Tilburg University Market-Consistent Valuation of Pension … · netspar industry series design 63 design 63 This is a publication of: Netspar P.O. Box 90153 5000 LE Tilburg the Netherlands

Tilburg University

Market-Consistent Valuation of Pension Liabilities

Pelsser, Antoon; Salahnejhad, Ahmad; van den Akker, Ramon

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Publication date:2016

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Citation for published version (APA):Pelsser, A., Salahnejhad, A., & van den Akker, R. (2016). Market-Consistent Valuation of Pension Liabilities.(Netspar Industry Paper; Vol. Design 63). NETSPAR.https://www.netspar.nl/assets/uploads/P20161000_des063_Pelsser.pdf

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design 63

design 63

This is a publication of:

Netspar

P.O. Box 90153

5000 LE Tilburg

the Netherlands

Phone 013 466 2109

E-mail [email protected]

www.netspar.nl

October 2016

Market-consistent valuation of pension liabilities

Due to the long maturity of its contracts, a pension fund or life-insurance

company is exposed to actuarial risks such as longevity risk and also to

market risks such as interest rate risk and inflation risk. The insurance and

pensions regulator in Europe (EIOPA) has also recognized the importance

of valuation methods that take financial risks and non-financial risks into

account. In this paper, Antoon Pelsser, Ahmad Salahnejhad (both UM) and

Ramon van den Akker (SNS/TiU) want to show that it is computationally

feasible to price pensions contracts in an incomplete market setting with

time-consistent and market-consistent (TCMC) pricing operators.

Market-consistent valuation

of pension liabilities

Antoon PelsserAhmad SalahnejhadRamon van den Akker

Antoon Pelsser, Ahmad Salahnejhad and Ramon van den Akker

Market-consistent valuation of pension liabilities

design paper 63

netspar industry paper series

Design Papers, part of the Industry Paper Series, discuss the design of a component of a pension system or product. A Netspar Design Paper analyzes the objective of a component and the possibilities for improving its efficacy. These papers are easily accessible for industry specialists who are responsible for designing the component being discussed. Design Papers are published by Netspar both digitally, on its website, and in print.

ColophonOctober 2016

Editorial BoardRob Alessie – University of GroningenRoel Beetsma (Chairman) - University of AmsterdamIwan van den Berg – AEGON NederlandBart Boon – AchmeaKees Goudswaard – Leiden UniversityWinfried Hallerbach – Robeco NederlandIngeborg Hoogendijk – Ministry of FinanceArjen Hussem – PGGMMelanie Meniar-Van Vuuren – Nationale NederlandenAlwin Oerlemans – APGMaarten van Rooij – De Nederlandsche BankMartin van der Schans – Ortec FinancePeter Schotman – Maastricht UniversityHans Schumacher – Tilburg UniversityPeter Wijn – APG

DesignB-more Design

Lay-outBladvulling, Tilburg

PrintingPrisma Print, Tilburg University

EditorsFrans KooymansNetspar

Design Papers are publications by Netspar. No reproduction of any part of this publication may take place without permission of the authors.

contents

Summary 7

1. Introduction 82. Time- and Market-Consistent Valuation 113. Valuation of Unit-linked Contract 184. Time- and Market Consistent Pension Valuation 235. Conclusions 31

References 33

6

AffiliationsAntoon Pelsser – Maastricht UniversityAhmad Salahnejhad – Maastricht UniversityRamon van den Akker – SNS / Tilburg University

-

SummaryPension funds and life insurance companies have liabili es on theirbooks with extremely long-dated maturi es that are exposed tonon-hedgeable actuarial risks and also to market risks. In thispaper, we show that it is computa onally feasible to price pensionscontracts in an incomplete market se ng with me-consistent andmarket-consistent (TCMC) pricing operators. Furthermore, wecompare the TCMC prices for life-insurance and pension contractsto alterna ve pricing methods that are currently used for pricingpension and life-insurance liabili es: the best es mate pricingmethod which is typically used for pension liabili es, and theEIOPA’s risk margin method that is used under Solvency II to valuelife-insurance liabili es. We show that the best es mate pricingmethod completely ignores the uncertainty in the non-hedgeablerisks. We also show that the risk margin method is a significantstep in the right direc on to reflect most of this uncertainty in thepricing. However, the risk margin price s ll ignores someuncertain es, and is therefore not fully me-consistent. Forlong-dated contracts this effect should not be ignored.

. Introduc on

Pension funds and life insurance companies have liabili es on theirbooks with extremely long-dated maturi es. People typically startsaving for their pension at age with the build-up phase las ngun l the re rement age and thus extending for a period of up toyears. Then they enter into the decumula on phase, whichtypically lasts for years, but this can last up to years, becausetheir life expectancy is years, with the oldest people living toage . Hence, pension funds and life insurance companies facecontractual obliga ons with maturi es of up to years. Thevalua on and risk management of these extremely long-datedcontracts is therefore a major and challenging problem.

Due to the long maturity of these contracts, a pension fund orlife-insurance company is exposed to actuarial risks (such aslongevity risk) and also to market risks (such as interest rate riskand infla on risk). In par cular for the actuarial risks, such aslongevity risk, it is generally impossible to hedge these risks sincehardly any contracts traded in financial markets that can be used tohedge these risks. In addi on, these long-dated contracts havesignificant exposure to market risks such as interest rate andinfla on risk. The pricing and risk management of pensionliabili es therefore requires valua on methods that take bothfinancial risks and non-financial risks into account.

The European Insurance and Occupa onal Pensions Authority(EIOPA) has recognised the importance of valua on methods thattake both financial and non-financial risks into account. Under theSolvency II framework, insurance companies are required to useso-calledmarket-consistent valua on methods to price theirliabili es which explicitly consider non-market risks. Furthermore,EIOPA has formulated in the IORP Direc ve the ambi on to assess

-

the risk posi on of pension funds also in a market-consistentframework . In par cular, EIOPA has proposed that the embeddedop ons in pension contracts should be explicitly valued using aHolis c Balance Sheet approach.

