Chapter 5 mean variance management in stochastic aggregated pension funds with nonconstant interest rate

Mean–Variance Management in Stochastic Aggregated Pension Funds 5.4 Optimal Contribution, Optimal Portfolio,... Keywords: pension funding, stochastic control, portfolio theory, mean–

Mean–Variance

Management in

Stochastic Aggregated Pension Funds

5.4 Optimal Contribution, Optimal Portfolio,

In t his ch a pter, w e study the optimal management of a defi ned

ben-efi t pension fund of aggregated type, which is common in the ment system We consider the case where the risk-free market interest rate

employ-is a time-dependent function and the benefi ts are given by a diff usion cess increasing on average at an exponential rate Th e simultaneous aims

pro-of the sponsor are to maximize the terminal value pro-of the expected fund’s assets and to minimize the contribution risk and the terminal solvency

risk Th e sponsor can invest the fund in a portfolio with n risky assets and

a riskless asset Th e problem is mathematically formulated by means of a continuous-time mean–variance portfolio selection model and solved by means of optimal stochastic control techniques

Keywords: pension funding, stochastic control, portfolio theory,

mean–variance, nonconstant interest rate

Th e analysis of DB pension plans from the dynamic optimization point

of v iew ha s be en w idely d iscussed i n t he l iterature; se e, f or ex ample, Haberman and Sung (1994), Chang (1999), Cairns (2000), Haberman et al

(2000), Josa-Fombellida and Rincón-Zapatero (2001, 2004, 2006, 2008a,b, 2010), Taylor (2002), and Chang et al (2002) It is generally accepted that managers’ objectives should be related to the minimization of the solvency risk and the contribution risk Th ese risk concepts are defi ned as quadratic deviations of fund wealth and amortization rates with respect to liabilities and normal cost, respectively Th us, the objective in a DB plan should be related to minimizing risks instead of only maximizing the fund’s assets

Th e main concern of the sponsor is, of course, the solvency risk, related to the security of the pension fund in attaining the comprised liabilities

Th e main aim of the plan manager is t he simultaneous maximization

of the terminal value of the fund and the minimization of the terminal solvency risk (identifi ed with the variance of the unfunded actuarial lia-bility) and t he contribution r isk along t he planning interval He or she thus co nsiders a m ultiobjective p rogramming o ptimization p roblem o f portfolio selection where the attainment of the highest possible expected return with the lowest possible variance is desired Th e problem is settled

in the familiar mean–variance framework, translating the static model of Markowitz (1952) to the continuous-time setting of a DB plan that evolves with time

Th ere are several previous papers in the literature dealing with the agement of pension funds, containing dynamic models of mean–variance; see, for example, Chiu and Li (2006), Josa-Fombellida and Rincón-Zapatero (2008b), Delong et al (2008), Chen et al (2008), and Xie et al (2008)

man-In this chapter, we provide an extension of a previous work by the authors, Josa-Fombellida and Rincón-Zapatero (2008b), in an attempt to incorporate a de terministic nonconstant riskless market rate of interest

to t he m odel a nd t o a ssume n onconstant pa rameters f or t he processes defi ning the risky assets and the benefi ts In Josa-Fombellida and Rincón-Zapatero (2008b), the minimization of risk in a DB pension plan is for-mulated as a m ean–variance problem where the manager can select the

contributions and the investments in a po rtfolio with n risky assets and

a r iskless a sset W ith r espect t o J osa-Fombellida a nd R incón-Zapatero (2008b), it is necessary to provide an extension of the defi nitions of the actuarial functions, because the technical rate of interest is not constant

To do so, we will adopt the framework of Josa-Fombellida and Zapatero (2010), where the riskless rate of interest is stochastic

