Optimum Centralized Portfolio Construction with. Decentralized Portfolio Management

Transcription

1 Optmum Centralzed ortfolo Constructon wth ecentralzed ortfolo Management Edwn J. Elton*and Martn J. Gruber* October 8, 00 * Nomura rofessors of nance, Stern School of usness, New York Unversty. e would lke to thank Stephen rown for hs helpful comments on our manuscrpt.

2 Many fnancal nsttutons employ outsde portfolo managers to manage part or all of ther nvestable assets. These nsttutons nclude penson funds, prvate endowments (e.g., colleges and chartes), and prvate trusts. In 999, the nvestment company nsttute estmated that these nsttutons managed 5. trllon dollars n assets. Most of these nsttutons employed outsde managers to nvest these funds. The relevancy of ths problem has been wdely recognzed n the practtoners lterature on portfolo? urthermore, t s recognzed n the prudent man law that spells out the responsbltes of the centralzed decson maker delegatng management responsblty. or example the New York State law n estate power and trust states. enson funds are the largest and most lkely organzatons to employ several outsde managers, each of whom manages a part of the overall portfolo. In ths paper we wll use the penson fund manager as the prototype of the centralzed decson-maker tryng to optmally manage a set of decentralzed portfolo managers but the analysts s general. If the centralzed decson-maker (CM) s a mean varance maxmzer, the CM could construct a portfolo usng standard portfolo theory and estmates of mean return, varances, and covarances between the portfolos constructed by a group of decentralzed managers. However, ths overall portfolo s unlkely to be optmum snce the ndvdually managed portfolos themselves were constructed wthout takng nto account the portfolos of the other managers. The purpose of ths artcle s to set up a structure that leads to the optmum portfolo from the vewpont of the CM when there are multple managers and ther portfolos are constructed wthout reference to each other. See, for example, artolomer (999), Grnald and Kahn (995), arrell (976), and osenberry (977). or a full dscusson of modern portfolo theory and the prudent man rule see Elton and Gruber n Longstaff ( ).

3 Ths paper can be vewed as a contrbuton to the extensve lterature n nancal Economcs developng condtons under whch a CM wll never make a worse decson than decentralzed managers provdng the nformaton s used optmally. 3 Ths lterature assumes the decentralzed managers are wllng to provde all nformaton to each other or to a centralzed manager. Ths case has been examned for the penson fund problem by osenberg (977) and. artolomeo (999). or example, osenberg (977) demonstrated that wth full nformaton, the decentralzed portfolo managers wll not make better decsons than that of the centralzed manager. Ths paper examnes a specal case of ths more general lterature: the case of a centralzed portfolo manager employng a set of ndvdual portfolo managers each of whom constructs hs or her own portfolo wthout communcatng wth other managers. Several authors have asserted that ths problem s too dffcult to solve (see osenberg (977) and. artolomeo (999)). They argue the only solutons are for each outsde manager to ether turn over all of ther estmates for ndvdual securty characterstcs to a centralzed manager or to supply all purchases and sales to the centralzed managers. In the latter case models are presented that allow the central manager to approxmate from ths nformaton, the ndvdual managers forecasts. hat makes ths a specal case s the realstc assumpton that a decentralzed manager s only wllng to share some nformaton wth the centralzed manager and none wth other managers. In ths artcle, we solve for sets of condtons under whch the centralzed manager can make optmum decsons despte partal nformaton through the use of gudelnes for the decentralzed managers. hle ths can be vewed as an extenson of the prevous lterature on centralzed versus decentralzed 3 See, for example, adner (96), Marshak and adner (97), and Ohlson (975 and 979). 3

