optimal design of the attribution of pension fund performance to employees

©

DOI: 10.1111/j.1539-6975.2013.01516.x

431

OPTIMAL DESIGN OF THE ATTRIBUTION OF PENSIONFUND PERFORMANCE TO EMPLOYEESHeinz MullerDavid Schiess

ABSTRACT

The article analyzes risk sharing in a defined contribution pension fund incontinuous time. According to a prespecified attribution scheme, the interestrate paid on the employees’ accounts is a linear function of the fund’s invest-ment performance. For each attribution scheme, the pension fund maximizesthe expected utility and the employees derive utility from their savings ac-counts. It turns out that all Pareto-optimal attribution schemes are character-ized by the same optimal participation rate. We derive the total welfare gainthat installs from replacing no participation with optimal participation. Thiswelfare gain can be quantified and is substantial for reasonable parametervalues.

INTRODUCTION

In pension finance one must distinguish between defined benefit and defined con-tribution plans. In a defined benefit plan, benefits are defined in advance and theplan sponsor and the employees must adjust their contributions. In the literature,optimal investment strategies for defined benefit plans are treated by Haberman andSung (1994), Boulier, Trussant, and Florens (1995), Cairns (2000), and Josa-Fombellidaand Rincon-Zapatero (2006) among others. Furthermore, the articles by Wilkie (1985),Sharpe and Tint (1990), Keel and Muller (1995), and Leippold, Trojani, and Vanini(2004) deal with portfolio optimization in the asset–liability context. Since the plansponsor has to bear substantial financial and actuarial risk and since the valuationof accrued retirement benefits of employees changing their plan is difficult, definedbenefit plans have become less and less popular. In a defined contribution plan inthe narrow sense, the accrued retirement benefits of employees are invested and thepositive or negative fund return is fully attributed to the employees’ accounts. Atretirement each employee obtains the final amount from his account either as a cashpayment or a corresponding pension. However, there is a wide span of different

Heinz Muller and David Schiess are at Group for Mathematics and Statistics, University ofSt. Gallen, Bodanstrasse 6, 9000 St. Gallen, Switzerland. The authors can be contacted viae-mail: [email protected] and [email protected], respectively. We would like tothank Dr. Roger Baumann for his valuable input and the two anonymous referees for their veryconstructive advice. We also gratefully acknowledge financial support from the University ofSt. Gallen’s research program “Wealth and Risk.”

The Journal of Risk and Insurance, 2013, Vol. 81, No. 2, 431–468

432 THE JOURNAL OF RISK AND INSURANCE

defined contribution plans. On the one hand, there exist pure defined contributionplans like the U.S. 401(k) plans where the employee is free to choose an individ-ual investment policy. The pension fund consequently bears no financial risk at alland acts only as a broker between employees and financial and life insurance mar-kets. Hence, employees may be considered as individual investors. Authors dealingwith this topic in continuous time are Merton (1969), Adler and Dumas (1983), andGao (2008, 2009); in discrete time there are articles by Solnik (1978) and Habermanand Vigna (2002) among others. On the other hand, there exist defined contributionpension plans with certain guaranteed benefits. The plan sponsor bears substantialfinancial risk in order to protect employees against fluctuations of financial markets.For example, in the mandatory part of Swiss pension funds, the interest paid on theemployees’ accounts may depend on the financial situation and the recent perfor-mance of the fund, but it has a lower bound fixed by central authorities. As thisminimum interest rate is typically above the riskless interest rate we will model therate attributed to the employees’ accounts as the riskless interest rate plus a premium.However, on top of this, the majority of the Swiss pension funds let their employeesparticipate in the fund’s investment performance in some way. The overmandatorypart allows them to go below the minimum rate in case of bad investment perfor-mances. Inspired by these facts, we will assume that the total rate attributed to theemployees’ accounts consists of a premium in addition to the riskless interest rateand participation in the fund’s investment performance. At retirement, the accountsare transformed into benefit streams. Of course, in such a framework the value of theassets may differ from the present value of the liabilities. In the literature, Boulier,Huang, and Taillard (2001) and Deelstra, Graselli, and Koehl (2003, 2004) analyzea defined contribution plan providing a guarantee. This article concentrates on theoptimal design of the attribution of the pension fund performance to its employ-ees. If a pension plan increases the funding ratio (ratio of the value of assets to thepresent value of net obligations) during prospering financial markets and decreasesit in declining financial markets, then a risk transfer between different generationsof employees can be established. This idea was formalized by Baumann and Muller(2008). In their model they introduced an intertemporal risk transfer by assumingthat the rate attributed to the employees’ accounts depends on the funding ratio butnot directly on the fund’s recent investment performance. In this way the attributionof investment performance is flattened over time. They showed that in such a frame-work the pension fund can choose investment policies such that all employees wouldbe worse off if they acted as individual investors. Furthermore, there are articlesderiving optimal investment strategies for defined contribution plans that put thefunding ratio into the focus of the optimization like Browne (1997), Denzler, Muller,and Scherer (2001), and Muller and Baumann (2006). In contrast to these articles,the liabilities in the papers by Muller and Baumann (2008) and Baumann and Muller(2008) are not exogeneous. Baumann and Muller (2008) model the return attributed tothe employees’ accrued retirement benefits as a function of the funding ratio. In thisarticle, we analyze the dependency on the fund’s investment performance. A similaridea has already been pursued in the work of Baumann (2005). In this article, however,we get closed-form solutions by restricting the fund’s performance attribution to lin-ear schemes. Substantial welfare gains result even for this special class of attributionschemes.

OPTIMAL DESIGN OF THE PENSION FUND’S PERFORMANCE ATTRIBUTION 433

This article deals with financial risk sharing between the defined contribution plansponsor and the employees in continuous time. We assume that the rate attributedto the employees’ accounts consists of a premium in addition to the riskless interestrate and a linear participation in the fund’s investment performance. The attributionscheme is part of the institutional setup and can be characterized by the premium andthe participation rate. One of the main purposes of the article is to find Pareto-optimalattribution schemes. Given an attribution scheme, the pension fund maximizes itsexpected utility by choosing a corresponding investment policy. Hence, for eachattribution scheme a corresponding expected utility is obtained. This defines theindirect utility function of the pension fund on the set of attribution schemes. Given theinvestment policies of the pension fund, one may calculate the expected utility of theemployees for each attribution scheme. In this way, the indirect utility function of theemployees can be derived as well. The set of Pareto-optimal attribution schemesare calculated and it is shown that all Pareto-optima are characterized by the sameparticipation rate. Since the optimal participation rate is constant, there is a conflictof interests about the premium only. The specific choice of the premium depends onthe interaction of employees, the firm or institution attached to the pension fund,and the regulator. On the one hand, the premium should attain a level such that anemployee is better off as a member of the fund than as an individual investor usingthe Merton solution. On the other hand, the premium should be low enough to ensurefinancial stability of the pension fund. We show that such levels for the premium canbe attained for reasonable parameter values. Furthermore, a formula for the optimalparticipation rate is derived. Afterward, we define the welfare of the pension fundand the employees, respectively, by making use of appropriate certainty equivalents.Finally, a comparison of optimal participation and no participation shows that thereis a substantial welfare gain for reasonable parameter values.

