MEAN-VARIANCE ASSET ALLOCATION FOR LONG HORIZONS. Isabelle Bajeux-Besnainou* James V. Jordan** January 2001

Transcription

1 MEAN-VARIANCE ASSE ALLOCAION FOR LONG HORIZONS Isabelle Bajeux-Besnainou* James V. Jordan** January 1 *Deparmen of Finance he George Washingon Universiy 3 G S., NW Washingon DC (fax 514) bajeux@gwu.edu **Naional Economic Research Associaes rd S. NW Washingon DC (fax 911) james.jordan@nera.com he auhors have benefied from he commens of paricipans in he Easern Finance Associaion 1998 Annual Meeing a which heir paper received he award for Ousanding paper in Invesmens.

2

3 MEAN-VARIANCE ASSE ALLOCAION FOR LONG HORIZONS Absrac We invesigae wheher mean-variance porfolio heory can produce he convenional wisdom ha invesors wih long horizons should make a large iniial allocaion o socks and hen decrease he allocaion as ime passes. For he case of a riskfree asse and a sock index following geomeric brownian moion, we derive closed-form soluions for he mean-variance porfolio problem allowing coninuous rebalancing based on realized prices and wealh (called a sochasic sraegy). his opimal sochasic sraegy is in general a convenional wisdom sraegy as i involves large iniial allocaion o socks which hen decreases wih ime. We relae his sraegy o he concave sraegies described by Perold and Sharpe and explain he role played by relaive risk aversion in his resul. We also derive he opimal deerminisic sraegy (predeermined schedule of weighs, independen of new price and wealh realizaions) and find i o be a consanweigh sraegy.

4 I. Inroducion he convenional wisdom, as defined in his paper, is ha invesors wih long horizons should allocae mos of heir liquid wealh o socks and hen decrease he sock allocaion as ime passes. No only is such advice very common in he pracical invesmen world (e.g., Del Pree (1997)), bu recen survey evidence from he IAA/CREF pension managemen organizaion shows ha individual invesors behave his way (Bodie and Crane (1997)). As invesmen in self-direced reiremen accouns coninues o grow, and as many governmens consider social securiy privaizaion, i becomes increasingly imporan o undersand he assumpions under which he convenional wisdom is raional. We ask wheher he convenional wisdom is raional in a mean-variance framework. Alhough he limiaions of mean-variance analysis are well esablished in porfolio heory, is relaive simpliciy and easy inuiion conribues o is coninued use among invesmen professionals, in heoreical and empirical sudies and in he classroom. I is of ineres o explore he raionaliy of he convenional wisdom o a mean-variance invesor. Much of he lieraure on porfolio heory does no suppor he convenional wisdom. he sandard Markowiz problem for wo asses (risk-free cash and a sock index following geomeric brownian moion) ypically dicaes a small iniial sock allocaion for long horizons. Moreover, in his buy-and-hold, or saic framework, he sock allocaion will ypically increase raher han decrease over ime due o he posiive excess reurn on socks. Dynamic porfolio heory provides mixed suppor for he convenional wisdom. Samuelson (1969) and Meron (1971) show ha a consan weigh sraegy, 1

5 raher han decreasing sock weighs, is opimal if he invesor s uiliy funcion displays consan relaive risk aversion (CRRA) and asse prices follow geomeric brownian moion. Samuelson has been able o jusify equiy weighs decreasing over ime for CRRA preferences by assuming mean-revering sock reurns (Samuelson (1991)) and by imposing a minimum wealh consrain for geomeric brownian moion (Samuelson (1989)), in boh cases wih a consan ineres rae. he lieraure on urnpike porfolios (e.g., Ross (1974) and more recenly Cox and Huang (199)) shows ha for a broad class of uiliy funcions (including he linear risk olerance, or hyperbolic absolue risk aversion (HARA) funcions), he opimal sraegy converges o consan weighs as he horizon increases. Bodie, Meron and Samuelson (199) have cerainly found he sronges case for he opimaliy of he convenional wisdom hrough he inroducion of human capial ino he Meron (1971) consumpion-invesmen model. In his model, an iniial large equiy allocaion balances a large, and less risky, iniial human capial allocaion. Over ime, as he human capial allocaion declines, so does he equiy allocaion. A recen addiional paper including consumpion as well as invesmen is Campbell and Viceira (1996). he invesor is here assumed o be infiniely-lived, and o use an Epsein-Zin-Weil recursive uiliy funcion. We do no consider he consumpion-invesmen problem in his paper, because we wish o focus only on he properies of he mean-variance problem. We consider he sylized problem, similar o he one considered in he urnpike lieraure, of an invesor concerned only abou he mean and variance of erminal wealh under self-financing consrains (no inermediae cash flows and consumpion).