Rincón-Th e chapter is organized as follows Section 5.2 defi nes t he elements

of the pension scheme and describes the fi nancial market where the fund

operates We consider that the fund is invested in a portfolio with n risky

assets and a riskless asset Section 5.3 is devoted to formulating the agement of the DB plan in a mean–variance framework, with the simulta-neous objectives of minimizing the expected unfunded actuarial liability,

man-as well man-as its variance at the fi nal time, and to minimize the contribution rate risk over the planning interval Th e problem is solved in Section 5.4

providing the optimal strategies and the mean–variance effi cient frontier Finally, Section 5.5 establishes some conclusions All proofs are developed

in Appendix A

5.2 THE PENSION MODEL

Consider a DB pension plan of aggregated type where, at every instant of time, active participants coexist with retired participants We suppose that the benefi ts paid to the participants at the age of retirement are fi xed in advance by the sponsor and are governed by an exogenous process whose source of randomness is correlated with the fi nancial market

Th e main elements intervening in a DB plan are the following:

T: Planning horizon or date of the end of the pension plan, with

0 < T < ∞.

to the salary at the moment of retirement

process

AL(t): Actuarial l iability a t t ime t, t hat i s, t otal l iabilities o f t he

sponsor

NC(t): Normal cost at time t; if t he f und assets match t he actuarial

liability, and if there are no uncertain elements in the plan, the normal cost is the value of the contributions allowing equality between asset funds and liabilities

UAL(t): Unfunded actuarial liability at time t, equal to AL(t) − F(t) SC(t): Supplementary cost at time t, equal to C(t) − NC(t).

age x ∈ [a,d], where a is the common age of entrance in the fund and d is the common age of retirement.

δ(t): Nonconstant rate of valuation of the liabilities, which can be

specifi ed by the regulatory authorities

5.2.1 Actuarial Functions

Following Josa-Fombellida and Rincón-Zapatero (2004), we suppose that disturbances exist that aff ect the evolution of benefi ts and hence the evolu-tion of the normal cost and the actuarial liability To model this randomness,

we consider a probability space (Ω,G,P), where P is a probability measure

on Ω and G = {G t}t≥0 is a complete and right continuous fi ltration generated

by t he (n + 1)-dimensional standard Brownian motion (w0, w1, …, w n)T,

that is to say, G t = σ{w0(s), w1(s), …, w n (s); 0 ≤ s ≤ t}.

Th e st ochastic ac tuarial l iability a nd t he st ochastic n ormal cost a re defi ned as in the more general case where the rate of valuation is stochastic (see Josa-Fombellida and Rincón-Zapatero, 2010):

s s d

t a

d s s

t a

for every t ≥ 0, where E(·|G t) denotes conditional expectation with respect to

the fi ltration G t Th us, to compute the actuarial functions at time t, the

man-ager makes use of the information available up to that time, in terms of the

conditional expectation In this way, AL(t) is the total expected value of the promised benefi ts accumulated according to M, discounted at the rate δ(t) Analogous comments can be given concerning the normal cost NC(t) with function M′ Note that previous defi nitions extend that of Josa-Fombellida and Rincón-Zapatero (2004, 2008b), where r and therefore δ are constants.

Using basic properties of the conditional expectation, the previous defi nitions can be expressed as

-( )d

t d x t

d s s

t a

d s s

t a

+ −

−∫ δ

A t ypical wa y o f m odeling P, f or a nalytical t ractability, i s t o co nsider

that the benefi ts are given by a d iff usion process increasing, on average,

at an exponential rate, extending the results obtained previously in

Josa-Fombellida and Rincón-Zapatero (2010), where P is a geometric Brownian motion, and in Bowers et al (1986), where P is an exponential determinis-

tic function Th is assumption is natural since, in general, benefi ts depend

on the salary and the population plan, which on average show exponential growth subject to random disturbances that may be supposed proportional

to the variables’ size Th is is the content of the following hypothesis

Assumption 5.1: Th e benefi t P satisfi es

d ( )P t ( ) ( )dt P t t ( ) ( )d ( ),t P t B t t 0,

a random variable that represents the initial liabilities.