The article is organized as follows. The second section presents the model setup bydefining the financial market as well as the objective function of both the pensionfund and the employees. Furthermore, linear attribution schemes are introduced andthe evolution of assets and liabilities are developed, which taken together yield thedynamics of the funding ratio. In the third section, we maximize the expected utilityof the pension fund for a given attribution scheme, which leads to a constant optimalinvestment strategy. We discuss the optimal investment rule and close the sectionwith the derivation of the optimally controlled funding ratio dynamics. In the fourthsection, we derive the indifference curves of the pension fund and the employees,respectively, which leads to a discussion of risk-adjusted rates of return. The fifthsection then presents the result that there is a unique Pareto-optimal participationrate α∗ ∈ (0, 1). The Pareto-optimal investment strategy is derived by exploiting thisparticipation rate. The section closes with a discussion of the parameter sensitivitiesof optimal participation and investment. The sixth section then narrows the set ofPareto-efficient attribution schemes by imposing individual rationality of the em-ployees and financial stability of the pension fund. The seventh section first definesthe welfare of the pension fund and the employees by making use of appropriatecertainty equivalents. Following this, we derive the welfare gain that results if noparticipation is replaced with optimal participation. It is shown that this welfare gainis substantial for reasonable parameter values. Finally, the eighth section summarizesand presents the main conclusions.


MODEL

Pension funds are typically allocated to firms or public institutions. Contributionsto pension funds represent a component of employees’ labor income. Hence, theultimate objective of a pension fund is the maximization of the employees’ welfare.However, there are different generations of employees with partly concurrent andpartly conflicting interests. Among others, Baumann and Muller (2008), Gollier (2008),and Cui, de Jong, and Ponds (2011) show that intergenerational risk transfer maylead to a Pareto improvement in this situation. In this sense the pension fund shouldrepresent the interests of different generations of employees. Moreover, risk sharingbetween the employees and the pension fund should take place. This obviouslyexcludes individual investment strategies of the employees. In our model this risksharing is accomplished by attributing to the employees’ accounts a rate of return thatdepends on the pension fund’s investment performance. Given the objective functionsof the pension fund and of the employees a set of Pareto-optimal attribution schemesresults.

The aim of this article is to analyze the Pareto-optimal set of attribution schemesand quantify the welfare gain that results from risk sharing. The choice of a partic-ular Pareto-optimal attribution scheme is considered to be part of the institutionalframework and is not dealt with in this article.

The objective function of the employees is the expected utility of the employee accountat retirement. Formulating an appropriate objective function of the pension fund is lessstraightforward. The value of the assets and the value of the liabilities are stochasticprocesses depending directly or indirectly through the attribution scheme on theinvestment policy of the fund. Therefore, either the surplus (value of assets minusliabilities) or the funding ratio (ratio of assets to liabilities) is the relevant key variablefor the pension fund. Practitioners and regulators (particularly in the Netherlandsand in Switzerland) use the funding ratio since it represents a measure of financialhealth that does not depend on the size of the pension fund (see also De Jong, 2008,p. 10). As in van Binsbergen and Brandt (2007, pp. 9-10) and Martellini (2006, p. 11),we shall also use the expected utility of the funding ratio as the objective function ofthe pension fund.

In the sense of an overlapping generation model at each point in time t ∈ R+ em-

ployees enter the plan. In order to simplify the analysis it is assumed that employeesentering at t immediately invest a fixed amount Xt,0. Of course, this assumption isnot satisfied in reality; it is clearly an assumption to keep the model tractable. Infact, paying contributions as a lump sum rather than spreading them over a wholeworking life has an impact on the exposure to financial risk. The pension fund at-tributes an interest rate on the employees’ accounts and after a time span τ at timet + τ employees leave the plan with their accrued retirement benefits Xt,τ .

Pension FundThe pension fund has a riskless investment opportunity i = 0 and risky opportunitiesi = 1, . . . , N, whose price processes are given by geometric Brownian motions, thatis,

dS0,t

S0,t= rdt


and

dSi ,t

Si ,t= (r + πi )dt +

N∑j=1

σijdZ j ,t , i = 1, . . . , N,

where r denotes the riskless interest rate, π ∈ RN is the expected excess returns, Zt is

an N-dimensional standard Brownian motion, and σ is a regular matrix determininghow the risky assets are driven by the Brownian motions.

At time t, the financial situation of the pension fund is given by

At = value of assets at time t

Lt = value of liabilities at time t

Ft = At

Ltfunding ratio at time t.

The pension fund chooses a portfolio xt ∈ RN in order to invest At . Hence, the dy-

namics of the value of assets are given by1

dAt

At= (

r + xTt π

)dt + xT

t σdZt. (1)

On the liability side, the fund attributes a return

dRt = (r + a + αxT

t π)dt + αxT

t σdZt

with 0 ≤ α < 1 to the accrued retirement benefits of the employees. The linear attribu-tion scheme (α, a ) ∈ D = [0, 1) × [0, ∞) is characterized by the participation rate α andthe premium a . The choice of the attribution scheme is part of the institutional setup.On the one hand, an attribution scheme should guarantee the financial stability of thepension fund. On the other hand, employees should be better off than under a stand-alone Merton solution for individual investment. These issues will be addressed later.In the following we work with a prespecified attribution scheme (α, a ).

Hence, the dynamics of the liabilities are given by

dLt

Lt= dRt

or

dLt

Lt= (

r + a + αxTt π

)dt + αxT

t σdZt. (2)

1In principle, net contributions Ct to the fund should be considered as well. This would lead to

dAt

At= (r + xT

t π )dt + Ctdt + xTt σdZt.

However, we simplify the analysis by assuming Ct = 0.


Applying Ito’s lemma to the funding ratio leads to

dFt

Ft= dAt

At− dLt

Lt+

(dLt

Lt

)2− dAt

At

dLt

Lt. (3)

Plugging (1) and (2) into (3) we get

dFt

Ft= [

(1 − α)xTt π − a + (α2 − α)xT

t Vxt]dt + (1 − α)xT

t σdZt (4)

with V = σσ T .

The pension fund maximizes the expected utility of the funding ratio FT at the plan-ning horizon T by choosing the investment strategy. Risk sharing between employeesand the pension fund obviously excludes individual investment strategies of the em-ployees. The relative risk aversion of the fund2 is Rp > 1. Hence, for a prespecifiedattribution scheme (α, a ) ∈ D, the objective function of the fund is given by

Up(FT ) = E[F

1−RpT

]1 − Rp

. (5)

EmployeesOn the employees’ accounts Xt,s at time t + s the pension fund attributes a return

dRt+s = (r + a + αxT

t+sπ)ds + αxT

t+sσdZt+s .

Hence, the dynamics of the employees’ accounts are given by

dXt,s

Xt,s= (

r + a + αxTt+sπ

)ds + αxT

t+sσdZt+s . (6)

All employees have the same constant relative risk aversion3 Re > 1 with respectto the accrued retirement benefits Xt,τ at the end of their working life. Hence, theobjective function of the employees is given by

Ue (Xt,τ ) = E[X1−Re

t,τ]

1 − Re. (7)

OPTIMAL PORTFOLIO AND OPTIMALLY CONTROLLED FUNDING RATIO

The optimal investment policy of the pension fund can be derived with stochasticcontrol.

2According to the results of Gollier (2001) it is realistic to assume Rp > 1.3According to the results of Gollier (2001) it is realistic to assume Re > 1.


Proposition 1: Maximizing the utility of the pension fund defined in (5)

Up(FT ) = E[F

1−RpT

]1 − Rp

subject to the funding ratio evolution given in (4)

dFt

Ft= [

(1 − α)xTt π − a + (α2 − α)xT


t σdZt

leads to the optimal investment policy

xt = x∗= 12α+Rp(1 − α)

V−1π , 0 ≤ t ≤ T. (8)

Proof: See the Appendix. Q.E.D.

