Available online at wwwsciencedirectcom Procedia Engineering 3 () 387 39 Power Electronics and Engineering Application A Simple Method for Solving Multiperiod Mean-Variance Asset-Liability Management Problem Hai-xiang Yao * School of Informatics, Guangdong University of Foreign Studies, Guangzhou 56, China Abstract his paper introduces the Lagrange duality method for solving the multiperiod mean-variance (M-V) asset-liability management (ALM) problem First, Using the Lagrange multiplier technique, the original problem is turned into a multi-period unconstrained Optimal Control Problem (OCP) that is separable in the sense of dynamic programming hen the dynamic programming approach is applied to solve the OCP Finally, closed form expressions of the efficient investment strategy and the M-V efficient frontier are obtained Published by Elsevier Ltd Open access under CC BY-NC-ND license Selection and/or peer-review under responsibility of [name organizer] Keywords: asset-liability management; multi-period mean-variance; Lagrange duality; dynamic programming Introduction Since the pioneer wor of Marowitz [], the mean variance (M V) model has became the foundation of modern finance theory and inspired hundreds of extensions and applications Among them, [] and [3] extend the model to cases of multi-period and continuous-time, respectively, by using an embedding technique that overcame the difficulty of non-separability in variance, and derived the analytical optimal solutions After that, many scholars adopt the dynamic M V model to study other portfolio selection or financial problems under some reality conditions (see [4] for example) On the other hand, it is well nown that asset-liability management (ALM) problem is of both theoretical interest and practical importance For example, ALM has extensive applications in bans, * el: +86--39358577; fax: +86--39383 E-mail address: yaohaixiang@mailgdufseducn 877-758 Published by Elsevier Ltd Open access under CC BY-NC-ND license doi:6/jproeng58
388 Hai-xiang Yao / Procedia Engineering 3 () 387 39 pension funds and insurance companies In recent years, using the M V criteria, [5] studied a multiperiod ALM problem [6] and [7] extended the wor of [5] to the cases of uncertain exit time and stochastic maret environment, respectively On the other hand, [8]-[9] investigated ALM problems under M V criterion in continuous-time setting o the best of our nowledge, Most the existing literatures about multiperiod M V model with liability apply the embedding techniques introduced in [] to solve the model hough embedding techniques overcome the difficulty of non-separability in variance, but it s quite complicated in procedure settings and calculation What is more, except for wealth, ALM problems need to consider another relevant state variable, liability, the inclusion of further state variable drastically enhances the computational complexity in obtaining closed form solutions (refer to [5]-[7]) For these reasons, this paper tries to introduce a new simple method, that named as Lagrange duality method, for solving multiperiod mean variance ALM problems Compared with the embedding techniques, Lagrange duality methods are simpler in procedure settings with less computational complexity As an application and a demonstration of Lagrange duality method, his paper adopt it to solve the multiperiod mean variance ALM problems of [5] he paper is structured as follows Section two sets up the mean-variance ALM problem as a multiperiod Optimal Control Problem (OCP) with equality constraint In section three, the original problem is turned into a separable multi-period unconstraint OCP by using Lagrange multiplier technique and the analytical solution is obtained by dynamic programming approach Finally, the efficient investment strategy and the efficient M-V frontier are obtained in section four Establishment of mean variance ALM