Proceedings of he 46h ISCIE Inernaional Symposium on Sochasic Sysems Theory and Is Applicaions Kyoo, Nov. 1-2, 214 A Liabiliy Tracking Porfolio for Pension Fund Managemen Masashi Ieda, Takashi Yamashia and Yumiharu Nakano Graduae School of Innovaion Managemen, Tokyo Insiue of Technology 2-12-1 Ookayama, Meguro-ku, Tokyo, Japan The Governmen Pension Invesmen Fund, Japan 1-4-1 Kasumigaseki, Chiyoda-ku, Tokyo Japan E-mail: ieda@craf.iech.ac.jp Absrac We sudy he long erm porfolio which is able o rack a liabiliy. The porfolio opimizaion problem is defined as he sochasic opimal conrol problem and he performance crierion is he lower mean square error beween he liabiliy and our wealh. We impose consrains for he porfolio weighs and obain he opimal porfolio sraegy numerically by solving he Hamilon-Jacobi-Bellman equaion applying he quadraic approximaion scheme. The numerical simulaions using he empirical daa provided by Japanese organizaions are run under he wo ypes of consrains: he no-shor-selling consrain; he upper bound consrain for he porfolio weighs. The former demonsraes ha he liabiliy racking abiliy of our opimal porfolio sraegy does no drop if we resric he shor selling. The laer implies he imbalance beween he growh rae of he liabiliy and he profiabiliy of he asses. 1 Inroducion A porfolio sraegy aking ino accoun he liabiliy is one of he significan issues in he pension fund managemen. Due o he populaion ageing especially observed in he developed counries, i is expeced ha he benefi of he welfare pension fund increases and his or her conribuion decreases. Moreover he low birh rae implies ha his relaion is no resolved for decades. Thus he pension funds face a problem o include he mehod for hedging heir liabiliies ino heir long erm porfolio managemen. In conras, a lo of praciioners in pension funds deermine heir porfolio by he radiional single period mean variance approach. I is difficul o adop his approach o include a view of hedging he liabiliy since he single ime period mehod is unable o allow us o change he porfolio afer he iniial ime. Alhough he muli ime period mehod allows he change afer he iniial ime, we face a problem of he compuaional cos: we are usually unable o obain he opimal long erm porfolio sraegy in realisic ime. The lieraure [1] sudies he long erm pension fund managemen aking ino accoun he liabiliy employing he LQG (Linear, Quadraic cos, Gaussian) conrol problem which is a class of sochasic conrol problem. Since he analyical soluion of he LQG conrol problem is available, he opimal porfolio sraegy is obained in realisic ime. I demonsraes ha he opimal porfolio sraegy racks he liabiliy wih he low racking error by he numerical simulaion using he empirical daa provided by he Japanese organizaions. In he presen paper, we develop he porfolio sraegy from he sraegy proposed by Ieda e. al. [1] menioned above. The previous sudy remains wo poins which should be improved: (i) penalising he wealh of he invesor exceed he liabiliy; (ii) permiing he large shor selling. To resolve hese poins, we employ a lower mean square error from he racked liabiliy as he performance crierion and resric he porfolio weigh o a posiive and bounded value. In his case we are no able o obain an analyical soluion and hus we employ he numerical mehod o solve he corresponding Hamilon-Jacobi-Bellman (HJB) equaion. Since we have currenly consider he porfolio consruced by muli asses which correlae each oher, i is difficul o obain a soluion in general: for insance, he finie difference mehod is failed in his case (see e.g., Kushner and Dupuis [2]). To cope wih his problem, we employ he quadraic approximaion scheme proposed by Nakano [3]. The advanage of his mehod is he low compuaional cos, and hence applying his mehod, we are able o obain a proxy of he opimal porfolio sraegy in realisic ime. Main implicaions from he presen work are as follows. The firs remarkable one is ha he liabiliy racking abiliy of our opimal porfolio sraegy does no drop if we resric he shor selling. The mean hedging error is approximaely 4.5% of he liabiliy and i is he equivalen level wihou he no-shor-selling consrain sudied in [1]. The hedge is realized by borrowing he money a lo and invesing he lowrisk asses. We