Remark 1:

(1) The optimal investment policy does not depend on the planning horizon T andis constant over time. Hence, the funding ratio follows a geometric Brownianmotion.

(2) The optimal investment strategy is proportional to the Merton investment rulexM = 1

RpV−1π . Moreover, we naturally obtain the Merton rule in the special case

of α = 0. Introducing risk sharing (assuming α > 0), the pension fund investsmore aggressively than the Merton investor if and only if

2α+Rp(1 − α) < Rp

⇔

Rp > 2. (9)

This result looks somewhat surprising. In fact, there are two opposite effects.To see this we must distinguish between the drift and the diffusion part of thefunding ratio. For the diffusion part, risk sharing is highly beneficial and woulditself lead to a more aggressive investment strategy. For the drift part, however,the positive correlation between assets and liabilities has an additional negativeimpact, resulting in a less aggressive investment strategy. Which effect dominatesdepends on the fund’s risk aversion Rp.

1. The optimal investment rule naturally depends on the variance–covariance matrixV, the expected excess returns π , the relative risk aversion of the pension fund Rp,and, finally, on the participation rate α. V−1π is the well-known growth-optimumportfolio that maximizes the median. From the denominator 2α + Rp(1 − α) it canbe seen that participation draws the effective risk aversion toward 2, a rather low


level in the pension fund context. Since the premium a of the attribution schemeis a prespecified constant it has no influence on optimal investment.

2. Finally, we give the sensitivity of the optimal investment rule with respect to theparticipation rate

∂x∗

∂α= Rp − 2

[2α+Rp(1 − α)]2 V−1π. (10)

Obviously, the pension fund invests more aggressively the higher the participationrate if Rp > 2. The interpretation of this result would follow the same lines as inRemarks 2 and 3.

Plugging the optimal investment rule (8) into the general funding ratio dynamics (4)we obtain the evolution of the optimally controlled funding ratio

dFt

Ft= μpdt + σpd Z∗

t (11)

with

μp = (1 − α)[

12α + Rp(1 − α)

− α

[2α + Rp(1 − α)]2

]πT V−1π − a

= (1 − α)α+Rp(1 − α)[

2α+Rp(1 − α)]2 πT V−1π − a

(12)

and

σp = 1 − α

2α+Rp(1 − α)(πT V−1π )0.5. (13)

The denominator in Equation (13) reflects the fact that the investment policy be-comes more aggressive with increasing α for Rp > 2. Formula (12) is more difficult tointerpret and will be dealt with in the fourth section.

PREFERENCES ON THE SET OF ATTRIBUTION SCHEMES

Indifference Curves of the Pension FundExploiting (11) we obtain

d ln Ft =(

μp − σ 2p

2

)dt + σpd Z∗

t .

Integration then yields

ln FT = ln F0 +(

μp − σ 2p

2

)T + σp Z∗

T .


Inserting this result into the objective function of the pension fund given in (5) leadsto

Up(FT ) = (1 − Rp)−1 E[F

1−RpT

] = (1 − Rp)−1 E[e(1−Rp) ln FT

]= (1 − Rp)−1 exp{(1 − Rp)E[ln FT ] + 0.5(1 − Rp)2var[ln FT ]}

= (1 − Rp)−1 exp

{(1 − Rp)

[ln F0 +

(μp − σ 2

p

2

)T + 0.5(1 − Rp)σ 2

pT

]}

= (1 − Rp)−1 exp{

(1 − Rp) ln F0 + (1 − Rp)T[μp − Rp

2σ 2

p

]}.

Since only μp and σp depend on the attribution scheme (α, a ) ∈ D, the preferences ofthe pension fund can be represented by the indirect utility function

Vp(α, a ) = μp − Rp

2σ 2

p ,

which can be interpreted as the risk-adjusted drift term of the funding ratio. Plugging(12) and (13) into the above we obtain

Vp(α, a ) (14)

= (1 − α)α+Rp(1 − α)

[2α+Rp(1 − α)]2 πT V−1π − a − Rp

2(1 − α)2

[2α+Rp(1 − α)]2 πT V−1π

= (1 − α)α+0.5Rp(1 − α)[2α+Rp(1 − α)]2 πT V−1π − a

= 1 − α

2[2α+Rp(1 − α)]πT V−1π − a .

(15)

Thus, the indifference curves of the pension fund Vp(α, a ) = C p on the set of attribu-tion schemes D are given by

a = f p(α) = 0.51 − α

2α+Rp(1 − α)πT V−1π−C p, C p ∈ R. (16)

Obviously, the indifference curves of the pension fund are downward sloping4 anddiffer from each other by vertical parallel shifts (choice of the constant C p). As arguedin the third section for Rp > 2 a higher participation rate α leads to a more aggressiveinvestment strategy. However, for any Rp > 1, the exposure to financial risk of the

4Formally, this follows from f ′p(α) < 0 (see 36 in the Appendix). Intuitively, one can argue as

follows. Since the positive correlation between assets and liabilities leads to the negative term(α2 − α)xT

t Vxt in Formula (4), simply leveraging away the participation of the employees witha more aggressive investment policy would lead to a lower drift term of the funding ratio.


FIGURE 1Indifference Curves of the Pension Fund for Rp = 3 and π T V−1π = 4%

Note: The dashed curve represents Cp = μp − Rp2 σ 2

p = −0.1%, whereas the solid curve corre-sponds to Cp = μp − Rp

2 σ 2p = 0.0%.

pension fund decreases with a higher participation rate α. Therefore, it correspondsto intuition that the premium a , which is offered by the pension fund, decreases aswell with a higher α. Figure 1 shows the indifference curves on the set of attributionschemes D for Rp = 3 and πT V−1π = 4%.5

Indifference Curves of the EmployeesInserting the optimal investment policy (8) into the dynamics of the employees’accounts (6) leads to

dXt,s

Xt,s= μeds + σed Z∗

t+s , (17)

with

μe = r + a + α

2α+Rp(1 − α)πT V−1π (18)

5Note that πT V−1π can be interpreted as the expected excess return of the growth optimalportfolio xopt = V−1π . Thus, assuming πT V−1π = 4% seems quite realistic.


and

σe = α

2α+Rp(1 − α)(πT V−1π )0.5. (19)

Similar to the “Indifference Curves of the Pension Fund” section, it is possible toderive the indirect utility function of the employees as the risk-adjusted return on theemployees’ accounts

Ve (α, a ) (20)

= μe − Re

2σ 2

e

= r + a + α

2α+Rp(1 − α)πT V−1π − Re

2α2

[2α+Rp(1 − α)]2 πT V−1π

= r + a + α2α+Rp(1 − α) − 0.5Reα

[2α+Rp(1 − α)]2 πT V−1π.

(21)

Thus, the indifference curves of the employees Ve (α, a ) = Ce on the set of attributionschemes D are given by

a = fe (α) = −α2α+Rp(1 − α) − 0.5Reα[

2α+Rp(1 − α)]2 πT V−1π − r+Ce , Ce ∈ R. (22)

Again, the indifference curves differ from each other only by vertical shifts (choice ofCe ). Figure 2 shows the indifference curves of the employees on the set of attributionschemes D for Rp = Re = 3 , r = 2% and πT V−1π = 4%.

