with aninterest rate swap contract and an inflationrate swap contract. Entering into theseswaps contracts fixes the funding gap ofthe plan at its current level, eliminatingthe unrewarding interest rate and inflationrisks.

Optimal dynamic portfolio strategy

Portfolio strategy optimization

Market views to obtain the optimal Sharpe portfolio

Objectives definition

Key performance indicator Time horizon Funding level

objectiveFunding level constraints

Liability hedging portfolio

Liability buyout market value Liability risk removal

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From this point, plans to bridge thepension plan deficits can be developed. Todefine such a strategy, we focus on a givenkey performance indicator (KPI). For ex-ample, the funding ratio or the funding gapcan be considered. We set a clear objec-tive for this KPI, such as canceling out thedeficits, and also impose a minimum ac-ceptable KPI value in order to control thelosses in the worst-case scenarios. The in-vestment period during which the objec-tives are expected to be reached, i.e. thetime horizon of the strategy, must also bedefined. Note that this horizon should bedistinguished from the term of the liabili-ties. Such clear objectives, together withsome market views, can be translated intoa mathematically solvable problem, thanksto the optimal control theory.

Within this framework, an optimalstrategy can be found. This strategy will

be the most adapted to our objectives andconstraints. In particular, only rewardingrisks are taken, for the sake of efficiency.These strategies constantly adapt the quan-tity of necessary risks to the present situ-ation, in a self-consistent and predictablemanner. For example, when the objectiveis already reached, no more risks are taken,as they are not needed anymore. The samefunding level will be maintained until theinvestment horizon, thus achieving the ob-jective in every case. As we give up the pos-sibilities of performing significantly abovethe objective, we greatly increase the prob-ability of reaching it. These strategies haveproven to be a lot more efficient than clas-sical constant mixes through a set of sim-ulations. In particular, we see that IAS 19charges are very low compared to classicalpractices, thanks to efficient risk budgeting.



From this point, plans to bridge thepension plan deficits can be developed. Todefine such a strategy, we focus on a givenkey performance indicator (KPI). For ex-ample, the funding ratio or the funding gapcan be considered. We set a clear objec-tive for this KPI, such as canceling out thedeficits, and also impose a minimum ac-ceptable KPI value in order to control thelosses in the worst-case scenarios. The in-vestment period during which the objec-tives are expected to be reached, i.e. thetime horizon of the strategy, must also bedefined. Note that this horizon should bedistinguished from the term of the liabili-ties. Such clear objectives, together withsome market views, can be translated intoa mathematically solvable problem, thanksto the optimal control theory.

Within this framework, an optimalstrategy can be found. This strategy will

be the most adapted to our objectives andconstraints. In particular, only rewardingrisks are taken, for the sake of efficiency.These strategies constantly adapt the quan-tity of necessary risks to the present situ-ation, in a self-consistent and predictablemanner. For example, when the objectiveis already reached, no more risks are taken,as they are not needed anymore. The samefunding level will be maintained until theinvestment horizon, thus achieving the ob-jective in every case. As we give up the pos-sibilities of performing significantly abovethe objective, we greatly increase the prob-ability of reaching it. These strategies haveproven to be a lot more efficient than clas-sical constant mixes through a set of sim-ulations. In particular, we see that IAS 19charges are very low compared to classicalpractices, thanks to efficient risk budgeting.

Table of Contents

1 Introduction 8

2 Description and recent history of defined benefit pension plans 92.1 Players in a pension plan . . . . . . . . . . . . . . . . . . . . . . 92.2 Liability structure . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Recent pension fund issues . . . . . . . . . . . . . . . . . . . . . 10

3 The liability-hedging portfolio provides a consistent valuation 103.1 Liability-driven investment: an increasingly widespread approach 113.2 Typical liability structure . . . . . . . . . . . . . . . . . . . . . 113.3 Liability valuation and hedging methodology . . . . . . . . . . . 123.4 Implementation of the liability-hedging portfolio . . . . . . . . . 13

4 Translating pension fund needs into a solvable problem 154.1 General form of a key solvency indicator . . . . . . . . . . . . . 154.2 Questions raised by the choice of a clear objective . . . . . . . . 164.3 Grouping objectives and constraints into one framework . . . . . 16

5 Fundamental case studies 185.1 Bridging the funding gap . . . . . . . . . . . . . . . . . . . . . . 185.2 Optimizing the funding ratio . . . . . . . . . . . . . . . . . . . . 23

6 Further discussions 296.1 Consequences of non hedgable liability factors . . . . . . . . . . 296.2 Minimization of regulatory sponsor contributions . . . . . . . . 31

Appendix 32

A Modeling the pension fund environment 32

B Elements of proof for optimal dynamic strategies 35



Beyond Liability-driven Investment: New Perspectiveson Defined Benefit Pension Fund Management∗

Benjamin BruderResearch & Development

Lyxor Asset Management, [email protected]

Guillaume JametResearch & Development


Guillaume LasserreQuantitative Management


March 2010

Abstract

In recent years, liability-driven investment (LDI) has become a popular conceptamong defined benefit pension plans. However, it does not provide a clear solutionto the portfolio choice problem. After some details about liability hedging and valu-ation methods, we will show that formulating clear objectives leads to a well-definedinvestment strategy. Through fundamental examples, we will compare these alternativestrategies with widespread practices in terms of performance and risk management.

Keywords: Liability-driven investment, dynamic portfolio strategies, pension funds, strat-egy optimization.

JEL classification: G11, G23, C61.

∗We are grateful to Alain Dubois, Nicolas Gaussel and Thierry Roncalli for their helpful comments.

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1 IntroductionMany companies offer their employees the possibility of taking part in a pension plan. Thesepensions are paid through a pension fund managed by a board of trustees, who are, to someextent, independent from the sponsoring company itself. Contributions are made into thepension fund by future pensioners (i.e. employees) and by the company, which is thereforereferred as the sponsor. These plans fall into two main categories: defined contribution ordefined benefit pension plans. In the defined contribution case, the pension level is unknownat inception and depends on many factors, such as the performance of the fund’s investments.As such, delivered pensions are linked to the fund’s net asset value, leaving most of the risksto the pensioners. In the defined benefits case, the pension level is guaranteed at inception.This guarantee introduces a considerable amount of risk to the pension fund and the sponsor.The objective of the trustee will be to achieve sufficient performance to match the guaranteedpension without additional contributions from the sponsoring company.

In recent years, liability-driven investment (LDI) has become a popular concept for thespecific problems of defined benefit pension plans. This concept is the counterpart of theasset/liability management practice in the banking and insurance business. The idea is toconsider the fund’s asset value compared with the fund’s liabilities, rather than in absoluteterms, and to compare the sensitivities of the assets and liabilities with market factors. Thisleads to designing portfolio strategies that will match the future pension plan payments asclosely as possible. To this end, the present market value of the pension plan liabilities mustfirst be computed. Then, the sensitivity of this value to market parameters, such as theinterest rate curve or the inflation rate swap prices, must be calculated. These calculationsallow the fund manager to construct the liability-hedging portfolio (LHP). This portfoliois designed to hedge variations in the prices of its liabilities with the appropriate marketinstruments. Note that this hedging strategy is not intended to provide the whole portfoliostrategy by itself. Indeed, we will show that, once the hedge is completed, the focus canshift to generating pure performance, while taking into account the current funding ratiolevel and the fund’s objectives in future years. In fact, the main insight from LDI is that,for a pension fund, the lowest risk strategy is not the cash account but the liability-hedgingportfolio.

The first section will introduce the players of a pension plan and their recent history. Thesecond section describes both the construction of the liability-hedging portfolio (LHP) andthe liability valuation method. Starting from a typical liability structure, we will show thatthe stream of future payments and contributions can be replicated with financial assets. Theidea is that the liability present value is nothing more that the value of these assets. Thisliability market value enables the construction of unambiguous key performance indicators,which will be examined in section 4. Combining some objectives and constraints for a keyperformance indicator with a given time horizon will precisely define the framework of amathematical problem. Fundamental examples of solutions will be presented in section 5.After the solution descriptions, we will compare their behavior with widespread practicesin terms of performance and risk management. To conclude, we will review some solvablefeatures that are direct applications of our method. For the sake of simplicity, all technicalconsiderations are given in the appendix.

Throughout this paper, our examples will refer to the same fictitious pension plan. Theliability present value of the plan will be fixed at around e2 billion, and the funding ratio ofthe associated pension plan is 85% as of today. The objective of the managers is to cancelout the deficits at a five-year horizon.



1 IntroductionMany companies offer their employees the possibility of taking part in a pension plan. Thesepensions are paid through a pension fund managed by a board of trustees, who are, to someextent, independent from the sponsoring company itself. Contributions are made into thepension fund by future pensioners (i.e. employees) and by the company, which is thereforereferred as the sponsor. These plans fall into two main categories: defined contribution ordefined benefit pension plans. In the defined contribution case, the pension level is unknownat inception and depends on many factors, such as the performance of the fund’s investments.As such, delivered pensions are linked to the fund’s net asset value, leaving most of the risksto the pensioners. In the defined benefits case, the pension level is guaranteed at inception.This guarantee introduces a considerable amount of risk to the pension fund and the sponsor.The objective of the trustee will be to achieve sufficient performance to match the guaranteedpension without additional contributions from the sponsoring company.

In recent years, liability-driven investment (LDI) has become a popular concept for thespecific problems of defined benefit pension plans. This concept is the counterpart of theasset/liability management practice in the banking and insurance business. The idea is toconsider the fund’s asset value compared with the fund’s liabilities, rather than in absoluteterms, and to compare the sensitivities of the assets and liabilities with market factors. Thisleads to designing portfolio strategies that will match the future pension plan payments asclosely as possible. To this end, the present market value of the pension plan liabilities mustfirst be computed. Then, the sensitivity of this value to market parameters, such as theinterest rate curve or the inflation rate swap prices, must be calculated. These calculationsallow the fund manager to construct the liability-hedging portfolio (LHP). This portfoliois designed to hedge variations in the prices of its liabilities with the appropriate marketinstruments. Note that this hedging strategy is not intended to provide the whole portfoliostrategy by itself. Indeed, we will show that, once the hedge is completed, the focus canshift to generating pure performance, while taking into account the current funding ratiolevel and the fund’s objectives in future years. In fact, the main insight from LDI is that,for a pension fund, the lowest risk strategy is not the cash account but the liability-hedgingportfolio.