4 decson makng t s of at least equal mportance because t offers a soluton to a problem whch s at the heart of nvestment allocaton today. In the frst secton we wll present a more detaled dscusson of the problem. e wll then solve the problem for one actve manager and multple passve portfolos. The model s then generalzed to multple actve managers. Next, we present solutons under a smplfed structure of the return-generatng process. nally, we dscuss the complcatons when short sales are not allowed. I. ackground In ths secton we dscuss some background materal on the penson nvestment problem and revew the relevant lterature. The same consderatons hold for prvate endowments and trusts. Most penson plans are managed by a centralzed decson maker at a frm. Most frms have one person who s prncpally n charge, although the ultmate responsblty rests wth a commttee, usually the board. Ths CM normally employs outsde portfolo managers to construct actve portfolos. Index funds are generc products and we wll assume the centralzed decson maker can potentally select one or more of these. The centralzed decson maker s task s fourfold: ) decde how much to nvest n each portfolo, ) gve the outsde managers nstructons that wll result n ther makng optmum securty allocatons from the pont of vew of the overall plan, 3) desgn ncentve systems so that the managers wll behave optmally 4, and 4) evaluate and select the portfolo managers. In ths paper we deal only wth the frst two of these problems although our solutons have maor mplcatons for the thrd and fourth problems. Throughout the paper, we assume that the portfolo managers wll not provde the 4 No one has addressed the mult-perod ncentve problem outlned here. However, there are a number of related artcles. See for example, yvg, arnsworth and Carpenter (00), Khlstrom (988), Stoughton (993), and ender (988). 4

5 centralzed decson maker wth ther return forecasts for ndvdual securtes, but wll provde aggregate nformaton about the portfolos they hold. spects of ths problem have prevously been addressed by Treynor and lack (973) and n more detal by Sharpe (98). The Treynor lack artcle dscussed the actve passve splt when the CM descrbed the returns on the passve portfolo, short sales are allowed and the sngle-ndex model descrbes the return generatng process. The clear antecedent to ths artcle s Sharpe s (98) resdental address. Sharpe develops, wth one actve and one passve manager, the nstructons for the actve manager that wll result n the actve manager producng a globally optmal portfolo for a partcular utlty functon. He assumes short sales are allowed and the varance covarance matrx s agreed on by all partes. He also solves for the nstructons to be gven to the managers that results n a global optmal for the case of two managers followng exactly the same set of securtes where the centralzed decson maker beleves the best forecast of a securtes alpha s a weghted average of the two managers alphas and where these weghts add to one. In solvng ths problem he mantans the assumpton of short sales allowed and agreement on the varance covarance matrx. Sharpe could not obtan an exact soluton for the case of managers followng non-overlappng securtes. Our analyss extends Sharpe n that we generalze to N managers, have no requrement that each manager holds the same securtes, and, by employng a multfactor model, can arrve at smple rules for formng myopc optmum portfolos, understandng the weght placed on each securty n these portfolos, and the amount to allocate to each actve and passve portfolo. e also extend the analyss to the case 5

6 where short sales are not allowed and show condtons under whch optmal decentralzed management s possble and when t s not. II. Separaton wth a sngle actve and multple passve managers In ths secton of the paper we wll assume that a centralzed decson maker (CM) exsts who hres a sngle actve manager. e wll shortly expand the case to several actve managers. e wll assume the followng: ) the CM s a mean varance decson maker, ) the CM beleves a mult-ndex model descrbes the return structure for securtes and all ndexes n the mult-ndex model are tradable. The second pont requres some clarfcaton. The CM beleves that returns can be descrbed as beng generated by a set of ndexes (not necessarly orthogonal) that the CM can take postons n as passve portfolos 5. or example, ths s consstent wth a belef that the return on securtes s a functon of the market return, the return on a portfolo of small stocks, and/or the return on a portfolo of value or growth stocks. The CM wshes to consder these sources of rsk n makng the optmum mean varance decson. or expostonal reasons we wll analyze the CM s problem wth a two ndex model though the soluton easly generalzes to any number of ndexes.. The CM s problem e start by examnng the optmum decson the CM would make f the CM had all the nformaton that s avalable to the actve managers. s mentoned earler, we beleve the CM would not be able to obtan rsk adusted return forecasts for ndvdual securtes from the actve manager, but for the moment we examne the optmum decson 5 Index funds, many of them exchange traded, exst for almost any ndex a manager mght want to use n a return generatng process. 6

7 as f the CM has such nformaton. e wll also assume that the CM does not have perfect fath n the return forecasts of the actve manager. Ths mples that the CM wll take postons n the passve portfolos for two reasons, to obtan dversfcaton across securtes so that the aggregate portfolo s mean varance effcent, and to elmnate some of the lack of relablty n the analyst s estmates. In order to specfy the return generatng process, defne. s the return on stock. s the rsk free rate of nterest 3., s the return on ndex and ndex respectvely, s the senstvty of stock to ndexes and s the varance of the return on ndexes and, e s the resdual rsk of stock from the two-ndex model 7. s the rsk adusted return on securty 8. e s the resdual return for securty 9. The superscrpt desgnates that the decson s from the pont of vew of the CM. Then the return generatng process s ( ) ( ) e () ssume that the CM had access to the excess return forecasts ( ) of the actve manager. urthermore, assume the CM beleves that the best estmate of rsk-adusted excess return s an average of the analysts forecasts and the value that would occur n 7