Remark 4:

(1) With an increasing participation rate α the employee’s exposure to financial riskincreases. Since an employee with a relative risk aversion Rp = 3 is willing to takesome risk for low values of α this is beneficial for him and his required premium a ,therefore, decreases with α. However, as α approaches 1 his exposure to financialrisk tends to the Merton solution for R = 2. Since he is not willing to take asmuch risk the required premium a increases for high values of α. This explainsthe U-shape of the indifference curves in Figure 2.

(2) The indifference curves attain the minimum at a participation rate α∗ which leadsto the same exposure to financial risk of employees as in the stand-alone Mertoncase. In the Appendix, it is shown that

α∗ = Rp

Re + Rp − 2and


FIGURE 2Indifference Curves of the Employees for Rp = Re = 3, r = 2% and π T V−1π = 4%

Note: The dashed curve represents Ce = μe − Re2 σ 2

e = 2.7%, whereas the solid curve correspondsto Ce = μe − Re

2 σ 2e = 2.8%.

α∗x∗ = 1Re

V−1π

holds.

PARETO-EFFICIENCY AND PARETO-OPTIMAL PARTICIPATION RATE

As has already been pointed out, the attribution scheme (α, a ) ∈ D is assumed tobe predetermined (negotiated by the employees, the firm or institution behindthe pension fund, and possibly the regulator). This section restricts this choice toPareto-efficient attribution schemes Dp. It emerges that all Pareto-efficient attributionschemes are characterized by the same participation rate α∗. The sixth section ad-ditionally imposes individual rationality on the Pareto-efficient attribution schemes(α∗, a ) ∈ Dp. On the one hand, this means that employees must not be worse off thanunder individual investment according to the Merton solution. On the other hand,financial stability of the pension fund must be guaranteed. This leads to the set ofPareto-efficient and individually rational attribution schemes, which is denoted byDp,i = {(α∗, a )|amin ≤ a ≤ amax}.The set of Pareto-efficient attribution schemes Dp is calculated in this section.Due to the special properties (vertical parallel shifts) of the indifference curves,


Pareto-optimal attribution schemes (α∗, a∗) ∈ D are characterized by

α∗ = arg maxα∈[0,1)

[ f p(α) − fe (α)].

Remark 5:

(1) f p(α) denotes the premium the pension fund is willing to pay under the partici-pation rate α, whereas fe (α) denotes the premium required by the employees fora given α.

(2) The Pareto-optimal set is obtained by different choices of the premium6 a∗ =f p(α∗) = fe (α∗); that is, the Pareto frontier is a vertical line in the α–a plane.

(3) α∗ will be called the optimal participation rate in the remainder.

(4) Evaluating f ′p(α) and f ′

e (α) given in the Appendix in (A2) and (A3), respectively,we obtain the following in the no participation case:

f ′p(0) = − 1

R2pπT V−1π > f ′

e (0) = − 1Rp

πT V−1π. (23)

(5) Figure 3 shows different indifference curves of the pension fund and the em-ployees on the set of attribution schemes D for Rp = Re = 3, r = 2% andπT V−1π = 4%.

Assumption 1:

Rp(2 − Re ) < 2

Remark 6: By imposing Assumption 1 the possibility that employees with a ratherlow relative risk aversion Re belong to a pension plan with a high relative risk aversionRp is excluded. Figure 4 shows the pairs (Re , Rp) of the relative risk aversion of theemployees and the pension fund, respectively, where Assumption 1 holds. Obviously,imposing Assumption 1 is not really restrictive.

Proposition 2: Let Assumption 1 be given. The optimal participation rate α∗ then satisfies0 < α∗ < 1 and is given by

α∗ = Rp(Rp − 1)(Re+Rp − 3)Rp + 2

(24)

while the Pareto-optimal investment strategy is given by

xPa∗ = (Re+Rp − 3)Rp + 2R2

p ReV−1π. (25)

6The constants Cp and Ce in (16) and (22) must be chosen accordingly.


FIGURE 3Indifference Curves of the Pension Fund (Solid Curves) and the Employees (DashedCurves) for Rp = Re = 3, r = 2% and π T V−1π = 4%

FIGURE 4The Shaded Area Displays the Pairs of the Relative Risk Aversion Levels (Re, Rp) ThatSatisfy Assumption 1



Remark 7:

(1) The restriction to linear attribution schemes leads to an optimal value of theparticipation rate α∗, which exclusively depends on the relative risk aversionlevels of both the pension fund and the employees.

(2) The Pareto-optimal portfolio (25) is more aggressive than the Merton portfolio ifand only if

(Re+Rp − 3)Rp + 2R2

p Re>

1Rp

⇔ (Rp − 3)Rp + 2>0

⇔ (Rp − 2)(Rp − 1)>0

⇔ Rp > 2.

Unsurprisingly, the question of whether the optimal investment strategy is moreaggressive than the Merton portfolio does not depend on the relative risk aversionof the employees. This stems from our model setup, in which the pension fundchooses the investment rule for an externally given attribution scheme (α, a ).Moreover, the comparison to the Merton investment rule naturally depends onthe optimal participation rate α∗ (indirect effect via Rp) but it does not depend onthe premium a . This is of course a special case of the result in Remark 2.

The following proposition deals with the parameter sensitivity of the optimal partic-ipation rate and the Pareto-optimal investment strategy.

Proposition 3: Let Assumption be given.

The following is then obtained for the parameter sensitivities of the optimal participationrate:

∂α∗

∂ Rp> 0 ⇔ Re > 2

(Rp − 1

Rp

)2.

Specifically, this implies

Re ≥ 2 ⇒ ∂α∗

∂ Rp> 0.


∂α∗

∂ Re= − R2

p(Rp − 1)

[(Re+Rp − 3)Rp + 2]2 < 0.

The following is then obtained for the parameter sensitivities of the Pareto-optimal invest-ment strategy:

∂xPa∗

∂ Rp=− (Re − 3)Rp + 4

R3p Re

V−1π.

Hence, increasing the relative risk aversion of the pension fund leads to a less aggres-sive Pareto-optimal investment strategy if and only if

Re > 3 − 4Rp

.

In particular, increasing the relative risk aversion of the pension fund leads to a lessaggressive Pareto-optimal investment strategy if Rp ≤ 2.

∂xPa∗

∂ Re=− (Rp − 2)(Rp − 1)

R2p R2

eV−1π.

Hence, increasing the relative risk aversion of the employees leads to a less aggressivePareto-optimal investment strategy if and only if Rp > 2.


Remark 8:

(1) According to (1) in part (a) of Proposition 3, we naturally get a higher optimalparticipation rate for a higher relative risk aversion of the pension fund in mostparameter settings. These parameter settings are displayed by the shaded area inFigure 5.

(2) According to (2) in part (a) of Proposition 3, as expected, we always get a loweroptimal participation rate for a higher relative risk aversion of the employees.

(3) According to (1) in part (b) of Proposition 3, we obtain the intuitive result thatincreasing the relative risk aversion of the pension fund leads to a less aggressivePareto-optimal investment strategy in most parameter settings. These parametersettings are displayed by the shaded area in Figure 6.


FIGURE 5The Shaded Area Displays the Pairs of the Relative Risk Aversion Levels (R p, Re), forWhich an Increase in the Relative Risk Aversion of the Pension Fund Naturally Leads toa Higher Optimal Participation Rate

INDIVIDUAL RATIONALITY AND FINANCIAL STABILITY OF THE PENSION FUND

An attribution scheme should not only be Pareto-efficient but also individually ratio-nal.7 On the one hand, employees should be at least as well off as under individualinvestment with the Merton solution. On the other hand, financial stability of the pen-sion fund should be guaranteed. In our framework this is established by imposing anincreasing median of the funding ratio.