The first section will introduce the players of a pension plan and their recent history. Thesecond section describes both the construction of the liability-hedging portfolio (LHP) andthe liability valuation method. Starting from a typical liability structure, we will show thatthe stream of future payments and contributions can be replicated with financial assets. Theidea is that the liability present value is nothing more that the value of these assets. Thisliability market value enables the construction of unambiguous key performance indicators,which will be examined in section 4. Combining some objectives and constraints for a keyperformance indicator with a given time horizon will precisely define the framework of amathematical problem. Fundamental examples of solutions will be presented in section 5.After the solution descriptions, we will compare their behavior with widespread practicesin terms of performance and risk management. To conclude, we will review some solvablefeatures that are direct applications of our method. For the sake of simplicity, all technicalconsiderations are given in the appendix.

Throughout this paper, our examples will refer to the same fictitious pension plan. Theliability present value of the plan will be fixed at around e2 billion, and the funding ratio ofthe associated pension plan is 85% as of today. The objective of the managers is to cancelout the deficits at a five-year horizon.

2 Description and recent history of defined benefit pen-sion plans

2.1 Players in a pension planIn many companies, employees are given the opportunity to contribute to a pension plan inorder to receive defined pensions after retirement. Pension plan members can therefore bedivided into three categories:

• Active members, who are employees of the company, and are currently contributingto the pension plan.

• Deferred pensioners, who are not employed by the company anymore, but are notretired yet. They do not contribute, but will receive future payments.

• Pensioners, who are currently receiving pension payments.

A fundamental player is the trustee or board of trustees in charge of managing the pen-sion fund. Trustees are often assisted in making their decisions by consultants, who areexperienced in actuarial techniques and portfolio management. Apart from strategic deci-sions, day-to-day portfolio management is often delegated to asset managers, or sometimesto investment banks via dedicated products.

The other main player of a pension plan is the employer, i.e. the sponsoring company.The sponsor traditionally contributes to the pension fund together with the active membersof the plan and may need to make extraordinary contributions in case of substantial pensionplan deficits. As long as the sponsor is solvent, pensioner benefits remain guaranteed.

In many countries, a government agency can guarantee a part or the total amount ofpensions to the members in the event that both the sponsor and the pension fund becomeinsolvent. These agencies are financed through levies, defined by a given set of rules, onevery pension fund. Well-known examples are the Pension Benefit Guaranty Corporation inthe United States and the Pension Protection Fund in the United Kingdom.

2.2 Liability structureThe liability structure of a pension fund can be broken down into a long series of uncertaincash flows. On one side, contributions are made to the fund by the sponsor and its employees,depending on their number and salaries, among other factors. On the other side are pensionpayments, computed as the product of the number of pensioners and their average pension.These pensions are tied to the pensioners’ past salaries (sometimes their last salary beforeretirement) and the number of years of membership in the plan. They are often indexed forinflation by using, for example, the consumer price index as a reference.

Pension plans also often embed optional items, such as early retirement or pension cashcommutation, among others. These optional behaviors introduce difficulties to the liabilitypricing problem. One other major difficulty comes from the fact that pension cash flowstypically hold for very long times, such as 80 years or more. Nevertheless, many efforts havebeen made to estimate, at least approximately, the present value of these cash flows viaactuarial or accounting techniques. The most popular indicators for measuring the health ofpension plans are the funding ratio and the funding gap. The funding ratio is defined as thepension fund’s asset value divided by the present value of its liabilities, while the fundinggap (or surplus) is the difference between these two quantities. Ideally, pension funds havepositive funding surpluses and funding ratios greater than 100%.

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2.3 Recent pension fund issues

Until the 2000s, investment strategies were mostly focused on absolute performance. Toillustrate this, many pension plans benchmarked their investments to the average returnsof their peer group. Thus, they looked away from the sensitivity of the value of their ownliabilities to market parameters such as interest rates. This led to increasing proportionsof equity in pension funds portfolios, as each manager tried to outperform the peer groupusing the superior returns from equity. At that time, pension fund allocations were veryconcentrated around a 60/40 equity/bond mix (see Watson Wyatt’s study [17] or the PensionRegulator’s Purple Book [15] for current figures). These strategies provided good resultsduring equity bull markets. However, in the early 2000s, these kind of strategies suffered adouble negative effect: pension fund assets depreciated as equity markets fell, and liabilitypresent values went up due to lower interest rates. This had disastrous effects on pensionplan funding ratios, as described in [14] and [17] and illustrated in Figure 1.

Figure 1: Global defined benefits estimated asset/liability indicator (Source: Watson WyattWorldwide/Bloomberg)

3 The liability-hedging portfolio provides a consistentvaluation

In this chapter, we will describe how to construct the liability-hedging portfolio. To someextent, this portfolio will lead to a unique consistent liability valuation method. Indeed,we find that any liability should be valued at its buyout price. In particular, we will showthat a valuation method must be consistent with market prices given by the interest rateand inflation swap curves. Other valuation methods, such as discounting with respect tothe expected returns of the portfolio strategy (with a significant proportion of equity forexample), would hide most of the future risks.



2.3 Recent pension fund issues

Until the 2000s, investment strategies were mostly focused on absolute performance. Toillustrate this, many pension plans benchmarked their investments to the average returnsof their peer group. Thus, they looked away from the sensitivity of the value of their ownliabilities to market parameters such as interest rates. This led to increasing proportionsof equity in pension funds portfolios, as each manager tried to outperform the peer groupusing the superior returns from equity. At that time, pension fund allocations were veryconcentrated around a 60/40 equity/bond mix (see Watson Wyatt’s study [17] or the PensionRegulator’s Purple Book [15] for current figures). These strategies provided good resultsduring equity bull markets. However, in the early 2000s, these kind of strategies suffered adouble negative effect: pension fund assets depreciated as equity markets fell, and liabilitypresent values went up due to lower interest rates. This had disastrous effects on pensionplan funding ratios, as described in [14] and [17] and illustrated in Figure 1.

Figure 1: Global defined benefits estimated asset/liability indicator (Source: Watson WyattWorldwide/Bloomberg)

3 The liability-hedging portfolio provides a consistentvaluation

In this chapter, we will describe how to construct the liability-hedging portfolio. To someextent, this portfolio will lead to a unique consistent liability valuation method. Indeed,we find that any liability should be valued at its buyout price. In particular, we will showthat a valuation method must be consistent with market prices given by the interest rateand inflation swap curves. Other valuation methods, such as discounting with respect tothe expected returns of the portfolio strategy (with a significant proportion of equity forexample), would hide most of the future risks.

3.1 Liability-driven investment: an increasingly widespread ap-proach

The concept of liability-driven investment is derived from asset/liability management, whichhas been developed in the banking industry. This kind of approach is also widely used byinsurance companies. Indeed, a pension fund may decide to offload all of its risks to aninsurance company in exchange for a lump sum. In such cases, insurers use a liability-hedging approach.

The fundamental idea of LDI is that particular attention should be given to the connec-tions between the value of one’s assets and liabilities. Indeed, the situation of a pension fundis based heavily on its funding position. In other words, investment performance should notbe viewed in absolute terms, but should be measured with respect to changes in the liabil-ities. As we will see, these liabilities contain risk due to, for example, changes in interestrates, and these risks should be taken into account in portfolio construction. On the otherhand, a somewhat common idea is that LDI portfolios turn away from the most profitableassets such as equities. This is actually not the case. The point of LDI is that the lowestrisk strategy is the liability-hedging portfolio (LHP), essentially focused on bonds. How-ever, exhibiting the riskless portfolio does not preclude investment in risky assets, as theclassical portfolio theory does not claim that the cash account is the only relevant invest-ment. The goal of the method is to use the LHP to drive away unwanted risks and to investsimultaneously in rewarding risky assets if additional performance is needed.

Since the beginning of the 2000s, more and more pension plans have implemented LDIstrategies. Investment banks have stepped into this growing market, developing in-housecustomized solutions for pension funds and improving the market depth for related derivativeproducts. Asset managers have also built an increasing offering of liability-matching funds.This offering now contains customized funds for large pension plans, and generic pooled fundsfor smaller ones (see [14]). These funds offer exposure to interest rates and/or inflation, withvarying ranges of maturity, in order to hedge liabilities. Some of them are leveraged, leavingavailable cash to the pension plan in order to proceed with other investments at the sametime.

In practice, LDI may refer to many different strategies, ranging from precise liabilityhedging with an inflation-linked and nominal bond portfolio, to an overlay interest rate swapbacked by an equity/cash portfolio that may even contain hedge fund shares. However, theconcept of LDI does not answer several questions, such as what proportion of liabilitiesshould be hedged or what the exposure level to risky assets such as equity should be.

3.2 Typical liability structure

The liability structure of a pension fund can be broken down into series of future pay-ments and contributions. These future financial streams can be estimated with actuarialtechniques, which will not be explained here. Pension payments are typically indexed forinflation, but it may be considered that contributions, based on salaries, also follow theinflation rate up to some extent. Of course, as we consider lifetime pensions, the mortalityrate of the pension plan’s members, must be taken into account . This kind of problemis well known by actuaries and can be solved by using mortality tables, which are goodestimates for future mortality rates. Figure 2 represents our standard example of pensionfund liabilities used throughout this paper, broken down into five-year buckets.

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Paym

ents

Date

Figure 2: Total payments, net of pension plan contributions

Only actual members of the plan and not future members are considered, as the sponsormay arbitrarily close or modify the pension plan for newcomers. Therefore, future membersshould not be considered to imply actual liabilities.

More precisely, the stream of payments (net of contributions) can be broken down intoseries of inflation-linked and fixed amounts:

Year 2010 2015 ... 2085 2090Inflation-linked 180 195 ... 173 59payment (Me) ×CPI(2015)

CPI(2010) ×CPI(2085)CPI(2010) ×CPI(2090)

CPI(2010)

Table 1: Payments tables

3.3 Liability valuation and hedging methodology

The liability-hedging portfolio is designed to remove the unwanted risks caused by liabilities.It opens up the possibility of investing more in rewarding assets while maintaining the sameamount of total risks, hence performing a more efficient strategy.

We first introduce two financial instruments that are fundamental for understanding theliability-hedging portfolio construction. They may not be directly available on the market,but can be easily synthesized through a portfolio of liquid assets.

The fundamental financial instrument to analyze the hedging methodology is the zero-coupon bond. A zero-coupon bond with maturity T is a default-free bond that delivers afixed notional, for example 1 euro, at time T , without delivering any coupon in the meantime.A zero-coupon strip with various maturities is therefore the natural portfolio for hedginga stream of fixed future payments. The zero-coupon is priced without ambiguity as thediscounted value of the notional, using the market rate curve.

