7. Case study ALM

European Journal of Operational Research 201 (2010) 211–221

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Stochastics and Statistics

Optimal asset allocation for aggregated defined benefit pensionfunds with stochastic interest rates q

Ricardo Josa-Fombellida a, Juan Pablo Rincón-Zapatero b,*

a Departamento de Estadística e Investigación Operativa, Universidad de Valladolid, Paseo Prado de la Magdalena s/n, 47005 Valladolid, Spainb Departamento de Economía, Universidad Carlos III de Madrid, C/Madrid 126, 28903 Getafe, Madrid, Spain

a r t i c l e i n f o

Article history:Received 25 March 2008Accepted 9 February 2009Available online 26 February 2009

Keywords:Pension fundsStochastic controlOptimal portfolioStochastic interest rate

0377-2217/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.ejor.2009.02.021

q Both authors gratefully acknowledge financial supp04, and Spanish Ministerio de Ciencia y Tecnología andcomments.

* Corresponding author. Tel.: +34 91 6248666; fax:E-mail addresses: [email protected] (R. Josa-Fombel

a b s t r a c t

In this paper we study the optimal management of an aggregated pension fund of defined benefit type, inthe presence of a stochastic interest rate. We suppose that the sponsor can invest in a savings account, ina risky stock and in a bond with the aim of minimizing deviations of the unfunded actuarial liability fromzero along a finite time horizon. We solve the problem by means of optimal stochastic control techniquesand analyze the influence on the optimal solution of some of the parameters involved in the model.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Pension funds currently represent one of the most important institutions in financial markets because of their high investment capacityand because they complement the role of the Government, allowing those workers who have reached retirement age to maintain their stan-dard of living. These two aspects justify the interest generated over recent years in the study of the optimum management of pension plans.

There are two principal alternatives in pension plan designs with respect to the assignment of risk. In a defined contribution (DC) planthe risk derived from the fund management is borne by the beneficiary. However, in a defined benefit (DB) plan, where the benefits arenormally related to the final salary level, the financial risk is assumed by the sponsor agent.

Our aim in this paper is to analyze a BD pension fund of aggregated type, which is a common model in the employment system. Weprovide here an extension of the previous work of the authors, Josa-Fombellida and Rincón-Zapatero (2001, 2004, 2006, 2008a,b), in anattempt to incorporate more realistic assumptions to the model, dropping the hypothesis of a constant riskless rate of interest. Thus, inour model, there are three sources of uncertainty: (i) the fund assets returns; (ii) the instantaneous riskless rate of interest; and (iii) theevolution of benefits, based on the behavior of salaries and/or other main components of the pension plan.

There are several previous papers dealing with the management of DC funds in the presence of a stochastic rate of interest. Some ofthem are Boulier et al. (2001), Battocchio and Menoncin (2004), Cairns et al. (2006) and Menoncin (2005), where the interest rate is as-sumed to be of the Vasicek type. In Deelstra et al. (2003), the interest rate has an affine structure, as in Duffie and Kan (1996) which in-cludes as a special case the CIR and the Vasicek models. Other interesting papers where the interest rate is random, though in a discretetime are Vigna and Haberman (2001) and Haberman and Vigna (2002). The importance of DB funds calls for the completion of the theorystudying this case. Moreover, the differences in both types of pension plans makes it impossible to transfer the results from DC to DB plans.

The objective of the shareholder in a DC pension fund is to maximize the expected utility obtained from fund accumulation at a fixeddate. The contribution rate is exogenous to this optimization process, since it is generally determined by salary. However, in a DB plan theamortization effort is a control variable. The fund assets could be artificially increased with high contributions. Obviously, this makes nosense, since benefits are fixed in advance. Thus, the objective in a DB plan should be related with risk minimization instead of themaximization of fund assets. Of course, the main concern of the sponsor is the solvency risk, related to the security of the pension fundin attaining the comprised liabilities. Similar objectives have been considered in other works, such as Haberman and Sung (1994),

ll rights reserved.

ort from Consejería de Educación y Cultura de la Junta de Castilla y León (Spain) under projects VA004B08 and VA099/FEDER funds under project MTM2005-06534. We are particularly thankful to two anonymous referees for many useful

+34 91 6249329.lida), [email protected] (J.P. Rincón-Zapatero).

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212 R. Josa-Fombellida, J.P. Rincón-Zapatero / European Journal of Operational Research 201 (2010) 211–221

Haberman et al. (2000) and Josa-Fombellida and Rincón-Zapatero (2001, 2004). The optimal management of DB plans in the presence of arandom interest rate is found, but in discrete time, in Haberman and Sung (1994), Chang (1999) and Chang et al. (2003).

We make the contribution rate endogenous and dependent on the main variables of the fund, by adopting a spread method of amor-tization, as in Owadally and Haberman (1999). In this way, the contributions are proportional to the unfunded liabilities, requiring moreamortization effort when the plan is underfunded. The pension plan is stochastic, supposing that benefits follow a geometric Brownianmotion as in Josa-Fombellida and Rincón-Zapatero (2004). It is then shown that both the stochastic actuarial liability and the normal costare also geometric Brownian motions, and a relationship between these variables is found. The riskless rate of interest is supposed to begiven by a mean-reverting process, as in Vas�icek (1977). An interesting question addressed in the paper is the selection, according to a val-uation criterion, of the technical rate of actualization to value the liabilities. The financial market also comprises a family of zero-couponbonds of fixed maturity and a risky stock, which are correlated with the source of uncertainty of the benefits.

The results obtained are based on the analytical solutions found by means of the dynamic programming approach. The optimal invest-ment in the bond has four summands: (i) the classical optimal one in Merton (1971); (ii) a positive term decreasing to zero with the ter-minal date of the plan involving parameters of the riskless rate of interest; (iii) the market price of risk multiplied by an expressioninvolving diffusion coefficients of the bond and the stock and the excess expected return of the stock; and (iv) a term proportional tothe actuarial liability that vanishes if there is no correlation in the financial instruments or if the benefits are deterministic. The optimalinvestment in the risky asset follows a similar pattern, but in this case there is no corresponding term to those described in (i) and (ii).