8 equlbrum namely zero. Thus, we defne the excess rsk adusted return that the CM would use as where s set by the CM between 0 and 6. To solve ths problem, assumng short sales, the CM can use the standard frst order condtons. The nvestments that can be selected are the N ndvdual securtes and the two ndexes. The frst order condton for securty and ndex s N for,,n () here. N s the number of securtes enterng nto the decson makng process. Securty N and N are ndexes whch we henceforth desgnate as and. 3. s a number proportonal to the optmal weght whch the CM would place n securty If the return generatng process descrbed n equaton () s an accurate descrpton of returns and we recognze that the ndexes need not be orthogonal, then we can defne the varance and covarance between ndvdual securtes as for,,n e for N, N 6 hle the optmum way to set s beyond the scope of ths paper, there have been a number of excellent artcles publshed n the past few years explanng optmum ways of changng alpha for estmaton rsk and bas. See awa, rown, and Kleen (979) for the fundamental applcaton of aysan analyss and aks, Metrck, and achter (00) and astor and Stambaugh (00) for recent applcatons of aysan analyss to estmatng the nputs for optmal portfolo allocaton. 8

9 or the N and N securtes (the ndexes), a smpler form exsts. or example, for ndex the varance s and the covarance wth ndex s and the covarance wth ndvdual securtes s Employng these relatonshps wth the frst order condton (), we get for securty ) ( ) ( e N N (3) N and for the ndexes N N (4) N N (5) Substtutng equatons (4) and (5) nto equaton (3) and smplfyng, we get 7 (6) e e To solve for the optmum amount n securty we consder the actve portfolo denoted by as a separate portfolo and look at the optmum composton of ths portfolo before we allocate across all three portfolos. e can treat the desgn of as a separate portfolo because from equaton (6), s not a functon of or. 7 smlar expresson but n a dfferent context can be found n Elton and Gruber and adberg (979). 9

10 The fracton to nvest n any stock, p, n the actve portfolo can be determned by recognzng that 8 p. Therefore, recognzng that the amount to nvest n an stock n the optmal actve portfolo from the vewpont of the CM s p e e (7) N N t e e Once portfolo s determned smple procedures exst for allocatng funds between the actve and passve portfolos. These are presented n Secton C below.. Optmum actve portfolo The CM can ensure that the actve manager wll hold the optmal actve portfolo from the pont of vew of the CM smply by nstructng the actve manager to compute by e for each stock and to hold them n that proporton. 9 Ths smple nstructon ensures that the actve manager wll turn over to the CM the same actve portfolo that the CM would hold f all the securty estmates were suppled drectly to the CM. Optmzaton for the actve portfolo s reached wthout the actve manager gvng up prvate nformaton. Of course the CM stll has the problem of decdng what fracton of funds to place n the actve portfolo and each of the passve portfolos. 8 See Elton, Gruber, and adburg (976) for a full exposton or Lntner (965) for the orgnal proof. 9. If the decentralzed manager were smply told to form the optmum actve portfolo assumng that he could hold the passve portfolo, he would get the same result as followng the drecton from the central manager. lthough ths rankng devce was derved n the pror secton usng two ndexes t s easy to show that the same rankng devse holds f there are N ndexes 0

11 C. Solvng the aggregate allocaton problem enote the characterstcs of the actve portfolo by the subscrpt. Then from the vewpont of the CM, gnorng for the moment any dffculty of gettng nformaton, the problem can be formulated and solved usng the followng frst order condtons 0. ) ( ) ( ( e ) ( ) ( ) These are standard frst order condtons. Snce everythng but the s are known, the equatons can be solved explctly for the optmal fracton of funds n each portfolo. To do ths we utlze a relatonshp we derve later. s shown n equaton (3), e e. Usng ths expresson, the soluton s ( ) ( ρ ) ( ) ρ ( ρ ) e ( ρ ) ( ) ( ρ ) (8) e e here ρ s the correlaton between passve portfolo and. These three equatons along wth the expresson normalzng the portfolo weghts to add to one whch s 0 The extenson to more than two ndexes s straghtforward. One new equaton would be added for each ndex, one for each new ndex would be added to each equaton and the varance and covarance terms would be modfed to account for the addtonal ndexes.