6 wo recen papers in a non-mean-variance seing are Brennan (1998) and Barberis (). Brennan assumes coninuous rading wih an uncerain mean reurn for he risky asse abou which he invesor learns over ime. He finds ha allocaion o he risky asse depends on he invesor s degree of risk aversion. Our paper is similar o he sudy by Barberis (), who invesigaes hree sraegies for power uiliy invesors: buy-and-hold, a deerminisic sraegy wih consan weighs, and an opimal rebalancing sraegyfor a rebalancing inerval of one year. Barberis focuses on wo issues, parameer uncerainy and predicabiliy in asse reurns. We do no deal wih hese issues because we are focusing in his paper solely on he implicaions of mean-variance crieria. hese oher issues could be considered in exensions of his research. In order o obain closed-form soluions and useful inuiion, we limi he asse allocaion problem o wo asses, a risk-free asse ( cash ) and a sock index. he riskfree rae is consan, and he sock index, which pays no dividends, follows geomeric brownian moion. hese assumpions are also found in he urnpike lieraure. In relaed mean-variance research, Richardson (1989) and Bajeux-Besnainou and Porai (1998) have invesigaed, in a general seing, opimally rebalanced mean-variance porfolios in coninuous ime. hey have no, however, considered he specific long-horizon issues raised here. Nguyen and Porai () have addressed a very similar mean variance model, bu wih a solvency consrain, ha wealh has o be posiive a all imes. In his sudy, we derive explicily he opimal sochasic sraegy in which porfolio revisions are based on realized prices and wealh. he sochasic sraegy shows ha he convenional wisdom is, in general, opimal for a mean-variance invesor. We noe ha his sochasic sraegy is concave in he sock price (see Perold and Sharpe 3

7 (1988)), in ha he sock allocaion is reduced (increased) afer unexpecedly high (low) realized sock prices. Anoher surprising resul is ha i produces a wealh disribuion which is bounded above and unbounded below, alhough he probabiliy of negaive wealh is almos negligible. We hen compare his sochasic opimal sraegy wih a deerminisic sraegy defined by an opimal predeermined schedule of weighs. Secion II presens he noaion and framework and provides he derivaion of he opimal sochasic mean-variance sraegy. Secion III conains numerical examples and inerpreaions. Secion IV is devoed o a comparison wih he deerminisic opimal sraegy. Secion V is he conclusion. II. Opimal Sochasic Porfolio Sraegy. he invesmen opporuniy se consiss of wo asses a dae, a risk-free asse (-bill) wih curren price B and a risky asse (sock index) wih curren price S. For simpliciy, boh prices are normalized o one. his allows wriing gross reurns as B and S raher han B /B and S /S. he insananeous reurn on he bill is given by (1) db B = rd where r is he consan insananeous risk-free rae. For any dae >, he gross reurn on he bill is given by e r. he insananeous reurn on he sock index is given by 4

8 () ds S = µ d + σ dw where µ is he consan insananeous reurn, σ is he consan insananeous sandard deviaion of reurn, and W is a sandard brownian moion. As in Bajeux-Porai (1998), he mean-variance invesor chooses he proporion of iniial wealh o allocae o he sock index in order o minimize he variance of erminal (=) wealh subjec o he consrain of desired expeced erminal wealh (represened by a desired annualized reurn α). he sochasic sraegy allows he invesor o rebalance he porfolio coninuously depending on realized prices and wealh. he problem can be saed as opimizaion program (P): minvar( X x ) (P) s.. E( X dx X ) = e = x α ds S db + (1 x ) B where x is deermined opimally a each dae. Noe here ha, for simpliciy, he iniial wealh has been normalized o one; hus, X is he gross reurn on wealh a any dae. Cox and Huang (1989) and Karazas, Lehoczky and Shreve (1987) have shown ha program (P) is equivalen in a sense specified below o he following opimizaion program (P*) which subsiues a single linear consrain for he self-financing consrains: 1 1 his equivalence is obained under he assumpion of complee markes, which is obviously saisfied in our framework conaining one brownian moion, wo non redundan securiies and coninuous rebalancing of he porfolio. 5