The paper is organized as follows: Section 2 defines the elements of the pension scheme of an employment system. We suppose thetechnical rate of interest is random. The actuarial functions are also introduced and we prove a relation between these functions whenthe benefits are given by a geometric Brownian motion. In Section 3, we explain the financial market structure. In Section 4, we find arisk-neutral valuation of the liabilities, giving rise to an expression for the technical rate of actualization, that relates it with the interestrate and the correlation parameters between the sources of uncertainty, as well as with the parameters defining the stochastic evolution ofliabilities. In Section 5, we consider that the fund is invested in a riskless asset (savings account) and in two risky assets (a bond and astock). We state the problem of minimizing the expected value of the terminal solvency risk and we explicitly solve it. In Section 6, theresults are illustrated with a numerical analysis of the problem, analyzing the investment time evolution pattern in the bond and in thestock. Finally, Section 7 is dedicated to establishing some conclusions and possible extensions. All proofs are in Appendix A.

2. The pension model

The pension plan we take into account is an aggregated pension fund of the DB type, thus the benefits are established in advance by themanager. With the objective of the delivery of retirement benefits to the workers, the plan sponsor continuously withdraws time-varyingfunds. The variables listed below refer to the total group of participants. The principal elements intervening in the funding process and theessential hypotheses allowing its temporary evolution to be determined are as follows.

Notation of the elements of the pension planT planning horizon or date of the end of the pension plan, with 0 < T <1FðtÞ value of fund assets at time tPðtÞ benefits promised to the participants at time t, which are related to the salary at the moment of retirementCðtÞ contribution rate made by the sponsor at time t to the funding processALðtÞ actuarial liability at time t, that is, total liabilities of the sponsorNCðtÞ normal cost at time t; if the fund assets match the actuarial liability, and if there are no uncertain elements in the plan, the nor-

mal cost is the value of the contributions allowing equality between asset funds and obligationsUALðtÞ unfunded actuarial liability at time t, equal to ALðtÞ � FðtÞMðxÞ � 100% percentage of the actuarial value of the future benefits accumulated until age x 2 ½a;d�, where a is the common age of entrance in

the fund and d is the common age of retirement for all participants. Function M is a differentiable distribution function on ½a;d�.In particular, MðaÞ ¼ 0 and MðdÞ ¼ 1

dðtÞ technical rate of actualization. It is the rate of valuation of the liabilities, which can be specified by the regulatory authoritiesrðtÞ risk-free market interest rate.

Josa-Fombellida and Rincón-Zapatero (2004) considers that there exist disturbances affecting the evolution of the benefits P and hence,the evolution of the normal cost NC and the actuarial liability AL, but the rate of valuation d of the plan is constant. In this paper we add amore general assumption: we suppose the short rate of interest r is random. This means that d is also random. In order to simplify, we willsuppose both processes have the same source of uncertainty. As we have commented in Section 1, three sources of randomness appear inthe problem: benefits, interest rate and stock. Thus, to model this situation, we consider a probability space ðX;F;PÞ, where F ¼ fFtgtP0

is a complete and right continuous filtration generated by the three-dimensional standard Brownian motion, Ft ¼ rfðwðuÞ;wBðuÞ;wSðuÞÞ : 0 6 u 6 tg and P is a probability measure on X. We assume that r and d satisfy stochastic differential equations depending onwB only. The benefits randomness is due to another Brownian motion wP . Given that the benefits P are conditioned by the increase in salaryof the sponsoring employees, we suppose the existence of correlation q1 2 ½�1;1� between the Brownian motions wP and wB, andq2 2 ½�1;1� between Brownian motions wP and wS, which can be explained by the effects of salary on inflation and the effects of the latter

on the asset prices. This means that wPðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

1 � q22

qwðtÞ þ q1wBðtÞ þ q2wSðtÞ, for all 0 6 t 6 T; supposing q2

1 þ q22 6 1. When

q21 þ q2

2 < 1, the risk in the benefits outgo cannot be eliminated by trading in the financial market.We consider that there r, d and P are diffusion processes given by the stochastic differential equations

drðtÞ ¼ lrðt; rðtÞÞdt þ grðt; rðtÞÞdwBðtÞ;ddðtÞ ¼ ldðt; dðtÞÞdt þ gdðt; dðtÞÞdwBðtÞ;dPðtÞ ¼ lPðt; PðtÞÞdt þ gPðt; PðtÞÞdwPðtÞ;

R. Josa-Fombellida, J.P. Rincón-Zapatero / European Journal of Operational Research 201 (2010) 211–221 213

for all t P 0, with rð0Þ ¼ r0, dð0Þ ¼ d0 and Pð0Þ ¼ P0 representing the initial values of the interest rates and the benefits. However, to obtain aclosed form solution we need a more concrete specification of these processes.

We extend the definitions of the actuarial functions from the constant rate of valuation case given in Bowers et al. (1986), for determin-istic benefits, and Josa-Fombellida and Rincón-Zapatero (2004), for stochastic benefits. The stochastic actuarial liability and the stochasticnormal cost are defined as follows:

ALðtÞ ¼ E

Z d

ae�R tþd�x

tdðsÞdsMðxÞPðt þ d� xÞdx jFt

!;

NCðtÞ ¼ E

Z d

ae�R tþd�x

tdðsÞdsM0ðxÞPðt þ d� xÞdx jFt

!;

for every t P 0, where Eð�jFtÞ denotes conditional expectation with respect to the filtration associated to the standard Brownian motionfðwðtÞ;wBðtÞ;wSðtÞÞgtP0. Thus, to compute the actuarial functions at time t, the manager makes use of the information available up to thattime, in terms of the conditional expectation. In this way, ALðtÞ is the total expected value of the promised benefits accumulated accordingto M, discounted at the rate dðtÞ, that we suppose is adapted to the filtration. Analogous comments can be given to the normal cost NCðtÞwithfunction M0. Note that benefits of retired participants are not a tradable asset and, in consequence, the inherent risk cannot be hedged and themarket is incomplete.