The other fundamental asset is the inflation-linked zero-coupon bond, which delivers attime T a fixed notional multiplied by the value of a consumer price index (divided by theCPI as of the issuance date) at time T . This is a perfect instrument for hedging a futureinflation-linked payment. The valuation of such instruments is determined entirely by themarket rate curve and the inflation swap curve.



Paym

ents

Date

Figure 2: Total payments, net of pension plan contributions

Only actual members of the plan and not future members are considered, as the sponsormay arbitrarily close or modify the pension plan for newcomers. Therefore, future membersshould not be considered to imply actual liabilities.

More precisely, the stream of payments (net of contributions) can be broken down intoseries of inflation-linked and fixed amounts:

Year 2010 2015 ... 2085 2090Inflation-linked 180 195 ... 173 59payment (Me) ×CPI(2015)

CPI(2010) ×CPI(2085)CPI(2010) ×CPI(2090)

CPI(2010)

Table 1: Payments tables

3.3 Liability valuation and hedging methodology

The liability-hedging portfolio is designed to remove the unwanted risks caused by liabilities.It opens up the possibility of investing more in rewarding assets while maintaining the sameamount of total risks, hence performing a more efficient strategy.

We first introduce two financial instruments that are fundamental for understanding theliability-hedging portfolio construction. They may not be directly available on the market,but can be easily synthesized through a portfolio of liquid assets.

The fundamental financial instrument to analyze the hedging methodology is the zero-coupon bond. A zero-coupon bond with maturity T is a default-free bond that delivers afixed notional, for example 1 euro, at time T , without delivering any coupon in the meantime.A zero-coupon strip with various maturities is therefore the natural portfolio for hedginga stream of fixed future payments. The zero-coupon is priced without ambiguity as thediscounted value of the notional, using the market rate curve.

The other fundamental asset is the inflation-linked zero-coupon bond, which delivers attime T a fixed notional multiplied by the value of a consumer price index (divided by theCPI as of the issuance date) at time T . This is a perfect instrument for hedging a futureinflation-linked payment. The valuation of such instruments is determined entirely by themarket rate curve and the inflation swap curve.

With these instruments, the construction of the liability-matching portfolio is straight-forward. We start with a set of figures, such as Table 1. For each fixed payment date Ti,a zero-coupon notional with maturity Ti that corresponds exactly to the payment amountmust be bought. If the payment is inflated by the CPI, then an inflation-linked zero-couponof maturity Ti is used. This is illustrated in Figure 3.

Figure 3: Construction of the liability-hedging portfolio

Now that the LHP is constructed, the valuation of the liabilities is straightforward. It isthe sum of the prices of all necessary zero-coupon bonds needed to replicate the liabilities.This method is actually equivalent to discounting fixed payments with the nominal interestrate market curve and inflation-indexed payments with the real interest rate market curve.The result is the buyout price of the liabilities, which should be viewed at their unique value.

On the other hand, things might not be so perfectly straightforward, as there may beunhedgeable factors driving the liabilities, such as an unpredicted change in the mortalitytables. Also, there might not be enough market instruments (for very long maturities forinstance) to hedge the liabilities perfectly. For the sake of simplicity, we will assume that theLHP provides perfect hedging. Nevertheless, a residual unhedgeable risk can be introducedinto our framework, as it is introduced into the framework of [5], without changing the spiritof our results. This is discussed in the last section of this paper.

3.4 Implementation of the liability-hedging portfolio

Buying the zero-coupon portfolio might actually be unsuitable for three main reasons:

• In case of an underfunded position, the fund does not hold enough cash to buy all ofthe zero-coupons bonds.

• The cash invested in bonds cannot be invested in performing assets, such as equity.

• The fund is exposed to counterparty risk, especially in case of a single issuer.

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We will show here that an equivalent financial position, in terms of hedging, to holdinga zero-coupon bond can be achieved through a swap contract. This will leave the cashavailable to the pension fund and eliminate the counterparty risk. Indeed, the importantfeature is to hedge movements in liability value, not to actually hold bonds.

To duplicate a zero-coupon bond with maturity Ti, one can invest the zero-coupon pricein rolling short-rate investments (such as 3-month Euribor) and exchange the interest earnedagainst a lump sum at time Ti through a swap. Note that a swap has no value at inception,thus it does not immobilize any cash. This mechanism is described in Figure 4 below.

Figure 4: Replication of a zero-coupon bond through a swap contract

Inflation swaps can be used the same way to synthesize inflation-linked zero-coupons.The same techniques can be performed for all maturities with different swaps. Finally, wefind that the liabilities can be matched with short-rate investments and a macro-swap, whichis the sum of all individual swaps, and will be called the liability hedging swap. This swapcan be taylor made, or approximated with a combination of liquid swap contracts.

The market value of this swap will replicate the liability value movements exactly. There-fore, in any case, future variations in the liabilities mark to market are hedged. For example,starting from an underfunded position and entering into the swap contract without enoughcash will make the funding gap change according to the riskless rate, as illustrated in Figure5. We see that the funding gap as of the payment date is equal to the difference betweenthe present value of the liability and the current asset value of the fund, capitalized at theshort rate.



We will show here that an equivalent financial position, in terms of hedging, to holdinga zero-coupon bond can be achieved through a swap contract. This will leave the cashavailable to the pension fund and eliminate the counterparty risk. Indeed, the importantfeature is to hedge movements in liability value, not to actually hold bonds.

To duplicate a zero-coupon bond with maturity Ti, one can invest the zero-coupon pricein rolling short-rate investments (such as 3-month Euribor) and exchange the interest earnedagainst a lump sum at time Ti through a swap. Note that a swap has no value at inception,thus it does not immobilize any cash. This mechanism is described in Figure 4 below.

Figure 4: Replication of a zero-coupon bond through a swap contract

Inflation swaps can be used the same way to synthesize inflation-linked zero-coupons.The same techniques can be performed for all maturities with different swaps. Finally, wefind that the liabilities can be matched with short-rate investments and a macro-swap, whichis the sum of all individual swaps, and will be called the liability hedging swap. This swapcan be taylor made, or approximated with a combination of liquid swap contracts.

The market value of this swap will replicate the liability value movements exactly. There-fore, in any case, future variations in the liabilities mark to market are hedged. For example,starting from an underfunded position and entering into the swap contract without enoughcash will make the funding gap change according to the riskless rate, as illustrated in Figure5. We see that the funding gap as of the payment date is equal to the difference betweenthe present value of the liability and the current asset value of the fund, capitalized at theshort rate.

Figure 5: Funding gap locking mechanism with a liability-hedging swap

On the other hand, while the swap contract has been established, the cash can be investedin any asset, including those with a strong risk premium, in order to generate additionalperformance, if needed. This additional performance can help to bridge the funding gap.

4 Translating pension fund needs into a solvable problem

4.1 General form of a key solvency indicatorThe most popular indicators for assessing the financial situation of a pension fund are thefunding ratio and the funding gap. Now that the present value of the liabilities is welldefined, we can write the funding ratio of a pension fund at time t as:

Funding ratio =Asset valueLiabilities

=At

Lt

Where A (t) represents the asset value of the fund (i.e. the result of the portfolio strategy, netof past payments and contributions), and L (t) represents the present value of the liabilities.Meanwhile, the funding gap, which is the difference between the asset value and the presentvalue of the liabilities, can be represented as:

Funding gap = Asset value − Liabilities= At − Lt

We can regroup these two measures into a more general framework. It can be said that bothformulas can be expressed as the difference between the asset value and the liabilities of thefund, divided by some reference numeraire Nt:

Performance indicator =Asset value − Liabilities

Reference numeraire(1)

=At − Lt

Nt

In the case of the funding gap, the reference is the cash numeraire Nt = 1. To obtain thefunding ratio, the liability value must be used as a numeraire, i.e. Reference numeraire =Liabilities. This results in the formula:

Asset value − LiabilitiesLiabilities

= Funding ratio − 100%

i.e. the difference between the actual and the “perfect” funding ratio (equal to 100%).

16

4.2 Questions raised by the choice of a clear objective

Separate from the modeling problem, the main issue for achieving portfolio optimization isformulating our objectives in a manageable fashion. We start by choosing a key indicatorIt, of the generic form (1), on which we will focus our efforts. This key indicator can be thefunding ratio or the funding gap, for instance.

Consideration must also be given to the time horizon of the problem. We may wantto optimize our indicator IT at some given future date T . This date may be introduced,for example, as the expected end of a recovery period, during which the fund attempts toclose its deficit. It can also be the end of the financial year, as the pension plan fundinggap is taken into account in the sponsoring company’s balance sheet. We can also extendour framework to work with several observation dates T1, ..., Tn corresponding, for instance,to the end of each financial year. We would therefore consider a joint optimization ofthe indicators IT1

, ..., ITnat those times. Note that the observation date should be clearly

distinguished from the term of the liabilities.

Then, we would define the values of this performance indicator that must be avoidedin all cases. In other words, we want our performance indicator to be above a thresholdImin, even in the worst-case scenario. For example, this may be introduced to avoid, in allcases, huge deficits that would possibly lead the sponsoring company to bankruptcy. Thismay also come from regulatory issues imposing a minimum funding ratio below which thesponsor would have to contribute immediately.

As symmetrical input, we want to consider our ideal objective Iobj for our key indicatorIT . In the following, we will devote particular attention to maximizing the probabilityto meet this objective at the observation date(s). As a counterpart, we will assume thatthe trustee will not have any incentive to achieve significantly better performance than theobjective. For example, we may consider a pension fund trying to fill its funding gap, withno interest in generating surpluses.

Now that we have defined the lower acceptable bound Imin for our performance indicatorand our objective (which can be viewed as the upper bound), we must assume the trustee’sbehavior between these bounds. This is linked to the trustee’s risk tolerance and willingnessto take some risks within the interval [Imin, Iobj ].

4.3 Grouping objectives and constraints into one framework

We can sum up our methodology as follows:

• Find a performance indicator I = A−LN , in other words, choose a good numeraire N

to express the funding gap.

• Choose an observation date T .

• Choose a minimal acceptable performance indicator level Imin.

• Choose an objective Iobj for our performance indicator.

• Describe our behavior towards risk within the interval [Imin, Iobj ].



4.2 Questions raised by the choice of a clear objective

Separate from the modeling problem, the main issue for achieving portfolio optimization isformulating our objectives in a manageable fashion. We start by choosing a key indicatorIt, of the generic form (1), on which we will focus our efforts. This key indicator can be thefunding ratio or the funding gap, for instance.

