CHAPTER Mean–Variance Management in Stochastic Aggregated Pension Funds with Nonconstant Interest Rate Ricardo Josa Fombellida CONTENTS 5.1 I ntroduction 5.2 The Pension Model 5.2.1 A ctuarial Functions 5.2.2 F inancial Market 5.2.3 F und Wealth 5.3 Pr oblem Formulation 5.4 Optimal Contribution, Optimal Portfolio, and Efficient Frontier 5.5 C onclusions Appendix Acknowledgments References 104 106 107 109 109 112 114 118 18 26 26 103 © 2010 by Taylor and Francis Group, LLC 104 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling I n t his ch a pter, w e study the optimal management of a defined benefit pension fund of aggregated type, which is common in the employment system We consider the case where the risk-free market interest rate is a time-dependent function and the benefits are given by a diffusion process increasing on average at an exponential rate The simultaneous aims of the sponsor are to maximize the terminal value of the expected fund’s assets a nd to m inimize t he contribution r isk a nd t he ter minal solvency risk The sponsor can invest the fund in a portfolio with n risky assets and a riskless asset The 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 5.1 INTRODUCTION The o ptimal ma nagement o f pens ion p lans i s a n i mportant sub ject o f study in the financial economic field, because fund managers make use of financial markets to assure t he f uture wealth of participants in t he pension funds during their retirement period There are two different ways to manage a pension plan: defined benefit (DB) plans and defined contribution (DC) plans In a DB plan, benefits are fi xed in advance by the sponsor and contributions are designed to maintain the fund in balance, that is to say, to amortize the fund according to a previously chosen actuarial scheme Future benefits due to participants are thus a liability for the sponsor, who bears the financial risk Thus, historically, D B p lans a re p referred b y pa rticipants This r isk is i ncreased with the formation of a risky portfolio, although it offers higher expected returns, w ith t he subseq uent pos sibility o f r educing t he a mortization quote It is the concern of the sponsor to drive the dynamic evolution of the f und, t aking i nto acco unt t he t rade-off be tween r isk a nd co ntribution In a DC plan, only the contributions are fixed in advance, while the benefits depend on the returns of the fund portfolio, administrated by the fund manager The associated financial risk is supported by the workers Our aim in this work is to study the optimal management of a DB pension plan of aggregated type using the dynamic programming approach The a nalysis of DB pension plans f rom t he dy namic optimization point of v iew 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 © 2010 by Taylor and Francis Group, LLC Mean–Variance Management ◾ 105 (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 These risk concepts are defined as quadratic deviations of fund wealth and amortization rates with respect to liabilities and normal cost, respectively Thus, the objective in a DB plan should be related to minimizing risks instead of only maximizing the fund’s assets The main concern of the sponsor is, of course, the solvency risk, related to the security of the pension fund in attaining the comprised liabilities The main aim of the plan manager is t he simultaneous maximization of t he t erminal val ue o f t he f und a nd t he minimization o f t he t erminal solvency risk (identified with the variance of the unfunded actuarial liability) a nd t he co ntribution r isk alo ng t he p lanning in terval H e o r 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 The 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 There are several previous papers in the literature dealing with the management 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) 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 n onconstant r iskless ma rket r ate o f i nterest to t he m odel a nd t o a ssume n onconstant pa rameters f or t he p rocesses defining the risky assets and the benefits In Josa-Fombellida and RincónZapatero (2008b), the minimization of risk in a DB pension plan is formulated a s a m ean–variance problem where t he ma nager c an select t he contributions and t he 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 i s necessary to provide a n ex tension of t he definitions of t he actuarial functions, because the technical rate of interest is not constant To so, we w ill adopt t he f ramework of Josa-Fombellida a nd R incónZapatero (2010), where the riskless rate of interest is stochastic The cha pter i s o rganized a s f ollows Section defines t he el ements of the pension scheme and describes the financial market where the fund operates We consider that the fund is invested in a portfolio with n risky © 2010 by Taylor and Francis Group, LLC 106 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling assets and a riskless asset Section 5.3 is devoted to formulating the management of the DB plan in a mean–variance framework, with the simultaneous objectives of minimizing the expected unfunded actuarial liability, as well as its variance at the final time, and to minimize the contribution rate risk over the planning interval The problem is solved in Section 5.4 providing the optimal strategies and the mean–variance efficient 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 benefits paid to t he participants at t he age of retirement are fixed in advance by the sponsor and are governed by an exogenous process whose source of randomness is correlated with the financial market The main elements intervening in a DB plan are the following: T: F(t): P(t): C(t): AL(t): NC(t): UAL(t): SC(t): M(x)%: δ(t): r(t): Planning horizon or date of the end of the pension plan, with < T < ∞ Value of fund assets at time t Benefits promised to the participants at time t They are related to the salary at the moment of retirement Contribution rate made by the sponsor at time t to the funding process Actuarial l iability a t t ime t, t hat i s, t otal l iabilities o f t he sponsor Normal cost at time t; i f t he f und a ssets match t he ac tuarial 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 Unfunded actuarial liability at time t, equal to AL(t) − F(t) Supplementary cost at time t, equal to C(t) − NC(t) Percentage of the value of the future benefits accumulated until age x ∈ [a,d], w here a is the common age of entrance in the fund and d is the common age of retirement Nonconstant r ate of va luation of t he l iabilities, wh ich c an be specified by the regulatory authorities Nonconstant risk-free market interest rate © 2010 by Taylor and Francis Group, LLC Mean–Variance Management ◾ 107 5.2.1 Actuarial Functions Following Josa-Fombellida and Rincón-Zapatero (2004), we suppose that disturbances exist that affect the evolution of benefits and hence the evolution 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 = {Gt}t≥0 is a complete and right continuous filtration generated by t he ( n + 1)-dimensional standard Brownian motion (w0, w1, …, wn)T, that is to say, Gt = σ{w0(s), w1(s), …, wn(s); ≤ s ≤ t} The st ochastic ac tuarial l iability a nd t he st ochastic n ormal cost a re defined as in the more general case where the rate of valuation is stochastic (see Josa-Fombellida and Rincón-Zapatero, 2010): ⎛ d ⎜ AL(t ) = E ⎜ e ⎜a ⎜⎝ t +d−x ⎛d NC(t ) = E ⎜ e ⎜ ⎜⎝ a t +d − x ∫ − ∫ δ ( s )ds t ∫ − ∫ δ ( s )ds t ⎞ ⎟ M(x ) P (t + d − x ) dx|Gt⎟ , ⎟ ⎟⎠ ⎞ M ′( x ) P (t + d − x ) dx |Gt⎟ , ⎟ ⎟⎠ for every t ≥ 0, where E(·|Gt) denotes conditional expectation with respect to the filtration Gt Thus, to compute the actuarial functions at time t, the manager 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 benefits 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 definitions 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 definitions can be expressed as d ∫ AL(t ) = e t +d−x − ∫ δ ( s )ds M (x )E (P (t + d − x )dx |Gt )dx , t a d ∫ NC(t ) = e t +d−x − ∫ δ ( s )ds t a © 2010 by Taylor and Francis Group, LLC M ′( x )E (P (t + d − x)dx |Gt )dx 108 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling A t ypical wa y o f m odeling P, f or a nalytical t ractability, i s t o co nsider that