CHAPTER Pension Funds under Inflation Risk Aihua Zhang CONTENTS 4.1 I 4.2 4.3 4.4 ntroduction Investment Problem for a DC Pension Fund How to Solve It Optimal Management of the Pension Fund without Liquidity Constraints 96 Acknowledgments References 86 87 92 00 01 T his ch a pter i nv esti gat es a n optimal investment problem faced by a defined contribution (DC) pension fund manager under inflationary risk It is assumed that a representative member of a DC pension plan contributes a fi xed share of his salary to the pension fund during the time horizon [0,T] The pension contributions a re i nvested continuously i n a risk-free bond, an index bond, and a stock The objective is to maximize the expected utility of the terminal value of the pension fund By solving this investment problem, we present a way to deal with the optimization problem, in case there is an (positive) endowment (or contribution), using the martingale method Keywords: Pension funds, inflation, optimal portfolios, martingale method JEL Classifications: C61; G11; G12; G31 85 © 2010 by Taylor and Francis Group, LLC 86 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 4.1 INTRODUCTION There a re t wo ba sic t ypes of pens ion sch emes: defined benefit (D B) a nd defined contribution ( DC) I n a D B p lan, t he p lan spo nsor p romises t o the plan beneficiaries a final l evel of pens ion ben efits This level is u sually defined according to a ben efit formula, as a f unction of a m ember’s (or employee’s) final salary (or average salary) a nd/or years of ser vice in the company Benefits are usually paid as a l ife annuity rather t han as a lump sum The main advantage of a DB plan is that it offers stable income replacement rates (i.e., pension as a p roportion of final salary) to retired beneficiaries and is subsequently indexed to inflation The financial risks associated with a p ure DB plan are borne by the plan sponsor, usually a large company, r ather t han t he plan member The sponsor i s obliged to provide adequate funds to cover the plan liabilities The major drawbacks include t he lack o f benefit portability when changing jobs and t he complex valuation of plan liabilities In a DB plan, when a worker moves jobs, he c an en d u p w ith a m uch l ower pens ion i n r etirement F or ex ample, a typical worker in the United Kingdom moving jobs six times in a career could end up with a pension of only 71%–75% of that of a worker with the same salary experience who remains in the same job for his whole career (Blake and Orszag, 1997) In a DB pension plan, the risk associated with future returns on a f und’s assets is carried out by the employer or sponsor and the contribution rate varies through time as the level of the fund fluctuates above a nd below its t arget level This fluctuation c an be de alt with through the plan’s investment policy (including asset allocation decision, i nvestment ma nager selec tion, and performance me asurement) (see Winklevoss, 1993) Cairns (2000) has considered an investment problem, in which the sponsor minimizes the discounted expected loss by selecting a contribution rate and an asset-allocation strategy A DC p lan s a defined a mount of c ontribution p ayable by b oth employee and employer, often as a fi xed percentage of salary The employee’s re tirement b enefit i s de termined by t he s ize of t he acc umulation a