9 (P*) minvar( X X s.. E( X E* ( X ) = e ) α ) = e r where E* denoes expecaion under he risk-neural probabiliy. he opimizaion programs are equivalen in he following sense: he soluion of program (P) is a porfolio sraegy (x *); his sraegy produces a erminal wealh disribuion X *; he soluion of program (P*) is he same erminal wealh disribuion X *. Afer solving (P*), he assumpion of complee markes is used in deriving he unique porfolio sraegy (x *) ha replicaes X *. I is shown in Appendix, using Proposiion 1.1 (equaion (1.7)) and Lemma 1. (equaion (1.)), ha he soluion o (P*) is given by { 1 ( )( ) } α β (3) X = e + k ( φ S ) where (4) k e = e α λ (5) φ( ) = exp[( βµ. 5λ. 5λσ) ] and where λ = (µ-r)/σ and β = λ/σ. Wealh a any ime can be calculaed as he discouned risk-neural expeced value of erminal wealh, ha is, e r 1 r (6) X = e ( ) E* [ X ] Proposiion 1. wih Lemma 1. in Appendix 1 leads o r( ) α λ ( ) λ ( ) β (7) X = e { e k( e 1) + ke ( 1 φ( )( S ) )} 6

10 where φ() is given by (5) evaluaed a. We can clarify (7) by wriing i as (8) X = E( X ) + Y where he expeced value componen is given by r( ) ( ) (9) E[ X ] = e { e k ( e )} α λ 1 and he sochasic componen is given by ( λ r)( ) β (1) Y = k e { 1 φ ()( S ) } he sandard deviaion of wealh is shown in Proposiion 1. o be ( λ )( ) (11) [ ] λ SD X = k e e 1 r Simple calculaion (using (4) and (9)) shows ha E[X ] is always posiive, and i can be shown ha E[X ] is increasing in. he sochasic componen is an increasing funcion of he (sochasic) sock price; herefore wealh is an increasing funcion of he sock price. Since S is lognormally disribued (and herefore unbounded above), i can be shown (from (1)) ha X is bounded above by (1) e r ( ) α { e + k } X is unbounded below; when S goes o zero, X goes o negaive infiniy. erminal wealh is a special case of X wih =. he expeced value of erminal wealh is he objecive e α ; is upper boundary is (e α +k ) and i is unbounded below. I may seem surprising ha wealh is possibly infiniely negaive. However, low values of wealh occur wih very low probabiliies. For example, he probabiliy of X < e r is N[-1.5λ ], which is approximaely.45 for he parameers assumed in he examplesin Secion III below. 7

11 Proposiion 1.3 in Appendix 1 shows ha he opimal allocaion o he sock index a any ime is given by (13) x 1 X e r k e ( = β ) α ( + ) 1 (he proporion x is undefined when wealh X is zero. Neverheless, he dollar amouns invesed in each securiy are readily calculaed afer (13) is muliplied hrough by X.) For he iniial allocaion, =, his reduces o (14) x { e r α = β ( k + e ) } 1 he iniial allocaion in (14) is increasing in he desired expeced reurn α, decreasing in he risk-free rae, and invarian wih respec o iniial wealh. he subsequen allocaion (13) shows ha as realized wealh X increases (decreases) he porfolio is rebalanced by allocaing less (more) o he sock index. his reallocaion is necessary o keep he opimal erminal wealh disribuion he same afer unexpeced realizaions of he sock price. III. Commens on he Opimal Sochasic Sraegy. he sock allocaion sars iniially high and decreases, in general, as ime passes. able 1 provides an example for =3 years, X =$1, r=5%, µ=1%, σ=% and α=6.5% (all annual raes), allowing he sock price o evolve along is expeced pah. he iniial allocaion is over 85% sock. he allocaion o sock decreases (along he average sock price pah) o less han 4% a =3 years. his porfolio sraegy has he paern of he convenional advice. [Inser able 1] 8