Consideration must also be given to the time horizon of the problem. We may wantto optimize our indicator IT at some given future date T . This date may be introduced,for example, as the expected end of a recovery period, during which the fund attempts toclose its deficit. It can also be the end of the financial year, as the pension plan fundinggap is taken into account in the sponsoring company’s balance sheet. We can also extendour framework to work with several observation dates T1, ..., Tn corresponding, for instance,to the end of each financial year. We would therefore consider a joint optimization ofthe indicators IT1

, ..., ITnat those times. Note that the observation date should be clearly

distinguished from the term of the liabilities.

Then, we would define the values of this performance indicator that must be avoidedin all cases. In other words, we want our performance indicator to be above a thresholdImin, even in the worst-case scenario. For example, this may be introduced to avoid, in allcases, huge deficits that would possibly lead the sponsoring company to bankruptcy. Thismay also come from regulatory issues imposing a minimum funding ratio below which thesponsor would have to contribute immediately.

As symmetrical input, we want to consider our ideal objective Iobj for our key indicatorIT . In the following, we will devote particular attention to maximizing the probabilityto meet this objective at the observation date(s). As a counterpart, we will assume thatthe trustee will not have any incentive to achieve significantly better performance than theobjective. For example, we may consider a pension fund trying to fill its funding gap, withno interest in generating surpluses.

Now that we have defined the lower acceptable bound Imin for our performance indicatorand our objective (which can be viewed as the upper bound), we must assume the trustee’sbehavior between these bounds. This is linked to the trustee’s risk tolerance and willingnessto take some risks within the interval [Imin, Iobj ].

4.3 Grouping objectives and constraints into one framework

We can sum up our methodology as follows:

• Find a performance indicator I = A−LN , in other words, choose a good numeraire N

to express the funding gap.

• Choose an observation date T .

• Choose a minimal acceptable performance indicator level Imin.

• Choose an objective Iobj for our performance indicator.

• Describe our behavior towards risk within the interval [Imin, Iobj ].

This whole arrangement of preferences can be summed up in a utility function U (I) of theperformance indicator. This function represents the fund manager’s “satisfaction” at obser-vation time T , and is designed to establish an order of preference among uncertain events.Indeed, each portfolio strategy will give an uncertain level of performance at the observationdate, and the utility function can put them in order to obtain an optimal portfolio strategy.To be in accordance with our objectives, we impose some constraints on that utility function:

• U is a function of our chosen performance indicator IT at observation date T .

• We look upon the expected value of U at the observation date(s). Following theframework of [8], we will optimize, over all portfolio strategies, the expected utility:

E (U (IT ))

• To make sure that a performance level under Imin is forbidden, we require the utilityfunction U (IT ) to be infinitely negative for IT ≤ Imin. This means that the fundmanager is “infinitely unhappy” if the performance indicator is below the constraint.Therefore, the investor will do everything in his power to avoid those outcomes.

• To take the objective into account, we require U (IT ) to be constant for IT ≥ Iobj .This considers the idea that the investor is not interested in taking risks in order toexceed the objective. He considers the fulfillment of the objective to be sufficient.

• Investor behavior within the interval [Imin, Iobj ] is given by the shape of the utilityfunction, namely its risk-aversion parameter. Risk aversion comes from the fact thatmanagers should be more reluctant to incur a given loss than they are incentivized forthe same amount of gains. The risk aversion parameter describes the magnitude ofthis effect.

Figure 6 illustrates an example of utility functions, with a funding ratio objective of 100%and a minimum acceptable ratio of 70%.

Finally, this complex problem comes down to the maximization of the expected utilityover all dynamic portfolio strategies. The expected utility concept for measuring preferenceswas introduced by Von Neumann and Morgenstern in 1944 (see [16]), and has been widelyused since then. The continuous time portfolio theory based on utility functions was de-veloped by Merton in [8]. The mathematical framework for solving these problems can befound in [4], [1], [10] and [11].

18

Figure 6: Utility function examples

5 Fundamental case studiesGoing forward, we will consider some fundamental examples. For the sake of simplicity, wewill refer to the appendix for all hypotheses and model assumptions. We will look at thefunding gap, then funding ratio optimization problems for the following three reasons:

• From the sponsor’s point of view, the funding gap is a relevant indicator, as it isconsistent with the amount that may need to be contributed.

• The funding ratio seems appropriate from the trustee’s perspective, as it may representthe proportion of pensions that can be paid without any external help.

• The accounting rules provided by IAS 19 take these two indicators into account tocompute the balance sheet charges resulting from the pension plan.

5.1 Bridging the funding gapThe first example is the case of a plan wishing to bridge the funding gap at some time T .The natural reference numeraire for this gap is the zero-coupon bond with maturity T , i.e.NT = 1. Thus, we must solve the problem:

Maximize E (U (AT − LT )) over all portfolio strategies (2)

Where the utility function U is given by:

U (I) = −∞ if I < −300 (the funding gap must be less than 300 Me) (3)

U (I) = (min(I, 0%) + 300)12 if I ≥ −300 (no incentive to create surpluses)



Figure 6: Utility function examples

5 Fundamental case studiesGoing forward, we will consider some fundamental examples. For the sake of simplicity, wewill refer to the appendix for all hypotheses and model assumptions. We will look at thefunding gap, then funding ratio optimization problems for the following three reasons:

• From the sponsor’s point of view, the funding gap is a relevant indicator, as it isconsistent with the amount that may need to be contributed.

• The funding ratio seems appropriate from the trustee’s perspective, as it may representthe proportion of pensions that can be paid without any external help.

• The accounting rules provided by IAS 19 take these two indicators into account tocompute the balance sheet charges resulting from the pension plan.

5.1 Bridging the funding gapThe first example is the case of a plan wishing to bridge the funding gap at some time T .The natural reference numeraire for this gap is the zero-coupon bond with maturity T , i.e.NT = 1. Thus, we must solve the problem:

Maximize E (U (AT − LT )) over all portfolio strategies (2)

Where the utility function U is given by:

U (I) = −∞ if I < −300 (the funding gap must be less than 300 Me) (3)

U (I) = (min(I, 0%) + 300)12 if I ≥ −300 (no incentive to create surpluses)

5.1.1 A two-fund separation result

In this situation, it can be shown that, the optimal allocation policy leads to 100% hedgingof the liabilities (2). Thus, the fund allocation is broken down as follows:

• The liability-hedging swap (denoted as LHS) which we recall is the “non-funded” im-plementation of the liability-hedging portfolio.

• The remaining asset, called the performance portfolio.

As we hedge 100% of the liabilities with the LHS, the funding gap A − L depends only onthe returns of the performance portfolio and not on liability price movements anymore.

We obtain the optimal portfolio formula:

Optimal Portfolio = Performance Portfolio + 100%× LHS (4)

This result is very clear: if the performance criterion is based on the funding gap, thenthe liabilities must be completely hedged, for example with an overlay strategy. Meanwhile,the focus can be placed on generating performance independently, to try to bridge the gap.

Once the liability risk is fully hedged, and nothing else is done, the funding gap remains(up to discounting terms) constant and equal to I0 = X0−L0 over time. This leaves us withthe objective of earning the difference between the initial funding gap I0 = −300Me andour target Iobj = 0 until investment horizon T , without giving any more attention to theliability risk. Note that this strong result is valid for any utility function or, more generally,any criterion concerning the funding gap only.

5.1.2 Description of the performance portfolio

As the liability-hedging portfolio has been already discussed, we will focus on the descriptionof the performance portfolio. As shown in [13] and [9], this portfolio is a dynamic allocationbetween two assets:

• The optimal return/risk reward basket, which will be referred to as the risky asset.

• The riskless asset relative to our numeraire at investment horizon T , which is thezero-coupon bond with maturity T .

The rule for allocation between these two assets depends only on the current value of thefunding gap and the remaining time horizon.

We can illustrate the exposure to the risky asset, assuming a volatility of 15% and aSharpe ratio of 50%, which gives us an average return of 7.5% p.a. above the risklessrate. We also assume that our preferences are described through the utility function (3).The investment horizon T is equal to five years. We have reported the results for severalremaining time horizons (T − t) in Figure 7.

20

0%

10%

20%

30%

40%

50%

60%

70%

80%

Expo

sure

to r

isky

ass

et

Funding Gap

5Y to maturity

3Y to maturity

1Y to maturity

Reduce the exposure when

close to the minimal level

Secure the performance once the performance

objective is reached

Risk budgeting according to maturity horizon and current

funding gap

Figure 7: Optimal exposure to the risky asset is a function of the funding gap depending onthe time horizon

Here the funding gap is expressed as its “forward” value, defined as the value of thefunding gap that would be attained at the considered time horizon T if the fund managerfollowed the riskless strategy. This riskless strategy is defined as contracting 100% of theLHS, and borrowing the remainder in zero-coupon bonds with maturity T , thus locking inthe funding gap value at time T .

The risky asset exposure is defined as the percentage of the total asset value of the fundinvested in the risky asset. The level of exposure to the risky asset has three states:

• When the funding gap approaches the worst acceptable level, the exposure is reducedin order to stay above that level and is driven by the distance between the asset valueand the value of this level.

• When both the objective and the constraint are far away, the exposure is high in orderto derive more benefit from the risk premium and raise the probability of reaching theobjective.

• When the objective is close, the exposure is lowered in order to secure the performanceand avoid drawdowns and is driven by the distance between the asset value and theobjective.

Overall, the exposure function increases as maturity approaches. When maturity is far away,we have a lot of time to take advantage of the risk premium. As such, we do not need tohave a high level of exposure. On the other hand, if we are still at the same funding gaplevel and maturity gets closer, this means that the risky asset has not performed in themeantime. Thus, we need to increase exposure so that we still have significant chances toreach the objective in a shorter term with the same risk premium anticipation (see Figure8).



0%

10%

20%

30%

40%

50%

60%

70%

80%

Expo

sure

to r

isky

ass

et

Funding Gap

5Y to maturity

3Y to maturity

1Y to maturity

Reduce the exposure when

close to the minimal level

Secure the performance once the performance

objective is reached

Risk budgeting according to maturity horizon and current

funding gap

Figure 7: Optimal exposure to the risky asset is a function of the funding gap depending onthe time horizon

Here the funding gap is expressed as its “forward” value, defined as the value of thefunding gap that would be attained at the considered time horizon T if the fund managerfollowed the riskless strategy. This riskless strategy is defined as contracting 100% of theLHS, and borrowing the remainder in zero-coupon bonds with maturity T , thus locking inthe funding gap value at time T .

The risky asset exposure is defined as the percentage of the total asset value of the fundinvested in the risky asset. The level of exposure to the risky asset has three states:

• When the funding gap approaches the worst acceptable level, the exposure is reducedin order to stay above that level and is driven by the distance between the asset valueand the value of this level.