From a theore cal perspec ve, the problem of pricing along-dated pension liability is a pricing problem in an incompletemarket, since a pension contract is exposed to hedgeable financialrisks, as well as non-hedgeable actuarial risks. When faced with anincomplete market, the standard machinery of risk-neutralBlack-Scholes pricing breaks down, because it is no longer possibleto construct a perfect replica ng por olio that hedges all risks. Wetherefore need to consider pricing methods that explicitly take thenon-hedgeable risks into account, but that remainmarket-consistent in the sense that the prices of “pure” financialcontracts are s ll consistent with risk-neutral pricing.

Another important requirement for a pricing operator isme-consistency. When calcula ng the price of a contract, we

cannot simply price a contract at t = 0 and then “forget” aboutthe contract. Instead, we should follow the contract over me and,when new informa on about the financial market or actuarial risksarrives, then update our pricing and the hedging posi on.

In this paper, we want to show that it is computa onally feasibleto price pension contracts in an incomplete market se ng withme-consistent and market-consistent (TCMC) pricing operators.

Furthermore, we compare the TCMC prices for life insurance andpension contracts against alterna ve pricing methods that arecurrently used to price pension and life insurance liabili es:

For more informa on we refer to Netspar Design Paper “European super-vision of pension funds: purpose, scope and design” and Design Paper “Thedesign of European supervision of pension funds”.

• the Best Es mate pricing method, which is currently theusual method to value pension liabili es;

• EIOPAs Risk Margin pricing method, which is used underSolvency II to value life-insurance liabili es.

-

. Time- and Market-Consistent Valua on

As stated in the introduc on, the problem of pricing a long-datedpension liability is that it involves an incomplete market, since apension contract is exposed to hedgeable financial risks as well asto non-hedgeable actuarial risks. We therefore look for pricingoperators that are bothmarket-consistent and me-consistent.

Before we proceed, let us introduce some nota on. We willdenote the collec on of market risks by the vector-valuedstochas c process xt . The collec on of non-market (or actuarial)risks is denoted by the vector-valued stochas c process yt . We willdenote a general pension contract payoff at me T as a func onf (xT , yT ), where the argument (xT , yT ) denotes that thecontract value at me T may depend on the whole path of theprocesses {xt}0≤t≤T and {yt}0≤t≤T .

A pricing operator is denoted byΠ[t, XT ]. This means that itassigns to any payoff (i.e. a random variable) observable at me Tan amount of money (“the price”) that is computable at me tgiven the state of the world (xt , yt).

Many of the pricing operators that we will discuss can beexpressed in terms of condi onal expecta on operators. We willmake use of the nota on: EP[XT ], EQ[XT ] to denote anexpecta on of the random variable XT under the real-worldmeasureP or the risk-neutral measureQ. In many cases, we wantto condi on on the market informa on available at me t . In thatcase we use the nota on E[XT |xt ]. Similarly, the nota onE[XT |yt ] means that we condi on on the non-market informa onavailable at me t .

Amarket consistent pricing operator has the property that forany “pure financial” payoff f (xT ) we get the same value as theBlack-Scholes price. Using our nota on we can express this as

Π[t, f (xT )] = e−r(T−t)EQ[f (xT )|xt ].A me consistent pricing operator has the property that the

price at me t for any payoff XT that is held to maturity T is equalto the price at me t of the same posi on that is held un l me sand then sold at the then-current s-price. Using our nota on, wecan express this as: for all t ≤ s ≤ T we haveΠ[t, XT ] = Π

[t,Π[s, XT ]

].

The risk-neutral pricing operator of (Black and Scholes, )given byΠBS[t, f (xT )] = e−r(T−t)EQ[f (xT )|xt ] is an example ofa me-consistent and market-consistent pricing operator for acomplete market. The me consistency of the Black-Scholes pricearises from the “tower property” of the condi onal expecta onoperator E[f (xT )|xt ]. The market-consistency arises from therisk-neutral probability measureQ.

In the remainder of this sec on we will discuss two types ofpricing operators used by prac oners: Best Es mate valua onand valua on with a Risk Margin. We will also introduce the meconsistent and market consistent (TCMC) pricing operator ofPelsser and Stadje ( ).

. . Best Es mate Valua onA pricing operator widely used to price pension liabili es is the BestEs mate pricing operator. This pricing operator is constructed asfollows: the actuarial risks (i.e. the non-market risks) are projectedwith the best possible model to obtain projected cash flows for thecontract. These projected cash flows are then priced using therisk-neutral Black-Scholes pricing operator. For fixed cash flows thisboils down to discoun ng the projected cash flows with therisk-free term-structure of interest rates observed in the market.

Using our nota on, we can define the best es mate pricing

-

operator as

ΠBE[t, f (xT , yT )] =e−r(T−t)EQ

[f(xT ,EP[yT |yt ]

) ∣∣ xt]

. ( )

Note that we have nested two expecta on operators inside eachother. The inner expecta on operator performs the projec on ofactuarial risks under the real-world measureP. The result of theinner expecta on opera on is a set of cash flows that depend onlyon the market-risks xT . In the outer expecta on, thesemarket-risks are then priced with the market consistentBlack-Scholes pricing operatorΠBS[].

The advantage of the best es mate pricing operator is that it iseasy to evaluate. We replace the uncertain actuarial randomvariable yT with its best es mate projec on EP[yT |yt ]. Such areplacement is rou nely performed by actuaries when they use amortality table to project cash flows, instead of a stochas cmortality process to project stochas c cash flows.