model Suppose that there are n + securities with return vector e (,,,, n = e e e e ) at time period, =,,, Here A represents the transpose of matrix A An investor, equipped with initial wealth x and initial liability l, enters the maret at time, and maes investments within period He (she) not only need to consider the investment strategy, but also consider the liability management Following [5], the liability are uncontrollable, and its dynamic process is l = + ql, () q is a exogenous random variable, it can be understood as the random growth rate of liability Let i x and l denote the value of wealth and liability he holds at period, respectively u, i =,,, n, is the amount invested in the i th security at period, then the amount invested in the th security is n i x u herefore, the wealth dynamics can be written as (see [5]) x+ = xe + Pu, () P (,,, n = e e e e e e ) his paper has the assumptions as [5] is the overall information sets till time hen the investment strategy u is admissible if u is adapted to he collection of all admissible investment strategy is defined as Θ he multiperiod mean-variance ALM problems is to find out the optimal admissible strategy to minimize the ris of the final surplus, defined as S = x l, under the condition that expectation is given as d, here the ris is measured by variance, ie VarS [ ] = ES [ ] E[ S] = ES [ ] d herefore, the multiperiod ALM model under the M-V framewor can be now formulated as OCP: min Var[ S ] = E[ S ] d, s t E[ S ] = d,() () (3) u Θ
Hai-xiang Yao / Procedia Engineering 3 () 387 39 389 he solution of this OCP is called efficient investment strategy he collection of all these points ( d, Var[ S ]) in coordinate plane M-V corresponded to efficient strategies is called as efficient frontier 3 ransformation and solution to the problem It is well nown that the equality constraint ES [ ] = d in OCP (3) can be dealt with by introducing a Lagrange multiplier We can turn to solve the following unconstrained OCP parameterized by min E[ S] d ( E[ S] d), st () () u Θ + (4) Since ES [ ] d + ( ES [ ] d) = E[( x l) + ( x l)] d d herefore, OCP (4) is equivalent to min E[ x + l x l + x l d d], st () () (5) u Θ In the following, we solve OCP (5) by using dynamic programming approach Let f( x, l ) denote the optimal value function associated with OC (5) starting from time with state: wealth x and liability l hen, according to the dynamic programming principle, the basic equations of OC (5) are as follows: f( x, l) = min E f+ ( xe + Pu, ql ), u Θ (6) f( x, l) = x + l xl + x l d d As a result, H ( x, l, λ ): = f( x, l) is the optimal value of OCP (5) For simplicity, let x = x, l = l We guess and subsequently verify that the expression of f ( x ) has the form as follows f(,) x l = wx + λxl + γl + hx + gl + α, (7) w >, λ, γ, h, g, α are series to be determined Substituting (7) into the first equation of basic equation (6) gives wx + λxl + γl + hx + gl + α = min E f+ ( xe + P u, ql) u = min E w+ ( xe + P u) + λ+ ( xe + P u) ql + γ+ ql + h+ ( xe + P u) + g+ ql + α+ } u = w+ E[( e) ] x + λ+ E[ eq] xl + γ+ E[ q] l + h+ E[ e] x+ g+ Eq [ ] l+ α+ + min w u EPP [ + ] u + ( w EeP [ + ] x+ λ EqP [ + ] l+ h+ EP [ ]) u u he first order condition (since w >, then is also sufficient condition ) gives λ h + + u = E [ PP ] Ee [ P] x+ Eq [ P] l+ E[ P], w+ w+ substituted bac into the above formula, it follows that (8)