• When both the objective and the constraint are far away, the exposure is high in orderto derive more benefit from the risk premium and raise the probability of reaching theobjective.

• When the objective is close, the exposure is lowered in order to secure the performanceand avoid drawdowns and is driven by the distance between the asset value and theobjective.

Overall, the exposure function increases as maturity approaches. When maturity is far away,we have a lot of time to take advantage of the risk premium. As such, we do not need tohave a high level of exposure. On the other hand, if we are still at the same funding gaplevel and maturity gets closer, this means that the risky asset has not performed in themeantime. Thus, we need to increase exposure so that we still have significant chances toreach the objective in a shorter term with the same risk premium anticipation (see Figure8).

Figure 8: Needed exposure is a function of the remaining time horizon

A good feature of this strategy is that, when we are closer to the objective than to theconstraint, the exposure is reduced when the risky asset price increases. This means thatwe sell at high prices and we buy at low prices, thus benefiting from volatility.

As a remarkable property, we can also predict, up to the limits of this Black-Scholesmodel, the funding gap at time T corresponding to the performance of our risky assetduring the time period [0, T ]. Note that this result would need to be adapted for the case ofstochastic volatility and would be less clear with portfolio constraints. Therefore, the resultis approximate but still provides good insight.

Figure 9: Optimal funding gap strategy payout for several time horizons

In Figure 9, the objective is achieved if the global performance of the risky asset ismore than 27% (i.e. 4.9% per annum), which is a very reasonable assumption for an equitymix. On the other hand, the worst-case funding gap is nearly attained if the risky asset

22

has performed below -50%. The break-even point between this dynamic strategy and theriskless strategy (investment in 100% LHP + zero-coupon bonds with maturity T ) is attainedif performance of the risky asset is 12.3% (i.e 2.4% per annum). Meanwhile, the 5Y zero-coupon rate is equal to 2.5% p.a. Therefore, due to the shape of the exposure function, thedynamic strategy performs better than the riskless investment, provided that the risky assetdoes not perform less than 10bps p.a. below the risk-free rate.

5.1.3 Behavior of our solution compared with popular practices

Here, we will compare the optimized strategy with some popular practices. First, we comparethe strategy’s typical payoff with a constant mix between the risky asset and the zero-couponbond with maturity T , with both strategies using the liability-hedging swap as an overlay,to clear out the liability risk. We denote the 60/40 constant mix as a strategy keeping 60%exposure in the risky asset and 40% in zero-coupon bonds with maturity T , together witha 100% liability hedge with the liability-hedging swap. Note that consideration of constantmixes without the liability-hedging swap would just bring more risks to the constant mixes,making things turn even more in favor of the optimized strategy. Starting from the samefunding gap at the initial time, we obtain the following funding gaps at T , with respect tothe risky asset final value, in Figure 10.

Figure 10: Comparison of the optimal strategy and constant mix payouts at a 5Y horizon

First, the optimized strategy fills the funding gap for lower terminal values of the riskyasset and has a limited worst-case scenario compared with constant mixes. The 60/40constant mix is nearly always below the optimal strategy, except when the risky asset valueis more than 146% of its initial value. In that case, the 60/40 mix brings surpluses, whilethe optimized strategy only fills the funding gap. This seems clearly suboptimal, as possiblesurpluses do not match the increased funding gap in all other cases.

Still using the same assumptions of 15% volatility and 10% mean return p.a. for ourrisky asset, we obtain the funding gap distribution at observation date T in Figure 11.



has performed below -50%. The break-even point between this dynamic strategy and theriskless strategy (investment in 100% LHP + zero-coupon bonds with maturity T ) is attainedif performance of the risky asset is 12.3% (i.e 2.4% per annum). Meanwhile, the 5Y zero-coupon rate is equal to 2.5% p.a. Therefore, due to the shape of the exposure function, thedynamic strategy performs better than the riskless investment, provided that the risky assetdoes not perform less than 10bps p.a. below the risk-free rate.


Here, we will compare the optimized strategy with some popular practices. First, we comparethe strategy’s typical payoff with a constant mix between the risky asset and the zero-couponbond with maturity T , with both strategies using the liability-hedging swap as an overlay,to clear out the liability risk. We denote the 60/40 constant mix as a strategy keeping 60%exposure in the risky asset and 40% in zero-coupon bonds with maturity T , together witha 100% liability hedge with the liability-hedging swap. Note that consideration of constantmixes without the liability-hedging swap would just bring more risks to the constant mixes,making things turn even more in favor of the optimized strategy. Starting from the samefunding gap at the initial time, we obtain the following funding gaps at T , with respect tothe risky asset final value, in Figure 10.


First, the optimized strategy fills the funding gap for lower terminal values of the riskyasset and has a limited worst-case scenario compared with constant mixes. The 60/40constant mix is nearly always below the optimal strategy, except when the risky asset valueis more than 146% of its initial value. In that case, the 60/40 mix brings surpluses, whilethe optimized strategy only fills the funding gap. This seems clearly suboptimal, as possiblesurpluses do not match the increased funding gap in all other cases.

Still using the same assumptions of 15% volatility and 10% mean return p.a. for ourrisky asset, we obtain the funding gap distribution at observation date T in Figure 11.

0%

10%

20%

30%

40%

50%

60%

Prob

abili

ty

Funding gap at observation date

Optimal strategy

60/40 constant mix

Figure 11: Probability distribution of the funding gap at a 5-year horizon

The optimal strategy has a 60% probability of reaching the objective and no chance offalling below the minimal level. Meanwhile, the constant mix has a 46% chance of beingat or above the objective. The probability distribution of the constant mix is much moredispersed than the optimal strategy.

From the sponsor’s perspective, a good strategy should preserve the sponsor’s balancesheet. The company wants to avoid extraordinary contributions or charges due to inappro-priate portfolio management in its pension fund. To reveal how both strategies deal withthis problem, we have simulated them using three sets of assumptions and calculated theIAS 19 charges that they both cause each year (see [2] for a precise definition). Several quan-tiles for our strategy and the constant mix are given in Table 12. Three kinds of scenariosare considered, depending on the risky asset average return: an optimistic, conservative, orpessimistic scenario. The mean return of the risky asset is respectively set at 10%, 2.5%and -5% in each scenario.

In the conservative scenario of 2.5% p.a., both strategies lead to quite similar results,with a slight advantage for the optimized strategy. On the other hand, in the optimistic andpessimistic scenarios, the optimized strategy clearly outperforms the constant mix, both interms of average and dispersion (i.e. risk). In the pessimistic scenario, these differences canbe explained by the fact that the IAS charges are capped at the constraint that we imposedon the maximum funding gap. The behavior of our strategy near the objective explains thegood results in the optimistic case, as it is designed to maximize the probability of reachingthe objective with controlled risks. Especially, when the funding gap is bridged, all risksources are cancelled out in order to maintain that level.

5.2 Optimizing the funding ratio

Our second example is the optimization of the funding ratio at time T . This particularexample in complete markets was examined in [6], but our framework covers a more generalrange of possible features. Here, T is a 5-year horizon. We start from a funding ratio of85%, and our objective is to reach a funding ratio of 100% within five years. Our constraintis that we must never finish with a funding ratio below 70%.

24

Figure

12:C

ompared

IAS

19charges

(inMe

)for

threesim

ulationassum

ptions



Figure

12:C

ompared

IAS

19charges

(inMe

)for

threesim

ulationassum

ptions

Focusing on the funding ratio can be interpreted as choosing the liability present valueas a numeraire in (1). The formula of the funding ratio is given by:

Funding ratio = 100% +Asset value − Liabilities

Liabilities

We must therefore solve the problem:

Maximize EU

AT − LT

LT

over all portfolio strategies

Where our utility function is given by, denoting I = A−LT

LT:

U (I) = −∞ if I < −30% (the funding ratio must be above 70%) (5)

U (I) = (min(I, 0%) + 30%)12 if I ≥ −30% (no incentive for creating surpluses)

To obtain a more manageable result, we assume that no payments occur during the invest-ment period. Introducing these payments would not change the results significantly butwould introduce more complexity into our presentation.

5.2.1 Composition of the optimal portfolio

The main difference between the solutions of funding gap optimization and funding ratiooptimization is that, in the funding gap case, the liability is always hedged with a 100% ratio,while in the funding ratio case, the liability-hedging ratio is dynamic. This means that wedo not cancel out all liability risk in this strategy. Indeed, as our reference numeraire is nowthe liability price, the liability-hedging portfolio becomes the riskless asset in our problem.Therefore, in the spirit of [9], we find that the optimal portfolio strategy is a balancedstrategy between two portfolios:

• The liability-hedging portfolio, referred to as the riskless asset.

• The optimal return/risk ratio basket, referred to as the risky asset.

As the optimal solution invests only in these two portfolios (and has no cash component,for example), and the sum of the two relative exposures is equal to 100%:

wealth in LHP + wealth in the risky asset = Asset value of the pension fund (6)

This is another difference with the funding gap problem, where the riskless asset consisted ofzero-coupon bonds maturing at our observation date T . Here, we have no such investment.

5.2.2 Description of the dynamic strategy

Roughly, the optimal investment rule for risky assets has the same behavior as in the fundinggap case. We obtain the same kind of bell-shape exposure as in Figure 7. The exposure ofthe funding ratio optimization strategy has roughly the same shape as the gap optimizationstrategy. We computed the funding ratio optimization strategy from the same set of param-eters as in the funding gap problem. In other words, the risky asset is supposed to have anaverage performance of 7.5% over the risk-free rate with a volatility of 15%. As liabilitiesare not entirely hedged, we must introduce the volatility and the mean return of the LHP.In our example, the mean duration of the liabilities is 27 years. Thus, assuming a volatilityof 0.5% for the real interest rate, we find that the liabilities have a volatility of 13.5%. Weassume that the LHP has no risk premium with respect to the riskless rate and that the

26

0%

10%

20%

30%

40%

50%

60%

Expo

sure

to r

isky

ass

et

Funding Ratio

5Y to maturity

3Y to maturity

1Y to maturity

Figure 13: Optimal exposure to the risky asset is a function of the funding ratio dependingon the time horizon

LHP is not correlated with the risky asset. We start with a funding ratio of 85% at thebeginning of the investment period. We obtain the exposure to the risky asset for variousremaining time horizons (T − t) in Figure 13.