12 he opimaliy of decreasing he sock allocaion on he average sock price pah can be explained in erms of invesor risk aversion. Problem (P) is isomorphic o he dynamic opimizaion of he expeced uiliy of erminal wealh for an invesor wih quadraic uiliy. he increasing relaive risk aversion of he quadraic uiliy funcion implies ha he increase in invesor s wealh, which accompanies he increase in sock prices, causes he invesor o allocae a lower proporion of wealh o he risky asse. We can also sudy he sensiiviy of he iniial sock allocaion x and he sandard deviaion of erminal wealh SD(X ) o changes in he drif and volailiy parameers µ and σ. able summarizes some of hese numerical examples. [Inser able ] Everyhing being equal, in paricular keeping consan he annualized goal reurn α, an increase (decrease) of he volailiy parameer induces a decrease (increase) in he iniial sock allocaion and an increase (decrease) in he oal porfolio risk (sandard deviaion of final wealh); an increase (decrease) in he drif parameer induces an increase (decrease) in he iniial sock allocaion and a decrease (increase) in he oal porfolio risk. his laer resul seems couner-inuiive, as when you increase he drif, you also increase your iniial sock allocaion. his can be explained by looking, for example, a he oal paern of sock allocaion on he average sock pah: as he convenional wisdom Quadraic uiliy also implies increasing absolue risk aversion and saiaion (decreasing uiliy as wealh increases beyond a cerain poin). Bajeux-Besnainou, Jordan and Porai () have shown ha increasing absolue risk aversion does no generae he decreasing sock allocaions, because hese levels of wealh are never opimally chosen. hese decreasing sock allocaions are also found in HARA uiliy funcions, which do no have increasing absolue risk aversion, bu which do have increasing relaive risk aversion. Saiaion is also no a facor in his resul, because in problem (P), uiliy is bounded from above a a level below he saiaion level. 9

13 is sill saisfied (decreasing sock allocaion hrough ime), his sock allocaion decreases much faser for higher values of he drif (from 118% o.3% for a drif of 15% and a volailiy of %), han for lower values (from 75% o % for a drif of 9% and a volailiy of %). Perold and Sharpe (1988) have defined concave sraegies as sraegies for which sock is sold (purchased) as he sock price increases (decreases). he sochasic sraegy is a concave sraegy requiring rebalancing afer unexpeced sock price changes. However, along he average sock price pah, his sraegy rebalances o weighs ha are decreasing over ime. hus, his sraegy will end o be a more concave sraegy han a consan-weigh sraegy (described in Perold and Sharpe (1988) as a dynamic sraegy designed o keep consan weighs in all of he differen securiies). he weigh process (x ) is unbounded above and bounded below by zero. As sock price falls, he invesor may find ha borrowing a he risk-free rae is necessary o obain he required allocaion o sock. As sock price increases, he allocaion will approach zero. As noed above, he resuling wealh process is bounded above and unbounded below. I is ineresing o compare hese resuls in he opimal sochasic case wih he resuls from he Markowiz-ype case when no rebalancing is allowed beween over he invesor s horizon. Any reasonable values of he parameers of he -bill and sock price processes resul in an opimal soluion of small iniial sock allocaions which increase over ime as he sock price increases faser han he -bill price over mos sock pahs. If we rerun he numerical example wih he previous se of parameer values as in able 1, by = 3, he porfolio sandard deviaion is 55% compared o 57% wih he sochasic rebalancing schedule. In he buy and hold sraegy, only by choosing a relaively small 1

14 iniial allocaion o he sock index (he example would provide a 7.83% iniial sock allocaion) can he invesor minimize he variance of erminal wealh. When sochasic rebalancing as described in secion II is allowed, such conservaism is unnecessary because he sock allocaion can be reduced (on average) over ime o keep he ending variance wihin desired bounds. Our resuls imply ha for mean-variance invesors, large gains in risk conrol are obainable hrough coninuous rebalancing. Of course, he applicaion of hese resuls would be limied by ransacion coss. An area for fuure research is o invesigae he pracicaliy of a rebalancing approach given realisic ransacion coss. Similar exercises in he lieraure include sudies of he coninuous rebalancing assumpion in opion pricing and dynamic hedging, such as Mercurio and Vors (1996) and Loewensein (). he general resuls of such sudies is ha coninuous ime models are approximaions of varying degrees of accuracy depending on how he much he frequency of rading mus be reduced in order o opimally rade off ransacion coss and racking error. However, as a pracical maer, opion pricing and dynamic hedging sraegies (such as porfolio insurance) coninue o be widely applied. IV. Comparison wih a deerminisic opimal schedule. he convenional advice migh also be inerpreed as recommending a pre-deermined rebalancing schedule. For example, he Fideliy Invesmens organizaion (Del Pree (1997)) has recommended a schedule of allocaions for invesors wih horizons of more han en years (1% socks), seven o en years (7% socks), four o seven years (5% socks), wo o four years (% socks), and less han wo years (no sock). Similar "rules 11