t retirement The benefit ultimately paid to the member is not known for certain until retirement The benefit formula is not defined either, as opposed to DB pension plans At retirement, the beneficiaries can usually take the money as a life annuity, a phased withdrawal plan, a lump sum payment, or some combination of these As the value of the pension benefits is simply determined as the market value of the backing assets, the pension benefits are easily transferable between jobs In a pure DC plan, plan members have ex tensive co ntrol o ver t heir acco unts’ i nvestment st rategy (usually © 2010 by Taylor and Francis Group, LLC Pension Funds under Inflation Risk ◾ 87 subject to the investment menu offered) While the employer or sponsor is only obliged to make regular contributions, the employees bear a range of risks In particular, they bear asset price risk (the risk of losses in the value of their pension fund due to falls in asset values) at retirement and inflation risk (the risk of losses in the real value of pensions due to unanticipated inflation) Generally speaking, a pure DC pension plan is more costly for employees than a pure DB pension plan (see also Blake et al., 2001) Nevertheless, pens ion p lans, i n t he w orld, ve be en u ndergoing a transition from DB plans toward DC plans, which involves enormous transfers of risks from taxpayers and corporate DB sponsors to the individual m embers o f DC p lans (see, e.g , W inklevoss, 993; Bla ke e t a l., 2001) It is therefore of interest to study a DC plan’s investment policy, under which the plan members can protect themselves against both asset price r isk a nd i nflation r isk It s been shown, i n Z hang et a l (2007), that protection against the risk of unexpected inflation i s p ossible by investing some part of the wealth in the index bond The way to reduce these risks to the minimum would then be to trade the DC pension funds in the index bond by following the optimal portfolio rule that is derived in this chapter Mathematically, the type of stochastic optimal control problem that we encounter here, is one with multiple assets and continuous income stream Such problems have been d iscussed i n va rious c ases B odie et a l (1992) derive analytic solutions for a problem of this type, but the setup is quite specific a nd d oes n ot a pply t o our c ase K aratzas (1997, p 4) a nd Koo (1995) consider a very general setup, but not derive analytic solutions The i nvestment p roblem t hat a DC pens ion f und ma nager fac es i s described in Section 4.2 We discuss how to solve this problem using the martingale method in Sections 4.3 and 4.4 4.2 INVESTMENT PROBLEM FOR A DC PENSION FUND We assume that a representative member of a DC pension plan makes contributions co ntinuously t o t he pens ion f und d uring a fi xed finite time horizon [0,T] The contribution rate is fi xed as a percentage c of his salary We will consider the investment problem, from the perspective of a representative DC p lan member, i n wh ich t he i nvestment decision is made through an insurance company or a pension manager The objective is to maximize the expected utility of the terminal value of the pension fund © 2010 by Taylor and Francis Group, LLC 88 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Let us define the stochastic price level as dP (t ) = idt + σ1dW1 (t ) P (t ) (4.1) P (0) = p > where the constant i is the expected rate of inflation and W1(t) is the source of uncertainty that causes the price level to fluctuate around the expected inflation with an instantaneous intensity of fluctuation σ1 Assume that there is a ma rket Μ consisting of three assets that are of interest for the pension fund manager These assets are a risk-free bond, an index bond, and a stock The risk-free bond pays a constant rate of nominal return of R and its price dynamics is given by dB(t ) = Rdt B(t ) (4.2) The index bond offers a constant rate of real return of r and its price process is given by dI (t ) dP (t ) = rdt + I (t ) P (t ) = (r + i)dt + σ1dW1 (t ) (4.3) The price process of the stock is given by dS(t ) = µdt + σ2 dW2 (t ) S(t ) (4.4) where µ is the expected rate of return on the stock σ2 is the volatility caused by the source of risk W2(t), wh ich i s assumed to be independent of W1(t) Let us assume that σ1 ≠ and σ2 ≠ Then the volatility matrix ⎛ σ1 σ≡⎜ ⎝0 © 2010 by Taylor and Francis Group, LLC 0⎞ σ2 ⎠⎟ (4.5) Pension Funds under Inflation Risk ◾ 89 is nonsingular As a consequence, there exists a (unique) market price of risk θ satisfying ⎛ r + i − R ⎞ ⎛ θ1 ⎞ θ = σ −1 ⎜ ⎟≡⎜ ⎟ ⎝ µ − R ⎠ ⎝ θ2 ⎠ (4.6) The market is t herefore arbitrage-free and complete We f urther assume that the salary of the pension plan member follows the dynamics: dY (t ) = κdt + σ3 dW1 (t ) Y (t ) (4.7) Y (0) = y > where κ is the expected growth rate of salary σ3 is the volatility of salary We assume that salary is driven by the source of uncertainty as inflation Both κ and σ3 are constants It can be verified using It ô’s Lemma that the following process is the solution to the stochastic differential Equation 4.7: ⎛ 2⎞ ⎜ κ − σ3 ⎟⎠ t +σ3W1 (t ) Y (t ) = ye⎝ (4.8) If we write σY ≡ (σ3, 0)T, then we can rewrite Equation 4.8 as* Y (t ) = ye ⎛ 2⎞ Τ ⎜⎝ κ − σY ⎟⎠ t +σY W (t ) (4.9) If the plan member contributes continuously to his DC pension fund with a fixed contribution rate (i.e., t he per centage o f t he m ember’s s alary) o f c(>0) and − π1(t) − π2(t), π1(t), π2(t) shares of the pension fund are invested in the riskless bond, the index bond, and the stock, respectively, then the corresponding wealth process with an initial value of x (0 ≤ x < ∞), which we denote by Xπ (t), is governed by the following equation: dX π (t ) = X π (t )[Rdt + πΤ (t )σ(θdt + dW (t ))] + cY (t )dt (4.10) where, cY(t) is t he a mount of money contributed to t he pension f und at time t and π(t) = (π1(t), π2(t))T Note that the contributions are assumed to be * It is possible to allow the salary process to be correlated with the stock price © 2010 by Taylor and Francis Group, LLC 90 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling invested continuously over time The contribution at time t, cY(t), can be viewed as the rate of a r andom endowment and is strictly positive With regards to the set of admissible portfolios, we first follow the classical line taken in Merton (1971), Karatzas (1997), and Koo (1995), which allows the agent to borrow against expected future income Definition 4.1: A portfolio process π is said to be admissible if the corresponding wealth process Xπ(t) in (4.10), for all t ∈ [0,T], satisfies ⎡ T H ( s) ⎤ X (t ) + Et ⎢ cY (s)ds ⎥ ≥ 0, almost surely, ⎢ H (t ) ⎥ ⎣t ⎦ π ∫ (4.11) where, E t is the conditional expectation with respect to the