The disadvantage of the best es mate approach is that theinherent uncertainty of the actuarial risks yT is swept under therug whenever the random variable yT is replaced by its bestes mate projec on EP[yT |yt ]. Hence, the uncertainty arisingfrom the non-market risks yT is not reflected by the best es matepricing operator.. . Valua on with a Risk Margin

In the design of the Solvency II supervisory framework, the need toreflect the uncertainty arising from non-market risks is explicitlyincorporated in the pricing methodology. Under Solvency II, thepricing of liabili es consists of two components: a best es mateand a risk margin.

The best es mate component is iden cal to the pricing operatorΠBE[] discussed in the previous sub-sec on.

The risk margin component was introduced as an adjustment tothe best es mate price to cover the uncertainty arising from thenon-hedgeable risks in the liabili es. The calcula on of the riskmargin is based on a cost-of-capital argument.

The only way that non-hedgeable risks can be absorbed is bymaintaining a buffer capital in the balance-sheet. The buffer capitalhas been provided by external stakeholders (e.g. the shareholdersin the case of a publicly listed insurance company). The capitalproviders know that they are inves ng risk capital they are willingto provide such capital because they receive compensa on in theform of a higher return than the risk-free rate. The return in excessof the risk-free rate is called the cost-of-capital.

Based on this argument, we can quan fy the risk margin as theNPV of all cost-of-capital payments that need to be made to thecapital-providers during the life of the liability. To make thecalcula on explicit, we need to determine the size of the buffercapital and the cost-of-capital percentage. EIOPA has set thefollowing rules:

• The buffer capital is calculated as the one-year Value-at-Risk(for the non-hedgeable risks) with a confidence level of

. %.

• The cost-of-capital is set at %.

The Value-at-Risk (VaR) is the . % worst-case quan le of thedistribu on of the non-hedgeable risks. For a normal probabilitydistribu on, the one-year . % VaR is equal to . mes theper-annum standard-devia on.

-

We can express the risk margin pricing operator as:

ΠRM[t, f (xT , yT )] = ΠBE[t, f (xT , yT )]

+T∑

s=t+1e−r(s−t)γVaR0.995

[ΠBE[s, fT ]

∣∣ EP[ys−1|yt ]]

, ( )

where the parameter γ denotes the cost-of-capital percentage.The summa on term can be interpreted as follows. Thesumma on takes annual me-steps from t + 1 un l the maturityT of the liability. For each me-step s , we consider the one-yearVaR along the best es mate path of y , hence the VaR-operator iscondi oned on the value EP[ys−1|yt ]. Taking this projected valueof ys−1 as a star ng-point, we then consider the impact of a . %worst-case shock in the non-market risks ys on the best es matepriceΠBE[s, f (xT , yT )] at me s . This is the projected buffercapital for me s . Over this buffer capital we have to pay thecost-of-capital γ to the capital-providers. The discounted sum of allthese cost-of-capital payments is the risk margin.

Although the EIOPA risk margin pricing operator is calculated ina mul -period se ng, it is not a me consistent pricing-operator.The risk margin pricing operator does take the uncertainty arisingfrom the non-market risks yT on the best es mate price intoaccount. However, there is a “second-order” effect: theuncertainty arising from the non-market risks yT on the futurebuffer capitals. What EIOPA’s risk margin pricing operatortherefore ignores is the “capital-on-capital” effect that a fully meconsistent operator would take into account.. . Time- and Market Consistent Valua on

The ques on of how to build a pricing operator that is fully marketconsistent and me consistent is addressed in the paper by Pelsser

and Stadje ( ). They extend the “backward induc on” methodproposed by Jobert and Rogers ( ) for crea ng me consistentpricing operators.

The backward induc on method can be explained as follows. Ifwe have a liability with a maturity T , then one year before thematurity (at T − 1) we have a one-year contract. For this one-yearcontract, the EIOPA risk margin pricing operator will yield thecorrect price. A er all, there is no “capital-on-capital” problem asthe contract expires one year later at me T .

So, for each state of the world (xT−1, yT−1) we can computethe priceΠ[T − 1, f (xT , yT )]. For the one-step case, the EIOPArisk margin pricing operator simplifies to

ΠRM [T − 1, f (xT , yT )] = ΠBE[T − 1, f (xT , yT )]+ γe−rVaR0.995 [f (xT , yT ) | yT−1] . ( )

Note that the pricing operatorΠRM [] prices the market risks in amarket consistent way, due to the embeddedQ-expecta on in thefirstΠBE[] term. The price of the non-market risks is reflected inthe second term.

To emphasise the dependence on the state of the world, we willdenote the price at me T − 1 by π(T − 1, xT−1, yT−1). Wecould sell the liability at me T − 1 for the priceπ(T − 1, xT−1, yT−1) in the state of the world (xT−1, yT−1).Hence, we can also interpret the price π(T − 1, xT−1, yT−1) as anew liability with maturity T − 1.

Therefore, we can take another one-year step, where wecompute the one-step price at me T − 2 of the liabilityπ(T − 1, xT−1, yT−1). We con nue this procedure un l wereach t = 0.

The contribu on of Pelsser and Stadje ( ) is that they prove

-

that every me consistent and market consistent pricing operatorcan be obtained by applying the backward induc on technique toa “simple” one-period pricing operator. For further details, werefer to Pelsser and Stadje ( ) and also to Pelsser andSalahnejhad ( ).

. Valua on of Unit-linked ContractIn this sec on we introduce a simplified example, in which weexplicitly calculate the three pricing operators that we haveintroduced in the previous sec on. The example is that of aunit-linked contract for an insurance company, or equivalently astylised DC pension contract without any form of guarantee orindexa on.