Comparison With the Merton Solution

For an individual investment according to the Merton rule xM = 1Re

V−1π the risk-adjusted return is given by

Ve ,M = r + xTMπ − Re

2xT

MVxM = r + 12Re

πT V−1π.

Therefore, individual rationality of the employees requires

Ve (α∗, a ) ≥ Ve ,M

7For this section we were inspired by the comments of an anonymous referee.


FIGURE 6The Shaded Area Displays the Pairs of the Relative Risk Aversion Levels (R p, Re), forWhich an Increase in the Relative Risk Aversion of the Pension Fund Naturally Leads toa Less Aggressive Pareto-Optimal Investment Strategy

or according to (20) and (21)

a + 2α∗2 + Rpα∗(1 − α∗) − 0.5Reα∗2

[2α∗ + Rp(1 − α∗)]2 πT V−1π ≥ 12Re

πT V−1π.

This leads to

a ≥ amin = [2α∗ + Rp(1 − α∗)]2 − 4Reα∗2 − 2Re Rpα∗(1 − α∗) + R2

e α∗2

2Re [Rp − (Rp − 2)α∗]2 πT V−1π

= (Re − 2)2α∗2 − 2Rp(Re − 2)α∗(1 − α∗) + R2p(1 − α∗)2

2Re [Rp − (Rp − 2)α∗]2 πT V−1π

= [Rp − (Rp + Re − 2)α∗]2

2Re [Rp − (Rp − 2)α∗]2 πT V−1π > 0,

(26)

which provides a lower bound for the risk premium a .


Financial Stability of the Pension FundFrom

d ln Ft =(

μp − σ 2p

2

)dt + σpd Z∗

t

in the fourth section, we get

d ln[med(Ft)] =(

μp − σ 2p

2

)dt.

Hence, the long-run criterion for financial stability is given by

μp − σ 2p

2> 0.

Inserting (12) and (13) leads to the upper bound for the risk premium

a ≤ amax ={

(1 − α∗)[Rp − (Rp − 1)α∗][Rp − (Rp − 2)α∗]2 − (1 − α∗)2

2[Rp − (Rp − 2)α∗]2

}πT V−1π

= (1 − α∗)[2Rp − 1 − (2Rp − 3)α∗]2[Rp − (Rp − 2)α∗]2 πT V−1π.

(27)

Of course, this analysis is only meaningful if

amax ≥ amin (28)

holds.

Figure 7 shows for which pairs (Re , Rp) condition (28) is satisfied. The correspondingcalculations can be found in the Appendix. Obviously, this condition is very similarto Assumption 1, which is displayed in Figure 4.

Furthermore, Table 1 shows the bounds amin and amax for the risk premium a andthe optimal participation rate α∗ for some pairs of the relative risk aversion of thepension fund and the employees (Rp, Re ), respectively.

The range [amin, amax] looks rather small in some parameter settings. However, onehas to take into account the fact that the risk premium a corresponds to a yearly rate,whereas an employee is typically active for approximately 40 years.

WELFARE GAIN MEASURED BY CERTAINTY EQUIVALENTS

In this section, we wish to measure the welfare gain that results from replacing anattribution scheme with no participation with an attribution scheme with optimal


FIGURE 7The Shaded Area Displays the Pairs (Re, R p), for Which Condition (28) Is Satisfied

TABLE 1Illustration of the Bounds amin and amax for the Risk Premium a and the Optimal Partici-pation Rate α∗ for Some Pairs of the Relative Risk Aversion of the Pension Fund and theEmployees (R p, Re), Respectively

(Rp , Re ) amin amax α∗

(3, 3) 0.074% 0.508% 54.5%(4, 2) 0.063% 0.148% 85.7%(2, 4) 0.125% 1.031% 25.0%

participation. The welfare gain is measured on the basis of appropriate certaintyequivalents. It is shown that the welfare gain does not depend on the particular choiceof the Pareto-optimal attribution scheme and the scheme with no participation.

Measures of WelfareWelfare of the Employees. For the employees the certainty equivalent Xc

t,τ of therandom variable Xt,τ is defined by

(1 − Re )−1(

Xct,τ

Xt,0

)1−Re

= (1 − Re )−1 E

[(Xt,τ

Xt,0

)1−Re]


or

exp{(1 − Re )

(ln Xc

t,τ − ln Xt,0)} = E[exp{(1 − Re )(ln Xt,τ − ln Xt,0)}].

Since Xt,τ is lognormal8 we obtain

exp{(1 − Re )

(ln Xc

t,τ − ln Xt,0)}

= exp

{(1 − Re )E

[ln Xt,τ − ln Xt,0

] + (1 − Re )2

2var

[ln Xt,τ − ln Xt,0

]}

or

ln Xct,τ − ln Xt,0 = E[ln Xt,τ − ln Xt,0] + 1 − Re

2var[ln Xt,τ − ln Xt,0]

=(

μe − σ 2e2

)τ + 1 − Re

2σ 2

e τ.

The welfare of the employees is measured by

We (μe , σe ) = ln Xct,τ − ln Xt,0

τ= μe − Re

2σ 2

e . (29)

Hence, the welfare of the employees is equal to the risk-adjusted rate of return on theemployees’ accounts.

Welfare of the Pension Fund. For the pension fund the certainty equivalent F cT of the

random variable FT is defined by

(1 − Rp)−1(

F cT

F0

)1−Rp

= (1 − Rp)−1 E

[(FT

F0

)1−Rp]

or

exp{(1 − Rp)

(ln F c

T − ln F0)} = E[exp{(1 − Rp)(ln FT − ln F0)}].

Since FT is lognormal, we obtain

exp{(1 − Rp)(ln F cT − ln F0)}

= exp

{(1 − Rp)E [ln FT − ln F0] + (1 − Rp)2

2var [ln FT − ln F0]

}

8See (17).


or

ln F cT − ln F0 = E [ln FT − ln F0] + 1 − Rp

2var [ln FT − ln F0]

=(

μp − σ 2p

2

)T + 1 − Rp

2σ 2

pT.

The welfare of the pension fund is measured by

Wp(μp , σp

) = ln F cT − ln F0

T= μp − Rp

2σ 2

p . (30)

Hence, the welfare of the pension fund is equal to the risk-adjusted growth rate ofthe funding ratio.

Welfare GainProposition 4: Replacing an attribution scheme with no participation (0, a0) with an attri-bution scheme with optimal participation (α∗, a∗) leads to a welfare gain given by

g = We (μe , σe ) + Wp(μp, σp

) |(α∗,a∗)

− (We (μe , σe ) + Wp

(μp , σp

)) |(0,a0)

= (Rp − 1)2

2R2p Re

πT V−1π.

(31)


Remark 9:

(1) The fact that risk sharing leads to a positive welfare gain is of course not surprisingat all. However, an explicit formula is obtained and in Remark 10 it is shown thata welfare gain results, which is substantial in the pension fund context for realisticparameter values.

(2) The welfare gain naturally depends on the relative risk aversion of both the pen-sion fund and the employees. Moreover, as expected, it depends on the parametersof the financial market π and V. However, the welfare gain does not depend onthe particular choice of the attribution scheme with no participation (0, a0) and theattribution scheme with optimal participation (α∗, a∗), as the indifference curvesof the pension fund and the employees only differ from each other by parallelshifts.