12 k k l,, and l l gve us the closed form soluton for the optmal weght to place n the actve and each passve portfolo. The optmal weghts depend on the fundamental characterstcs of each of the three portfolos n a way that makes ntutve sense. or each of the passve ndex funds, the hgher the excess return on the fund relatve to ts varance, the larger the allocaton of funds to that portfolo. Smlarly, for the actve portfolo, the larger the rsk adusted return for that portfolo relatve to ts unsystematc rsk, the greater the funds placed n t. The correlaton coeffcent between the ndexes also has a large effect on the relatve nvestment n each of the passve portfolos. The mpact of the correlaton coeffcent on allocaton depends on the rato of the excess return to standard devaton of ndex to that of ndex as well as the sze of the correlaton coeffcent tself. In order to determne the splt across portfolos, the CM needs to request the actve manager s estmate of the alpha for the actve portfolo, the resdual rsk of the actve portfolo and the actve portfolo senstvtes to the two ndexes. These are the types of estmates the actve manager should be wllng to supply snce they are aggregate portfolo values rather than ndvdual securty values. The CM needs to estmate the expected return above the rskless rate and rsk on the passve funds, the covarance between the passve s stated earler, we are assumng that the CM and the actve manager are employng dentcal estmates of the s and resdual rsks but not return characterstcs of each securty. Ths could come about naturally f the rsk parameters were estmated from the same commercal servce (e.g., or lshre). The CM could ether specfy that decentralzed managers use a partcular commercal servce or drectly supply the rsk parameters for the assumpton of our model to hold. The need for a common return generatng process mght partcularly explan the specfcaton of benchmarks n contracts wth managers.

13 3 funds, and the amount of weght () to put on the actve manager s estmates. If futures are avalable on the ndexes, the aggregate portfolo problem s smplfed.. The ggregate ortfolo roblem wth utures If futures are avalable on the ndexes, then senstvtes to the ndexes can be adusted wthout affectng the amount nvested n the actve portfolo. The expected return and rsk on the portfolo of the CM s, ( ) ( ) C e C where C s the overall portfolo held by the CM. The choce varables for the CM are how much to put n the actve portfolo, how much to place n the rskless asset and the level of senstvty of the overall portfolo to each of the factors. Takng dervatves of C C θ wth respect to and, respectvely results n the followng frst order condtons e e C C ( ) C C ( ) C C. The effcent fronter s the lne connectng the rskless asset wth the optmal rsky portfolo. If we can determne one pont on the lne, we can trace out the full effcent

14 fronter. Varyng traces out the lne. Thus, wth no loss of generalty we can solve for the portfolo wth. Settng equal to one we get that the optmum betas are ( ) ( ) ( ρ ) ( ) ( ρ ) ρ ρ e e where ρ s the correlaton between the two ndexes. The easest way to nterpret the results s to consder the case ρ equal to zero. th ths assumpton, s equal to the excess return to rsk of ndex dvded by the rato of the rsk adusted return to the resdual rsk of the actve portfolo. hen the ndexes are correlated, ths rato s modfed to take account the correlaton between the ndexes. e have now presented a set of condtons under whch a centralzed decsonmaker can optmze portfolo composton whle employng one actve manager. The next problem to solve s the case where the CM employs several actve managers. III. Multple actve managers The analyss generalzes to multple actve managers whether these managers follow some or all securtes n common or follow ndependent sectons of the market. or Sharpe (98) dd not reach an explct soluton n the case where only some securtes were n common across actve managers. 4