We are going the model the financial risk as a stock-price St .Please note, that we do not use xt to denote the financial marketprocess here, but instead the more familiar nota on St . Weassume that the stock-price St follows a Geometric BrownianMo on under the real-world measureP:

dSt = µSt dt + σSt dW St , ( )

where the growth rate µ of the stock and the vola lity σ areconstants. We also assume that the risk-free interest-rate r isconstant. Under these assump ons, we find that the stock-priceprocess under the risk-neutral measureQ is given by

dSt = rSt dt + σSt dW SQt . ( )

The actuarial risk is modelled as follows. We assume that ytdenotes the number of par cipants alive at me t . To obtain atractable example, we model the number of surviving par cipantsas a Geometric Brownian Mo on (GBM) under the real-worldmeasureP:

dyt = −ayt dt + byt dW yt , ( )

where the mortality rate a of the survivors and the vola lity b areconstants. We also assume that the financial market process Stand the actuarial process yt are independent processes.Note, that this way of modelling the number of survivors is not very realis c,

-

We now introduce the payoff of our contract. At the maturitydate T each surviving par cipant receives the value of thestock-price ST . Hence, the total liability at me T is given by theformula:

f (ST , yT ) = ST yT , ( )which is the stock-price mes the number of surviving par cipants.. . Best Es mate Valua on

First, we consider the price of this liability at me t = 0 using thebest es mate price operator, defined in equa on ( ).

For the best es mate valua on, we first project the number ofsurvivors as EP[yT ] = y0e−aT . The projected cash flow at me Tis therefore: ST y0e−aT . This projected cash flow is thencalculated in a market consistent way under the risk-neutralmeasureQ as

ΠBE[0, ST yT ] = e−rTEQ[ST ]y0e−aT = S0y0e−aT . ( )

Note, that this best es mate price is the same when we would payST to the determinis c number of survivors y0e−aT . Hence, theuncertainty in the number of survivors yT at me T is notreflected by the best es mate pricing operator.. . Valua on with a Risk Margin

Second, we consider the price of this liability at me t = 0 usingthe risk margin price operator, defined in equa on ( ).

To compute the risk margin, we need to consider the one-yearValue-at-Risk for each year t between 0 and T . Since we haveassumed the number of survivors to follow a GBM, we know thatbecause a GBM process can have paths that increase over me, even when themortality trend is nega ve.

the one-year probability distribu on of yt given yt−1 is alog-normal distribu on with dri term−a and vola lity b.Therefore, the . % worst-case value of yt is given byyWC

t = yt−1e−a+2.58b . Note that, for our par cular contract, theworst-case scenario is thatmore people than expected survive. Forthis reason we take the upward shock +2.58b.

We now have to compute the Value-at-Risk for each year alongthe best es mate path. The best es mate value for yt−1 is equal toEP[yt−1] = y0e−a(t−1). We then apply the worst-case one-yearshock: yWC

t = y0e−a(t−1)e−a+2.58b = y0e−ate2.58b . Using thisshocked value at me t , we then project the number of survivorsat me T , this leads to y0e−ate2.58be−a(T−t) = y0e−aT e2.58b .The Value-at-Risk for me t is the difference between the bestes mate price S0erty0e−aT and the price for the shockedprojec on. Hence, we can express the VaR for year t asS0erty0e−aT (e2.58b − 1).

When we subs tute these VaR expressions for each year t intothe risk margin pricing formula, we obtain

ΠRM[0, ST yT ] = S0y0e−aT (1 + γT (e2.58b − 1)

). ( )

In the summa on of the VaR-terms we encounter T mes thesame term S0y0e−aT (e2.58b − 1). Therefore the risk margin termsimplifies to γT S0y0e−aT (e2.58b − 1).. . Time- and Market Consistent Valua on

Finally, we consider the price of this liability at me t = 0 usingthe TCMC price operator.

When we apply the one-year risk margin pricing formula atT − 1 we obtain the priceΠRM [T − 1, ST yT ] =ST−1yT−1e−a (1 + γ(e2.58b − 1)

), which is the one-year best

es mate price mes the factor(1 + γ(e2.58b − 1)

).

-

Note that, for this par cular example, the factor is independentfrom ST−1 and yT−1. Hence, if we apply the one-year risk marginpricing formula once more to obtain the price at me T − 2, thenwe get the best es mate price mes the factor(1 + γ(e2.58b − 1)

)2.A er T backward induc on steps, we find at me t = 0 the

TCMC price of

ΠTCMC[0, ST yT ] = S0y0e−aT (1 + γ(e2.58b − 1)

)T . ( )

When we compare the TCMC price to EIOPA’s risk margin price, weclearly see the “capital-on-capital” effect that is missing from therisk margin price, and that is included in the TCMC price.

The risk margin price is equal to the best es mate price mesthe factor

(1 + γT (e2.58b − 1)

), whereas the TCMC price is

equal to the best es mate price mes the factor(1 + γ(e2.58b − 1)

)T . For T = 1 both factors are the same, butfor T > 1 the TCMC factor is larger than the risk margin factor,and the gap widens for larger values of T .

. . Numerical Illustra onIn this subsec on, we compare the different prices for specificvalues of the model-parameters. In our example we use for thestock-price process the values S0 = 1, r = 4% and σ = 16%. Forthe actuarial risk process we take y0 = 1000, a = 1% andb = 7%.

We report the three different prices for a range of maturi esfrom T = 1 to T = 30. The contract values under the threedifferent pricing operators are shown in Figure . The bestes mate prices are labeled as “BestEst”. EIOPA’s risk margin pricesas “EIOPA”, and the TCMC-price as “TC-MC”.