the benefits are given by a d iff usion process increasing, on average, at an exponential rate, extending the results obtained previously in JosaFombellida and Rincón-Zapatero (2010), where P is a geometric Brownian motion, and in Bowers et al (1986), where P is an exponential deterministic function This assumption is natural since, in general, benefits 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 This is the content of the following hypothesis Assumption 5.1: The benefit P satisfies dP (t ) = κ(t )P (t )dt + η(t )P (t )dB(t ), t ≥ 0, where κ(t) ∈ ℜ and η(t) ∈ ℜ+, for all t ≥ The initial condition P(0) = P0 is a random variable that represents the initial liabilities The beha vior o f t he ac tuarial f unctions A L a nd N C i s t hen g iven i n the following proposition, that can be seen as a particular case of Proposition 5.1 in Josa-Fombellida and R incón-Zapatero (2010) when κ, η are constants To this end, we define the following functions: d ∫ ψ AL (t ) = e t +d − x ∫ (κ ( s ) −δ( s )) ds t d ∫ M (x )dx , ψ NC (t ) = e t +d − x ∫ (κ ( s ) −δ( s )) ds t M ′(x )dx , a a d ∫ ξ AL (t ) = e t +d − x ∫ (κ ( s )−δ( s )) ds t (κ(t + d − x ) − δ(t + d − x ))M (x )dx a − (κ(t ) − δ(t ))ψ AL (t ) Proposition 5.1: Under Assumption 5.1 the actuarial functions satisfy AL = ψALP and NC = ψNCP, and they are linked by the identity (δ(t ) − κ(t ) − ξ AL (t ) ψ AL (t ))AL(t ) + NC(t ) − P (t ) = (5.1) for every t ≥ Moreover, the a ctuarial liability satisfies the stochastic differential equation © 2010 by Taylor and Francis Group, LLC Mean–Variance Management ◾ 109 dAL(t ) = (κ (t ) + ξ AL (t ) ψ AL (t ))AL(t )dt + η(t )AL(t )dB(t ), AL(0) = AL0 = ψ AL (0) P0 (5.2) 5.2.2 Financial Market In the rest of this section, we describe the financial market where the fund operates The plan sponsor manages the fund in the planning interval [0, T] by means of a portfolio formed by n + assets, as proposed in Merton (1971) One asset is a bond (riskless asset), whose price S0 evolves according to the differential equation dS0 (t ) = r (t )S0 (t )dt , S0(0) = 1, (5.3) and the remainder n assets are stocks (risky assets) whose prices {Si }ni=1 are modeled by the stochastic differential equations, generated by (w1, …, wn)T, n ⎛ ⎞ dSi (t ) = Si (t ) ⎜ bi (t )dt + ∑ σij (t )dw j (t )⎟ , Si (0) = si > 0, i = 1,2, , n (5.4) ⎝ ⎠ j =1 Here r(t) > denotes the short risk-free rate of interest, bi(t) > the mean rate of return of the ith risky asset, and σij(t) ≥ t he covariance between asset i and j, for all i, j = 1, 2, …, n It is assumed that bi(t) > r(t) for all i, so the sponsor has incentives to invest with risk We suppose that there exists the correlation qi ∈ [−1, 1] between B and wi, for i = 1, 2, …, n As a consequence, B is expressed in terms of {wi }in=0 as B(t ) = − q T qw0 (t ) + q T w (t ) , where qT q ≤ for q = (q1, q2, …, qn)T In this way, the influence of salary and inflation on the evolution of liabilities P is taken into account, as well as the effect of inflation on the prices of the assets 5.2.3 Fund Wealth To cover the liabilities in an efficient way, the manager creates a portfolio and designs a n a mortization scheme va rying w ith t ime The a mount of fund invested in time t in the risky asset Si is denoted by λi(t), i = 1, 2, …, n n The remainder, F (t ) − ∑ i =1 λ i (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 n sells a pa rt of his or her risky asset Si short while, if ∑ i =1 λ i is larger than F, © 2010 by Taylor and Francis Group, LLC 110 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling he or she then gets into debt to purchase the stocks, borrowing at the riskless interest rate r We suppose t hat t he investment strategy {Λ(t) :t ≥ 0}, with Λ(t) = (λ1(t), λ2(t), …, λn(t))T, is a control process adapted to filtration {Gt}t≥0, Gt-measurable, Markovian, and stationary, satisfying T ∫ E Λ(t )T Λ(t )dt < ∞, (5.5) where E is the expectation operator The contribution rate process C(t) is also an adapted process with respect to {Gt}t≥0 verifying T ∫ E SC (t )dt < ∞ (5.6) Therefore, the dynamic fund evolution under the investment policy Λ is* n dF (t ) = ∑ λ i (t ) i =1 n ⎞ dS (t ) dSi (t ) ⎛ F ( t ) ( t ) + − λ i ∑ ⎟⎠ S0 (t ) + (C(t ) − P(t ))dt (5.7) Si (t ) ⎜⎝ i =1 By substituting (5.3) and (5.4) in (5.7), we obtain n ⎛ ⎞ dF (t ) = ⎜ r (t )F (t ) + ∑ λ i (t )(bi (t ) − r (t )) + C(t ) − P (t )⎟ dt ⎝ ⎠ i =1 n n + ∑∑ λ i (t )σij (t )dw j (t ), (5.8) i =1 j =1 with initial condition F(0) = F0 > We w ill n ow a ssume t he ma trix n otation: σ(t) = (σij(t)), b(t) = (b1(t), b2(t), …, bn(t))T, = (1, 1, …, 1)T and Σ(t) = σ(t)σ(t)T We take a s g iven t he existence of Σ(t)−1, for all t, that is to say, σ(t)−1 Finally, the vector of standardized risk premia, or t he Sha rpe ratio of t he portfolio, is denoted by θ(t) = σ −1(t)(b(t) −r(t)1) So, we can write (5.8) as dF (t ) = (r (t )F (t ) + Λ(t )T (b(t ) − r (t )1) + C(t ) − P (t ))dt + Λ(t )T σ(t )dw(t ), (5 9) which, with the initial condition F(0) = F0, determines the fund evolution * This is the familiar equation obtained and justified in, for e xample, Merton (1990, p 124) The only difference is that the consumption is replaced here by P − C © 2010 by Taylor and Francis Group, LLC Mean–Variance Management ◾ 111 In order to obtain the optimal contribution and portfolio in explicit form, we give a neutral risk valuation of the technical rate of actualization 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: The technical rate of actualization i s δ(t) = r(t) + η(t) q Tθ(t), for all t ≥ Notice that if either benefits are deterministic or there is no correlation between benefits and t he financial market, t hen δ is t he risk-free rate of interest With positive (resp negative) correlation, the valuation of liabilities is r plus a positive (resp negative) term, weighted by the product of the instantaneous variance of P and the Sharpe ratio of the assets This is the right way t o price l iabilities, s ince w ith pos itive (resp negative) correlation it is expected that liabilities and assets should move in the same (resp opposite) direction Equation 5.9 in terms of X = F − AL and SC = C − NC is, by (5.2), dX (t ) = (r (t )F (t ) + Λ(t )T (b(t ) − r (t )1) + SC(t ) − NC(t ) − P (t ) − (κ(t ) + ξ AL (t ) ψ AL (t ))AL(t ))dt + Λ(t )T σ(t )dw(t ) − η(t )AL(t )dB(t ) By Proposition 5.1 and Assumption 5.2, the above can be written dX (t ) = (r (t ) X (t ) + Λ(t )T (b(t ) − r (t )1) + SC(t ) − η(t )q T θ(t )AL(t ))dt + Λ(t )T σ(t )dw(t ) − η(t )AL(t )dB(t ), and using the independent Brownian motions {wi }in=0, we obtain dX (t ) = (r (t ) X (t ) + Λ(t )T (b(t ) − r (t )1) + SC(t ) − η(t )q T θ(t )AL(t ))dt − η(t )AL(t ) − q T q dw0 (t ) +(Λ(t )T σ(t ) − η(t )AL(t )q T)dw(t ), (5.10) with the initial condition X(0) = X0 = F0 − AL0 © 2010 by Taylor and Francis Group, LLC 112 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 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 The same interpretation of the results is valid when the fund is overfunded, but then X is surplus 5.3 PROBLEM FORMULATION One o bjective of t he ma nager i s to ma ximize t he ex pected va lue of t he fund’s assets or, equivalently, to minimize t he ex pected u nfunded ac tuarial liability EUAL(T) = −EX(T) = −(EF(T) −EAL(T)) Note that, as we are supposing X < 0, we most often refer to X as debt Also, the aim is to minimize the variance of the terminal debt, Var X(T), and the contribution risk SC2 on the interval [0, T] This bi-objective problem reflects the promoter’s concern to increase fund assets to pay due benefits, but at the same time not subject the pension fund to large variations to provide stability to the plan The 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) Thus, we are considering a multi-objective optimization problem with two criteria: (SC, Λ )∈AX0 ,AL0 ( J1 (SC, Λ ), J (SC, Λ )) T ⎛ ⎞ = ⎜ − E X (T ), E SC (t )dt + Var X (T )⎟ , (SC, Λ )∈AX0 ,AL0 ⎜ ⎟⎠ ⎝ ∫ (5.11) subject to (5.10) and (5.2) Here AX0, AL0 is the set of measurable processes (SC,Λ), w here S C s atisfies (5.6), Λ satisfies (5.5) a nd such t hat (5.2) a nd (5.10) admit a unique solution Gt-measurable adapted to the filter {Gt}t≥0 Note that problems (5.2), (5.10), and (5.11) are the same mean–variance problems as in Josa-Fombellida and