15 smplcty we wll solve for the case of two actve managers, but the analyss easly generalzes. ssume that the CM has all the nformaton produced by each manager but dfferent confdence n the forecasts of each manager. urthermore, the CM beleves that all the managers estmates are too extreme but that the approprate estmate s some combnaton of them. 3 If we desgnate the weght the CM puts on the estmate prepared by manager as and manager as. Then. Once agan t s necessary for the CM to supply estmates of betas and resdual varances to all actve managers ether drectly or by specfyng that they use a common servce such as. Snce e s suppled by the CM to all managers, t s common and e (9) e e Earler we showed that e was proportoned to the optmum amount that the CM wshed to place n securty f all alphas were suppled to the CM. The ssue we address n ths secton s the nstructons to gve to the ndvdual managers and the correct proportons to nvest n each actve portfolo so that the CM, by combnng the portfolos of the actve managers, ends up wth a fracton n the actve portfolo proportonal to e for each securty. Summng both sdes of equaton (9) across all securtes 3 Implct n what follows s f only one manager follows a securty, the CM assumes the best estmate of the second manager s alpha f he/she followed t would be zero. 5

16 6 e e e (0) If the CM nstructs each manager to compute e for each securty and to place a fracton of money n each securty proportonal to ths rato we can defne the fracton any manager (e.g., manager ) places n any securty as e N e. e wll now show ths nstructon results n an overall optmum. However, before we do so, we need to derve some of the attrbutes of the portfolo whch manager (or any manager) wll hold. The rsk-adusted excess return on the portfolo held by manager s e e e e () and the resdual rsk of ths actve portfolo s

17 7 e e e e e e e () Takng the rato of () and () yelds e e (3) urthermore, K where the subscrpt K s a counter, ndcaton ether ndex or ndex equals e K e k K (4) earrangng and substtutng equaton (3) yelds e e K K (5) Havng developed these expressons, we can now show that there exsts an allocaton across the actve portfolos along wth the nstructon to the ndvdual managers to hold

18 8 stocks n proporton e, whch results n an overall optmum to the CM. Substtutng equaton (3) nto (0) yelds e e e (6) ecall that the ndvdual portfolo manager has been nstructed to form a portfolo by holdng securtes proportonal to the rato of excess return to resdual rsk. ecognzng ths nstructon and usng equaton (3) to smplfy the denomnator e e (7) vdng both sdes of equaton (9) by e, the correct amount n securty n the actve portfolos from the pont of vew of the CM s e e e e e e e e e e (8) e e e e

19 here the terms n brackets represent the proporton of the actve portfolo to nvest wth manager and manager, respectvely. Snce e can be computed from equaton (6), f the CM obtans and from manager, and e and from e manager, he or she can determne optmum proportons among actve managers. In addton, snce the CM knows the characterstcs of the aggregate actve portfolo, the CM can act n determnng the splt between the actve and passve portfolos as f there s a sngle portfolo. Thus, the allocaton between the actve portfolo and the two passve portfolos can be determned usng the equatons n Secton II C. IV. Orthogonal Indexes Up to ths pont we have assumed that the ndexes are not orthogonal. The advantage of ths s that t allows the passve portfolos to be portfolos that exst n the market such as small stocks, the S& Index, growth stocks, etc. However, f we are wllng to assume orthogonal ndexes the allocaton across actve and passve managers s smplfed. th orthogonal ndexes, the covarance among ndexes s zero, and there exsts a smple formula for the amount to nvest n the passve ndex. or passve ndex equaton (4) becomes Solvng for 9

20 0 N Substtutng for from equaton (6) yelds e N Expressng e n terms of the two actve portfolos e N e N nally, usng equaton (5): e e Thus, the centralzed decson maker can determne the total and the splt between each of the passve portfolos and each of the actve portfolos usng a smple formula f all managers provde ther estmates of and, e on each ndex, and the centralzed decson maker estmates the s and excess return and rsk on the ndex. The actve managers also need to have common rsk measures, s and e for all securtes under consderaton. In the case of orthogonal ndexes, characterstcs of ndexes other than the

21 one beng analyzed do not mpact the assocated wth any ndex. Thus, the equaton apples to any number of ndexes 4. IV. Short sales not allowed Let s start wth the case of a sngle actve manager where short sales of the ndexes are allowed but short sales of securtes are not. Ths case s realstc for some centralzed decson makers. utures exchange traded funds or future replcatons wth optons exst for many ndexes. In ths case a CM can effectvely short sell ndexes. To determne the optmum when securtes cannot be short sold, we need to use Kuhn, Tucker condtons. Ths smply nvolves addng the dual varables M s (one for each securty) to equaton (3) the frst order condtons for each securty when short sales are allowed. The soluton to the portfolo problem makes use of the complmentary condtons that the product of the dual and the prmal must be zero ( M 0 for all ) and that M and must be equal to or greater than zero for all. Snce there are no duals on the frst order condtons for ndexes, equatons (4) and (5) are unchanged. Equaton (6) holds wth the addton of the dual for the securty 5. ddng the dual, equaton (6) becomes M * e. If s postve, must be postve snce postve from the complementary correlaton, * M must be postve so that * M cannot be negatve. If s * M must be zero. If s negatve then s not negatve and from the complementary condton 5 * M s a transformaton of the M added to each equaton, but has the same sgn as M.