15 of humb" are common. Barberis () explores a similar idea; he defines and invesigaes a sraegy of myopic rebalancing, in which he same predeermined allocaion is rese a he sar of every year. We pose a more general quesion: Wha is he opimal predeermined schedule of allocaions? Noe ha in our coninuous-ime seing, such sraegies imply coninuous rebalancing: since x is consrained o have he same value whaever he value of S, he invesor almos surely will have o rebalance he porfolio. he problem can be saed as problem (P D ): (P D ) Min Var( X ) x s.. E( X ) = e dx X α x ds db = + ( 1 x ) S B where x is a deerminisic funcion of ime. he addiional consrains are he selffinancing consrains required in a model wih rebalancing. he derivaion of he opimal deerminisic weighs is provided in Appendix ; in paricular, Proposiion.4 shows ha he soluion is a consan proporion sraegy, defined by a consan weigh x in he sock index: (15) α r x = µ r From his resul, we can conclude ha an invesmen plan recommending a predeermined sraegy of decreasing weighs is a subopimal mean-variance sraegy. In addiion, his resul provides a new case of he opimaliy of consan weigh sraegies ha does no depend on CRRA uiliy wih a finie horizon (as in Meron (1971)) nor on more general uiliy funcions wih an infinie horizon as in he urnpike lieraure. 1

16 he resuls of he deerminisic porfolio sraegy are illusraed in able 3 for he same parameers as in he opimal sochasic case. [Inser able 3] he expeced erminal wealh is he same for boh sraegies because i is imposed by consrain. he sandard deviaion of erminal wealh of 185% is more han hree imes he sandard deviaion in he opimal sraegy of 57%. However, i is abou 1/3 he buyand-hold sandard deviaion of 55%. Wih his resricive mehod of predeermined rebalancing, he invesor has much less conrol of porfolio risk han an invesor following he opimal sochasic sraegy. V. Conclusion he convenional wisdom abou long-erm invesing is o sar wih large sock weighs which hen decrease over ime. In his paper, we ask wheher such a sraegy is opimal for a mean-variance invesor wih a long horizon. We derive closed-form soluions for wo dynamic mean-variance sraegies in a coninuous-ime, complee markes framework. In he opimal sraegy, he allocaions each period depend on realized prices and wealh. Only his sraegy enails large iniial weighs ha decrease (on he average sock price pah) over ime, an allocaion resembling he usual advice abou invesing for he long erm. he abiliy o opimally rebalance he porfolio as ime passes and wealh is deermined allows he invesor o ake large iniial posiions wihou geing locked ino large end-of-horizon porfolio risk. he opimaliy of decreasing sock allocaions over ime on he average sock price pah is due o he close relaionship beween he mean- 13

17 variance minimizaion problem and he problem of maximizing quadraic uiliy. he increasing relaive risk aversion of he quadraic uiliy funcion implies ha he invesor reduces he allocaion o he risky asse as wealh increases 3. his research reveals some general characerisics of an opimal sochasic sraegy. I is a concave sraegy in ha he invesor's sock allocaion is concave in he sock price. Wealh is bounded above and unbounded below. his seems counerinuiive given ha he lognormal sock price disribuion is bounded below and unbounded above. he counerinuiive resul is a consequence of coninually reducing he sock allocaion when wealh ges oo high and increasing he sock allocaion when wealh ges oo low in order o mee he expeced erminal wealh consrain. he second sraegy invesigaed is a deerminisic sraegy in which a predeermined schedule of porfolio weighs is defined and no adjused over ime for realized prices and wealh. Given ha an invesor would consider such as sraegy (as considered, for example, in Barberis ()), we show ha he opimal coninuouslyrebalanced deerminisic sraegy is a consan-weigh sraegy. Any invesmen plan recommending a predeermined sraegy of decreasing weighs is a subopimal meanvariance sraegy, a leas in he case of geomeric brownian moion wih consan parameers. Possible exensions of his research include parameer uncerainy, predicabiliy in asse reurns and he effecs of ransacion coss. 3 A he suggesion of an anonymous referee, we poin ou ha our heoreical suppor for he convenional wisdom does no preclude oher explanaions, including frequen, bu sub-opimal rebalancing due o adverse shor horizon porfolio managers incenives. 14