Rincón-Zapatero (2008b) and similar to the one studied in Zhou and Li (2000), but with the additional control variable SC in the state Equation 5.10, and an additional running cost in (5.11) An ad missible co ntrol p rocess (SC*, Λ*) i s Pareto e fficient (or s imply efficient) if there exists no admissible (SC, Λ) such that J1 (SC, Λ) ≤ J1 (SC*, Λ*), J (SC, Λ) ≤ J (SC*, Λ *), © 2010 by Taylor and Francis Group, LLC 114 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling The r elationship b etween problems (5.2), (5.10), (5.12) a nd (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 of (5.2), (5.10), (5.13) for γ = (2µ)−1 + EX*(T) The main consequence of Proposition 5.2 is that any optimal solution of problems (5.2), (5.10), and (5.12) can be f ound to solve problems (5.2), (5.10), and (5.13) This 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 efficient frontier for the original problem (5.2), (5.10), (5.11) is of quadratic type We first solve the problem (5.2), (5.10), (5.13), depending on the parameter γ THEOREM 5.1 The optimal rate of supplementary cost and the op timal investment in the risky assets are given by ⎛ −T r ( s )ds ⎞ ∫ ⎜ t SC *(t , X , AL) = f (t ) γe − X⎟ , ⎜ ⎟ ⎝ ⎠ (5.14) ⎛ −T r (s )ds ⎞ ∫ −1 ⎜ Λ *(t , X , AL) = Σ (t ) (b(t ) − r (r)1) γe t − X ⎟ + η(t )σ(t )− T qAL, ⎜ ⎟ ⎝ ⎠ (5.15) where f is the solution of the differential equation f (t ) = (−2r (t ) + θ(t )T θ(t )) f (t ) + f (t ), f (T ) = (5.16) Note that the function f satisfies f(t) ≥ 0, for all t ∈ [0, T], and if we assume that θTθ > 2r, then ≤ f(t) ≤ 1, for all t ∈ [0,T] In order to check this property, note that (5.16) implies (∂/∂t)ln f(t) = −2r(t) + θ(t)Tθ(t) + f(t) and then © 2010 by Taylor and Francis Group, LLC Mean–Variance Management ◾ 115 T T − ( − r +θ θ+ f )(s )ds f (t ) = f (t )/f (T ) = e ∫t ≥ On t he other hand, if we assume T that θ θ > 2r, b y i ntegrating t he d ifferential e quation ( 5.16), w e ob tain f (T ) − f (t ) = ∫ T t (−2r + θT θ)(s)ds + ∫ T t f (s)ds ≥ 0, and then f (t ) ≤ f (T ) = T − r ( s )ds − X (t ) , wh ich, The efficient st rategies depen d o n t he ter m γe ∫t by the definition of γ in Proposition 5.2, decomposes into three terms that we collect into two summands T e 2µ ∫ − r ( s )ds t ⎡ ⎛ −T r ( s )ds ⎤ ⎞ ⎢ ⎜ ∫t ⎥ + ⎢E e X (T )⎟ − X (t )⎥ ⎜ ⎟ ⎟⎠ ⎢ ⎜⎝ ⎥ ⎣ ⎦ The first summand is always positive, increasing with time, and depends inversely o n µ, t he pa rameter w eighing t he r elative i mportance o f t he objective of va riance m inimization w ith respect to t he objective of debt reduction The summand in brackets is the expected value of the planned 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 first 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 benefits If the actuarial liability AL is positively correlated with the financial 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 of the benefits, κ Theorem a lso g ives a l inear r elationship be tween t he su pplementary cost and investment strategies, whose vector coefficient is the optimal growth portfolio, Σ(t)−1(b(t) –r(t)1), multiplied by the inverse of f(t): Λ* = Σ(t )−1 (b(t ) − r (t )1)SC * + η(t )σ(t )− T qAL f (t ) (5.17) This c an be co nsidered a s a “ rule o f t humb” f or t he spo nsor: a t t ime t, each monetary unit of additional amortization with respect to the © 2010 by Taylor and Francis Group, LLC 116 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling computed normal cost, must be accompanied by an investment of Σ(t )−1 (b(t ) − r (t )1) monetary units in risky assets, plus η(t)σ(t)−T qAL f (t ) units due to the stochastic elements defi ning the pension plan The following result shows that the expected debt is linear with respect to the parameter γ and provides an explicit formula for the efficient frontier in terms of the expected returns and variance (disregarding the influence of the contribution risk) THEOREM 5.2 The expected optimal debt satisfies E X (T ) = α(T ) X0 + β(T )γ , (5.18) where T α(T ) = e ∫ − r (s ) d s f (0), T β(T ) = − e ∫ −2 r (s ) ds f (0), and the me an–variance efficient frontier of the problem (5.2), (5.10), (5.11) is given by Var X (T ) = ε (T ) + ε1 (T )EX (T ) + ε (T )(E X (T ))2 , (5 19) where the constants are ε (T ) = K (T ) T s ⎛ ⎞ T −2 ∫ r (u )du −2 ∫ r (u )du ⎛ ⎛ α ( T ) 2 T ⎜ ⎟ s × ⎜ X0 + X0 e f (s) − 2e f (0) + θ(s) θ(s) ⎜ ⎟ K (s) ⎜⎝ β2 (T ) ⎜⎝ ⎟⎠ ⎝⎜ ∫ T s − r ( s )ds ∫ (2(κ (u )+ξ AL(u )/ ψ AL(u ))+ η α(T ) ∫0 e X 02 + η2 (s)(1 − q Tq)AL20e −2 β(T ) © 2010 by Taylor and Francis Group, LLC ⎞ ⎞ ⎟ ds⎟ , ⎟ ⎟ ⎟⎠ ⎟ ⎠ (u )) du Mean–Variance Management ◾ 117 T T −2 ∫ r (u )du ⎛ −2α(T ) s X e ε1(T ) = K(T ) K(s) ⎜⎝ β2(T ) ∫ s T ⎛ ⎞ ⎞ −2 ∫ r (u )du − ∫ r ( s )ds T ⎜ ⎟ ⎟ 0 f (0) + θ(s) θ(s) + e X ds , × f (s) − 2e ⎜ ⎟ ⎟ ⎜⎝ ⎟⎠ β(T ) ⎟⎠ T s ⎛ ⎞ T −2 ∫ r (u )du −2 ∫ r (u )du K(T ) T ⎜ s f (0) + θ(s) θ(s)⎟ ds − e f (s) − 2e ε 2(T ) = ⎜ ⎟ β (T ) K(s) ⎝ ⎠ ∫ and t ∫ ( −2r +θTθ)( s )ds f 2(0) K(t ) = e , ∀t ∈[0, T ] f 2(t ) Remark 1: The op timal i nvestment d ecisions, c ontribution r ate a nd fund’s wealth evolution can be expressed in terms of the optimal expected debt at time T, EX*(T), instead of using the parameters γ or µ This provides a more clever interpretation of the results The substitution of γ may be d one f rom eq uality (5.18) Taking i nto acco unt (5.15) a nd (5.18), t he investment at instant t is T ⎛ ⎞ − ∫ r ( s )ds −1 ⎜ t Λ*(t , X , AL) = Σ(t ) (b(t ) − r (r )1) e (z − α(T ) X0 ) − X ⎟ ⎜ β(T ) ⎟ ⎜⎝ ⎟⎠ + η(t )σ(t )− T qAL, where z = EX*(T) This shows t he ex isting relation be tween t he de sired expected levels of debt at time T and the optimal composition of the portfolio at every instant of time t Analogously, (5.14) a nd (5.18) a llow u s t o r ewrite t he o ptimal r ate o f contribution at instant t as © 2010 by Taylor and Francis Group, LLC 118 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling T ⎛ ⎞ − ∫ r ( s )ds ⎜ t e C *(t , X ) = NC(t ) + f (t ) (z − α(T ) X0 ) − X ⎟⎟ ⎜ β(T ) ⎝ ⎠ By (5.27), the optimal fund satisfies T T dX(t ) = ((r(t ) − θ(t ) θ(t ) − f (t )) X(t ) + ∫ e t β(T ) − r ( s )ds (z − α(T ) X0 ) × (θ(t ) T θ(t ) + f (t )dt ) − η(t ) − q Tq AL(t )dw0(t ) + θ(t )T T ⎛ ⎞ − ∫ r ( s )ds ⎜ × e t (z − α(T ) X0 ) − X(t )⎟ dw(t) ⎜ β(T ) ⎟ ⎜⎝ ⎟⎠ with X(0) = X0 5.5 CONCLUSIONS We ve a nalyzed t he ma nagement o f a pens ion-funding p rocess o f a n aggregated D B pens ion p lan wh ere t he ben efits a re st ochastic a nd t he riskless market interest rate is nonconstant The objective is to determine the contribution rate and investment strategies, maximizing the expected terminal f und, a nd at t he s ame t ime m inimizing both t he contribution and t he so lvency r isk The p roblem i s f ormulated a s a m odified mean– variance optimization problem and has been solved by means of dynamic programming techniques We find that there is a linear relationship between the optimal supplementary cost a nd the vector of efficient investment strategies, with a co rrection term due to the random behavior of benefits The mean–variance efficient frontier is of a quadratic type, that is to say, the terminal solvency risk has a parabolic dependence on the expected terminal unfunded actuarial liability APPENDIX A Proof of Proposition 5.1: tation is By Assumption 5.1, the conditional expect +d−x E(P (t + d − x ) |Gt ) = (P t )e © 2010 by Taylor and Francis Group, LLC ∫ t κ ( s )ds , Mean–Variance Management ◾ 119 thus, recalling the definition of AL and ψAL, we get t +d − x d ∫ AL(t ) = e − ∫ δ( s)ds M (x )E( P (t + d − x ) Gt )dx t a d ∫ t +d − x = P(t ) e ∫ ( κ ( s ) −δ( s )) ds t M (x )dx = P(t )ψ AL(t ) a Analogously, NC(t) = ψNC(t)P(t) Now, by means of an integration by parts, and the definition of ξAL , we have t +d − x d ∫ ψ NC (t ) = e ∫ (κ ( s )−δ( s ))ds t dM(x ) a t +d − x =e ∫ (κ ( s )−δ( s ))ds t t +d − x d + ∫ M (x ) |xx == da e ∫ (κ ( s )−δ ( s )) ds t (κ(t + d − x ) − δ(t + d − x ))M(x )dx a = + ξ AL (t ) + (κ(t ) − δ(t ))ψ AL (t ) In consequence NC(t ) = ψ NC (t )P (t ) = P (t ) + ξ AL(t )P (t ) + (κ(t ) − δ(t ))ψ AL(t )P (t ) ⎛ ξ AL(t ) ⎞ = P (t ) + ⎜ κ(t ) − δ(t ) + AL(t ), ψ AL(t ) ⎟⎠ ⎝ which is (5.1) Finally, we deduce the stochastic differential equation that the actuarial liability satisfies Notice that dψAL(t) = ξAL(t)dt Thus, using Assumption 5.1 © 2010 by Taylor and