39 Hai-xiang Yao / Procedia Engineering 3 () 387 39 wx + λxl + γl + hx + gl + α λ λ h + + + h+ = w+ Ax + λ+ Gxl + γ+ E[ q] B l + h+ J x + g+ E[ q] M l + α+ D, w+ w+ w+ A = E[( e) ] EeP [ ] E [ PP ] EeP [ ], B = EqP [ ] E [ PP ] EqP [ ], D = E[ P ] E [ PP ] E[ P], G = Eeq [ ] [ ep ] E [ PP ] Eq [ P], (9) J = Ee [ ] [ ep ] E [ PP ] EP [ ], M = EqP [ ] E [ PP ] EP [ ] herefore, we obtain the recursion relationship about w, λ, γ, h, g, α as w = w+ A, λ = λ+ G, h = h+ J, λ+ λ+ h+ h+ γ = γ+ Eq [ ] B, g = g+ Eq [ ] M, α = α+ D w+ w+ w+ First, by means of repeatedly iteration and notice that w =, λ =, h = from (6), we obtain () w = F, λ = C, h = L, =,,,, () () F = A, C = G, L = J, =,,, i i i Here, we set that () i = It is nown from [] that A E e EeP E PP EeP = [( ) ] [ ] [ ] [ ] > hereby w >, which satisfies the previous assumption Substituting () into (), then the recursion formula about γ, g, α can be rewritten as γ = γ Eq [ ] CBF, g = g Eq [ ] + LM F, α = α LDF (3) + + + γ, g, α d d i = E[ qi ] Ci BF i i E[ qj], g = I, = d d N, j= After repeatedly iterating and note that = = = from (6), we get (4) γ α i i i i i j i i i j= (5) I = Eq [ ] + LMF Eq [ ], N = LDF, =,,, Here, we define () i = As a result, the solution to basic equation (6) is (, ) γ, f x l = Fx Cxl + l + Lx + Il d d N (6) and the optimal investment strategy is ( + + + + ) u = E [ P P ] E[ e P ] x C F E[ q P ] l+ L F E[ P ] (7) 4 he efficient investment strategy and efficient frontier From previous analysis, we now that when =, f( x, l ) is the optimal value of OCP (5), ie H ( x, l, ): = f ( x, l) = Fx Cxl + γ l + Lx + Il d d N (8)
Hai-xiang Yao / Procedia Engineering 3 () 387 39 39 It is well nown from the Lagrange duality theorem (see [3]) that the optimal value of OCP (3), namely minimum variance can be obtained by solving the maximum of H( x, l, ) about, ie Var[ x ] = max H ( x, l, ) = max N + ( L x + I l d) + F x C x l + γ l d (9) { } Obviously, N >, thereby, the maximum to OCP (9) exists, and its first-order condition gives the maximum point as * = N Lx + Il d () ( ), substituted into (7) and note that x = x, l = l, we obtain the optimal strategy to OCP (3), namely the efficient investment strategy as ( + + + + ) u = E [ PP ] EeP [ ] x C F EqP [ ] l + N ( Lx + Il d) L F EP [ ] () Again substituting () into (9), we obtain the optimal value of OCP (3), namely minimum variance as Var[ x ] = N ( Lx + Il d) + Fx Cxl + γ l d () = ( N) N d ( N) ( Lx + Il) + Fx Cxl + γ l ( N) ( Lx + Il) ( ) So far, we obtain the following results heorem : For given expected terminal surplus ES [ ] = d in the multi-period mean-variance ALM model, the efficient investment strategy can be obtained by (), while, the efficient M-V frontier can be given by (), here d ( N) ( Lx + Il) Acnowledgements his research is supported by the Humanity and Social Science Foundation of Ministry of Education of China (No YJC79339), Natural Science Foundation of Guangdong Province (No S553), and Philosophy and Social Science Foundation of Guangdong Province (No 9O-9) References [] Marowitz H Portfolio selection Journal of Finance 95;7():77-9 [] Li D, Ng WL Optimal dynamic portfolio selection: multiperiod mean-variance formulation Mathematical Finance ;:387-46 [3] Zhou XY, Li D Continuous-ime mean-variance portfolio selection: a stochastic LQ framewor Applied Mathematics Optimization ;4:9-33 [4] Oswaldo LVC, Michael VA A generalized multi-period mean variance portfolio optimization with Marov switching parameters Automatica 8;44:487 497 [5] Leippold M, rojani F, Vanini P A geomeric approach to multiperiod mean vaiance optimization of assets and liabilities Journal of Economic Dynamics and Control 4;8:79-3 [6] Yi L, Li ZF, Li D Mutli-period portfolio selection for asset-liability management with uncertain investment horizon Journal of industrial and management optimization 8;4(3):535-55 [7] Chen P, Yang HL Marowitz's Mean-Variance Asset-Liability Management with Regime Switching: A Multi-Period Model Applied Mathematical Finance ;8():9 5 [8] Chiu MC, Li D Asset and liability management under a continuous-time mean-variance optimization framewor Insurance: Mathematics and Economics 6;39:33-355 [9] Xie SX, Li ZF, Wang SY Continuous-time portfolio selection with liability: Mean variance model and stochastic LQ approach Insurance: Mathematics and Economics 8;4(3):943-953 []Luenberger DG Optimization by Vector Space Methods New Yor: Wiley; 968