18 References Bajeux-Besnainou, I., and R. Porai, Dynamic Asse Allocaion in a Mean-Variance Framework, Managemen Science, (1998), November. Bajeux-Besnainou I., J. Jordan and R. Porai, Dynamic Asse Allocaion for Socks, Bonds and Cash over Long Horizons, (), working paper. Barberis N., Invesing for he Long Run when Reurns Are Predicable, he Journal of Finance, Vol. LV, No 1, (), February. Bielecki,., Pliska S. and M. Sherris, Risk sensiive Asse Allocaion, o appear in Journal of Economics, Dynamics and Conrol (1999). Bodie, Z., R. C. Meron and W. F. Samuelson, Labor Supply Flexibiliy and Porfolio Choice in a Life Cycle Model, Journal of Economic Dynamics and Conrol, 16 (199), Brennan M.J., he Role of Learning in Dynamic Porfolio Decisions, European Finance Review, (1998), Campbell J. and L. Viceira, Consumpion and Porfolio Decisions when Expeced Reurns are ime Varying, NBER working paper No. 5857, December Cox, J.C., and C.F. Huang, Opimal consumpion and Porfolio Policies When Asse Prices Follow Diffusion Processes, Journal of Economic heory, 49 (1989). Cox, J.C., and C. F. Huang, A Coninuous-ime Porfolio urnpike heorem, Journal of Economic Dynamics and Conrol, 16 (199), Del Pree, Dom, "Piecing ogeher Your Reiremen Puzzle," Fideliy Focus, Spring (1997), Karazas, I., J. Lehoczky and S. Schreve, Opimal Porfolio and Consumpion Decisions for a Small Invesor on a Finie Horizon, SIAM Journal of Conrol and Opimizaion, 5 (1987), Krizman, M, Wha Praciioners Need o Know... Abou ime-diversificaion, Financial Analyss Journal, (January-February 1994), Loewensein, Mark, On Opimal Porfolio rading Sraegies for an Invesor Facing ransacions Coss in a Coninuous rading Marke, Journal of Mahemaical Economics, 33(), () 9-8. Mercurio, F. and Vors, CF. Opion Pricing wih Hedging a Fixed Daes, Applied Mahemaical Finance, 3, (1996)

19 Meron, R. C., Opimum Consumpion and Porfolio Rules in a Coninuous-ime Model, Journal of Economic heory, (December 1971): Meron, R.C., and P.A. Samuelson, "Fallacy of he Log-Normal Approximaion o Porfolio Decision-Making Over Many Periods," Journal of Financial Economics, Vol. 1, No. 1, (May 1974), Nguyen P. and R. Porai, Dynamic Mean-Variance Efficiency and Sraegic Asse Allocaion wih a Solvency Consrain, forhcoming, Journal of Economic Dynamics and Conrol. Perold, A. F., and W.F. Sharpe, "Dynamic Sraegies for Asse Allocaion," Financial Analyss Journal, (January-February 1988), Richardson, H, A Minimum Variance Resul in Coninuous rading Porfolio Opimizaion, Managemen Science, 35 (1989), Ross, Sephen A., Porfolio urnpike heorems for Consan Policies, Journal of Financial Economics, 1(), Rubinsein, Mark, "Coninuously Rebalanced Invesmen Sraegies," he Journal of Porfolio Managemen, (Fall 1991), Samuelson, P.A., "Risk and Uncerainy: A Fallacy of Large Numbers," Scienia, 6h Series, 57h Year, (April/May 1963), 1-6. Samuelson, P.A., "he Fallacy of Maximizing he Geomeric Mean in Long Sequences of Invesing or Gambling," Proceedings of he Naional Academy of Science, 68 (Ocober 1971), Samuelson, P.A., "he Judgemen of Economic Science on Raional Porfolio Managemen: iming and Long-Horizon Effecs," he Journal of Porfolio Managemen, 16 (Fall 1989), 4-1. Samuelson, P. A., Long-Run Risk olerance When Equiy Reurns Are Mean Regressing: Psuedoparadoxes and Vindicaion of Businessmen s Risk. In W.C.Brainard, W. D. Nordhaus, and H.W.Was, eds., Money, Macroeconomics, and Economic Policy, Cambridge, MA: he MI Press, 1991, Samuelson, P.A., he Long-erm Case for Equiies and How I Can Be Oversold, he Journal of Porfolio Managemen, 1 (Fall 1994),