nex noe ha resricing he large money borrowing by imposing he upper bound consrain for he porfolio weighs, we find a number of sample pahs which have a large racking error. I implies ha he imbalance beween he growh rae of he liabiliy and he profiabiliy of he asses. The presen paper is organized as follows. Secion 2 gives he mahemaical formulaion of he problem. The processes describing he asses and he liabiliy is defined as he sochasic differenial equaions, and we adop he lower mean square error beween he liabiliy and our wealh as he performance crierion. The numerical mehod is presened in Secion 3. We describe he procedure o obain he opimal porfolio sraegy according o he quadraic ap- - 112 -
proximaion scheme wih regression mehod. In Secion 4, numerical simulaions using he empirical daa provided by he Japanese organizaions are served. We regard he shorfall of income of he pension fund as he liabiliy and hedge i under he following wo ypes of consrains: (i) he noshor-selling consrain in Secion 4.1; (ii) he upper bound consrain for he porfolio weighs in Secion 4.2. 2 Model Le (Ω, F, {F },P) be a filered probabiliy space and le {W } be a n-dimensional Brownian moion. We denoe by S, S = (S 1,, Sn ) and Y prices of a risk-free asse, risky asses and a liabiliy a ime which are governed by he following sochasic difference equaions: ds ds i n S = r()d, S i = b i ()d + σ i j ()dw j, S j=1 = s >, S i = si >, i = 1,2,,n, dy = (A()Y + B()) d, Y = y R, where r, A, B : [,T] R, b : [,T] R n, σ : [,T] R n n are deerminisic coninuous funcions and T < sands for he mauriy. We assume ha all of he reurn rae of risky asse are larger han he risk-free rae, i.e., b i () r() >, i {1,2,,n}, [,T]. The wealh of an invesor X saisfies dx n = π i ds i n X S i + 1 π i ds i=1 S, {π } T A π, i=1 X = x = s + s 1, where π = (π 1,, π n ) sands for he porfolio weigh vecor and 1 = (1,,1) R n. A class of porfolio sraegies A π is he collecion of R n -valued F -adaped process {u } T which saisfies u i π, i = 1,2,,n. The hear of he definiion of he class A π is prohibiion of he shor selling. We define he performance crierion J π based on he downward side of he mean square error beween he liabiliy and our wealh which improve he negaive poin in he previous sudy [1] menioned in he inroducory secion: J π (x, y) [ 1 = E 2 (Y T X T ) + 2 + 1 2 T (Y s X s ) +ds ] 2 X = x,y = y. Here we use a noaion ( ) + s.. (x) + = x1 {x>}, x R. The value funcion of our problem V is defined by V (x, y) = inf π A Jπ π (x, y), wih erminal condiion V T (x, y) = 1 2 (y x)2 +, and he corresponding HJB equaion is V (x, y) + min π A π { L π V (x, y) } =, (1) where L π is an infiniesimal generaor of (X,Y ) : L π ϕ(x, y) = ( r() + (b() r()1) π ) x x ϕ(x, y) + (A()y + B()) y ϕ(x, y) + 1 2 x2 π σ()σ() π x x ϕ(x, y) + 1 2 (y x)2 +, for ϕ : R 2 R. We define he Hamilonian H as follow: H(, x, y, V, 2 V ) = min π A π { L π V (x, y) }. (2) 3 Numerical Mehod Under he curren conrol se A π, i is difficul o obain he analyical soluion for he HJB equaion (1). Hence we employ he quadraic approximaion scheme proposed by Nakano [3]. This mehod sars wih he following ime sepping V i (x, y) V i+1 + hh( i, x, y, V i+1, 2 V i+1 ), (3) and our goal is o obain V i, he approximaed value funcion around a poin ( x, ȳ) a ime i, in he following quadraic form: V i (x, y) =c 1,i + c 2,i (x x) + c 3,i (y ȳ) + 1 2 c 4,i (x x) 2 + c 5,i (x x)(y ȳ) + 1 2 c 6,i (y ȳ) 2, where i = ih, i =,1,, M, h = T/M, c j,i R, j = 1,,6. Hence he key o obain he approximae soluion is he approximaion of he Hamilonian. Before we inroduce he procedure o obain he approximae soluion, we ransform he Hamilonian (2) ino he form wihou he minimum. We assume ha x x V >, and hen he Karsh-Kuhn-Tacker condiion implies ha where H(, x, y, V, 2 V ) = L ˆπ V (x, y). {( ) } x V ˆπ i = min (π ) i, π, (4) x x x V + and π () = ( σσ ) 1 (b r1) (). We now show he procedure o obain he approximae soluion. Since erminal condiion is given in he quadraic funcion of x and y, he coefficiens { } c j, M are deermined as j=1,,6 follows: c 1, M = 1 2 (ȳ x)2, c 2, M = (ȳ x), c 3, M = ȳ x c 4, M = 1,c 5, M = 1,c 6, M = 1. Nex we define q R 2 and P R 2 2 as follows: ( ) q = (c 2, M,c 3, M ) c4,, P = M c 5, M, (5) c 5, M c 6, M - 113 -