850

900

950

1000

1050

1100

Pri

cePrice of Unit-Linked Contract

BestEst

EIOPA

700

750

800

850

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Maturity

EIOPA

TC-MC

Figure : Comparison of Prices for a Unit-Linked Contract

We see that the best es mate prices simply reflect the expectednumber of survivors with a % mortality rate, ranging from forT = 1 to for T = 30. The uncertainty surrounding thisprojected number of survivors is not reflected in the best es mateprice.

The risk margin prices do reflect the uncertainty surrounding theprojected number of survivors, hence the EIOPA prices are higherthan the best es mate prices. For longer-dated contracts, theunhedgeable uncertainty becomes larger; therefore the gap withthe best es mate prices becomes consistently larger.

The TCMC prices reflect the “capital-on-capital” effect, and aretherefore s ll a bit higher than the EIOPA prices. We also see thatthe risk margin prices do reflect a significant adjustment in theright direc on. Hence, the remaining gap to the TCMC price isrela vely small. However, for longer-dated contracts thecapital-on-capital effect becomes rela vely more important.

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. Time- and Market Consistent Pension Valua onThe example from the previous sec on allowed us to calculate allprices explicitly. But the payout and the modelling of the mortalityprocess were not very realis c. In this sec on we demonstrate thatwe can compute our pricing operators also for a realis c type ofcontract. The purpose of this sec on is twofold. First, we wish toshow that the computa on of a me consistent and marketconsistent pricing operator is feasible for a realis c contract.Second, we wish to compare the prices of the three pricingoperators for a realis c contract.

. . Indexa on MechanismWe focus our a en on on a path-dependent contract withprofit-sharing (i.e. indexa on) introduced by Grosen and Jorgensen( ). This contract is similar to a typical Dutch pension contractwith condi onal indexa on.

The Grosen and Jorgensen ( ) indexa on mechanism worksas follows. At me t = 0 a par cipant obtains one unit of thecontract with nominal value P0. The pension fund invests the fullamount in the financial market. Let St be the market value of theinvested amount in the financial market and let Pt be the nominalpension claim in year t .

At the beginning of each year t , the nominal pension claim ofeach par cipant grows by the following formula:

Pt = Pt−1

(1 + max

{rG ,α

(St−1

Pt−1− (1 + β)

)}). ( )

This formula can be interpreted as follows. The ra o St−1/Pt−1 isthe ra o of the assets over the nominal pension claims, hence thisra o can be interpreted as the nominal funding ra o of the pension

fund. Each year, the ra o St−1/Pt−1 is compared to a targetfunding ra o (1 + β). Also each year, a frac on α of the excessfunding ra o St−1/Pt−1 − (1 + β) is credited to the par cipants.However, if the credited amount falls below the minimumguarantee rG , then each par cipant receives the minimumguarantee. This credi ng mechanism is comparable to thecondi onal indexa on mechanism used by Dutch pension funds.. . Lee-Carter Model

To model the evolu on of the survival probabili es in a realis cway, we use the model introduced by Lee and Carter ( ). In thismodel, the force-of-mortality mk,t for age k at me t is given by

ln mk,t = αk + βkκt ( )

where κt is the stochas c mortality trend, αk is the averageage-specific mortality and βk is the age-specific sensi vity of themortality to change of κt . The stochas c process κt is a latentprocess to model the mortality trend specified by

dκt = µκdt + σκdW κt , ( )

where W κt a standard Brownian Mo on under the real-world

measureP.To implement a Lee-Carter model, we must es mate the

parameter vectors αk and βk for all ages k , and the parametersκ0,µκ,σκ from historical mortality data.. . Contract Payoff

We assume that the financial market process St follows the GBMprocess defined in ( ) under the risk-neutral measureQ.

Figure illustrates five simulated paths of the market value ofinvested assets St and the policy reserve Pt for a -year pension

-

0 5 10 15 200

100

200

300

400

500

600

t: Contract Year

At ,

Pt

Asset Value At

Policy Reserve Pt

Figure : Simula on of the Asset Value St and Policy Reserve Pt of thePension Contract. Parameter set: S0 = P0 = 100, r = 4%,σS = 0.15,rG = 2%.

contract with guaranteed interest rate rG = 2%. We see that theGrosen-Jorgensen credi ng mechanism smoothes thedevelopment of the policy reserve. We add the actuarial risk ofmortality/longevity to the above financial se ng to get a realis cpension payoff at me T . This requires that the policyholder isalive to get the policy reserve PT .

When we use the Lee-Carter to model the force of mortality, thenumber of survivors NT at me T is given by a Poisson processwith stochas c intensity−mk,t at me t , where the jump-processis assumed to be independent of the Brownian Mo ons W S andW κ.

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Figure : Simula on of the Policy reserve Pt and Survival event 1{Tx>T}for an individual of age 50 for maturityT = 20. Other parameters: P0 =100, r = 4%, σS = 0.15, rG = 2%.

The final contract payoff will be

f (ST ,κT ) = PT (ST )NT (κT ), ( )

where use ST to denote the en re history of the financial riskprocess, and κT to denote the en re history of the actuarial riskprocess.

Figure shows a simula on of the policy reserve and themortality events for an individual with age k = 50, up tore rement age where T = 20 and with guaranteed interestrate rG = 2%. The simula on is performed for scenarios insome of which the death event shi s the evolu on of Pt from thele side of the graph to the right side. In each of those cases thepayoff at age is zero.

. . Numerical Computa onFor a realis c payoff, we can no longer use analy cal pricingformulas, as in Sec on ; instead we have to use numerical

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approxima on methods. To implement the calcula ons, we useMonte-Carlo simula on to generate paths for the financial andactuarial risk drivers. To simulate paths for the financial riskprocess, we use equa on ( ). To simulate paths for the mortalityrisk driver we use equa on ( ) combined with a simula on for thePoisson process Nt . Along each path, we can move forward in meand update Pt using the Grosen-Jorgensen credi ng formula ( ).Then, for each path, we determine the payoff f (ST ,κT ) at meT using the payoff formula ( ). We now have for each simulatedpath the contract payoff at me T .