Francis Group, LLC 120 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling AL(t ) = d(ψ AL P )(t ) = ψ AL (t )dP (t ) + dψ AL (t )P (t ) = ψ AL (t )P (t )(κ(t )dt + η(t )dB(t )) + ξ AL (t )P (t )dt ⎛ ξ (t ) ⎞ = ⎜ κ(t ) + AL ⎟ AL(t )dt + η(t )AL(t )dB(t ) ψ AL (t ) ⎠ ⎝ with the initial condition AL(0) = AL0 = ψAL(0)P0 Proof of Proposition 5.2: See proof of Proposition 5.2 in Josa-Fombellida and Rincón-Zapatero (2008b) Proof of Theo rem 5.1: In order to prove this result, we use the dynamic programming approach, see Fleming and Soner (1993) Consider the value function of the control problem (5.2), (5.10), (5.13), Vˆ(t , X , AL) = (SC,Λ )∈AX, AL {J ((t , X , AL);SC, Λ) : s.t (5.2),(5.10)} ˆ i s t he solution of t he Ha milton–Jacobi–Bellman It is well known that V equation: Vt + {SC + (rX + Λ T (b − r1) + SC − ηq T θAL)VX (SC, Λ ) + (κ + ξ AL/ψ AL )ALVAL + (Λ T ΣΛ − η AL Λ T σq + η2 AL)VXX ⎫ + η2 AL2VAL,AL + (η AL Λ T σq − η2 AL2 )VX,AL⎬ = 0, (5.20) ⎭ V(T , X , AL) = X − 2γX (5.21) Note that in (5.20) we have used (5.10) and the stochastic differential equation of AL as a function of the Brownian motions {wi }in=0, obtained from (5.2), that is © 2010 by Taylor and Francis Group, LLC Mean–Variance Management ◾ 121 dAL(t ) = (κ(t ) + ξ AL (t ) ψ AL (t ))AL(t )dt + η(t )AL(t ) − q T q dw0(t ) + η(t )AL(t )q T dw(t ) If there exists a s mooth solution V of this equation, strictly convex with respect t o X, t hen t he m inimizer va lues o f t he su pplementary cost a nd investments are given by SC(VX ) = − ⎛ V ⎞ VX V , Λ(VX,VXX ,VX,AL) = −Σ −1 (b − r1) X + ηALσ − T q ⎜ − X, AL ⎟ ⎝ VXX VXX ⎠ (5.22) ˆ to satisfy After substitution of these values in (5.20), we get V 1 V2 Vt + rXVX − VX2 − θT θ X + (κ + ξ AL /ψ AL )ALVAL + η2 AL2VAL,AL VXX VX,AL + η2 AL2 (1 − q Tq)VXX − η2 AL2 (1 − q Tq)VX,AL − η AL θT qVX VXX VX,AL − η2 AL2q Tq = 0, VXX with the final condition (5.21) We try a quadratic solution of the form Vˆ (t , X , AL) = β0(t ) + β X (t ) X + β AL (t )AL + β XX (t ) X + β AL,AL (t )AL2 + β X,AL(t ) XAL, so that, from (5.22), the optimal controls must be ⎛ −β X ⎞ ⎛ β β Λ = Σ −1 (b − r1) ⎜ − X − X,AL AL⎟ + η AL σT q ⎜ − X,AL β XX 2β XX ⎝ 2β XX ⎠ ⎝ ⎞ ⎟⎠ , (5.23) ⎛ −β X ⎞ β − X − X, AL AL⎟ SC = − (β X + 2β XX X + β X,AL AL) = β XX ⎜ 2β XX ⎝ 2β XX ⎠ © 2010 by Taylor and Francis Group, LLC 122 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling The following ordinary differential equations are obtained for t he above coefficients appearing in previous identities: β X = (−r + θ Tθ) β X + β X β XX, β X (T ) = −2γ , β XX = (−2r + θT θ) β XX + β2XX, (5.24) β XX (T ) = 1, (5 25) β X,AL = (−r − κ + ξ AL /ψ AL + η + θT θ) β X,AL + β XX β X,AL , β X,AL(T ) = (5.26) We a ssume t hat E quation 25, o f R icatti t ype, s a u nique so lution βXX(t) = f(t) From Arnold (1974 p 139), we obtain the solution to (5.24): T β X (t ) = −2 γe ∫ − r ( s )ds t f (t ) Substituting in (5.26) is given by β X,AL = (−r − κ + ξ AL /ψ AL + η + θT θ + f )β X,AL, β X,AL(T ) = 0, that is to say, βX,AL = I nserting these expressions into (5.23), we obtain (5.14) and (5.15), respectively Proof of Theo rem 5.2: Under the optimal feedback control (5.14), (5.15), the stochastic differential equation for process X, (5.10), is T ⎛ ⎞ − ∫ r ( s )ds T T ⎜ ⎟ dt t dX (t ) = (r (t ) − θ(t ) θ(t ) − f (t ))X (t ) + (θ(t ) θ(t ) + f (t ))γe ⎜ ⎟ ⎜⎝ ⎟⎠ ⎛ −T r ( s)ds ⎞ ∫ T T⎜ −η(t ) − q q AL(t )dw0 (t ) + θ(t ) γe t − X (t )⎟ dw(t ), ⎜ ⎟ ⎜⎝ ⎟⎠ (5.27) © 2010 by Taylor and Francis Group, LLC Mean–Variance Management ◾ 123 with X(0) = X0 Applying Ito’s formula to X2 we obtain T ⎛ − ∫ r ( s )ds dX 2(t ) = ⎜ (r (t ) − θ(t )T θ(t ) / − f (t ))X 2(t ) + f (t )γe t X (t ) ⎜ ⎜⎝ T ⎞ −2 ∫ r ( s )ds 1 T T t + θ(t ) θ(t )γ e + η (t )(1 − q q)AL (t )⎟ dt ⎟ 2 ⎟⎠ ⎛ −T r ( s )ds ⎞ ∫ T T⎜ t X (t ) − X (t )⎟ dw(t ), − η(t ) − q q AL(t )dw0 (t ) + 2θ(t ) γe ⎜ ⎟ ⎜⎝ ⎟⎠ with X 2(0) = X02 Taking expectations on both previous stochastic differential equations, we get functions m1(t) = EX(t) and m2(t) = EX2 (t) to satisfy the linear ordinary differential equations T m1(t ) = (r (t ) − θ(t )Tθ(t ) − f (t ))m1(t ) + (θ(t )T θ(t ) + f (t ))γe ∫ − r ( s )ds