20 APPENDIX 1 Lemma 1.1: he Radon-Nikodym derivaive a dae, Z, defining he change of probabiliy from he original o he risk-neural probabiliy, saisfies he following dynamics: dz /Z = -λdw, where λ = (µ-r)/σ is he insananeous price of risk. hen, Z e e = 1 λ λ and Z e e = 1 λ λ W W (1.1) (1.) Defining now Z - as he Radon-Nikodym derivaive a dae of he change in probabiliy we ge Z = e e 1 λ ( ) λ ( W W ) (1.3) hen, Z = Z Z (1.4) E Z ( ) = 1 (1.5) and E ( Z ) = e λ ( ) (1.6) Proposiion 1.1: he risk-neural erminal wealh disribuion ha solves Problem (P*) is α X = [ e + k ( 1 Z )] (1.7) where k α r e e = (1.8) λ e 1 17

21 Proof of Proposiion 1.1: Using he Radon-Nikodym derivaive, he second consrain in Problem (P*) can be wrien in erms of he acual probabiliy, E( X Z ) = e he Lagrangian for Problem (P*) is hen 4 r r L = E[ X ( ω) ] k { E[ X ( ω)] e α } + k { E[ X ( ω) Z ( ω)] e } (1.9) 1 where for mahemaical convenience he firs Lagrangian muliplier is defined o be (-k 1 ), and he second Lagrangian muliplier is defined o be (k ). In (1.9) he dependence of X on he sae of naure ω is made explici by wriing X (ω). he firs order condiions can be wrien X ( ω) = k1 kz ( ω) = (1.1) α E[ X ( ω)] = e (1.11) r E[ X ( ω) Z ( ω )] = e (1.1) Evaluaing he expecaion E(X ) from (1.1) and subsiuing ino (1.11) produces k k e 1 = α (1.13) Subsiuing (1.1) ino (1.1) and simplifying produces λ k k e = e 1 r (1.14) where in he simplificaion we use (.6) wih =. From (1.13) and (1.14) we have (1.8). From (1.8), (1.1) and (1.13) we have (1.7). Proposiion 1.: he wealh a ime is given by X = e r( ) ( ) { e k ( e 1) k e (1 Z )} α λ ( ) λ + (1.15) Is expeced value and sandard deviaion are given by 18

22 E[ X ] = e r( ) α λ ( ) { e k ( e 1) } [ ] ( λ r )( ) = k e SD X Proof of Proposiion 1.: e λ 1 Using he risk-neural probabiliies, wealh a ime is he discouned risk-neural expeced value of erminal wealh, condiional on informaion a ime, r( ) * r( ) X = e E ( X ) = e E ( X Z ) (1.16) where E is he condiional expecaions operaor. Subsiuing from (1.7) for X produces Using (1.4) and (1.5) we have α [( ( )) ] r( ) X = e E e + k 1 Z Z (1.17) Using (1.6) his can be wrien α { ( )} r( ) X = e e + k k Z E Z { ( ) ( )} α λ λ 1 1 r X = e ( ) e + k e ( ) + k e ( ) Z (1.18) (1.19) he compuaion of expeced values and sandard deviaion are derived from he previous expression and (1.5) and (1.6) from lemma 1.1. Lemma 1.: he Radon-Nikodym derivaive can be wrien in erms of he sock price process as where β Z = φ( )( S ) (1.) φ( ) = exp[( βµ. 5λ. 5λσ) ] (1.1) 4 Minimizing E[X ] is equivalen o minimizing Var[X ] because E[X ] is a consan, X e α. 19