and hen we find ha V M = P(x x, y ȳ) + q and 2 V M = P. Applying he regression mehod, we approximae he Hamilonian by he quadraic funcion: H ( n 1, x, y, P(x x, y ȳ) + q, P ) L 1 + L 2 (x x) + L 3 (y ȳ) + 1 2 L 4(x x) 2 + L 5 (x x)(y ȳ) (6) + 1 2 L 6(y ȳ) 2, where { } L j R, j = 1,,6. To deermine he coefficiens L j, we employ he leas square mehod wih K j=1,,6 sample poins (x k, y k ), k = 1,, K which are randomly chosen from he circle area wih radius r and he cenral poin ( x, ȳ). By (3) and (6), we have c j, M 1 = c j, M + hl j, j = 1,,6. Replacing M 1 ino M, going back o equaion (5) and { } repeaing his procedure, we obain he coefficiens cj,i which is equivalen o find he approximaed j=1,,6 value funcion V i. We are able o find he opimal sraegy subsiue V i ino he equaion (4). We remark ha he above procedure o obain he opimal sraegy does no conain processes which need a high compuaional cos. This is he advanage employing he quadraic approximaion mehod. Furhermore, if we only impose he no-shor-selling consrain, i.e., π =, we find he explici form of ˆπ, ˆπ = ( ( c2,i + c 4,i (x x) + c 5,i (y ȳ) )) + xc 4,i π 1 {( x<ȳ) (x<y)}, and he explici recursive formula for { c j,i } j=1,,6 : c 1,i =c 1,i+1 + h { r( i ) xc 2,i+1 + (ȳa( i ) + B( i )) c 3,i+1 2 (i ) c2 2,i+1 1 {c2,i+1 } + 1 (ȳ x)2 2c 4,i+1 2, c 2,i =c 2,i+1 + h { r( i )c 2,i+1 + r( i ) xc 4,i+1 + (ȳa( i ) + B( i )) c 5,i+1 2 (i )c 2,i+1 1 {c2,i+1 } (ȳ x) c 3,i =c 3,i+1 + h { r( i ) xc 5,i+1 + A( i )c 3,i+1 + (ȳa( i ) + B( i )) c 6,i+1 2 (i ) c } 2,i+1c 5,i+1 1 {c2,i+1 } + ȳ x, c 4,i+1 { c 4,i =c 4,i+1 + h 2r( i ) } 2 (i )c 4,i+1 1 {c2,i+1 } + 1, c 5,i =c 5,i+1 + h {r( i ) + A( i ) 1 2 (i )c 5,i+1 1 {c2,i+1 } }, c 6,i =c 6,i+1 + h { 2A( i )c 6,i+1 + 1 2 (i ) c2 5,i+1 1 {c2,i+1 } c 4,i+1, }, under he condiion c 4,i >, i = 1,,n. Therefore we are able o obain a proxy of he opimal sraegy wih remarkable high speed. 4 Numerical Resuls In his secion, we serve numerical simulaions base on our mehod using he empirical daa provided by he Japanese organizaions. The usage of daa is he same as in he lieraure [1]. Le C and B be he esimaed income and expense of he pension fund which are deermined by he daa published by he Japanese Minisry of Healh, Labour and Welfare [4]. We se = as he year 24 when he esimaed shorfall of he pension fund sars o expand drasically (see Figure 1). We se A() = and B() as he numerical differeniaion fo B C and hen we regard he esimaed shorfall of he pension fund as he liabiliy. [rillion yen] 16 12 8 4 225 25 275 21 [year] income expence Fig. 1: Esimaions of he income and expense of he Japanese welfare pensions. Red and blue lines indicae C, he esimaed income, and B, he esimaed expense respecively. We nex deermine he risk-free rae and he expeced reurn raes and volailiies of risky asses. We inves he following four asses: indices of he domesic bond, he domesic sock, he foreign bond and he foreign sock; we se n = 4 and number hem sequenially. According o he esimaions of reurn rae and volailiies by he Governmen Pension Invesmen Fund, Japan [5], we consruc b() and σ S () as follows: b 1 () = 3%, b 2 () = 4.8%, b 3 () = 3.5% and b 4 () = 5.%; σ() is he Cholesky decomposiion of he following variance-covariance marix of he asses: 29.7 18.2 4.39 5.41 Σ = 18.2 495 77.8 119 4.39 77.8 181 147 1 4. 5.41 119 147 394 We choose a money marke accoun as he risk-free asse and we se r() =.%. In he following wo subsecions, we consider he cases: (i) he no-shor-selling consrain; (ii) he upper bound consrain for he porfolio weigh. We calculae he opimal porfolio weigh by he mehod menioned in Secion 3-114 -