Th e behavior of t he ac tuarial f unctions A L a nd NC is t hen g iven i n the following proposition, that can be seen as a particular case of Propo-sition 5.1 in Josa-Fombellida and Rincón-Zapatero (2010) when κ, η are constants To this end, we defi ne the following functions:

d s s s a

Proposition 5.1: Under Assumption 5.1 the actuarial functions satisfy

AL = ψALP and NC = ψNCP, and they are linked by the identity

( ( )t ( )t ( )t ( ))AL( ) NC( )t t t P t( ) 0 (5.1)

for every t ≥ 0 Moreover, the actuarial liability satisfi es the stochastic ferential equation

sponsor has incentives to invest with risk We suppose that there exists

the correlation q i ∈ [−1, 1] between B and w i , for i = 1, 2, …, n As a quence, B is expressed in terms of { }n=0

fund invested in time t in the risky asset S i is denoted by λi (t), i = 1, 2, …, n

Th e remainder, F t( )−∑n i=1λi( )t , i s i nvested i n t he bo nd B orrowing

and shortselling is allowed A negative value of λi means that the sponsor

sells a pa rt of his or her risky asset S i short while, if ∑=1λ

n i

i is larger than F,

he or she then gets into debt to purchase the stocks, borrowing at the

risk-less interest rate r We suppose that the investment strategy {Λ(t) : t ≥ 0}, with Λ(t) = (λ1(t), λ2(t), …, λ n (t))T, is a control process adapted to fi ltration

{G t}t≥0 , G t-measurable, Markovian, and stationary, satisfying

∫ T 0

with initial condition F(0) = F0 > 0

We w ill n ow a ssume t he ma trix n otation: σ(t) = (σ ij (t)), b(t) = (b1(t),

b2(t), …, b n (t))T, 1 = (1, 1, …, 1)T and Σ(t) = σ(t)σ(t)T We take as given the

existence of Σ(t)−1, for all t, that is to say, σ(t)−1 Finally, the vector of dardized risk premia, or the Sharpe ratio of the portfolio, is denoted by

stan-θ(t) = σ −1(t)(b(t) − r(t)1) So, we can write (5.8) as

T T

which, with the initial condition F(0) = F0, determines the fund evolution

* Th is is the familiar equation obtained and justifi ed in, for e xample, Merton (1990, p 124)

Th e only diff erence is that the consumption is replaced here by P − C.

In order to obtain the optimal contribution and portfolio in explicit form, we give a neutral risk valuation of the technical rate of actualiza-tion δ, as in Josa-Fombellida and Rincón-Zapatero (2008b) We suppose

the value of δ is a modification of the short rate of interest r, taking into

account the sources of uncertainty and the stock coefficients

Assumption 5.2: Th e technical rate of actualization is δ(t) = r(t) + η(t)

To fi x t he nomenclature, t hroughout t his work, we w ill suppose t hat

the fund is underfunded at time 0, X0 < 0, so that X has the meaning of

debt Th e same interpretation of the results is valid when the fund is

over-funded, but then X is surplus.

SC2 on the interval [0, T] Th is bi-objective problem refl ects the promoter’s

concern to increase fund assets to pay due benefi ts, but at the same time not subject the pension fund to large variations to provide stability to the plan Th e minimization of the contribution risk (related to the stability of the plan) has been considered in other works such as Haberman and Sung (1994), Haberman et al (2000), and Josa-Fombellida and Rincón-Zapatero (2001, 2004)

Th us, we are considering a multi-objective optimization problem with two criteria:

An ad missible control process (SC*, Λ*) i s Pareto e ffi cient (or simply

1(SC, ) 1(SC*, *), 2(SC, ) 2(SC*, *),

with at least one of the inequalities being strict Th e pairs ( (SC*, *),J1 Λ

2

2(SC*, *))

J Λ ∈ℜ for m t he Pareto f rontier We will call SC* an effi cient supplementary cost, C* = SC* + NC an effi cient contribution rate and Λ*

an effi cient portfolio Th roughout this chapter, the term optimal must be

understood in t he sense o f effi ciency Actually, we a re not interested in the representation and properties of the Pareto frontier, but in the pairs

(−EX(T), Var X(T)) for optimal X(T), that we call the mean–variance effi cient frontier.