Since P is a diffusion process, it satisfies the Markov property (see Øksendal (2003)), hence, conditional expectation with respect to thefiltration equals conditional expectation with respect to the current values of P at time t. It is plausible to think that, in the task of com-puting the ideal values of the fund, the information given by the evolution of the random source will be used. Using basic properties of theconditional expectation, the previous definitions can be rewritten:

ALðtÞ ¼Z d

aE e�

R tþd�x

tdðsÞdsPðt þ d� xÞ jFt

� �MðxÞdx;

NCðtÞ ¼Z d

aE e�

R tþd�x

tdðsÞdsPðt þ d� xÞ jFt

� �M0ðxÞdx:

For analytical tractability, we will need a more concrete specification for P. A typical way of modelling P in the certain case is to pos-tulate exponential growth, see Bowers et al. (1986). The stochastic counterpart is to consider the benefits outgo as a geometric Brownianmotion. This is the content of the following hypothesis.

Assumption A. The benefits P satisfies

dPðtÞ ¼ lPðtÞdt þ gPðtÞdwPðtÞ; t P 0;

where l 2 R and g 2 Rþ. The initial condition Pð0Þ ¼ P0 is a random variable that represents the initial liabilities.

Hence, we are supposing that the benefits increase or decrease on average at a constant exponential rate. The behavior of the actuarialfunctions AL and NC is then given in the following proposition. To this end, we define the following random functions:

wALðtÞ ¼Z d

aeR tþd�x

tðl�dðsÞÞdsMðxÞdx;

wNCðtÞ ¼Z d

aeR tþd�x

tðl�dðsÞÞdsM0ðxÞdx;

nALðtÞ ¼Z d

aeR tþd�x

tðl�dðsÞÞdsðl� dðt þ d� xÞÞMðxÞdx� ðl� dðtÞÞwALðtÞ:

Proposition 2.1. Under Assumption A the actuarial functions satisfy AL ¼ wALP and NC ¼ wNCP, and they are linked by the identity

lþ nALðtÞwALðtÞ

� �ALðtÞ þ NCðtÞ � PðtÞ ¼ 0; ð1Þ

for every t P 0. Moreover, the actuarial liability satisfies

dALðtÞ ¼ lþ nALðtÞwALðtÞ

� �ALðtÞdt þ gALðtÞdwPðtÞ; ALð0Þ ¼ AL0 ¼ wALð0ÞP0: ð2Þ

Thus, AL is a geometric Brownian motion with random drift. Processes AL and P differ in the drift term nALðtÞ=wALðtÞð ÞAL. Notice that wALðtÞcan be interpreted as the discounted value, at rate dðsÞ � l, of a security paying an amount of MðxÞ continuously in the interval ½t; t þ d� a�.Since nALðtÞ is the derivative of wALðtÞ, the quotient nALðtÞ=wALðtÞ is the stochastic rate of growth of the above discounted value. Thus, the driftof dAL given in (2) takes into account not only the mean growth of the liabilities, l, but the random fluctuations due to the stochastic d.

We will now use a spread method of fund amortization, as mentioned in the Introduction. Thus, we will assume that the supplementarycontribution rate (difference between contribution rate and normal cost) is proportional to the unfunded actuarial liability, that is

CðtÞ ¼ NCðtÞ þ kðALðtÞ � FðtÞÞ; ð3Þ

where k is a constant selected by the employer, representing the rate at which surplus or deficit is amortized. Though actuarial practice takes1=k equal to a continuous annuity with amortization over m years, we consider more flexibility in the selection of k than actuarial practicesuggests, as in Haberman and Sung (1994) or Josa-Fombellida and Rincón-Zapatero (2001, 2004).


3. The financial market

In this section we describe the underlying financial market in our model. The plan sponsor manages the fund by means of a portfolioformed by a riskless R, a coupon zero bond B and a stock S.

First we assume the following hypothesis.

Assumption B. The instantaneous riskless interest rate rðtÞ satisfies the stochastic differential equation:

1 GivYðtÞ ¼ f

drðtÞ ¼ aðb� rðtÞÞdt þ rdwBðtÞ; rð0Þ ¼ r0; ð4Þ

where a; b and r are strictly positive constants.

This process of type mean-reverting and known as the Orstein–Uhlenbeck process, has been introduced in Vas�icek (1977) to explaininterest rate behavior.

We assume the price process of the riskless asset R is given by

dRðtÞ ¼ rðtÞRðtÞdt; Rð0Þ ¼ R0; ð5Þ

where the evolution of rðtÞ is given by (4). This asset can be interpreted as a bank account paying the instantaneous interest rate rðtÞwithoutany risk.