The next step is to evaluate our pricing operators. For the bestes mate pricing operator, we can simply take the discountedaverage of all payoffs at me T to compute the Monte-Carloapproxima on of the price.

For the risk margin and the TCMC pricing operators, we need toevaluate numerically the condi onal expecta on operators at eachannual me-point t . An efficient method to perform thesecomputa ons is the Least Squares Monte Carlo (LSMC) method.LSMC was introduced by Carriere ( ) and Longstaff andSchwartz ( ) to price American-style op ons. The LSMCmethod uses regressions across all simulated paths to es mate thecondi onal expecta ons at all the me-points. The condi onalexpecta on at me t of any general payoff π(t + 1, St+1,κt+1)at me t + 1 can be approximated by a series of basis func ons inSt and κt as,

E[π(t + 1, St+1,κt+1) | St ,κt ] =K∑

i ,j=0atij ei(κt)ej(St) ( )

where we can choose different types of the basis func ons such asei(z) = z i . The coefficients atij are then es mated for me t by

regressing across all simulated paths the dependent variableπ(t + 1, St+1,κt+1) onto the explanatory variablesei(κt)ej(St). The numerical es mate for the condi onalexpecta on is then obtained by evalua ng the right-hand sideof ( ) with the es mated coefficients atij .

To evaluate the risk margin and the TCMC pricing operator, wealso need to evaluate numerically the condi onal Value-at-Risk atdifferent points in me t . For our calcula ons, we approximatethe VaR as . mes the condi onal standard devia on. Thecondi onal standard devia on is the square-root of the condi onalvariance. The condi onal variance can be computed from thecondi onal expecta on

E[π(t + 1, St+1,κt+1)2 ∣∣ St ,κt

]=

K∑i ,j=0

btij ei(κt)ej(St),

( )where the coefficients btij for me t can be es mated from across-sec onal regression.. . Comparison of the Different Pricing Methods

In this subsec on, we compare the different prices for the pensioncontract. We consider a cohort of par cipants aged k = 40,and compute the value of the pension contract for a range ofmaturi es from T = 1 to T = 30. The Grosen-Jorgensencredi ng parameters are α = 0.50, β = 0.15, hence the targetfunding ra o is 1 + β = 1.15, and the minimum guarantee is setDue to the path-dependency of the contract, we can also add addi onal ex-

planatory variables, such as Pt(St) or Nt(κt ), that capture the history of theprocesses St and κt .In principle, it is possible to es mate the VaR directly from the cross-sec onal

sample by using quan le regressions.

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at rG = 2%. The financial market process has parametersS0 = 100, r = 4%, σ = 15%. The parameters of the Lee-Cartermodel are es mated from Dutch mortality data, which areavailable from www.mortality.org.

We compare the prices for the pension contract using the threedifferent pricing operators:

• the best es mate price;

• EIOPA’s risk margin price;

• the Time Consistent and Market Consistent (TCMC) price.

The contract values under the three different pricing operators areshown in Figure . The best es mate prices are labeled as“Expected”. EIOPA’s risk margin prices as “EIOPA” and theTCMC-price as “Time Consistent”.

We see that the best es mate prices simply reflect the expectednumber of survivors in the Lee-Carter model, ranging frome

for payoff at age toe for payoff at age . Theuncertainty surrounding the projected number of survivors is notreflected in the best es mate price.

The risk margin prices do reflect the uncertainty surrounding theprojected number of survivors, hence the EIOPA prices are higherthan the best es mate prices, ranging frome for payoff atage toe for payoff at age . For longer-datedcontracts, the unhedgeable uncertainty becomes larger, andtherefore the gap with the best es mate prices becomes everlarger. At payoff age the EIOPA price is % higher than theEIOPA price.

The TCMC prices reflect the “capital-on-capital” effect, and aretherefore s ll somewhat higher than the EIOPA prices. We also see

Figure : Comparison of Prices for a Pension Contract.

that the risk margin prices do reflect a significant adjustment in theright direc on. Hence, the remaining gap to the TCMC price isrela vely small. However, for longer-dated contracts thecapital-on-capital effect becomes rela vely more important. Atpayoff age the TCMC price ise , which is % higher thanthe EIOPA price.

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. ConclusionsThe pricing of long-dated pension liabili es is an significantproblem. In order to reflect both market and non-markets risks inthe pricing operator in an arbitrage-free way, we need pricingoperators that are both me consistent and market consistent(TCMC). However, current pricing methods for pension liabili es donot properly reflect the non-market risks.

In this paper, we have demonstrated that it is computa onallyfeasible to price pension contracts in an incomplete market se ngwith TCMC pricing operators. Furthermore, we have compared theTCMC prices for life-insurance and pension contracts to alterna vepricing methods that are currently used for pricing pension andlife-insurance liabili es:

• the Best Es mate pricing method, which is used by mostpension funds;

• EIOPA’s Risk Margin pricing method, which is used for thepricing of life-insurance liabili es.

Our main findings can be summarised as follows:

• Best es mate prices simply reflect the expected value of theactuarial risk drivers. The uncertainty surrounding thisprojec on is not incorporated in the best es mate price.

• Risk margin prices do reflect the uncertainty surroundingthe projected actuarial risk drivers, hence EIOPA prices arehigher than best es mate prices. However, the uncertaintyin the projected capital requirements is not incorporated inthe risk margin price. This implies that risk margin prices arenot fully me consistent.

• TCMC prices do reflect the “capital-on-capital” effect, andare therefore fully me- and market consistent. Hence,TCMC prices are higher than risk margin prices.