(3) For Rp = Re = 3 and πT V−1π = 4% the welfare gain from optimal participationis

g = 0.296%.


FIGURE 8Illustration of the Welfare Gain Measured by the Increase in the Risk-Adjusted Rate ofReturn for Rp = Re = 3, r = 2% and π T V−1π = 4%

At first sight such a welfare gain seems to be rather small. However, over a workinglife of 40 years it amounts to e40g − 1 = 12.58%, which is quite substantial.

(4) We next describe an alternative way to measure the welfare gain that resultsfrom replacing an attribution scheme with no participation with a scheme withthe optimal participation rate. As shown in Figure 8 we start with an attributionscheme (0, a0). This scheme is compared to the attribution scheme (α∗, a ) leadingto the same indirect utility level for the pension fund. In other words, (0, a0) and(α∗, a ) are on the same indifference curve of the pension fund, that is,

a0 − a = f p(0) − f p(α∗). (32)

For the employees, the attribution scheme (α∗, a ) leads to the same indirect utilitylevel as the scheme (0, a∗) with no participation, that is,

a∗ − a = fe (0) − fe (α∗). (33)

Exploiting (32) and (33), the welfare gain measured by the risk-adjusted rate of returncan be obtained as

g = a∗ − a0 = f p(α∗) − fe (α∗) − f p(0) + fe (0). (34)


This alternative approach using the change in the risk-adjusted rate of return leads toexactly the same expression for the welfare gain as the one we obtained in (31).

The following proposition discusses the parameter sensitivity of the welfare gain.

Proposition 5: Let Assumption 1 be given. The following is then obtained for the parametersensitivities of the welfare gain given in (31):(1)

∂g∂ Rp

= Rp − 1

R3p Re

πT V−1π > 0.

(2)

∂g∂ Re

= − (Rp − 1)2

2R2p R2

eπT V−1π < 0.

(3)

∂g∂ R

|R=Re=Rp

=−12

(R − 3) (R − 1)R4 πT V−1π.

Proof: The proof is simple and therefore omitted for the sake of brevity. Q.E.D.

Remark 11:

(1) According to (1) in Proposition 5, the welfare gain naturally increases in the relativerisk aversion of the pension fund as risk sharing then becomes more valuable.

(2) The parameter effect of (2) in Proposition 5 is less obvious at first sight. It resultsfrom the fact that the welfare gain is zero by definition in the case of no partici-pation and the fact that a higher relative risk aversion of the employees leads to alower optimal participation rate.9

(3) Finally, (3) in Proposition 5 shows the parameter sensitivity of the welfare gain foridentical relative risk aversion levels of the pension fund and the employees. ForR < 3, the effect that a higher relative risk aversion makes risk sharing principallymore valuable dominates. For R > 3, however, the effect that a higher relative riskaversion decreases the aggressiveness of the Pareto-optimal investment strategyovercompensates the aforementioned effect. In fact, it can be seen from (24) and(25) that for R close to 1, the optimal participation rate is close to 0, whereas forlarge values of R there is almost no risk taking. Fortunately, the highest welfaregains occur for R around 3, a rather realistic domain for the relative risk aversionof the pension fund and the employees. Figure 9 displays this result for πT V−1π =4%.

9See (ii) in part (a) of Proposition 3.


FIGURE 9The Dependency of the Welfare Gain (in %) on Identical Relative Risk Aversion of Boththe Pension Fund and the Employees for π T V−1π = 4%

CONCLUSIONS

We analyzed a continuous-time model where the employees participate in the pen-sion fund’s investment performance. The expected utility of the funding ratio at someplanning horizon was taken as the objective function of the pension fund. Employeesderive their utility from their savings accounts at the time they leave the pension fund.The interest rate they get on their savings accounts consists of the riskless interest rateplus a premium and a participation in the fund’s investment performance. Hence, anattribution scheme was characterized by the premium and the participation rate. Fora given predetermined attribution scheme, the fund chooses the optimal investmentstrategy. First, the set of predetermined attribution schemes was narrowed by requir-ing Pareto-efficiency. Subsequently, we additionally imposed individual rationality.Employees could not be worse off than under a standalone investment with the Mer-ton investment rule, while financial stability had to be guaranteed for the pensionfund.

In order to obtain closed-form solutions several simplifying assumptions wereneeded. Asset prices were assumed to follow geometric Brownian motions and CRRAutility functions were applied on the funding ratio and on the employees’ accounts.Moreover, we had to assume that employees start the employment phase with alump-sum payment to the pension fund rather than paying their contributions overtheir entire working life.


Under these assumptions the optimal investment rule was constant over time. Pareto-efficiency led to a uniform optimal participation rate. Individual rationality of theemployees and a stability requirement of the pension fund narrowed down the in-terval for the risk premium. Furthermore, it could be shown that the individuallyrational Pareto-efficient attribution schemes allow for a substantial welfare gain incomparison to solutions where the pension fund bears the entire financial risk.

Since the optimal participation rate is smaller than one, the financial risk of theemployees is flattened over time and, given realistic parameter values, employeescan be made better off than under individual investment. In fact, some Europeanpension funds bear a part of the financial risk by lowering the funding ratio in thecase of bad investment results and increasing it when financial markets recover.

For future research our simplifying assumptions should be relaxed and numericalcalculations conducted. Modeling the employees’ contributions to the pension fundmore accurately would involve contribution rates over the entire working life takinginto account typical wage profiles. Moreover, the interest rate on the employees’accounts could depend on both the recent investment performance and the fundingratio. Finally, actuarial risk could be integrated into our pension fund model.

APPENDIX

Proof of Proposition 1

Proof: The pension fund maximizes its utility defined in (5)

Up(FT ) = E[F

1−RpT

]1 − Rp

subject to the funding ratio evolution given in (4)

dFt

Ft= [

(1 − α)xTt π − a + (α2 − α)xT


t σdZt.

The corresponding HJB equation is

0 = J t + max{xt}∈S

{J F Ft

[(1 − α)xT

t π − a + (α2 − α)xTt Vxt

]+ 0.5JFF F 2

t (1 − α)2xTt Vxt

} (A1)

with the terminal condition

J (T , FT ) = F1−RpT

1 − Rp,

where S denotes the set of admissible investment strategies.


We try the solution

J (t, Ft) = F1−Rpt

1 − Rp

and get the following derivatives

J t = 0,

J F = F−Rpt ,

J F F = −Rp F−Rp−1t .

Inserting the above derivatives into the HJB Equation (A1) yields

0 = max{xt}∈S

{F

−Rpt Ft

[(1 − α)xT

t π − a + (α2 − α)xTt Vxt

]−0.5Rp F

−Rp−1t F 2

t (1 − α)2xTt Vxt

}.

The necessary and sufficient condition in the above maximization is

0 = F−Rp+1t [(1 − α)π + (α2 − α)2Vxt] − 0.5Rp F

−Rp+1t (1 − α)22Vxt.

Solving for the optimal investment rule we obtain

−(1 − α)π = 2Vxt[(α2 − α) − 0.5Rp(1 − α)2]

xt = − 1 − α

2[α(α − 1) − 0.5Rp(1 − α)2]V−1π.

Thus, the optimal investment strategy is given by

x∗= 12α+Rp(1 − α)

V−1π ,

which completes the proof. Q.E.D.

Proof of Proposition 2

Proof:

(a) We first prove that α∗ given in (24) is the optimal participation rate. Recalling (16)we have

a = f p(α) = 0.51 − α

2α+Rp(1 − α)πT V−1π−C p.