Brownian filtration {F(t)}t and H (t ) ≡ e − Rt − θ t −θΤW (t ) (4 12) is the stochastic discount factor that adjusts for the nominal interest rate and the market price of risk We denote the class of admissible portfolio processes by AY Note t hat, i n t he p resence o f a pos itive r andom en dowment st ream, the wealth is allowed to become negative, so l ong as the present value of future en dowments i s l arge en ough to offset such a n egative va lue The ⎡ T H ( s) ⎤ term Et ⎢ cY (s)ds ⎥ is what Bodie et al (1992) call human capital, ⎣ t H (t ) ⎦ or more precisely a share c of it This setup therefore takes into account the fact that agents base their investment decision not only on their current financial wealth but also on their individual human capital The objective of the representative plan member is then to maximize expected u tility f rom t he ter minal va lue* o f t he pens ion f und g iven a n initial investment of x > 0, that is, ∫ π max E[u ( X (T ))] π∈A( x ) (4.13) * It has been shown in Zhang (2008) that, when the index bond is a traded asset, maximizing the expected utility of real terminal wealth is equivalent to maximizing the expected utility of nominal terminal wealth For t he for mer, t he corresponding objective is determined by dividing the terminal wealth in (4.13) by the price level at date T © 2010 by Taylor and Francis Group, LLC Pension Funds under Inflation Risk ◾ 91 subject to dX π (t ) = X π (t )[Rdt + π Τ (t )σ(θdt + dW (t ))] + cY (t )dt π X (0) = x (4.14) where A(x ) ≡ {π ∈ AY (x ) : E[u −( X π (T ))]< ∞} (4.15) The function u− is defined as u−(·) ≡ max{−u−(·),0} The utility function is assumed to be of constant relative risk aversion (CRRA) type, that is, u(z ) = z 1−γ 1− γ (4.16) The problem of (4.13) through (4.15) is a classical terminal wealth optimization problem except the fact that there is an additional term cY(t)dt in the st ochastic d ifferential equation (4.14) for the wealth process In the literature, t here a re t wo ma in approaches dealing w ith continuous-time stochastic o ptimization p roblems These a re t he dynamic pr ogramming approach (see Merton, 1969, 1971) and the martingale method (see Karatzas et al., 1986; Pliska, 1986; Cox and Huang, 1989) The economic literature is dominated by t he stochastic dy namic programming approach, wh ich has the advantage that it identifies the optimal strategy automatically as a function of the underlying observables, which is sometimes called the feedback form However, it often turns out that the corresponding Hamilton– Jacobi–Bellman eq uation, wh ich i n g eneral i s a seco nd-order n onlinear partial d ifferential eq uation, d oes n ot ad mit a cl osed-form so lution I n contrast, by utilizing the martingale method, a closed-form solution can often be obtained without solving any partial differential equation This at least is true when asset prices follow a geometric Brownian motion (Cox and Huang, 1989) We will use the martingale approach to solve the problem of (4.13) through (4.16) and will be able to derive a closed-form solution Note that this fund is not self-financing in the classical context due to t he (continuous) contributions to t he f und Therefore, t he discounted wealth process will in general not be a ma rtingale under the risk neutral measure, a nd t he ma rtingale m ethod i s n ot d irectly a pplicable H ence, © 2010 by Taylor and Francis Group, LLC 92 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling before applying the existing results on the terminal wealth optimization problem, we need to get rid of the contribution term.