• Risk margin prices do represent a significant adjustment inthe right direc on over the best es mate price, as theremaining gap to the TCMC price is rela vely small.However, for longer-dated contracts the capital-on-capitaleffect becomes rela vely more important.

• Using the Least-Squares Monte-Carlo method it iscomputa onally feasible to compute TCMC prices forrealis c pension and life-insurance contracts.

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ReferencesBlack, F. and Scholes, M. ( ). The pricing of op ons and

corporate liabili es. Journal of Poli cal Economy, : – .

Carriere, J. ( ). Valua on of the early-exercise price for op onsusing simula ons and nonparametric regression. Insurance:Mathema cs and Economics, ( ): – .

Grosen, A. and Jorgensen, P. L. ( ). Fair valua on of lifeinsurance liabili es: The impact of interest rate guarantees,surrender op ons, and bonus policies. Insurance: Mathema csand Economics, : – .

Jobert, A. and Rogers, L. ( ). Valua ons and dynamic convexrisk measures. Mathema cal Finance, ( ): – .

Lee, R. D. and Carter, L. R. ( ). Modeling and forecas ng U.S.mortality. Journal of the American Sta s cal Associa on,

( ): – .

Longstaff, F. A. and Schwartz, E. S. ( ). Valuing Americanop ons by simula on: A simple least-squares approach. TheReview of Financial Studies, ( ): – .

Pelsser, A. and Salahnejhad, A. ( ). Time-consistent actuarialvalua on. Insurance Mathema cs and Economics,

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Pelsser, A. and Stadje, M. ( ). Time-consistent andmarket-consistent evalua ons. Mathema cal Finance,

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1 Naar een nieuw pensioencontract (2011)

Lans Bovenberg en Casper van Ewijk2 Langlevenrisico in collectieve

pensioencontracten (2011) Anja De Waegenaere, Alexander

Paulis en Job Stigter3 Bouwstenen voor nieuwe pensi-

oencontracten en uitdagingen voor het toezicht daarop (2011) Theo Nijman en Lans Bovenberg

4 European supervision of pension funds: purpose, scope and design (2011) Niels Kortleve, Wilfried Mulder and Antoon Pelsser

5 Regulating pensions: Why the European Union matters (2011) Ton van den Brink, Hans van Meerten and Sybe de Vries

6 The design of European supervision of pension funds (2012) Dirk Broeders, Niels Kortleve, Antoon Pelsser and Jan-Willem Wijckmans

7 Hoe gevoelig is de uittredeleeftijd voor veranderingen in het pensi-oenstelsel? (2012) Didier Fouarge, Andries de Grip en Raymond Montizaan

8 De inkomensverdeling en levens-verwachting van ouderen (2012) Marike Knoef, Rob Alessie en Adriaan Kalwij

9 Marktconsistente waardering van zachte pensioenrechten (2012) Theo Nijman en Bas Werker

10 De RAM in het nieuwe pensioen-akkoord (2012) Frank de Jong en Peter Schotman

11 The longevity risk of the Dutch Actuarial Association’s projection model (2012) Frederik Peters, Wilma Nusselder and Johan Mackenbach

12 Het koppelen van pensioenleeftijd en pensioenaanspraken aan de levensverwachting (2012) Anja De Waegenaere, Bertrand Melenberg en Tim Boonen

13 Impliciete en expliciete leeftijds-differentiatie in pensioencontracten (2013)

Roel Mehlkopf, Jan Bonenkamp, Casper van Ewijk, Harry ter Rele en Ed Westerhout

14 Hoofdlijnen Pensioenakkoord, juridisch begrepen (2013) Mark Heemskerk, Bas de Jong en René Maatman

15 Different people, different choices: The influence of visual stimuli in communication on pension choice (2013) Elisabeth Brüggen, Ingrid Rohde and Mijke van den Broeke

16 Herverdeling door pensioenregelingen (2013) Jan Bonenkamp, Wilma Nusselder, Johan Mackenbach, Frederik Peters en Harry ter Rele

17 Guarantees and habit formation in pension schemes: A critical analysis of the floor-leverage rule (2013) Frank de Jong and Yang Zhou

overzicht uitgaven in de design paper serie

18 The holistic balance sheet as a building block in pension fund supervision (2013) Erwin Fransen, Niels Kortleve, Hans Schumacher, Hans Staring and Jan-Willem Wijckmans

19 Collective pension schemes and individual choice (2013) Jules van Binsbergen, Dirk Broeders, Myrthe de Jong and Ralph Koijen

20 Building a distribution builder: Design considerations for financial investment and pension decisions (2013) Bas Donkers, Carlos Lourenço, Daniel Goldstein and Benedict Dellaert

21 Escalerende garantietoezeggingen: een alternatief voor het StAr RAM-contract (2013)

Servaas van Bilsen, Roger Laeven en Theo Nijman

22 A reporting standard for defined contribution pension plans (2013) Kees de Vaan, Daniele Fano, Herialt Mens and Giovanna Nicodano

23 Op naar actieve pensioen consu-men ten: Inhoudelijke kenmerken en randvoorwaarden van effectieve pensioencommunicatie (2013) Niels Kortleve, Guido Verbaal en Charlotte Kuiper

24 Naar een nieuw deelnemergericht UPO (2013)

Charlotte Kuiper, Arthur van Soest en Cees Dert

25 Measuring retirement savings adequacy; developing a multi-pillar approach in the Netherlands (2013) Marike Knoef, Jim Been, Rob Alessie, Koen Caminada, Kees Goudswaard, and Adriaan Kalwij

26 Illiquiditeit voor pensioenfondsen en verzekeraars: Rendement versus risico (2014) Joost Driessen

27 De doorsneesystematiek in aanvullende pensioenregelingen: effecten, alternatieven en transitie-paden (2014) Jan Bonenkamp, Ryanne Cox en Marcel Lever