0%

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30%

40%

50%

60%

70%

80%

90%

100%

110%

Liab

ility

hed

ging

rati

o

Funding ratio

5Y to maturity3Y to maturity1Y to maturity

Liabilities are hedged at 70% to keep this minimal acceptable

funding ratioThe funding ratio of

100% is secured

Investment in the liability hedging portfolio gets

lower, as exposure in the risky asset increases

Figure 14: Proportion of the liabilities hedged by the LHP in the optimal strategy



0%

10%

20%

30%

40%

50%

60%

Expo

sure

to r

isky

ass

et

Funding Ratio

5Y to maturity

3Y to maturity

1Y to maturity

Figure 13: Optimal exposure to the risky asset is a function of the funding ratio dependingon the time horizon

LHP is not correlated with the risky asset. We start with a funding ratio of 85% at thebeginning of the investment period. We obtain the exposure to the risky asset for variousremaining time horizons (T − t) in Figure 13.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

110%

Liab

ility

hed

ging

rati

o

Funding ratio

5Y to maturity3Y to maturity1Y to maturity

Liabilities are hedged at 70% to keep this minimal acceptable

funding ratioThe funding ratio of

100% is secured

Investment in the liability hedging portfolio gets

lower, as exposure in the risky asset increases

Figure 14: Proportion of the liabilities hedged by the LHP in the optimal strategy

We get lower exposures than in the funding gap problem. This is due to the fact that themain parameter is the volatility of the risky assets returns with respect to the riskless assetreturn. Here, the riskless asset is volatile; therefore, the volatility of the risky asset/risklessasset ratio is higher, and exposure to the risky asset introduces more risks. These risks canalso be derived from the fact that investing in the risky asset reduces the investment in theLHP (as seen in Equation (6)), thus increasing the liability risk. When the funding ratio isabove 85%, we still have the good property that exposure decreases as the risky asset priceincreases. Thus, according to the strategy, the risky asset is bought at low prices and soldat high prices, taking advantage of the market volatility in that region.

This strategy implies a dynamic exposure to the LHP. The proportion of liabilities hedgedby investment in the LHP as a function of the funding ratio for several remaining timehorizons are given in Figure 14. When the funding ratio is between the objective and theconstraint, the investment in the LHP is reduced, as we invest more in the risky asset, dueto Equation (6). This strategy may leave up to 55% of the liability risk unhedged. Figure15 shows the typical payoff of our strategy for different time horizons, supposing that ourmodel assumptions are realized. Note that the relevant quantity here is the performance ofthe risky asset with respect to the LHP.

Figure 15: Optimal funding ratio strategy payout for several time horizons

The objective is attained if the risky asset performs 16% more than the liability-hedgingportfolio during the 5-year period. This is equivalent to an outperformance of 3% p.a.,which is a reasonable assumption. We also see that if the risky asset has exactly the sameperformance as the LHP during the investment period, the funding ratio goes from 85% to87%, due to the dynamic exposure profile.


This strategy is now compared with 60/40 risky asset/LHP and 40/60 risky asset/LHPconstant mixes. With the same assumptions, the funding ratio at investment horizon T forthese three strategies can be represented as a function of the outperformance of the riskyasset with respect to the LHP (see Figure 16)

28

The optimal strategy brings a systematically higher funding ratio than the 60/40 constantmix, except when the constant mix yields a funding ratio greater than the objective of 100%.Meanwhile, the 40/60 constant mix needs enormous performance of the risky asset withrespect to the liabilities to attain a funding ratio of 100%.

We still use the assumption of an average performance of the risky asset of 7.5% morethan the LHP performance, with a volatility of 15%. We assume that the LHP has no riskpremium with respect to the riskless rate and that the LHP volatility is 13%. Finally, we stillassume that the LHP and the risky asset are not correlated. Simulating our strategy andthe 60/40 risky asset/LHP constant mix with those parameters, we obtain the probabilitydistribution of the funding ratio after 5 years in Figure 17.


0%

10%

20%

30%

40%

50%

60%

70%

80%

Prob

abili

ty

Funding ratio at observation date

Optimal strategy

60/40 constant mix




The optimal strategy brings a systematically higher funding ratio than the 60/40 constantmix, except when the constant mix yields a funding ratio greater than the objective of 100%.Meanwhile, the 40/60 constant mix needs enormous performance of the risky asset withrespect to the liabilities to attain a funding ratio of 100%.

We still use the assumption of an average performance of the risky asset of 7.5% morethan the LHP performance, with a volatility of 15%. We assume that the LHP has no riskpremium with respect to the riskless rate and that the LHP volatility is 13%. Finally, we stillassume that the LHP and the risky asset are not correlated. Simulating our strategy andthe 60/40 risky asset/LHP constant mix with those parameters, we obtain the probabilitydistribution of the funding ratio after 5 years in Figure 17.


0%

10%

20%

30%

40%

50%

60%

70%

80%

Prob

abili

ty

Funding ratio at observation date

Optimal strategy

60/40 constant mix


The optimized strategy has a 72% probability of reaching a funding ratio of 100%. Mean-while, the constant mix has a 64% probability of being at or above this level. Also, theconstant mix exhibits a far more dispersed probability distribution, with a 5% probabilityof being strictly below our critical level of a 70% funding ratio. Note that our simulation as-sumptions are rather optimistic. More conservative hypotheses would handicap the constantmix more than our strategy, as shown in Figure 18.

As in the funding gap case, Figure 18 gives insights about the probability distribution ofIAS 19 charges for the 60/40 risky asset/LHP constant mix and the funding ratio optimiza-tion strategies. Three scenarios are represented again, depending on the average return thatwe assume for the risky asset. These results are represented in Figure 18. The funding ratiooptimization strategy gives great results compared with the 60/40 constant mix, in termsof both the average and standard deviation of the IAS 19 charges. This is explained by thedynamic allocation rule between the risky asset and the LHP. This strategy only takes riskswhen necessary, and the minimal funding ratio constraint is successful in limiting losses inthe worst-case scenarios.

6 Further discussionsHere, we elaborate on several points that were not discussed for the sake of simplicity. Ourframework is adaptable to imperfectly hedgeable liabilities for more realistic modeling. Wecan also take into account portfolio constraints and still find an adapted optimal strategy.Lastly, different objectives can be introduced, such as the minimization of extraordinarysponsor contributions, due to regulation, in a long-term horizon.

6.1 Consequences of non hedgable liability factorsIn this analysis, we assumed that the liabilities could be perfectly hedgeable with availablemarket instruments. This might not be the case in the real world. For example, as the futurepension valuation method depends heavily on mortality assumptions, unpredicted changesin the mortality tables can influence the liability present value. These risks cannot be hedgedby any market instrument, although over-the-counter contracts with insurance companies,for example, could be considered. Other unhedgeable risks could be linked with behavioralrisk, such as the number of members who choose a lump sum rather than a lifetime pension,when this option is available. Also, in this study, we assumed that the average salary wasdriven by inflation. This might not be exactly the case. In [5], the problem of partiallyhedgeable liabilities is examined, but without the notion of objective or constraint.

Nevertheless, these facts can be integrated into our framework. A financial model with apartially hedgeable liability factor is given in the appendix. Similar results can be derived.The optimal solution does not differ significantly, but there are two important differences:

• We required that our strategies never end up below a given constraint. This might notbe possible to implement with partially hedgeable liabilities. We have to reformulateour problem, transforming the constraint into a strong incentive to stay above thatsame level.

• We saw that when the objective is attained before the time horizon, our strategies areentirely invested in the liability-hedging portfolio, thus leading to a 100% chance ofreaching the objective at maturity. This is not possible with an imperfect hedge, andthe strategy would always keep a small proportion of risky asset in order to use itsrisk premium to prevent unhedgeable adverse movements of liabilities.

30

Figure

18:C

ompared

IAS

19charges

(inMe

)for

threesim

ulationassum

ptions



Figure

18:C

ompared

IAS

19charges

(inMe

)for

threesim

ulationassum

ptions

6.2 Minimization of regulatory sponsor contributionsHere, we have formulated our objectives regarding the funding ratio or funding gap at a givenfuture time T . A wider range of problems can be considered. Instead of considering a singletime horizon T , we could consider series of observation dates T1, T2, ..., Tn (for instance, theend of each financial year). We could use this arrangement to minimize the sum of regulatorysponsor contributions at those dates. For example, this could lead to the optimization ofthe results of Figures 12 and 18 with a new dedicated strategy.

Concluding remarksThis paper essentially gives a systematic process and a clear investment solution for man-aging the assets of a DB pension plan:

1. Assess its liabilities using markets instruments, when possible, in order to understandthe embedded risks of the plan.

2. Define a business plan for its asset: time horizon, key performance indicator (KPI),risk aversion, KPI objectives, and constraints.

3. Choose a global portfolio (PP) to generate performance.

4. Manage the assets by implementing a dynamic portfolio depending on the choice ofthe KPI

(a) Funding Gap: enter into the LHS and dynamically allocate between a zero couponand PP

(b) Funding Ratio: dynamically allocate between LHP and PP

The two examples of situations presented in this paper involve under-funded plans. Thischoice has been driven by recent studies (see [15] and [7]), which show that 89% of S&P500 DB plans and 85% of DB plans monitored by the Pension Protection Fund (UK) areunderfunded. Nevertheless, our approach may still be applied in situations of over-fundedplans.

To conclude, we would like to present two examples of how our solution fits with observedbehaviors in pension plan investment decisions.

• In section 7.2 of [15], a clear need for flexible asset allocations over time can be ob-served. Our solution provides clear allocation rules for managing market changes.

• Section 7.5 of [15] points out the existence of a relationship between the shape of theasset allocation and the level of the funding ratio. Our analysis brings up some formalrules to explain this phenomenon in terms of the distance to the objectives and thedistance to a floor level (see Figures 7 and 13).

32

Appendix

A Modeling the pension fund environment

Market modelWe assume that the financial markets are described by m risky assets which prices at somegiven time t will be denoted as

�S1t , ..., S

mt

and a cash account evolving as the risk free

rate rt. These risky assets are supposed to describe the entire possible investment universe:every bond, equity, swap, commodity, mutual fund and hedge fund. Therefore, there may bea huge number of available assets in the market. Some may think that this would make ouranalysis unmanageable; however, we will see below that decomposition theorems will enableus to work with a limited set of portfolios without any loss of efficiency. Namely we willhave to find a good balance between a liability-hedging portfolio, an optimal Sharpe ratioportfolio and eventually a reference numeraire asset. The practical methods to constructthese portfolios from our whole market universe would be a separate problem which is notthe main topic of this work.