* We will consider this in the next section 4.3 HOW TO SOLVE IT In order to express the first equality in the budget constraint of (4.14) in a form that is linear in the corresponding wealth, let us examine the expectation of the plan member’s future contribution (in Bodie’s terms a share of the individuals human capital), which is defined below Definition 4.2: is defined as The present value of expected future contribution process ⎡ T H (s ) ⎤ D(t ) = Et ⎢ cY (s)ds ⎥ ⎢ H (t ) ⎥ ⎣t ⎦ ∫ (4.17) By i nspecting t he Ma rkovian st ructure o f t he ex pression o n t he r ighthand side of Equation 4.17, we note that it is possible to express D(t) i n terms of the instantaneous contribution cY(t) The following proposition shows how this is done Proposition 4.1: The present value of expected future contribution D(t) is proportional to the instantaneous contribution cY(t), that is, D(t ) = (eβ(T −t ) − 1) cY (t ), for all t ∈[0, T ] β (4.18) with β ≡ κ − R − σ3θ1 In particular d ≡ D(0) = (eβT − 1)cy β (4.19) D(T ) = Proof By definition we have * Note that, for the dynamic programming approach, there is no need for the fund to be selffinancing © 2010 by Taylor and Francis Group, LLC Pension Funds under Inflation Risk ◾ 93 ⎡ T H ( s) ⎤ D(t ) = Et ⎢ cY (s)ds ⎥ ⎢ H (t ) ⎥ ⎣t ⎦ ∫ ⎡ T H ( s) Y ( s) ⎤ = cY (t )Et ⎢ ds ⎥ ⎢ H (t ) Y (t ) ⎥ ⎣t ⎦ ∫ (4.20) Both processes H(·) a nd Y(·) are geometric Brownian motions and thereH (s ) Y (s ) fore it follows easily that is independent of F(t) for s ≥ t As a H (t ) Y (t ) consequence, t he co nditional ex pectation co llapses t o a n u nconditional expectation and we obtain D(t ) = cY (t ) g (t , T ) (4 21) with the deterministic function g(t, T) being defined by ⎡T − t Y (s) ⎤⎥ g (t , T ) ≡ E ⎢ H (s) ds Y (0) ⎥ ⎢ ⎣ ⎦ ∫ (4.22) Noting that H (s ) 2 Y (s ) κ − R s ( σ3 −θ1 )W1 ( s ) −θ2W2 ( s ) − (PθP +σ3 ) s = e( ) e Y (0) Τ 2 κ − R )s ( σY −θ ) W ( s ) − (PθP + PσY P ) s = e( = e βs e e ( σY −θ)Τ W ( s ) − (PσY −θP2 )s (4.23) we obtain ⎡ (σY −θ)Τ W (s )− (PσY −θP2 ) s ⎤ Y (s ) ⎤ ⎡ βs ⎡ ⎤ E ⎢ H (s ) = E e E ⋅ ⎥ ⎥ ⎣ ⎦ ⎢e Y (0) ⎣ ⎦ ⎣ ⎦ = eβs (4.24) The last equality is obtained by the fact that a stochastic exponential martingale s ex pectation o ne I ntegrating bo th s ides o f E quation 4.24 gives © 2010 by Taylor and Francis Group, LLC 94 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling T −t ∫ Y (s ) ⎤ ⎡ E ⎢ H (s ) ds = Y (0) ⎥⎦ ⎣ T −t ∫e βs ds = (eβ(T −t ) − 1) β (4.25) The left-hand side of Equation 4.25 is equal to g(t, T) by the Fubini theorem Differentiating bo th s ides o f E quation 18 a nd u sing E quation 7, we get ⎛1 ⎞ dD(t ) = d ⎜ (eβ(T −t ) − 1)cY (t )⎟ ⎝β ⎠ 1 = (eβ(T −t ) − 1)cdY (t ) + cY (t )d(eβ(T −t ) − 1) β β 1 = (eβ(T −t ) − 1)cY (t ) (κdt + σYΤ dW (t )) − cY (t )eβ(T −t )βdt β β Collecting terms, we obtain dD(t ) = (eβ(T −t ) − 1)cY (t ) ((κ − β)dt + σYΤ dW (t )) − cY (t )dt β and using the equality in (4.18) and the definition of β in Proposition 4.1, we then have dD(t ) = D(t ) ⎡⎣(R + σ3 θ1 )dt + σYΤ dW (t )⎤⎦ − cY (t )dt (4 26) If we add E quation 4.27 and the first equality in Equation 4.14 together, the term cY(t) cancels out We will define a