28 EIOPA: bevoegdheden en rechts-bescherming (2014) Ivor Witte

29 Een institutionele beleggersblik op de Nederlandse woningmarkt (2013) Dirk Brounen en Ronald Mahieu

30 Verzekeraar en het reële pensioencontract (2014) Jolanda van den Brink, Erik Lutjens en Ivor Witte

31 Pensioen, consumptiebehoeften en ouderenzorg (2014) Marike Knoef, Arjen Hussem, Arjan Soede en Jochem de Bresser

32 Habit formation: implications for pension plans (2014) Frank de Jong and Yang Zhou

33 Het Algemeen pensioenfonds en de taakafbakening (2014) Ivor Witte

34 Intergenerational Risk Trading (2014) Jiajia Cui and Eduard Ponds

35 Beëindiging van de doorsnee-systematiek: juridisch navigeren naar alternatieven (2015) Dick Boeijen, Mark Heemskerk en René Maatman

36 Purchasing an annuity: now or later? The role of interest rates (2015) Thijs Markwat, Roderick Molenaar and Juan Carlos Rodriguez

37 Entrepreneurs without wealth? An overview of their portfolio using different data sources for the Netherlands (2015) Mauro Mastrogiacomo, Yue Li and Rik Dillingh

38 The psychology and economics of reverse mortgage attitudes. Evidence from the Netherlands (2015)

Rik Dillingh, Henriëtte Prast, Mariacristina Rossi and Cesira Urzì Brancati

39 Keuzevrijheid in de uittreedleeftijd (2015) Arthur van Soest

40 Afschaffing doorsneesystematiek: verkenning van varianten (2015) Jan Bonenkamp en Marcel Lever

41 Nederlandse pensioenopbouw in internationaal perspectief (2015) Marike Knoef, Kees Goudswaard, Jim Been en Koen Caminada

42 Intergenerationele risicodeling in collectieve en individuele pensioencontracten (2015) Jan Bonenkamp, Peter Broer en Ed Westerhout

43 Inflation Experiences of Retirees (2015) Adriaan Kalwij, Rob Alessie, Jonathan Gardner and Ashik Anwar Ali

44 Financial fairness and conditional indexation (2015) Torsten Kleinow and Hans Schumacher

45 Lessons from the Swedish occupational pension system (2015) Lans Bovenberg, Ryanne Cox and Stefan Lundbergh

46 Heldere en harde pensioenrechten onder een PPR (2016) Mark Heemskerk, René Maatman en Bas Werker

47 Segmentation of pension plan participants: Identifying dimensions of heterogeneity (2016) Wiebke Eberhardt, Elisabeth Brüggen, Thomas Post and Chantal Hoet

48 How do people spend their time before and after retirement? (2016) Johannes Binswanger

49 Naar een nieuwe aanpak voor risicoprofielmeting voor deelnemers in pensioenregelingen (2016) Benedict Dellaert, Bas Donkers, Marc Turlings, Tom Steenkamp en Ed Vermeulen

50 Individueel defined contribution in de uitkeringsfase (2016) Tom Steenkamp

51 Wat vinden en verwachten Neder-landers van het pensioen? (2016) Arthur van Soest

52 Do life expectancy projections need to account for the impact of smoking? (2016) Frederik Peters, Johan Mackenbach en Wilma Nusselder

53 Effecten van gelaagdheid in pensioen documenten: een gebruikersstudie (2016) Louise Nell, Leo Lentz en Henk Pander Maat

54 Term Structures with Converging Forward Rates (2016) Michel Vellekoop and Jan de Kort

55 Participation and choice in funded pension plans (2016) Manuel García-Huitrón and Eduard Ponds

56 Interest rate models for pension and insurance regulation (2016) Dirk Broeders, Frank de Jong and Peter Schotman

57 An evaluation of the nFTK (2016) Lei Shu, Bertrand Melenberg and Hans Schumacher

58 Pensioenen en inkomens ongelijk-heid onder ouderen in Europa (2016) Koen Caminada, Kees Goudswaard, Jim Been en Marike Knoef

59 Towards a practical and scientifi-cally sound tool for measuring time and risk preferences in pension savings decisions (2016) Jan Potters, Arno Riedl and Paul Smeets

60 Save more or retire later? Retire ment planning heterogeneity and perceptions of savings adequacy and income constraints (2016) Ron van Schie, Benedict Dellaert and Bas Donkers

61 Uitstroom van oudere werknemers bij overheid en onderwijs. Selectie uit de poort (2016) Frank Cörvers en Janneke Wilschut

62 Pension risk preferences. A personalized elicitation method and its impact on asset allocation (2016) Gosse Alserda, Benedict Dellaert, Laurens Swinkels and Fieke van der Lecq

63 Market-consistent valuation of pension liabilities (2016) Antoon Pelsser, Ahmad Salahnejhad and Ramon van den Akker

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This is a publication of:

Netspar

P.O. Box 90153

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E-mail [email protected]

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October 2016

Market-consistent valuation of pension liabilities

Due to the long maturity of its contracts, a pension fund or life-insurance

company is exposed to actuarial risks such as longevity risk and also to

market risks such as interest rate risk and inflation risk. The insurance and

pensions regulator in Europe (EIOPA) has also recognized the importance

of valuation methods that take financial risks and non-financial risks into

account. In this paper, Antoon Pelsser, Ahmad Salahnejhad (both UM) and

Ramon van den Akker (SNS/TiU) want to show that it is computationally

feasible to price pensions contracts in an incomplete market setting with

time-consistent and market-consistent (TCMC) pricing operators.

Market-consistent valuation

of pension liabilities

Antoon PelsserAhmad SalahnejhadRamon van den Akker