22 must be zero. Thus, all ether equal e or zero. The optmum portfolo for the CM s obtaned by havng the manager nvest n all securtes for whch > 0 and as before n proporton to e. The equatons n secton II C, then defne the optmal splt between the actve and passve portfolos. If there are multple actve managers, the condton under whch an optmal soluton can be reached are more restrctve. To understand the problem, consder the case where manager forecasts > 0 and manager forecasts < 0 where the absolute value of s greater than and the CM puts equal weghts on the estmates of each manager. In ths case the CM would want to hold zero n securty. However, manager wll hold postve proportons and wthout short sales, manager wll hold zero rather than short sell. No combnaton wll provde an optmum to the CM. The only excepton to ths scenaro s the case where the centralzed manager wshes to place no weght on a forecast of a negatve alpha. Ths mples that the CM beleves the managers have no ablty to forecast below normal returns but have some ablty on the upsde. In the case where > 0 and < 0, the CM would want to use and, provdng all passve portfolos are held long or short sales of passve portfolos are allowed, the analyss outlned above goes through wth each actve manager not allowed to have short sales.

23 V. Concluson In ths artcle we have shown that under realstc condtons when short sales are allowed, t s possble, and ndeed qute easy, for a centralzed decson maker to form an optmal overall portfolo whle employng multple outsde portfolo managers 6. Ths s on contrast to the assertons n the practtoner lterature that argue ths s not possble or possble only wth full nformaton. Outsde managers should be wllng to supply the nformaton the CM needs n our models snce t does not requre them to reveal prvate nformaton on ndvdual securtes. Managers should be hestant to reveal nformaton on ndvdual securtes, snce t s useful for multple portfolos and to reveal t opens up the possblty of resale or drect use of the nformaton. hen short sales are not allowed and f there s a sngle actve manager to combne wth passve ndexes, a soluton exsts f t s optmum for the manager to place some funds n each ndex and/or the ndexes (as opposed to the securtes) can be sold short 7. hen short sales are not allowed and there are multple actve managers, the prevous analyss holds as long as a forecast of a negatve alpha by a manager s taken to convey no nformaton and the manager s smply told not to hold securtes wth negatve alpha. e have shown that n the case of multple managers, f short sales are not allowed and the centralzed manager makes use of estmates of negatve alphas as well as postve alphas, a general optmum soluton does not exst. 6 llowng short sales s an ncreasngly realstc case wth the ablty to use futures to short and wth funds lke hedge funds routnely shortng. 7 The assumpton that ndexes can be sold short becomes ncreasngly realstc over tme as exchange traded funds and futures have been created for an ncreasng number of ndexes. It can be shown f the ndexes cannot be sold short, a soluton stll exsts as long as one and only one ndex s not held long 3

24 blography aks, Klaas, ndrew Metrck and Jessca achter (00), Should nvestors abort all actvely managed mutual funds? Journal of nance, Vol. 56, pp away, Vay, Stephen rown and oger Klen (979), Estmaton rsk and optmal portfolo choce, (North Holland, msterdam). artolomeo (999), radcal proposal for the operaton of mult-manager nvestment funds, Northfeld Informaton Servces. ypvg, hlp, Heber arnsworth, and Jennfer Carpenter (000), ortfolo performance and agency, unpublshed manuscrpt, New York Unversty. Elton, Edwn J., Martn J. Gruber and Manfred adberg (976). Smple crtera for optmal portfolo selecton. Journal of nance Vol., pp Elton, Edwn J., Martn J. Gruber, and adberg Manfred (979), Smple crtera for optmal portfolo selecton, n Edwn J. Elton and Martn J. Gruber ortfolo Theory: Twenty-ve Years Later. msterdam, North Holland. 4

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