Given r we assume that there exists a market for zero-coupon bonds with a fixed maturity T1 > T. Following Vas�icek (1977) (see alsoBattocchio and Menoncin (2004)) the price at instant t of a zero-coupon bond with maturity T1, with t < T < T1, is given by

Bðt; T1Þ ¼ ecðt;T1Þ�bðt;T1ÞrðtÞ;

where

bðt; T1Þ ¼1að1� e�aðT1�tÞÞ;

cðt; T1Þ ¼ �Rð1ÞðT1 � tÞ þ bðt; T1Þ Rð1Þ � r2

2a2

� �þ r2

4a3 ð1� e�2aðT1�tÞÞ;

and Rð1Þ ¼ bþ rfa� r2=ð2a2Þ represents the return of a zero-coupon bond with maturity equal to infinite, and f is the constant marketprice of risk. Applying Itô’s formula,1 the price of the bond process verifies the stochastic differential equation

dBðt; T1Þ ¼ Bðt; T1Þ rðtÞ þ rfbðt; T1Þð Þdt � rbðt; T1ÞdwBðtÞð Þ; BðT1; T1Þ ¼ 0: ð6Þ

Finally, we consider a stock whose dynamic is given by the stochastic differential equation

dSðtÞ ¼ SðtÞ lSðrðtÞÞdt þ rrdwBðtÞ þ rSdwSðtÞ� �

; Sð0Þ ¼ s0; ð7Þ

where rr;rS are positive constants defining the stock volatility, that isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

r þ r2S

q, and the drift parameter lSðrÞ is the instantaneous mean

having the form lSðrÞ ¼ r þmS, with mS a constant representing the expected excess return from investing in the stock, as in Deelstraet al. (2003), Battocchio and Menoncin (2004) or Menoncin (2005).

4. Risk-neutral valuation of the liabilities

The accurate valuation of liabilities is of the utmost importance for the sponsor. We address here the question of how to perform a fairvaluation, keeping in mind that the promised benefits to participants is not a tradeable asset. To circumvent this problem we resort to avaluation based on a concept of equilibrium, see e.g. Constantinides (1978). The risk-neutral valuation of liabilities offers a univocally de-fined value for the technical rate of actualization, d. This value is a modification of the short rate of interest, r, to take into account the driftand diffusion components of the financial instrument and benefits as well as the several correlations existing between them.

For age x, let Yxðt; PÞ be the asset, valued at time t, consisting in a payment of P monetary units at the age of retirement, d, to a participantwith current age x. Process P is a geometric Brownian motion according to Assumption A. As P is not tradeable, it cannot be used to form aportfolio to hedge the risk. Thus we form a portfolio formed by Yx, B and S, with two tradeable assets and three independent Brownianmotions. To obtain a risk-neutral valuation it is assumed that the risk uncorrelated with the two freely traded financial instruments, i.e.the stock and the bond, is not priced. Notice that the actuarial liability of the fund is

ALðtÞ ¼Z d

aYxðt; PðtÞÞMðxÞdx:

Once Yx is found, and after matching this expression of AL with that given in Section 2, the value of d is determined. To this end, consider theasset at any intermediate time, Yxðt þ s; PÞ, for 0 6 s 6 d� x. Forming a portfolio P ¼ Yx þ pBBþ pSS with one unit of asset Yx, pB units of Band pS units of S, and applying Itô’s formula, see Footnote 1, we have

en an scalar Itô process dXðtÞ ¼ lðt;XðtÞÞdt þ rðt;XðtÞÞdWðtÞ with W a standard Brownian motion, and a function f that is twice continuously differentiable on ½0;1Þ � R,ðt;XðtÞÞ is again an Itô process and (Itô’s formula)

dYðtÞ ¼ ftðt;XðtÞÞ þ fxðt;XðtÞÞlðt;XðtÞÞ þ12

fxxðt;XðtÞÞr2ðt;XðtÞÞ� �

dt þ fxðt;XðtÞÞrðt;XðtÞÞdWðtÞ:


dP ¼ dYx þ pB dBþ pS dS

¼ YxplP þ 1

2Yx

ppg2P2 þ Yx

s

� �dt þ Yx

pgP dwP þ pB ðr þ rfbÞBdt � rbBdwBð Þ þ pS lSðrÞSdt þ rrSdwB þ rSSdwS� �

¼ YxplP þ 1

2Yx

ppg2P2 þ Yx

s þ pBðr þ rfbÞBþ pSlSðrÞS� �

dt þ Yxpg

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

1 � q22

qP dwþ Yx

pgPq1 � pBrbBþ pSrrS� �

dwB

þ YxpgPq2 þ pSrSS

� �dwS:

The first equality is due to the self-financing property of the strategies pB, pS. In the following equalities we use Assumption A, (6) and (7).Now we select pB and pS in order to eliminate the risks related with wB and wS, that is pBB ¼ ðYx

pgPq1 þ pSrrSÞ=ðrbÞ andpSS ¼ �Yx

pgPq2=rS. We also disregard the risk orthogonal to them, that is, the risk related with w is not priced. The total return of the hedgeportfolio must be equal to the rate of interest at time t þ s, rðt þ sÞ. Thus, we obtain

rðt þ sÞðYx þ pBBþ pSSÞ ¼ Yxs þ Yx

plP þ 12

Yxppg

2P2 þ pBðr þ rfbÞBþ pSlSðrÞS;

that, with the expressions for pB and pS found above and using lSðrÞ ¼ r þmS, becomes the pricing partial differential equation

rðt þ sÞYx ¼ Yxs þxPYx

p þ12g2P2Yx

pp ð8Þ

with boundary conditions Yxðt þ d� x; PÞ ¼ P, Yxðt þ s;0Þ ¼ 0, and where

x ¼ lþ fgq1 �mS þ frr

rSgq2:

The solution to (8) is Yxðt þ s; PÞ ¼ Pe�R tþd�x

tþsðrðsÞ�xÞds hence, Yxðt; PÞ ¼ Pe�

R tþd�x

tðrðsÞ�xÞds. Consequently,

ALðtÞ ¼Z d

aYxðt; PðtÞÞMðxÞdx ¼ PðtÞ

Z d

ae�R tþd�x

tðrðsÞ�xÞdsMðxÞdx:

On the other hand, by Proposition 2.1 the actuarial liability satisfies

ALðtÞ ¼ PðtÞwALðtÞ ¼ PðtÞZ d

aeR tþd�x

tðl�dðsÞÞdsMðxÞdx:

Comparing both values of AL, dðtÞ must be chosen equal to rðtÞ þ l�x in order to attain a risk-neutral valuation.Thus we will assume throughout the paper, in a similar way to that found in Josa-Fombellida and Rincón-Zapatero (2004, 2008b), that

the technical interest rate coincides with the rate of return of the bond modified to get rid of the sources of uncertainty, that is:

Assumption C. The technical rate of actualization is chosen to be

dðtÞ ¼ rðtÞ � fgq1 þmS þ frr

rSgq2:

Besides the risk-neutral valuation it provides, this selection of d allows us to solve explicitly the problem in the following section.