The cash account S0t evolves according to the risk free rate rt:

dS0t

S0t

= rt dt

Where rt may be a stochastic process driven by any kind of randomness. The risky assetprices S1

t , ..., Smt change, under the manager’s view of the market, according to the dynamics:

dSit

Sit

= rtdt+

mj=1

σijt

dW j

t + θjt dt

Where θt =�θ1t , ..., θ

mt

is the vector of instantaneous Sharpe ratios and Σ =

σijt

i,j

is the

volatility matrix, which can be stochastic processes, and Wt =�W 1

t , ...,Wmt

is a standard

Brownian motion.

In principle, this market model is not supposed to be complete. By definition, theincomplete case corresponds to the existence of unhedgeable claims. The prices of theseclaims are not fixed by arbitrage conditions. Thus, there could be an infinite number ofpricing rules for these unhedgeable claims. Translated into mathematical language, thismeans that there are an infinite number of risk-neutral probabilities.

Summing up the market into the Sharpe-optimal portfolioIn the following, we will assume that the optimal Sharpe ratio portfolio has already beenbuilt, and we will denote Mt as the value of the associated strategy, which itself will beconsidered an asset. Note that, in the CAPM theory, this optimal Sharpe ratio portfoliois nothing more than the “market portfolio” (see [13], for example). Nevertheless, we donot need to assume that Mt is the market portfolio here. Under standard assumptions,as shown in [9], the optimal Sharpe portfolio Mt is the solution to the logarithmic utilitymaximization problem any reference numeraire N :

supπ

EP

ln

Aπ

T

NT

= sup

π

EP [ln (Aπ

T )]− EP [ln (NT )]



Appendix

A Modeling the pension fund environment

Market modelWe assume that the financial markets are described by m risky assets which prices at somegiven time t will be denoted as

�S1t , ..., S

mt

and a cash account evolving as the risk free

rate rt. These risky assets are supposed to describe the entire possible investment universe:every bond, equity, swap, commodity, mutual fund and hedge fund. Therefore, there may bea huge number of available assets in the market. Some may think that this would make ouranalysis unmanageable; however, we will see below that decomposition theorems will enableus to work with a limited set of portfolios without any loss of efficiency. Namely we willhave to find a good balance between a liability-hedging portfolio, an optimal Sharpe ratioportfolio and eventually a reference numeraire asset. The practical methods to constructthese portfolios from our whole market universe would be a separate problem which is notthe main topic of this work.

The cash account S0t evolves according to the risk free rate rt:

dS0t

S0t

= rt dt

Where rt may be a stochastic process driven by any kind of randomness. The risky assetprices S1

t , ..., Smt change, under the manager’s view of the market, according to the dynamics:

dSit

Sit

= rtdt+

mj=1

σijt

dW j

t + θjt dt

Where θt =�θ1t , ..., θ

mt

is the vector of instantaneous Sharpe ratios and Σ =

σijt

i,j

is the

volatility matrix, which can be stochastic processes, and Wt =�W 1

t , ...,Wmt

is a standard

Brownian motion.

In principle, this market model is not supposed to be complete. By definition, theincomplete case corresponds to the existence of unhedgeable claims. The prices of theseclaims are not fixed by arbitrage conditions. Thus, there could be an infinite number ofpricing rules for these unhedgeable claims. Translated into mathematical language, thismeans that there are an infinite number of risk-neutral probabilities.

Summing up the market into the Sharpe-optimal portfolioIn the following, we will assume that the optimal Sharpe ratio portfolio has already beenbuilt, and we will denote Mt as the value of the associated strategy, which itself will beconsidered an asset. Note that, in the CAPM theory, this optimal Sharpe ratio portfoliois nothing more than the “market portfolio” (see [13], for example). Nevertheless, we donot need to assume that Mt is the market portfolio here. Under standard assumptions,as shown in [9], the optimal Sharpe portfolio Mt is the solution to the logarithmic utilitymaximization problem any reference numeraire N :

supπ

EP

ln

Aπ

T

NT

= sup

π

EP [ln (Aπ

T )]− EP [ln (NT )]

Where π represents the portfolio strategy that we want to optimize. As we see, the optimalresult Mt does not depend on the reference numeraire N .

Our main assumption in this paper is that the optimal Sharpe ratio strategy evolves as:

dMt

Mt= rt dt+ σt

�θ dt+ dWM

t

Where σt is a random volatility process, and θ is the highest attainable Sharpe ratio. In ourwork, we assume that the optimal expected Sharpe ratio is constant, or at least deterministic.Using random θ would introduce severe complexity into our problem.

Dynamics of self-financing portfoliosOnce we have introduced the market assets, we can define the set of self financing strategies.Such a strategy, denoted as Aπ

t , is characterized by its initial value A0 and its portfoliocomposition over time. This composition will be defined by the vector πt =

�π1t , ..., π

mt

,

where πit denotes the amount of money invested in each risky asset. To ensure the self-

financing condition, we impose that the remainder At −m

i=1 πit is invested in (or borrowed

from, if negative) the cash account delivering the risk-free rate rt. In other words, if wewant to invest more than its available wealth in market assets, then we will have to borrowthe remaining cash at the market short rate rt. In the opposite case, if we do not invest allof our cash in the market, we will lend the remaining cash at the rate rt. As a conclusion,we obtain the dynamics of any self-financing strategy:

dAπt =

Xπ

t −mi=1

πit

rt dt+

mi=1

πit

dSit

Sit

This is well known as the self-financing condition. It ensures that the wealth of the portfoliostrategy is only generated by the performance of its investments.

Modeling portfolios with payments and contributionsAs we deal with pension funds, we must also consider non-self-financing portfolios. Pen-sion funds make pension payments and receive contributions from future pensioners and thesponsor. We define any stream of payments and contributions by a process Pt, which rep-resents the accumulated amount of cash entering/exiting the fund. In other words, positivevariations of Pt represent pension payments, while negative variations represent contribu-tions. For instance, for continuous pension payments pt dt and continuous contributions ctdtduring the time interval [t, t+ dt] we have:

dPt = (pt − ct) dt

On the other hand, we can consider, at fixed dates t1, ...tn series of discrete paymentsp1, ..., pn and contributions c1, ..., cn. In that case, we will introduce:

dPt =

ni=1

(pi − ci) 1t=ti

This formula means that the pension fund will pay pi − ci at each considered date ti. If wesum these variations, we obtain the accumulated value of past payments:

Pt =

ni=1

(pi − ci) 1t≥ti

34

Now we must combine this payment model with our portfolio strategies in order to obtainthe change in the fund asset values. We find that the dynamics of a portfolio strategydelivering a stream of payments P can be written as:

dAπ,Pt =

Aπ

t −mi=1

πit

rt dt+

mi=1

πit

dSit

Sit

− dPt

How to model liabilities in line with the market

Perfectly matching liability-hedging portfolio

Suppose that the stream of payments and contributions can be matched exactly throughan investment strategy π̂, starting at time 0 for a given wealth Lπ̂,P

0 . Mathematically, thismeans that the portfolio strategy πL making the stream of payments P evolves as:

dLπ̂,Pt = Lπ̂,P

t rt dt+

mi=1

π̂it

dSi

t

Sit

− rt dt

− dPt (7)

Such that Lπ̂,Pτ = 0 in every case (i.e. with a probability equal to 1). The existence of such

an investment strategy would ensure that, starting from initial endowment Lπ̂,Pt at time

t, the pension fund could meet all of its liabilities exactly without any risk by using theinvestment strategy π̂ as above. Therefore, through a no-arbitrage argument, Lπ̂,P

t shouldbe viewed as the price of the remaining liabilities at time t.

If such a strategy exists, then, taking the expected value of (7) under any risk-neutralprobability Q leads to the expression:

LPt = EQ

τ

t

e− ut

rs ds dPu|Ft

(8)

Where LPt is the unique price of the future liabilities, and τ is the term of the last payment.

This price coincides with the price of the liability-hedging portfolio. Note that the liabilityprice defined as above does not depend on the portfolio strategy actually chosen by the fundmanager.

Partially hedgeable liabilities

Now, we abandon the assumption that the liabilities can be perfectly hedged by a portfoliostrategy. This may be a more realistic framework, as many factors, such as mortality ratesor early retirement issues, cannot be properly tied to any tradable asset. Therefore, anyportfolio strategy will leave some residual risks that must be taken into account. We assumethat there is a portfolio strategy π, starting with some initial wealth Xπ,P

0 , providing theliability stream P :

dAπ̂,Pt = Aπ̂,P

t rt dt+mi=1

π̂it

dSi

t

Sit

− rt dt

− dPt

such that the remainder of the investment strategy at time τ is given by:

Aπ̂,Pτ =

τ

0

e τt

rs dsσt dW

t



Now we must combine this payment model with our portfolio strategies in order to obtainthe change in the fund asset values. We find that the dynamics of a portfolio strategydelivering a stream of payments P can be written as:

dAπ,Pt =

Aπ

t −mi=1

πit

rt dt+

mi=1

πit

dSit

Sit

− dPt

How to model liabilities in line with the market

Perfectly matching liability-hedging portfolio

Suppose that the stream of payments and contributions can be matched exactly throughan investment strategy π̂, starting at time 0 for a given wealth Lπ̂,P

0 . Mathematically, thismeans that the portfolio strategy πL making the stream of payments P evolves as:

dLπ̂,Pt = Lπ̂,P

t rt dt+

mi=1

π̂it

dSi

t

Sit

− rt dt

− dPt (7)

Such that Lπ̂,Pτ = 0 in every case (i.e. with a probability equal to 1). The existence of such

an investment strategy would ensure that, starting from initial endowment Lπ̂,Pt at time

t, the pension fund could meet all of its liabilities exactly without any risk by using theinvestment strategy π̂ as above. Therefore, through a no-arbitrage argument, Lπ̂,P

t shouldbe viewed as the price of the remaining liabilities at time t.

If such a strategy exists, then, taking the expected value of (7) under any risk-neutralprobability Q leads to the expression:

LPt = EQ

τ

t

e− ut

rs ds dPu|Ft

(8)

Where LPt is the unique price of the future liabilities, and τ is the term of the last payment.

This price coincides with the price of the liability-hedging portfolio. Note that the liabilityprice defined as above does not depend on the portfolio strategy actually chosen by the fundmanager.