process based on this observation below Definition 4.3: Let us define a process V (t ) ≡ X π (t ) + D(t ) (4 where Xπ(t) and D(t) satisfy Equations 4.14 and 4.17, respectively © 2010 by Taylor and Francis Group, LLC 27) Pension Funds under Inflation Risk ◾ 95 In terms of Bodie et al (1992) V(t) corresponds to the total wealth process, comprising financial wealth a nd a sha re of human c apital Taking differentials on both sides of Equation 4.28 and using Equations 4.14 and 4.27, we have dV (t ) = dX π (t ) + dD(t ) = X π (t ) ⎡⎣ Rdt + π Τ (t )σ (θdt + dW (t ))⎤⎦ + cY (t )dt + D(t ) ⎡⎣(R + σ3θ1 )dt + σYΤ dW (t )⎤⎦ − cY (t )dt Collecting terms gives us the following: ⎡ ⎤ X π (t )π Τ (t )σ + D(t )σYΤ dV (t ) = V (t ) ⎢ Rdt + θdt + dW (t ))⎥ ( V (t ) ⎣ ⎦ From Equation 4.29, we can see t hat dV(t) is proportional to V(t) Next, we check whether the discounted process of V(t) is a P-local martingale Multiplying V(t) by H(t) in (4.12) and taking differentials, we can get d (H (t )V (t ))= H (t )dV (t ) + V (t )dH (t ) + dH (t )dV (t ) ⎡ ⎤ X π (t )π Τ (t )σ + D(t )σYΤ = H (t )V (t ) ⎢ Rdt + θdt + dW (t ))⎥ ( V (t ) ⎣ ⎦ − H (t )V (t ) ⎡⎣ Rdt + θΤ dW (t )⎤⎦ − H (t )V (t ) X π (t )π Τ (t )σ + D(t )σYΤ θdt V (t ) After canceling out terms, we obtain ⎡ X π (t )π Τ (t )σ + D(t )σYΤ ⎤ d (H (t )V (t ))= H (t )V (t ) ⎢ − θΤ ⎥ dW (t ) V (t ) ⎣ ⎦ Τ ⎡ X π (t )σΤ π(t ) + D(t )σY ⎤ = H (t )V (t ) ⎢ − θ ⎥ dW (t ) V (t ) ⎣ ⎦ © 2010 by Taylor and Francis Group, LLC 96 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling This shows t hat H(t)V(t) is a P -local martingale as it can be w ritten as a stochastic integral with respect to the Brownian motion W(t) Moreover, we know that V (T ) = X π (T ) + D(T ) = X π (T ) (4 28) V (0) = X π (0) + D(0) = x + d (4 29) and So we can conclude that the plan member’s optimization problem of (4.13) through (4.15) is equivalent to maximizing E[u(V(T))] over a cla ss of admissible portfolio processes π, subject to the constraint of (4.29) and (4.32) [or (4.31)] We will discuss this formally in Section 4.4 4.4 OPTIMAL MANAGEMENT OF THE PENSION FUND WITHOUT LIQUIDITY CONSTRAINTS In the previous section, we have derived that the plan member’s optimization problem of (4.13) t hrough (4.15) w ith i nitial i nvestment x c an be solved by solving the corresponding problem with (4.29) and initial wealth x + d, that is, π max E[u( X (T ))] = max E[u(V (T ))] (4 π∈A( x ) π∈A1 ( x + d ) 30) subject to ⎡ ⎤ X π (t )π Τ (t )σ + D(t )σYΤ dV (t ) = V (t ) ⎢ Rdt + θdt + dW (t ))⎥ ( V (t ) ⎣ ⎦ (4.31) V (0) = x + d where A1 (x + d ) ≡ {π ∈ A(x + d ) : E ⎡⎣u − (V (T ))⎤⎦ < ∞} (4.32) and A(x + d) is t he class of admissible portfolio processes w ith an initial value x + d, such that the corresponding portfolio value process satisfies V (t ) = X π (t ) + D(t ) ≥ 0, for all t ∈[0, T ] © 2010 by Taylor and Francis Group, LLC Pension Funds under Inflation Risk ◾ 97 It is easy to check that π ∈A(x ) if and only if p ∈ A1(x + d) Applying the martingale method to the problem of (4.33) through (4.35), we know that the optimal wealth V*(t) is given by* V * (t ) = Et ⎡(H (T ))B* ⎤ ⎦ H (t ) ⎣ (4.33) and B * = (u ′)−1 (λH (T )) where λ is the Lagrangian multiplier to be determined by the constraint E[H (T )B * ] = x + d For the choice of CRRA utility, we can get B * = (x + d ) − γ (H (T )) γ −1 ⎡ ⎤ E ⎢(H (T )) γ ⎥ ⎢⎣ ⎥⎦ (4.34) and the corresponding