5. The optimal portfolio

In this section we analyze how the sponsor may select, in an optimal way, the proportion of fund assets put into a savings account, orinvested in a bond and in a risky stock. Thus, the sponsor faces three elements of randomness: one due to the benefits, which is inherent tothe pension plan, and the other two are financial market variables, specifically the stochastic interest rate and a risky stock.

The plan sponsor invests the fund in a portfolio formed by the savings account R, given by (5), the zero-coupon bond B with maturityT1 > T , given by (6), and the stock S whose dynamic is given by (7).

The amounts invested in the bond B and the stock S are denoted by kB and kS, respectively. The remainder, F � kB � kS, goes to the savingsaccount R. Borrowing and shortselling is allowed. A negative value of kB (resp. kS) means that the sponsor sells shares of B (resp. S) short,while, if kB þ kS is greater than F, then he or she gets into debt to purchase the stocks, borrowing money at the interest rate r.

We suppose fðkBðtÞ; kSðtÞÞ : t P 0g is a Markovian control process adapted to filtration fFtgtP0 and satisfying

E

Z T

0ðk2

BðtÞ þ k2S ðtÞÞdt <1: ð9Þ

Therefore, the fund’s dynamic evolution under the investment policy ðkB; kSÞ is:

dFðtÞ ¼ kBðtÞdBðtÞBðtÞ þ kSðtÞ

dSðtÞSðtÞ þ FðtÞ � kBðtÞ � kSðtÞð Þ dRðtÞ

RðtÞ þ ðCðtÞ � PðtÞÞdt: ð10Þ

By substituting (5), (6) and (7) in (10), and taking into account Proposition 2.1 and (3), we obtain:

dFðtÞ ¼ bðtÞrfkBðtÞ þ kSðtÞðlSðrðtÞÞ � rðtÞÞ þ ðrðtÞ � kÞFðtÞ þ kþ lþ nALðtÞwALðtÞ

� dðtÞ� �

ALðtÞ� �

dt

þ ð�bðtÞrkBðtÞ þ rrkSðtÞÞdwBðtÞ þ rSkSðtÞdwSðtÞ; ð11Þ

with initial condition Fð0Þ ¼ F0 > 0. By (2), and the relation wp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

1 � q22

qwþ q1wB þ q2wS in terms of X ¼ F � AL Eq. (11) is


dXðtÞ ¼ bðtÞrfkBðtÞ þmSkSðtÞ þ ðrðtÞ � kÞXðtÞ þ ðrðtÞ � dðtÞÞALðtÞð Þdt � gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

1 � q22

qALðtÞdwðtÞ

� bðtÞrkBðtÞ � rrkSðtÞ þ gq1ALðtÞð ÞdwBðtÞ þ rSkSðtÞ � gq2ALðtÞð ÞdwSðtÞ; ð12Þ

with the initial condition Xð0Þ ¼ X0.Let us now turn to the preferences of the controller. We assume that he or she wishes to minimize the terminal solvency risk. Thus, the

objective functional to be minimized over the class of admissible controls AX0 ;AL0 ;r0 , is given by

JððX0;AL0; r0Þ; ðkB; kSÞÞ ¼ EX0 ;AL0 ;r0 X2ðTÞ: ð13Þ

Here, AX0 ;AL0 ;r0 is the set of measurable processes fðkBðtÞ; kSðtÞÞgtP0 where ðkB; kSÞ satisfies (9) and where X, AL and r satisfy (12), (2) and (4),respectively. In the above, EX0 ;AL0 ;r0 denotes conditional expectation with respect to the initial conditions ðX0;AL0; r0Þ.

In the following developments we will suppose only the underfunded case where X0 < 0, so we will refer to X as debt. The overfundedcase, X0 > 0, leads to similar results.

The dynamic programming approach is used to solve the problem. To make the process work, some properties of the value functionneed to be established. The value function is defined as
bV ðt;X;AL; rÞ ¼ min
ðkB ;kSÞ2At;X;AL;r

Jðt; ðX;AL; rÞ; ðkB; kSÞÞ : subject to ð12Þ; ð2Þ; ð4Þf g: ð14Þ

The connection between value functions in optimal control theory (deterministic or stochastic) and optimal feedback controls is accom-plished by the HJB equation, see Fleming and Soner (1993).

We have the following result.

Theorem 5.1. Suppose that Assumptions A, B and C hold. Then the optimal investments are given by

k�Bðt;X;ALÞ ¼ �arð1� e�aðT1�tÞÞ f� 2

ra

1� e�aðT�tÞ� �þmSrr þ fr2

r

r2S

� �X þ q1 �

rr

rSq2

� �gAL

� �ð15Þ

k�Sðt;X;ALÞ ¼ �mS þ frr

r2S

X þ q2

rSgAL: ð16Þ

Remark 5.1. From (15) in Theorem 5.1, the optimal investments do not depend on r and the investment in the bond is of the form

k�Bðt;X;ALÞ ¼ 11� e�aðT1�tÞ

�afr

X þ 2 1� e�aðT�tÞ� �X � a

rr2S

ðmSrr þ fr2r ÞX �

agrrSðrSq1 � rrq2ÞAL

� �:

Leaving aside the common factor, it is the sum of four terms. The first term coincides with the classical optimal one in Merton (1971) whenthe coefficients are deterministic. The second term is proportional to X and depends on the time horizon planned, vanishing at the terminaldate T. The third term is also proportional to the debt, with a coefficient that depends on several of the elements defining the prices of thebond and the stock. The fourth term is quite different, as it involves random liability instead of debt. The summand is now proportional to AL,with a coefficient that depends on the volatilities of the processes AL, B and S and their respective correlations. Thus, this last term caresabout the random evolution of liabilities. In fact it vanishes when benefits are deterministic or when the relation: q1rS ¼ q2rr , between vari-ances and covariances holds. In both cases the optimal investment in the bond is proportional to the unfunded liability, UAL ¼ �X.