and run invesmen simulaions using he Euler-Maruyama scheme. We se he relaed parameers as in he Table 1. 16 12 resul Symbol Descripion Value T erminal ime 15 h ime sepping for calculaion.1 of opimal sraegy N number of sample pahs 1 ime sepping for simulaion.25 (quarerly rebalance) r radius of he regression region 2h K number of sample poins 1 for regression Table 1: Parameers Le us inroduce he mean hedging error ( ) N Y X i + E =, N i=1 o discuss he performance of he sraegy, where X i is he i-h sample pah of he invesmen simulaion. 4.1 Case of no-shor-selling consrain In his subsecion, le us consider he case of no-shor selling consrain only, i.e., π =. Figure 2 displays a sample pah of a invesmen simulaion. We find ha our opimal porfolio is mainly consruced by he domesic bond, he mos low-risk asse. The foreign bond is he second weighed asse and i is included in he opimal porfolio wih a considerable weigh. The weigh of domesic and foreign socks are lile and hus he high-risk asses do no play an imporan role in our hedging sraegy. These facs imply ha he opimal hedging mehod for he liabiliy is borrowing he money a lo and invesing he bonds, he low-risk asses. Since here does no exis he limi for he amoun of money borrowing, we are able o choose he sraegy ha invesing he low-risk asses a lo and earning he desired profi wih low-risk. Anoher feaure of our opimal sraegy is ha he large hedging error occurred in earlier sage of invesmen is quickly hedged. In Figure 2, we observe ha he porfolio weighs near he sar and erminal imes are differen despie he racking errors are no much differen. I is quie reasonable behavior: he racking error which is occurred in earlier ime and is no hedged expands by he reinvesmen effec, and hus we need o cover i quickly. Figure 3 describes he saisical resuls of he invesmen simulaion. We observe ha he mean wealh indicaed by he blue line racks he liabiliy indicaed by he red line well. The mean hedging error E described in he lower panel of Figure 3 is approximaely 4.5 % of he liabiliy and i is he equivalen level wihou he no-shor-selling consrain sudied in [1]. Hence we find ha he liabiliy racking abiliy of our opimal porfolio sraegy does no drop if we resric he shor selling. 8 4 1 1 2 weigh ime X B C asse_ype domesic bond domesic sock foreign bond foreign sock money marke accoun Fig. 2: A sample of he invesmen simulaion in he case of π =. Red and blue lines in he upper panel indicae he ime evoluion of X, our wealh, and B C, he liabiliy, respecively. Red, yellow, green, blue and purple bars in he lower panel display he porfolio weighs of he domesic bond, he domesic sock, he foreign bond, he foreign sock and money marke accoun respecively. 15 125 1 75 5 8 6 4 2 mean pah error ime E B C mean(x) Fig. 3: Saisical resuls of he invesmen simulaion in he case of π =. Red and blue lines in he upper panel represen he ime evoluion of mean(x ) = N X i i=1 N, he mean wealh, and B C, he liabiliy, respecively. Red bar in he lower panel indicae he ime evoluion E, he mean hedging error. 4.2 Cases of upper bound consrain for he porfolio weighs The opimal porfolio invesigaed in he previous subsecion is hard o apply he real business since he large money borrowing is quie difficul o realize. Therefore we sudy he case ha we impose he upper bound for he porfolio weighs. We se π = 1, i.e., he weigh of each asse is bounded by 1% and hence he minimum weigh of he money marke accoun is -3%. Alhough he bound of he money borrowing is no lile, resuls of invesmen simulaions imply ha he pension fund faces he severe siuaion under he curren resricion. Figure 4 displays a sample pah invesmen simulaion which is a case ha he liabiliy is well hedged. We find ha he porfolio weighs are no concenraed on he domesic bond: all of he weighs closes up he upper bound around = 1. We also noe ha he weigh rises up in he increasing - 115 -
order of he risk, i.e, he domesic bond is he mos weighed asse, he foreign bond is second one, and he domesic and foreign socks follows sequenially. 16 12 8 4 1 1 2 3 resul weigh ime X B C asse_ype domesic bond domesic sock foreign bond foreign sock money marke accoun Fig. 4: A sample of he invesmen simulaion under he consrain π = 1: he well hedged case. Red and blue lines in he upper panel indicae he ime evoluion of X, our wealh, and B C, he liabiliy, respecively. Red, yellow, green, blue and purple bars in he lower panel display he porfolio weighs of he domesic bond, he domesic sock, he foreign bond, he foreign sock and money marke accoun respecively. Under he consrain π = 1, he poor hedged resuls as shown in Figure 5 appear frequenly, hough hey are rarely found in he case of π =. Figure 6 implies ha he resul displayed in Figure 5 is no a special case. The mean hedging error E described in he lower panel of Figure 6 is approximaely 21% of he liabiliy and which is much higher han 4.5% sudied in he previous subsecion. Almos all of he porfolio weighs in Figure 5 reach he upper limi however he profi earned by hose asses is no enough o recover he racking error. 