t , m1(0) = X , T m2 (t ) = (2r (t ) − θ(t )T θ(t ) − f (t ))m2 (t ) + f (t )γe ∫ − r ( s )ds t m1(t ) T + θ(t )T θ(t )γ e ∫ −2 r ( s )ds t + η2 (t )(1 − q Tq)E AL2(t ), m2(0) = X02, (5.28) t −2 ∫ (2(κ ( s )+ξAL ( s )/ ψ AL ( s ))+ η2 ( s ))ds where E AL2 (t ) = AL20 e Following Arnold (1974, p 139) t , by (5.2) m1(t ) = E X (t ) = ∫ (r −θTθ− f )( s )ds e0 s T ⎛ ⎞ t − (r −θTθ− f )(v ) dv ∫ ∫ r (u)du T ⎜ s × X0 + γ e ds ⎟ , (θ(s) θ(s) + f (s))e ⎜ ⎟ ⎜⎝ ⎟⎠ ∫ © 2010 by Taylor and Francis Group, LLC 124 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling which, after some calculations, and by (5.16), is t E X (t ) = e ∫ − r ( s )ds s T ⎛ ⎞ t ∫ r (u )du ∫ r (u )du T f (0) ⎜ f (s ) ⎟ s X0 + γ e e ds (θ θ + f )(s) f (t ) ⎜⎜ f (0) ⎟⎟ ⎝ ⎠ ∫ t =e ∫ − r ( s )ds T f (0) X0 + e f (t ) ∫ − r ( s )ds t t t γ e f (t ) ∫ ∫ −2 r (u )ds s (θTθ + f )(s) f (s)ds (5.29) = α(t ) X + γβ(t ), where* t α(t ) = e ∫ − r ( s )ds T β(t ) = e ∫ − r ( s )ds t f (0) , f (t ) T ⎛ ⎞ −2 ∫ r ( s )ds f (0) ⎜1 − e t ⎟, ⎜ f (t ) ⎟ ⎝ ⎠ for all t ∈ [0,T] For t = T, using the final condition in (5.16), we have (5.18) Analogously, t ⎛ ⎞ m2(t ) = EX (t ) = K(t ) ⎜ X0 + N (s)ds⎟ , K(s) ⎜⎝ ⎟⎠ ∫ where for all s ∈ [0,T], s ∫ (2r −θTθ−2 f )(u )du K (s ) = e s ∫ ( −2r +θTθ)(u )du f 2(0) , = e0 f (s ) * By (5.16), θTθf + f = f + 2rf and t t ⎛ −2 t r(u)du ⎞ −2 ∫ r(u)du −2 ∫ r (u )du ∫ ⎜ ⎟ s s f (s ) = e ( f + 2rf )(s) = e s (θTθf + f )(s) (∂ / ∂s )⎜ e ⎟ ⎜⎝ ⎟⎠ © 2010 by Taylor and Francis Group, LLC Mean–Variance Management ◾ 125 T N (s) = f (s)γe ∫ − r ( u )du s T m1 (s) + θ(s)T θ(s)γ e ∫ −2 r ( u)du s + η2 (s)(1 − q T q)E AL2 (s) By 5( 29), N (s) = a0 (s) + a1 (s)γ + a2 (s)γ 2, where a0 (s) = η2 (s)(1 − q T q)E AL2 (s), T a1 (s) = f (s)e ∫ − r ( u )du s T α(s) = 2e T a2 (s) = f (s)e ∫ − r (u )du s ∫ − r ( u )du X0 , T β(s) + e ∫ −2 r (u )du θ(s)T θ(s) T =e ∫ −2 r (u )du s T (2 f (s) + θ(s)T θ(s)) − 2e ∫ −2 r (u )du For t = T, if we define T bi (T ) = ∫ K(s) a (s)ds, i for i = 1, 2, 3, the terminal solvency risk is Var X (T ) = E X (T ) − (EX (T ))2 = K (T )( X02 + b0 (T ) + b1 (T )γ + b2 (T )γ ) − (E X (T ))2 ⎛ α (T ) α(T ) b2 (T ) X02 − b1 (T ) X0 + b0 (T ) + X02 = K (T ) ⎜ β(T ) ⎝ β (T ) ⎛ −2α(T ) ⎞ b2 (T ) X0 + b1 (T )⎟ E X (T ) +⎜ β(T ) ⎝ β (T ) ⎠ ⎛ 1 ⎞ (E X (T ))2 , b2 (T ) − +⎜ ⎟ K ( T ) ⎝ β (T ) ⎠ by (5.18), that is (5.19) © 2010 by Taylor and Francis Group, LLC f (0) 126 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling ACKNOWLEDGMENTS The aut hor g ratefully a cknowledges financial su pport f rom t he Reg ional Government o f C astilla y L eón (Spain) u nder p roject VA004B08 a nd t he Spanish Ministerio de Ciencia e Innovación under project ECO2008-02358 REFERENCES Arnold, L., 1974 Stochastic Differential Equations Theory and Applications John Wiley & Sons, New York Bowers, N.L., G erber, H.U., Hickman, J.C., Jones, D.A., a nd Nesbitt, C.J., 1986 Actuarial Mathematics The Society of Actuaries, Ithaca, NY Cairns, A.J.G., 2000 Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time Astin Bulletin 30, 19–55 Chang, S.C., 1999 Op timal pension funding through dynamic simulations: The case of Taiwan public employees retirement system Insurance: Mathematics and Economics 24, 187–199 Chang, S.C., Tsai, C.H., Tien, C.J., and Tu, C.Y., 2002 Dynamic funding and 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selection with liability: Mean–variance model and stochastic LQ approach Insurance: Mathematics and Economics 42, 943–953 Zhou, X.Y and Li, D., 2000 Continuous-time mean–variance portfolio selection: A st ochastic L Q f ramework Applied M athematics a nd O ptimization 42, 19–33 © 2010 by Taylor and Francis Group, LLC ... market interest rate is nonconstant The objective is to determine the contribution rate and investment strategies, maximizing the expected terminal f und, a nd at t he s ame t ime m inimizing both... say, βX,AL = I nserting these expressions into (5. 23), we obtain (5. 14) and (5. 15) , respectively Proof of Theo rem 5. 2: Under the optimal feedback control (5. 14), (5. 15) , the stochastic differential... (5. 2), (5. 10), (5. 12) a nd (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