The optimal investment in the stock given by (16) is simpler. It is the sum of two terms, one proportional to debt X and the otherproportional to the actuarial liability AL. The latter is zero, that is, k�S is proportional to debt X, if either the benefits are deterministic orwhen there is no correlation between stock and benefits.

Substituting (15) and (16) in (12) we obtain that the optimal debt satisfies the stochastic differential equation:

dXðtÞ ¼ �f2 � ðmS þ frrÞ2

r2S

þ 2að1� e�aðT�tÞÞfrþ rðtÞ � k

!XðtÞdt � g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

1 � q22

qALðtÞdwðtÞ

þ f� 2að1� e�aðT�tÞÞr

� �XðtÞdwBðtÞ �

mS þ frr

rSXðtÞdwSðtÞ; ð17Þ

with the initial condition Xð0Þ ¼ X0 and where AL is given by (2) and r by (4). In the following section we will numerically integrate the linearsystem of SDEs formed by (2), (4) and (17) to illustrate the results.

6. A numerical illustration

In this section we consider a numerical application in order to illustrate the dynamic behavior of the debt and its expected value, and theoptimal portfolio strategy. The parameters defining the financial market have been taken from Boulier et al. (2001). Thus, the initial valuefor the interest rate r0 ¼ 0:05 coincides with its equilibrium value b, the maturity is T1 ¼ 10 and the market price of risk is f ¼ 0:15. Theseand the remainder parameter values are shown in Table 1.

We consider a contribution period before retirement of T = 6 years and that benefits are random with l ¼ 0:04 and g ¼ 0:08. The effortof amortization is k ¼ 0:06. The initial values for the actuarial liability and the fund wealth are taken to be AL0 ¼ 100 and F0 ¼ 80, respec-tively, so X0 ¼ �20, that is the fund is 20% underfunded. Initial benefits are supposed to be 1% of AL0, that is, P0 ¼ 1. It is supposed thatbenefits are accumulated uniformly, MðxÞ ¼ ðx� aÞ=ðd� aÞ.

The correlation between benefits and short rate is selected as q1 ¼ 0:2 and the correlation between benefits and stock as q2 ¼ 0:2. Fig. 1shows the evolution of debt, fund assets, actuarial liability and its expected values along the planning interval. First, we have run a sample

0 2 4 6−45

−40

−35

−30

−25

−20

−15

−10

0 2 4 640

50

60

70

80

90

100

110

0 2 4 6−20

−18

−16

−14

−12

−10

−8

−6

0 2 4 680

90

100

110

120

130

E X(t)

EF(t)EAL(t)

F(t)AL(t)

X(t)

Fig. 1. Debt, fund, actuarial liability and their expected values.

Table 1Values of parameters.

Interest rateMean reversion, a 0.2Mean rate, b 0.05Volatility, r 0.02Initial rate, r0 0.05

Maturity bondMaturity, T1 10Market price of risk, f 0.15

StockRisk premium, mS 0.06Interest rate source risk, rr 0.06Stock own volatility, rS 0.19


path of r in the interval ½0; T þ d� a�, which is needed to obtain wALðtÞ and nALðtÞ for t 2 ½0; T�. To obtain r, we have used the Euler method, seee.g. Kloeden and Platen (1999),2 and to obtain both wALðtÞ and nALðtÞ, the composed trapezoidal rule has been used to compute the integrals.With this data at hand, the Euler scheme has been employed again to find the solution of the system (2), (4) and (17), of course with the samesample path used for r, restricted to ½0; T�. This particular sample path drawn in Fig. 1 shows that the debt takes values in the range �43 to �12and it attains the value of �16.14 at instant t ¼ 6, from an initial value of �20 at t ¼ 0. Fig. 1 also shows the evolution of F and AL for the samesimulation. Obviously, growth of expected fund assets and expected liabilities over time is observed, since benefits present a positive meanincrease. This trend is seen in the next graph, where the expected values of debt, fund and actuarial liability are shown. These curves havebeen computed with Monte Carlo simulation, see e.g. Kloeden and Platen (1999). The expected value of X is increasing, that is, the expecteddebt decreases. In our example, mean debt is reduced from �20 to �7.18, that is, 64% of its value. This fact is better appreciated in the fourthgraph, where EFðtÞ gets closer to EALðtÞ as t increases.

2 It should be possible to apply other methods with higher order of convergence, as the Milstein scheme, see e.g. Kloeden and Platen (1999). For our purposes it suffices theEuler scheme. The calculations have been done with MATLAB

�.

0 2 4 6−3

−2

−1

0

1

2

3q1<0, q2<0

0 2 4 6−4

−3

−2

−1

0

1

2

3q1<0, q2>0

0 2 4 6−3

−2

−1

0

1

2q1>0, q2<0

0 2 4 6−3

−2

−1

0

1

2q1>0, q2>0

λB /FλS /F

cash/F

λB /FλS /F

cash/F

λB /Fλ

S/F

cash/F

λB /FλS /F

cash/F

Fig. 2. Investment proportions for four cases of correlations.