16 12 8 4 1 1 2 resul weigh ime X B C asse_ype domesic bond domesic sock foreign bond foreign sock money marke accoun Fig. 5: A sample of he invesmen simulaion under he consrain π = 1: he poor hedged case. Red and blue lines in he upper panel indicae he ime evoluion of X, our wealh, and B C, he liabiliy, respecively. Red, yellow, green, blue and purple bars in he lower panel display he porfolio weighs of he domesic bond, he domesic sock, he foreign bond, he foreign sock and money marke accoun respecively. Le us discuss he implicaion of hese resuls. Our simulaions imply he imbalance beween he growh rae of he liabiliy and he profiabiliy of he asses. We denoe by α he growh rae of he liabiliy and i is 7.6% which is higher han all of b i. Since our purpose is minimising he downside racking error, i is no enough ha he expeced reurn raes of he asses close o α: he high volailiy asses possibly disurb he racking. Furhermore he curren resricion which arrows he 3% money borrowing is even weak for he pracice. Therefore he implicaion is summarized ha he imbalance of he growh rae of he liabiliy and he profiabiliy of he invesed asses should be resolved. 15 125 1 75 5 3 2 1 mean pah error ime E B C mean(x) Fig. 6: Saisical resuls of he invesmen simulaion in he case of π = 1. Red and blue lines in he upper panel represen he ime evoluion of mean(x ) = N X i i=1 N, he mean wealh, and B C, he liabiliy, respecively. Red bar in he lower panel indicae he ime evoluion E, he mean hedging error. 5 Summary In he presen paper, we develop he porfolio sraegy from he sraegy proposed by Ieda e. al. [1]. The previous sudy remains wo poins which should be improved: (i) penalising he wealh of he invesor exceed he liabiliy; (ii) permiing he large shor selling. To resolve hese poins, we employ a lower mean square error from he racked liabiliy as he performance crierion and resric he porfolio weigh o a posiive and bounded value. In his case we are no able o obain an analyical soluion and hus we employ he numerical scheme proposed by Nakano [3] o solve he corresponding HJB equaion. We run numerical simulaions using he empirical daa provided by Japanese organizaions under he wo ypes of consrains: (i) he no-shor-selling consrain; (ii) he upper bound consrain for he porfolio weighs. The implicaions obained by hese simulaions are summarized as follows. We find ha he liabiliy racking abiliy of our opimal porfolio sraegy does no drop if we resric he shor selling. The resuls sudied in Secion 4.1 indicaes ha he mean hedging error is approximaely 4.5% of he liabiliy and i is he equivalen level wihou he no-shor-selling consrain sudied in [1]. Under he no-shor-selling consrain, he hedge is realized by borrowing he money a lo and invesing - 116 -
he low-risk asses. According o his resul, we resric he large money borrowing by imposing he upper bound consrain for he porfolio weighs in Secion 4.2. Then we find a number of sample pahs which have a large racking error. I implies ha he imbalance beween he growh rae of he liabiliy and he profiabiliy of he asses. Acknowledgemens This work is suppored by a collaboraion research projec wih he Governmen Pension Invesmen Fund in Japan in 213-214. References [1] Masashi Ieda, Takashi Yamashia, and Yumiharu Nakano. A Liabiliy Tracking Approach o Long Term Managemen of Pension Funds. Journal of Mahemaical Finance, 3(3):392 4, 213. [2] Harold J Kushner and Paul G Dupuis. Numerical mehods for sochasic conrol problems in coninuous ime, volume 24. Springer, 2. [3] Yumiharu Nakano. A quadraic approximaion scheme for Hamilon-Jacobi-Bellman equaions. Working Paper, Tokyo Insiue of Technology, Graduae School of Innovaion Managemen, pages 1 23, 213. [4] The Japanese Minisry of Healh, Labour and Welfare. hp://www.mhlw.go.jp/seisakunisuie/ bunya/nenkin/nenkin/zaisei-kensyo/index. hml [accessed 1 Ocober 214]. [5] The Governmen Pension Invesmen Fund in Japan. hp://www.gpif.go.jp/operaion/commiee/ pdf/h1911_appendix_5.pdf [accessed 1 Ocober 214]. - 117 -