optimal wealth process is then given by γ −1 ⎡ ⎤ Et ⎢(H (T )) γ ⎥ ⎥⎦ ( x + d) ⎢⎣ V *(t ) = γ −1 H (t ) ⎡ ⎤ E ⎢(H (T )) γ ⎥ ⎢⎣ ⎥⎦ (4.35) Let us write Z1 (t ) = e 1−γ Τ ⎛ 1−γ ⎞ θ W (t ) − ⎜ PθP2t γ ⎝ γ ⎠⎟ (4.36) and f1 (t ) = e ⎞ 1−γ ⎛ R + PθP2 ⎟ t γ ⎝⎜ 2γ ⎠ * The superscript (*) denotes the corresponding optima hereafter © 2010 by Taylor and Francis Group, LLC (4.37) 98 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling We then obtain (H (t )) γ−1 γ = f1 (t )Z1 (t ) (4.38) (x + d ) Z1 (t ) H (t ) (4.39) and Equation 4.38 now becomes V *(t ) = Multiplying both sides of Equation 4.42 by H(t) and then differentiating both sides, we get d(H (t )V *(t )) = H (t )V *(t ) 1− γ Τ θ dW (t ) γ (4.40) As Equation 4.30 should also hold at the optimum, we have Τ ⎡ X *(t )σ Τ π *(t ) + D(t )σY ⎤ − θ ⎥ dW (t ) d(H (t )V *(t )) = H (t )V *(t ) ⎢ V *(t ) ⎥⎦ ⎣⎢ where X *(t ) ≡ X π*(t ) A comparison of Equation 4.44 with Equation 4.43 leads to X *(t )σ Τ π *(t ) + D(t )σY 1− γ −θ = θ V *(t ) γ (4.41) from which we can solve for π*(t) V *(t ) D(t ) π *(t ) = (σ Τ )−1 θ − (σ Τ )−1 σY X *(t ) X *(t ) γ (4.42) This formula depends on the optimal portfolio value V*(t), which consists of the optimal pension fund level X*(t) and the expected future contributions D(t) We have seen in Proposition 4.1 that the expected future contributions of t he plan member is observable given t he member’s current salary So it will be more convenient for the fund manager to implement the optimal strategy if we express it in terms of D(t) Substituting V*(t) in Equation 4.46 by X*(t) + D(t), we get ⎛1 ⎞ D(t ) π *(t ) = (σ Τ )−1 θ + (σ Τ )−1 ⎜ θ − σY ⎟ γ ⎝γ ⎠ X *(t ) © 2010 by Taylor and Francis Group, LLC (4.43) Pension Funds under Inflation Risk ◾ 99 We obtain the following result: Proposition 4.2: The solution of problem (4.13) through (4.15) is to invest according to the strategy (4.47) with an initial investment of x At the initial date t = 0, we have ⎛1 ⎞d π *(0) = (σ Τ )−1 θ + (σ Τ )−1 ⎜ θ − σY ⎟ , for x > γ ⎝γ ⎠x (4.44) To understand the structure of (4.47) we hypothetically assume now that a n ini tial in vestment o f x + d were made We then obtain the following: Proposition 4.3: The optimal investment strategy of (4.47) is made up of two parts: Τ −1 (σ ) θ (and corresponding position in the bond) with an initial γ in vestment of x is the classical optimal portfolio rule (i.e., Merton rule) • ⎛1 ⎞ D(t ) • (σ Τ )−1 ⎜ θ − σY ⎟ (and corresponding pos ition i n t he bond) ⎝γ ⎠ X *(t ) with a n ini tial in vestment o f d re plicates t he f uture c ontributions (i.e., share of the human capital) The la tter c an be e asily v erified, t he co mputation h owever i s sl ightly lengthy, which is why it is omitted here We can further express the optimal portfolio strategy π*(t) obtained in Equation (4.47) in terms of the asset prices at time t(I(t) and S(t)) and the plan member’s current salary Y(t) S ince X*(t) = V*(t) − D(t) a nd D(t) is proportional to Y(t) (recall Equation 4.18), we only need to write V*(t) in terms of the observable variables I(t), S(t), and Y(t).