Fig. 2 represents the proportions of the fund invested in the portfolio in order to minimize the terminal solvency risk. The paths corre-spond to the same sample as in Fig. 1. These functions depend on the individual values of correlations q1 and q2, see (15) and (16), whereasthe processes in Fig. 1 depend on the aggregate value q2

1 þ q22, see (17). Thus we consider four possible scenarios: ðq1; q2Þ ¼ ð�0:2;�0:2Þ,

ð�0:2;0:2Þ, ð0:2;�0:2Þ, ð0:2; 0:2Þ.The four graphs in Fig. 2 show a similar pattern. In the first years where debt is large, the optimal strategy is to take more risk, borrowing

money to invest in the bond and in the stock. The higher mean returns they provide compared with the bank account is the factor that mayexplain this behavior. In fact, the time when debt takes the maximum value, that is, when X is minimum, is just when k�B þ k�S also attains itsmaximum value. At this point the strategy is quite aggressive indeed, requiring the borrowing of money for the amount of approximately278% of the fund’s wealth, or 2:78F to invest with risk (first graph in Fig. 2, with negative value of both correlations). In the final part of thetime interval the amount held in cash increases and the amount invested in the stock and the bond diminishes, considerably reducing therisky composition of the portfolio. The behavior described is similar in the four cases of correlations considered.

The relative weight of the stock and the bond in the portfolio is highly influenced by the signs of correlations, at least in the sampleshown. In the first two graphs where q1 < 0, the bond participates in the portfolio in a larger proportion than the stock, independentlyof the sign of q2. When q1 > 0 the situation is reversed, except the last year. Thus, the feature observed in the model studied in Menoncin(2005), where the bond’s share in the portfolio is larger than the share of the stock, is not maintained in our model. This different behaviormay be due, on the one hand, to the existence of correlations and on the other hand, to the aim of the sponsor to minimize the expectedsquare of debt, instead of maximizing expected utility from surplus.

7. Conclusions

We have analyzed the management of a pension funding process of a DB pension plan when the short interest rate is the Vasicek model.The problem of the minimization of the terminal solvency risk has been solved analytically when the benefits process is a geometricBrownian motion under a suitable selection of the technical interest rate. The components of the optimal portfolio (investments in thebond, in the stock and in the cash) are the sum of two terms, one proportional to the unfunded actuarial liability, and another to the actu-arial liability, depending on parameters of the randomness of benefits and its correlations with the interest rate and the stock.

We have done a numerical simulation showing some properties of the model. Though there are three sources of randomness, the debt isreduced by means of risky investment in the first years and with a more conservative investment policy in the last years of the plannedperiod.


Further research should include other dynamics for the interest rate processes, such as the Cox–Ingersoll–Ross (CIR) model, the Ho–Leemodel or affine models in general. As for benefits, it would also be interesting to consider the possibility of jumps, such as in Ngwira andGerrard (2006), or some more general Lévy process.

Appendix A

Proof of Proposition 2.1. The process DtðuÞ ¼ e�R u

tdðsÞds satisfies

dDtðuÞ ¼ �dðuÞDtðuÞdu; DtðtÞ ¼ 1;

hence, by Assumption A, DtP is a geometric Brownian motion with non-constant coefficients satisfying

dðDtPÞðuÞ ¼ DtðuÞdPðuÞ þ dDtðuÞPðuÞ ¼ ðDtPÞðuÞ ðl� dðuÞÞduþ gdwPðuÞð Þ:

This follows from the integration by parts formula since dDt has no diffusion term, see e.g. Karatzas and Shreve (1997). Then, the conditionalexpectation is

E Dtðt þ d� xÞPðt þ d� xÞ jFtð Þ ¼ DtðtÞPðtÞeR tþd�x

tðl�dðuÞÞdu

;

thus, recalling the definition of AL and wAL we get:

ALðtÞ ¼ PðtÞZ d

aeR tþd�x

tðl�dðuÞÞduMðxÞdx ¼ PðtÞwALðtÞ;

because DtðtÞ ¼ 1. Analogously, NCðtÞ ¼ PðtÞwNCðtÞ.Now, by means an integration by parts, and the definition of nAL we have

wNCðtÞ ¼Z d

aeR tþd�x

tðl�dðsÞÞdsdMðxÞ ¼ e

R tþd�x

tðl�dðsÞÞdsMðxÞjx¼d

x¼a þZ d

aeR tþd�x

tðl�dðsÞÞdsðl� dðt þ d� xÞÞMðxÞdx ¼ 1þ nALðtÞ þ ðl� dðtÞÞwALðtÞ:

In consequence

NCðtÞ ¼ wNCðtÞPðtÞ ¼ PðtÞ þ nALðtÞPðtÞ þ ðl� dðtÞÞwALðtÞPðtÞ ¼ PðtÞ þ l� dðtÞ þ nALðtÞwALðtÞ

� �ALðtÞ;

which is (1). Finally we deduce the stochastic differential equation that the actuarial liability satisfies. Notice that dwALðtÞ ¼ nALðtÞdt. Thus,using Assumption A,

dALðtÞ ¼ dðwALPÞðtÞ ¼ wALðtÞdPðtÞ þ dwALPðtÞ ¼ wALðtÞPðtÞðldt þ gdwPðtÞÞ þ nALðtÞPðtÞdt ¼ lþ nALðtÞwALðtÞ

� �ALðtÞdt þ gALðtÞdwPðtÞ;

with the initial condition ALð0Þ ¼ AL0 ¼ wALð0ÞP0. �

Proof of Theorem 5.1. Consider the value function (14) of the control problem (2), (4), (12), (13). This function so defined is non-negativeand strictly convex. Under some sufficient conditions, including smoothness, bV is a solution of the HJB equation, see Fleming and Soner(1993):

Vt þminkB ;kS

ðbrfkB þmSkS þ ðr � kÞX þ ðr � dÞALÞVX þ ðlþ nAL=wALÞALVAL þ aðb� rÞVr :