Differentiating f p(α) with respect to α we get

f p′(α) = 0.5−[2α+Rp(1 − α)] − (1 − α)(2 − Rp)

[2α+Rp(1 − α)]2 πT V−1π

= − 1[Rp + α(2 − Rp)]2 πT V−1π < 0.

(A2)

Recalling (22) we have

a = fe (α) = 0.5Reα2 − 2α2−Rpα(1 − α)

[2α+Rp(1 − α)]2 πT V−1π+Ce − r.

Calculating f ′e (α) yields

f ′e (α) = [2α+Rp(1 − α)]2[Reα − 4α−Rp(1 − α) + Rpα]

[2α+Rp(1 − α)]4 πT V−1π

× −2[2α+Rp(1 − α)](2 − Rp)[0.5Reα2 − 2α2−Rpα(1 − α)]

[2α+Rp(1 − α)]4

=

{[Rp + α(2 − Rp)][Reα − 2α(2 − Rp)−Rp]

−(2 − Rp)α[Reα − 2α(2 − Rp)−2Rp]

}[2α+Rp(1 − α)]3 πT V−1π.

We next develop the numerator separately and obtain

[Rp + α(2 − Rp)][Reα − 2α(2 − Rp)−Rp]

− (2 − Rp)α[Reα − 2α(2 − Rp)−Rp] + Rp(2 − Rp)α

= Rp[Reα − 2α(2 − Rp)−Rp] + Rp(2 − Rp)α

= Rp[Reα − α(2 − Rp)−Rp].

Hence, the derivative f ′e (α) is given by

f ′e (α) = −Rp

Rp + α(2 − Rp) − Reα

[Rp + α(2 − Rp)]3 πT V−1π. (A3)


We next develop the necessary condition for Pareto-optima f ′p(α) − f ′

e (α) = 0.Exploiting (A2) and (A3) gives

f ′p(α) = f ′

e (α)

− 1[Rp + α(2 − Rp)]2 = − Rp[Rp + α(2 − Rp) − Reα]

[Rp + α(2 − Rp)]3

−[Rp + α(2 − Rp)] = Rp Reα − Rp[Rp + α(2 − Rp)]

Rp(Rp − 1) = α{Rp Re + (1 − Rp)(2 − Rp)}.

Solving for α yields our claim in (24)

α∗ = Rp(Rp − 1)(Re+Rp − 3)Rp + 2

.

According to Assumption 1 we obviously have

α∗ = Rp(Rp − 1)Rp(Rp − 1) + Rp(Re − 2) + 2

∈ (0, 1).

It remains to be shown that α∗ represents a global maximum of f p(α) − fe (α).Since f p(α) − fe (α) is twice differentiable on [0, 1] and since the equation f ′

p(α) −f ′e (α) = 0 has the unique root α∗, we only need to show that f ′′

p (α∗) − f ′′e (α∗) < 0.

Exploiting (A2) and (A3) we obtain

f ′′p (α∗) − f ′′

e (α∗)

= 2(2 − Rp)[Rp + α∗(2 − Rp)]3 πT V−1π

− πT V−1π

[Rp + α∗(2 − Rp)]6 (Rp[Rp + α∗(2 − Rp)]3(Rp + Re − 2)

−3[Rp + α∗(2 − Rp)]2(2−Rp)Rp{Reα∗ − [Rp + α∗(2 − Rp)]}).

Thus, we have

f ′′p (α∗) − f ′′

e (α∗) < 0

⇔ 2(2 − Rp)[Rp + α∗(2 − Rp)]−Rp(Re + Rp − 2)[Rp + α∗(2 − Rp)]

+3(2−Rp)Rp{

Reα∗ − [Rp + α∗(2 − Rp)]

}< 0

⇔ [Rp + α∗(2 − Rp)]{2(2 − Rp)−Rp(Re + Rp − 2) − 3(2−Rp)Rp

}+3(2−Rp)Rp Reα

∗ < 0.


Plugging the optimal participation rate (24) into the above yields

⇔ {[(Re + Rp − 3)Rp + 2]Rp + Rp(Rp − 1)(2 − Rp)}

·{2R2p − 6Rp + 4−Rp Re

} + 3(2−Rp)R2p(Rp − 1)Re < 0

⇔ {Re Rp + R2

p − 3Rp + 2 − R2p + 3Rp − 2

}{2R2

p − 6Rp + 4−Rp Re}

+3(2−Rp)Rp(Rp − 1)Re < 0

⇔ Re Rp(2R2

p − 6Rp + 4−Rp Re) + ( − 3R2

p + 9Rp − 6)Rp Re < 0

⇔ −R2p + 3Rp − 2−Rp Re < 0

⇔ Rp(Rp + Re − 3) + 2 > 0

⇔ Rp(Rp − 1) − Rp(2 − Re ) + 2 > 0.

According to Assumption 1 this last condition is satisfied.

(b) Finally, we prove our claim in (25).Plugging the optimal participation rate (24 ) into the general expression of theoptimal investment rule (8) yields

xPa∗ = 12α∗+Rp(1 − α∗)

V−1π

= 1

2Rp(Rp − 1)

(Re+Rp − 3)Rp + 2+Rp

(1 − Rp(Rp − 1)

(Re+Rp − 3)Rp + 2

)V−1π

= (Re+Rp − 3)Rp + 22Rp(Rp − 1)+Rp[(Re+Rp − 3)Rp + 2 − Rp(Rp − 1)]

V−1π

= (Re+Rp − 3)Rp + 2R2

p ReV−1π ,

which completes the proof. Q.E.D.

Proof of Proposition 3Proof:

(a) We first prove the parameter sensitivities of the optimal participation rate.


(1) Differentiating the optimal participation rate given in (24) with respect to therelative risk aversion of the pension fund yields

∂α∗

∂ Rp

= [(Re+Rp − 3)Rp + 2](2Rp − 1) − (Re+2Rp − 3)Rp(Rp − 1)[(Re+Rp − 3)Rp + 2]2

=

((Re+Rp − 3)[Rp(2Rp − 1) − Rp(Rp − 1)]

+2(2Rp − 1) − R2p(Rp − 1)

)[(Re+Rp − 3)Rp + 2]2

= (Re+Rp − 3)R2p − R3

p + R2p + 4Rp − 2[

(Re+Rp − 3)Rp + 2]2

= (Re − 2)R2p + 4Rp − 2[

(Re+Rp − 3)Rp + 2]2 .

Concentrating on the sign of the above derivative we obtain

(Re − 2)R2p + 4Rp − 2 > 0

Re R2p > 2 − 4Rp + 2R2

p

Re >2(

1 − 2Rp + R2p

)R2

p

Re > 2(

Rp − 1Rp

)2.

The right-hand side of the above inequality is obviously smaller than 2. Hence,we obtain ∂α∗

∂ Rp> 0 if Re > 2.

(2) Differentiating the optimal participation rate given in (24) with respect to therelative risk aversion of the employees immediately yields

∂α∗

∂ Re= − R2

p(Rp − 1)

[(Re+Rp − 3)Rp + 2]2 < 0.

(b) Finally, we prove the parameter sensitivities of the Pareto-optimal investmentstrategy.