-According to Da Cunha and Polak (1967), when the objective als defi ning t he multiobjective program a re convex, t he Pareto optimal points can be found to solve a scalar optimal control problem, where the dynamics remain the same and the objective functional is a convex com-bination of the original cost functionals In our case, Equations 5.2 and

function-5.10 are linear, so both J1 and J2 are obviously convex Th erefore, the nal problems (5.2), (5.10), and (5.11) are equivalent to the scalar problem

0

X AL X

A

T A

subject to (5.2) and (5.10), with µ > 0 being a weight parameter As µ varies in the interval (0, ∞), the solutions of (5.12) describe the Pareto frontier Notice that µ serves the manager to linearly transfer units of risk to units of expected return, and vice versa Th e size of µ indicates which one of the objectives is of greater concern to the manager, to reduce risk or to reduce debt

Problems (5.2), (5.10), a nd (5.12) a re not st andard stochastic optimal

problems d ue t o t he ter m ( EX(T))2 i n t he va riance, a nd t he dy namic programming approach cannot be a pplied here Following Zhou and Li (2000) or Josa-Fombellida and Rincón-Zapatero (2008b), we propose an auxiliary problem that turns out to be a stochastic problem of linear qua-dratic type:

Th e relationship between problems (5.2), (5.10), (5.12) and (5.2), (5.10), (5.13) is shown in the following result.

Proposition 5.2: For any µ > 0, if (SC*, Λ*) is an optimal control of (5.2),

(5.10), (5.12) with associated optimal debt X*, then it is an optimal control

Th e main consequence of Proposition 5.2 is that any optimal solution

of problems (5.2), (5.10), and (5.12) can be found to solve problems (5.2), (5.10), and (5.13) Th is will be done in Section 5.4

5.4 OPTIMAL CONTRIBUTION, OPTIMAL PORTFOLIO, AND EFFICIENT FRONTIER

In this section, we show that the mean–variance effi cient frontier for the original problem (5.2), (5.10), (5.11) is of quadratic type We fi rst solve the problem (5.2), (5.10), (5.13), depending on the parameter γ

THEOREM 5.1

risky assets are given by

Note that the function f satisfi es f(t) ≥ 0, for all t ∈ [0, T], and if we assume

that θTθ > 2r, then 0 ≤ f(t) ≤ 1, for all t ∈ [0,T] In order to check this erty, note that (5.16) implies (∂/∂t)ln f(t) = −2r(t) + θ(t)Tθ(t) + f(t) and then

prop-− ∫ − +θ θ+

that θTθ > 2r, b y i ntegrating t he d iff erential e quation (5.16), w e ob tain

by the defi nition of γ in Proposition 5.2, decomposes into three terms that

we collect into two summands

debt reduction, valued at time t Notice from the expression of SC* t hat,

if this reduction is positive, then the amortization rate is higher than the

normal cost, because f is a nonnegative function In the same way, the fi rst

summand in Λ* is also positive Of course, this behavior is also observed for small values of µ, even if there is no reduction of the expected debt As the control of variance becomes less important for the sponsor, that is, µ decreases, the investment strategies are riskier

In contradistinction to the supplementary cost, the optimal investment also depends on AL and on the elements giving the randomness of assets and benefi ts If the actuarial liability AL is positively correlated with the

fi nancial market (an extreme case being uncorrelated, where q = 0), then

the investment in the risky assets is greater than that if the correlation is negative It is also remarkable that it does not depend on the rate of growth

Th is can be co nsidered as a “ rule of t humb” for t he sponsor: at t ime t,

each monetary unit of additional amortization with respect to the