þ 12ð1� q2

1 � q22Þg2AL2 þ ðbrkB � rrkS þ gq1ALÞ2 þ ðrSkS � gq2ALÞ2

� �VXX

þ 12g2AL2VAL;AL þ ð�g2AL2 þ gq1ðrrkS � brkBÞALþ gq2rSkSALÞVX;AL

þ12r2Vrr þ rgq1ALVr;AL þ r �brkB þ rrkS � gq1ALð ÞVrX

¼ 0; ð18Þ

VðT;X;AL; rÞ ¼ X2: ð19Þ
If there exists a smooth solution V of this equation, strictly convex with respect to X, then the optimal values of the investments are given by ckBðVX ;VXX ;VX;AL;VrXÞ ¼
1brr2

S VXX� fðr2

r þ r2S Þ þmSrr

� �VX þ rr2

S VrX þ grSðrrq2 � rSq1ÞALðVXX � VX;ALÞ� �

; ð20Þ

bkSðVX ;VXX ;VX;ALÞ ¼1

r2S VXX

�ðmS þ frrÞVX þ grSq2ALðVXX � VX;ALÞð Þ: ð21Þ

After substitution of these values in (18) we obtain that bV satisfies

Vt þ ðr � kÞX þ ðr � dÞALþ grS

fðq2rr þ q1rSÞ þmSq2ð ÞAL� �

VX

þ lþ nAL

wAL

� �ALVAL þ aðb� rÞVr þ

12ð1� q2

1 � q22Þg2AL2VXX þ

12g2AL2VAL;AL þ

12r2Vrr � ð1� q2

1 � q22Þg2AL2VX;AL þ gq1rALVr;AL

� 12r2

S

ðmS þ frrÞ2 þ f2r2S

� � V2X

VXXþ fgq1AL

VXVX;AL

VXXþ frVXVrX

VXX� 1

2r2 V2

rX

VXX� 1

2g2ðq2

1 þ q22ÞAL2 V2

X;AL

VXX� gq1rAL

VrXVX;AL

VXX¼ 0; ð22Þ


with the final condition (19). We will use a guessing method3 to solve (22), trying a quadratic solution of the form

3 Oncexistencfulfilledfor r an

bV ðt;X;AL; rÞ ¼ fXXðt; rÞX2 þ fAL;ALðt; rÞAL2 þ fX;ALðt; rÞXAL; ð23Þ

and the following ordinary differential equations are obtained for the above coefficients:

ðfXXÞt þ �f2 � ðmS þ frrÞ2

r2S

þ 2ðr � kÞ !

fXX þ ð2frþ aðb� rÞÞðfXXÞr � r2 ðfXXÞ2rfXX

þ r2

2ðfXXÞrr ¼ 0; f XXðT; rÞ ¼ 1: ð24Þ

ðfAL;ALÞt �14

f2 þ ðmS þ frrÞ2

r2S

� 2fgq1 þ g2ðq21 þ q2

2Þ !

f 2X;AL

fXXþ 2ðlþ nAL=wALÞfAL;AL

þ r � d� fgq1 þ gq2mS þ frr

rS� ð1� ðq2

1 þ q22ÞÞg2

� �fX;AL þ

r2

f� gq1ð Þ fX;ALðfX;ALÞrfXX

þ aðb� rÞðfAL;ALÞr þ ð1� q21 � q2

2Þg2fXX �r2

4ðfX;ALÞ2r

fXXþ g2fAL;AL þ

r2

2ðfAL;ALÞrr

þ 2gq1rðfAL;ALÞr ¼ 0; f AL;ALðT; rÞ ¼ 0:

ðfX;ALÞt þ �f2 � ðmS þ frrÞ2

r2S

þ fgq1 þ r � kþ lþ nAL

wAL

!fX;AL þ

r2

2ðfX;ALÞrr

þ 2 r � d� fgq1 þ gq2mS þ frr

rS

� �fXX þ frþ gq1rþ aðb� rÞð ÞðfX;ALÞr

þ ðf� gq1ÞrfX;ALðfXXÞr

fXX� r2 ðfX;ALÞrðfXXÞr

fXX¼ 0; f X;ALðT; rÞ ¼ 0: ð25Þ

In order to solve (24), we try fXXðt; rÞ ¼ gðtÞecðtÞr , with the final conditions gðTÞ ¼ 1 and cðTÞ ¼ 0, and after simplification we obtain

_g þ _c� acþ 2ð Þrg þ �r2

2c2 þ ð2frþ abÞc� 2k� f2 � ðmS þ frrÞ2

r2S

!g ¼ 0:

Choosing c, such that _c� acþ 2 ¼ 0, function g is given by _g þ hg ¼ 0, where

hðtÞ ¼ �ðr2=2Þc2ðtÞ þ ð2frþ abÞcðtÞ � 2k� f2 � ðmS þ frrÞ2=r2S :

With the final conditions we obtain

cðtÞ ¼ 2að1� e�aðT�tÞÞ

and gðtÞ ¼ eHðTÞ�HðtÞ with H a primitive of h. Hence we obtain

fXXðt; rÞ ¼ eHðTÞ�HðtÞþð2=aÞð1�e�aðT�tÞÞr:

Using Assumption C, it is easy to prove that function fX;AL, satisfying (25), is fX;AL ¼ 0. Inserting (23) into (20), (21) we obtain that the optimalinvestments are given by

k�Bðt;X;AL; rÞ ¼ 1b� fðr2

r þ r2S Þ þmSrr

rr2S

þ ðfXXÞrfXX

� �X þ gðrrq2 � rSq1Þ

brrSAL;

k�Sðt;X;AL; rÞ ¼ �mS þ rrf

r2S

X þ gq2

rSAL;

that is to say, (15) and (16), respectively, because fXXðt; rÞ ¼ gðtÞecðtÞr and fX;AL ¼ 0. �

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Documents

7. Case study ALM