Partially hedgeable liabilities

Now, we abandon the assumption that the liabilities can be perfectly hedged by a portfoliostrategy. This may be a more realistic framework, as many factors, such as mortality ratesor early retirement issues, cannot be properly tied to any tradable asset. Therefore, anyportfolio strategy will leave some residual risks that must be taken into account. We assumethat there is a portfolio strategy π, starting with some initial wealth Xπ,P

0 , providing theliability stream P :

dAπ̂,Pt = Aπ̂,P

t rt dt+mi=1

π̂it

dSi

t

Sit

− rt dt

− dPt

such that the remainder of the investment strategy at time τ is given by:

Aπ̂,Pτ =

τ

0

e τt

rs dsσt dW

t

The term τ

0e τt

rs dsσt dW

t represents the residual liability hedging error for the optimal

liability-hedging portfolio strategy π. We assume that W t is a Brownian motion under the

historical probability P and is independent from the Brownian vector Wt =�W 1

t , ...,Wmt

governing the movements of the financial asset prices. This assumption comes from theobservation that, if there were any correlation between W

t and the tradable assets,thiscorrelation could be used to build another liability-hedging portfolio π̃ that would introducea smaller variance for the hedging error.

As W t is not correlated with asset prices, there is a risk-neutral probability Q̂ under

which W t is a Brownian motion. Using this probability, we define the present value of the

pension fund liabilities at time t as:

LPt = EQ̂

τ

t

e− ut

rs ds dPu|Ft

We can differentiate this expression with respect to time to obtain:

dLPt = rtL

Pt dt+

mi=1

π̂t

dSi

t

Sit

− rt dt

− σ

t dWt − dPt

We obtain the instantaneous error, during the time interval [t, t+ dt], between the presentvalue of the liabilities and the hedging portfolio, equal to σ

t dWt . This probability Q̂ does

not depend on our performance indicator I. A pricing rule could also be derived directlyfrom our utility maximization problem (see [12] for instance).

B Elements of proof for optimal dynamic strategies

Separation result for the funding gap problemLet us look at the dynamics of the fund asset value minus the funding gap. We have:

dAπ,P

t − LPt

= (Aπ

t − Lt) rt dt+

mi=1

�πit − π̂i

t

dSit

Sit

− rt dt

Note that in this expression, both the payment terms for the fund value and the liabilitycancel each other out, leaving us with the dynamics of a self-financing strategy. If we denotethe funding gap at time t as:

Gπ̃t = Aπ

t − Lt

and the difference between the fund’s portfolio and the liability-hedging portfolio as:

π̃t = πt − π̂t.

We find, with these notations, that the funding gap Gt evolves as a self-financing portfolio,with a composition defined by π̃:

dGπ̃t = Gπ̃

t rt dt+

mi=1

π̃it

dSi

t

Sit

− rt dt

We also find, as Gπ̃T = Aπ,P

T −LPT , that solving problem (2) is exactly equivalent to solving

the classical utility maximization problem without liabilities:

u (0, G0) = supπ̃

EP0

�U�Gπ̃

T

(9)

36

This change of variables turned the LDI problem into a classical problem of portfolio strategyπ̃ optimization, starting from initial wealth G0 = A0−L0, i.e. the initial value of the fundinggap. Everything here is standard except for the particular form of the utility function U .

Once this problem has been solved, we have the optimal portfolio π∗ for problem (2).

Optimal gap-bridging strategy

Result for a general utility function

Now, we solve problem (9), which will give us the composition of our portfolio. We derivethe solution in the complete market case, as we can use the martingale optimization methodintroduced in [3] (see [11] for an overview). In incomplete markets, we would have tointroduce the more tedious framework of the Hamilton Jacoby Bellman equations (see [11]or [4] for an introduction to these methods).

As we are in complete market, the self-financing condition is equivalent to the fact thatthe expectation of our strategy, under the T -forward probability (QT ), is equal to the T -forward value of our portfolio today. This is written as:

G0

B (0, T )= EQT

0 (GT ) (10)

Any terminal payoff is the result of a self financing strategy if and only if that equality holdstrue at time 0. The problem can be therefore written as:

supEP0 (U (GT )) such that

G0

B (0, T )= EQT

0 (GT )

We transform this problem using the Lagrangian:

supGT

infλ>0

EP0 (U (GT ))− λ

EQT

0 (GT )−G0

B (0, T )

Introducing the change of probabilty ZT = dQT

dP (the value of which will be discussed later)leads to:

supGT

infλ>0

EP0 (U (GT )− λZTGT ) + λ

G0

B (0, T )

We can invert the supremum and the infimum to obtain:

infλ>0

supGT


G0

B (0, T )

Taking the supremum over GT gives the optimal payout for our utility, up to λ:

GT = (U )−1

(λZT ) (11)

and the constant term λ must be found such that constraint (10) holds. If U−1 does notexist, we refer to [1] to adjust this result.



This change of variables turned the LDI problem into a classical problem of portfolio strategyπ̃ optimization, starting from initial wealth G0 = A0−L0, i.e. the initial value of the fundinggap. Everything here is standard except for the particular form of the utility function U .

Once this problem has been solved, we have the optimal portfolio π∗ for problem (2).

Optimal gap-bridging strategy

Result for a general utility function

Now, we solve problem (9), which will give us the composition of our portfolio. We derivethe solution in the complete market case, as we can use the martingale optimization methodintroduced in [3] (see [11] for an overview). In incomplete markets, we would have tointroduce the more tedious framework of the Hamilton Jacoby Bellman equations (see [11]or [4] for an introduction to these methods).

As we are in complete market, the self-financing condition is equivalent to the fact thatthe expectation of our strategy, under the T -forward probability (QT ), is equal to the T -forward value of our portfolio today. This is written as:

G0

B (0, T )= EQT

0 (GT ) (10)

Any terminal payoff is the result of a self financing strategy if and only if that equality holdstrue at time 0. The problem can be therefore written as:

supEP0 (U (GT )) such that

G0

B (0, T )= EQT

0 (GT )

We transform this problem using the Lagrangian:

supGT

infλ>0

EP0 (U (GT ))− λ

EQT

0 (GT )−G0

B (0, T )

Introducing the change of probabilty ZT = dQT

dP (the value of which will be discussed later)leads to:

supGT

infλ>0


G0

B (0, T )

We can invert the supremum and the infimum to obtain:

infλ>0

supGT


G0

B (0, T )

Taking the supremum over GT gives the optimal payout for our utility, up to λ:

GT = (U )−1

(λZT ) (11)

and the constant term λ must be found such that constraint (10) holds. If U−1 does notexist, we refer to [1] to adjust this result.

The probability change is linked to the Sharpe-optimal portfolio

Once we have found the optimal payoff, we should focus the study on ZT . Applying theoptimal payoff formula to the logarithmic utility function U (x) = ln (x) gives:

GlnT =

1

λZT

On the other hand, the optimal payoff for the logarithmic utility is the Sharpe optimalportfolio (i.e. Gln

T = MT

M0), hence we find that:

ZT =λ

MT

Up to some constant value λ, where MT is the value at time T of the Sharpe-optimalportfolio.

Therefore, the terminal value of the optimal strategy is given for any utility function, upto a constant λ, by:

GT = (U )−1

λ

MT

Once λ has been calibrated using Equation (10), the optimal strategy value at time t isgiven by the function:

G (t,Mt, B (t, T )) = B (t, T )EQT

t

(U )

−1

λ

MT

The optimal allocation rule can be calculated using the sensitivities of G with respect to Mt

and B (t, T ). Market assumptions of volatility and Sharpe ratio need to be introduced forthese computations.

Some explicit formulas can be found for special utility functions such as in Equation (3).

The funding ratio problemThe funding ratio can be solved in the same way as the funding gap problem. We will focuson only the differences here. We start from nearly the same point, except that we considerAL instead of A− L:

supA

EPU

AT

LT

such that EQT

(AT ) =A0

B (0, T )

Using the same arguments, we get the optimal payout:

AT = LTU−1

λLT

MT

In addition, we must still find the value λ that satisfies the constraints. The importantquantity is therefore the value of the market portfolio with respect to the liabilities: LT

MT.

Lastly the optimal strategy value at time t can be computed with the formula:

A (t,Mt, Lt) = B (t, T )EQT

t

LT (U )

−1λLT

MT

= LtEQL

t

(U )

−1λLT

MT

38

where QL is the L-forward probability. To calculate this expectation, there must be anassumption of the expected return and volatility of M with respect to the liabilities. Oncethis has been calculated, the sensitivies of A with respect to L and M must be computed toobtain the optimal allocation.



where QL is the L-forward probability. To calculate this expectation, there must be anassumption of the expected return and volatility of M with respect to the liabilities. Oncethis has been calculated, the sensitivies of A with respect to L and M must be computed toobtain the optimal allocation.

References[1] B. Bouchard, N. Touzi, and A. Zeghal, Dual Formulation on the Utility Maximization

Problem: The Case of Nonsmooth Utility, Stochastic Processes and their Applications111 (2004), 175–206.

[2] IASB, International Accounting Standards IAS 19 Revised 2004, (2004).

[3] I. Karatzas, J. P. Lehoczky, and Shreve S. E., Optimal Portfolio and ConsumptionDecision for a Small Investor on a Finite Horizon, SIAM Journal on Control andOptimization 25 (1987), 1557–1586.

[4] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer, 2ème édition,1999.

[5] L. Martellini, Managing Pension Assets: From Surplus Optimization to Liability DrivenInvestment, EDHEC Risk and Asset Management Research Centre (2006).

[6] L. Martellini and V. Milhau, Measuring the Benefits of Dynamic Asset AllocationStrategies in the Presence of Liability Constraints, EDHEC Risk and Asset ManagementResearch Centre (2009).

[7] McKinsey & Company, The Asset Management Industry in 2010, (2010).

[8] R. Merton, Lifetime Portfolio Selection under Uncertainty: the Continuous Time Case,The Review of Economics and Statistics 51 (1969), 247–257.

[9] , Optimal Consumption and Portfolio Rules in a Continuous-Time Model, Jour-nal of Economic Theory 3 (1971), 373–413.

[10] , Continuous Time Finance, Blackwell Publishers, 1990.

[11] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Appli-cations, Springer Verlag, 2009.

[12] R. Rouge and N. El Karoui, Pricing via Utility Maximization and Entropy, Mathemat-ical Finance 10 (2000), 259–276.

[13] W. F. Sharpe, Capital Asset Prices: A Theory of Market Equilibrium under Conditionsof Risk, Journal of Finance 19 (1964), 425–442.

[14] The Finance, Investment and Risk Management Board Working Party, Practical Im-plementation of Liability Driven Investment, (2007).

[15] The Pensions Regulator, The Purple Book, DB Pensions Universe Risk Profile, (2007).

[16] J. Von Neumann and O. Morgenstern, Theroy of Games and Economic Behaviour,Springer, 2nd edition, 1999.

[17] Watson Wyatt, 2007 Global Pension Asset Study, (2007).



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