(1) Differentiating the Pareto-optimal investment rule given in (25) with respect tothe relative risk aversion of the pension fund gives

∂xPa∗

∂ Rp

= R2p Re (Re + 2Rp − 3) − 2Rp Re [(Re+Rp − 3)Rp + 2][

R2p Re

]2 V−1π

=

(R2

p Re (Re + Rp − 3) + R3p Re

−(Re+Rp − 3)2R2p Re − 4Rp Re

)R4

p R2e

V−1π

= R3p Re − (Re+Rp − 3)R2

p Re − 4Rp Re

R4p R2

eV−1π

= −(Re − 3)R2p Re − 4Rp Re

R4p R2

eV−1π

= − (Re − 3)Rp + 4

R3p Re

V−1π.

Concentrating on the sign of the scalar in the above derivative we obtain

(Re − 3)Rp + 4 > 0

Re >3Rp − 4

Rp= 3 − 4

Rp.

This inequality is clearly satisfied for Rp ≤ 2.(2) Differentiating the Pareto-optimal investment rule given in (25) with respect to

the relative risk aversion of the employees yields

∂xPa∗

∂ Re= R2

p Re Rp − R2p[(Re+Rp − 3)Rp + 2][

R2p Re

]2 V−1π

= Rp Re − (Re+Rp − 3)Rp − 2R2

p R2e

V−1π

= − R2p − 3Rp + 2

R2p R2

eV−1π

= − (Rp − 2)(Rp − 1)R2

p R2e

V−1π.

Q.E.D.


Derivation of Figure 7According to (26) and (27), the condition (28)

amax ≥ amin

is equivalent to

Re (2Rp − 1)(1 − α∗) − Re (2Rp − 3)α∗(1 − α∗) >[Rp − (Rp + Re − 2)

]2

or

2Re Rp − Re − Re(4Rp − 4

)α∗ + Re (2Rp − 3)α∗2 > R2

p − 2Rp(Rp + Re − 2)α∗+ (Rp + Re − 2)2α∗2.

Further simplifying finally yields

Re (2Rp − 1) − R2p − 2(Rp − 2)

(Re − Rp

)α∗ − [

R2p + R2

e − 4Rp − Re + 4]α∗2 > 0.

(A4)

Recalling the optimal participation rate α∗ from (24) we have

α∗ = Rp(Rp − 1)(Re + Rp − 3)Rp + 2

.

Finally, inserting α∗ into (A4) allows us to plot Figure 7.

Obviously, for a given value of Rp the condition is satisfied for large enough valuesof Re .

Proof of Proposition 4Proof: Recalling (21) and (29) as well as (15) and (30) we have Q.E.D.

We (μe , σe ) = μe − Re

2σ 2

e

= r + a + α2α+Rp(1 − α) − 0.5Reα

[2α+Rp(1 − α)]2 πT V−1π

(A5)

and

Wp(μp , σp) = μp − Rp

2σ 2

p

= 0.5(1 − α)2α+Rp(1 − α)

πT V−1π − a .

(A6)


Thus, the welfare sum can be written as

We (μe , σe ) + Wp(μp , σp)

= r + 2α2+Rp(1 − α)α − 0.5Reα2 + 0.5(1 − α)[2α+Rp(1 − α)]

[2α+Rp(1 − α)]2 πT V−1π

= r + 2α2 − 0.5Reα2 + (1 − α)α + 0.5(1 − α)Rp(1 + α)

[2α+Rp(1 − α)]2 πT V−1π

= r + −0.5Reα2 + (1 + α)[α + 0.5(1 − α)Rp]

[2α+Rp(1 − α)]2 πT V−1π

= r +{

1 + α

2[2α+Rp(1 − α)]− Reα

2

2[2α+Rp(1 − α)]2

}πT V−1π.

(A7)

We first compute the sum of the welfare of the employees and the pension fund inthe attribution scheme with no participation (0, a0) and obtain

We (μe , σe ) + Wp(μp, σp

) |(0,a0)

= r + 12Rp

πT V−1π. (A8)

We next compute the welfare sum in the attribution scheme with optimal participation(α∗, a∗)

We (μe , σe ) + Wp(μp , σp

) |(α∗, a∗)

= r +{

1 + α∗

2[2α∗+Rp(1 − α∗)

] − Re (α∗)2

2[2α∗+Rp(1 − α∗)

]2

}πT V−1π.

Plugging the optimal participation rate given in (24)

α∗ = Rp(Rp − 1)(Re+Rp − 3)Rp + 2


into the above yields

We (μe , σe ) + Wp(μp, σp) |(α∗,a∗)

= r +1 + Rp(Rp − 1)

(Re+Rp − 3)Rp + 2

2[

2Rp(Rp − 1)


(1 − Rp(Rp − 1)

(Re+Rp − 3)Rp + 2

)]πT V−1π

−Re

(Rp(Rp − 1)

(Re+Rp − 3)Rp + 2

)2

2[

2Rp(Rp − 1)


(1 − Rp(Rp − 1)

(Re+Rp − 3)Rp + 2

)]2 πT V−1π

= r + (Re+Rp − 3)Rp + 2 + Rp(Rp − 1)2[2Rp(Rp − 1)+Rp{(Re+Rp − 3)Rp + 2 − Rp(Rp − 1)}]π

T V−1π

− Re (Rp(Rp − 1))2

2[2Rp(Rp − 1)+Rp{(Re+Rp − 3)Rp + 2 − Rp(Rp − 1)}]2 πT V−1π

= r + (Re+2Rp − 4)Rp + 22[2Rp(Rp − 1)+(Re+Rp − 3)R2

p + 2Rp − R2p(Rp − 1)]

πT V−1π

− Re R2p(Rp − 1)2

2[2Rp(Rp − 1)+(Re+Rp − 3)R2p + 2Rp − R2

p(Rp − 1)]2 πT V−1π

= r +⎧⎨⎩ (Re+2Rp − 4)Rp + 2

2[(Re+Rp)R2

p − R3p] − Re R2

p(Rp − 1)2

2[(Re+Rp)R2

p − R3p]2

⎫⎬⎭πT V−1π

= r +⎧⎨⎩ (Re+2Rp − 4)Rp + 2

2Re R2p

− Re R2p(Rp − 1)2

2[Re R2

p]2

⎫⎬⎭πT V−1π

= r + (Re+2Rp − 4)Rp + 2 − R2p + 2Rp − 1

2Re R2p

πT V−1π

= r + (Re+Rp − 2)Rp + 12Re R2

pπT V−1π.

(A9)

Replacing a scheme with no participation (0, a0) with a scheme with a Pareto-optimalparticipation (α∗, a∗) therefore leads to the following change in welfare

We (μe , σe ) + Wp(μp, σp) |(α∗, a∗)

− We (μe , σe ) + Wp(μp , σp) |(0, a0)

= r + (Re+Rp − 2)Rp + 12Re R2

pπT V−1π−

(r + 1

2RpπT V−1π

)


= (Re+Rp − 2)Rp + 1 − Re Rp

2Re R2p

πT V−1π

= (Rp − 1)2

2Re R2p

πT V−1π.

(A10)

Finally, we note that the above derivation of the welfare gain does not depend on theparticular choice of the Pareto-optimal attribution scheme and the scheme with noparticipation.

Proof of (2) of Remark 4Proof: Setting the derivative in (A3) equal to zero yields

α∗ = Rp

Re + Rp − 2.

According to (8) we obtain

α∗x∗ = α∗ 1α∗(2 − Rp) + Rp

V−1π

= Rp

Re + Rp − 2· 1

Rp(2 − Rp)Re + Rp − 2

+ Rp

V−1π

= 12 − Rp + Re + Rp − 2

V−1π = 1Re

V−1π.

Q.E.D.

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