* * In fact, it will be sufficient to e xpress the optimal investment strategy in terms of on ly two variables from the combination of S(t) and any one of the variables Y(t), I(t) and the current price level P(t) (see Zhang et al., 2007 for details) © 2010 by Taylor and Francis Group, LLC 100 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Dividing both sides of Equation 4.41 by H(t)f1(t) gives us − 1 (H (t )) γ = Z1 (t ) f1 (t ) H (t ) (4.45) Equation 4.42 then can be rewritten as V *(t ) = − (x + d ) (H (t )) γ f1 (t ) (4.46) and now we only need to write H(t) in terms of the observable variables It can be shown that a ⎛ I (t ) ⎞ ⎛ S(t ) ⎞ H (t ) = eαt ⎜ ⎝ I (0) ⎟⎠ ⎜⎝ S(0) ⎟⎠ b (4.47) where ⎛ r + i − R 1⎞ ⎛ µ − R 1⎞ α ≡ (r + i) ⎜ − ⎟ + µ⎜ − ⎟− θ ⎝ σ1 ⎝ σ2 2⎠ 2⎠ a≡− r +i −R σ1 b≡− µ−R σ2 (4.48) Therefore, we have α (x + d ) − γ t ⎛ I (t ) ⎞ V *(t ) = e ⎜ ⎝ I (0) ⎟⎠ f1 (t ) − a γ ⎛ S(t ) ⎞ ⎜⎝ S(0) ⎟⎠ − b γ (4.49) ACKNOWLEDGMENTS This cha pter i s t aken f rom t he a uthor’s P hD t hesis, a nd sh e g ratefully acknowledges help and support from her PhD supervisor Professor Ralf Korn She would also like to thank Francesco Menoncin for the discussion at the 6th International Workshop on Pension and Saving: Consequences of Longevity Risks on Pension Systems and Labor Markets held at Université Paris-Dauphine where this chapter has been presented Support from the May and Stanley Smith Charitable Trust is also being acknowledged © 2010 by Taylor and Francis Group, LLC Pension Funds under Inflation Risk ◾ 101 REFERENCES Blake, D and Orszag, J.M (1997) Annual estimates of personal wealth holdings in the U.K since 1948 Technical report, Pensions Institute, Birkbeck College, University of London, London, U.K Blake, D., Cairns, A.J.G., and Dowd, K (2001) Pension metrics: Stochastic pension plan design a nd val ue-at-risk d uring t he acc umulation p hase Insurance: Mathematics and Economics 29, 187–215 Bodie, Z., Merton, R.C., and Samuelson, W.F (1992) Labor supply flexibility and portfolio c hoice in a lif e c ycle mo del Journal o f Ec onomic D ynamics & Control 16, 427–449 Cairns, A.J G (2000) S ome no tes o n t he d ynamics a nd o ptimal co ntrol o f stochastic pension fund models in co ntinuous time ASTIN Bulletin 30, 19–55 Cox, J a nd H uang, C.F (1989) Op timal co nsumption a nd p ortfolio p ractices when ass et p rices f ollow a diffusion process Journal o f Ec onomic Theor y 49, 33–83 Karatzas, I (1997) Lectures o n t he M 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optimization in finance and life insurance: Applications of the martingale method PhD thesis, Technische Universität Kaiserslautern, Kaiserslautern, Germany Zhang, A (2008) The t erminal r eal w ealth o ptimization p roblem wi th index bond: Equivalence of real and nominal portfolio choices for CRRA utility Working paper University of St Andrews, St Andrews, U.K Zhang, A., Korn, R., and Ewald, C (2007) Optimal management and inflation protection for defined contribution pension plans Blätter der D GVFM, 28(2), 239–258 © 2010 by Taylor and Francis Group, LLC ... Equation 4. 46 by X*(t) + D(t), we get ⎛1 ⎞ D(t ) π *(t ) = (σ Τ )−1 θ + (σ Τ )−1 ⎜ θ − σY ⎟ γ ⎝γ ⎠ X *(t ) © 2010 by Taylor and Francis Group, LLC (4. 43) Pension Funds under Inflation Risk ◾ 99... processes π, subject to the constraint of (4. 29) and (4. 32) [or (4. 31)] We will discuss this formally in Section 4. 4 4. 4 OPTIMAL MANAGEMENT OF THE PENSION FUND WITHOUT LIQUIDITY CONSTRAINTS In... ) (4 where Xπ(t) and D(t) satisfy Equations 4. 14 and 4. 17, respectively © 2010 by Taylor and Francis Group, LLC 27) Pension Funds under Inflation Risk ◾ 95 In terms of Bodie et al (1992) V(t)