23 and β = λ/σ. Proof of Lemma 1.: From he assumed sochasic process for he sock index, we have S = e e 1 µ σ σ W (1.) or e σw = S e 1 ( σ µ ) (1.3) From (1.), λw σw β Z = e e = e ( e ) (1.4) 1 1 λ λ Subsiuing (1.3) ino (1.4) proves Lemma. Proposiion 1.3: he allocaion o he sock index as a proporion of iniial wealh is given by: x 1 X e r k e = β ( ) α + 1 ( ) (1.5) Proof of Proposiion 1.3: Firs, from (1.19), he dynamics of X are given by [] ( λ r)( ) dx = d k e dz (1.6) (he drif erm is no needed explicily for he proof of his Proposiion.) Subsiuing dz /Z = -λdw ino (1.1) produces [] ( λ r)( ) dx = d + λk Z e dw (1.7) (1.19) can also be wrien Subsiuing (1.8) ino (1.7) yields ( λ r)( ) r( ke Z e ) α = ( e + k) X (1.8)

24 r( α [ ] λ{ ) ( ) } dx = d + e e + k X dw (1.9) Second, i is also rue ha he process for X can be wrien dx X x ds = + ( 1 x ) rd (1.3) S where x is he proporion of marke value of he porfolio allocaed o he sock index. (1.3) can also be wrien dx = X ( r + x ( µ r)) d + σ x X dw (1.31) Equaing he volailiy erms in (1.9) and (1.31) implies his can be wrien r α σxx = λ{ e ( ) ( e + k ) X} (1.3) x 1 X e r( ) e α = β( ( + k ) 1) (1.33) 1

25 APPENDIX Proposiion.1: he wealh process follows dx X = α d + σ dw (.1) where erminal wealh is given by α = r + x ( µ r) and σ = σx (.) [ α 1 σ ] X = exp d ( / ) d e (.3) σ dw Proof of Proposiion.1: he sochasic process for X is given by dx X x ds db = + ( 1 x ) (.4) S B Subsiuing (1) and () yields dx X [ ( )] = r + x µ r d + σ x dw (.5) Subsiuion of (.) ino (.5) and applicaion of Io's Lemma yields he proposiion. ( Proposiion.: E[ X ] = e α (.6) d Proof of Proposiion.: aking he expecaion of (.3), and requiring ha x, and hus α and σ, be deerminisic, yields [ ] = exp [ α ( / ) σ ] σ dw E X d d E e 1 (.7) he disribuion of σ dw is normal wih zero mean and variance σ d. herefore i can be wrien as a linear funcion of he sandardized normal variable,

26 where n ~ N(,1). herefore, ( σ ) σdw = d n (.8) σ dw σ d n E e E e = sn (/ ) s I is well-known ha for a consan s, Ee [ ] = Ee [ ] E e σdw = E e (/ 1) σ d 1. herefore, (.9) (.1) Subsiuion ino (.7) yields he proposiion. ( Proposiion.3: E[ X ] = [ d + d] Proof of Proposiion.3: he expecaion of X is exp α σ (.11) [ ] = exp[ α σ ] E X d d E e σ dw (.1) From here he proof proceeds as in Proposiion.. ( Proposiion.4: he soluion of Problem (P U ) is α = α (.13) hus, x = α r x µ r = (.14) Proof of Proposiion.4: Using Proposiion (.), Problem (P) can be wrien Min E( X ) x s.. αd e = e α Minimizing his objecive funcion is equivalen o minimizing he variance of X because E(X ) is consrained o be a consan in Problem (P). he new consrain is derived from 3

27 he previous one by subsiuing for E(X ) from Proposiion (.). Subsiuing (.11) for he objecive funcion and using he consrain resuls in Min x e σ d s.. α d= α Because r, µ, and σ are consans, from (.) his opimizaion program has he same soluion as he following problem (P'): Min x α d s.. α d= α (P ) Define y = α α. hen from he consrain in (P'), yd= α d α = (.15) he objecive funcion in (P') can be wrien α d = y d + α y d + α (.14) of which he las wo erms on he righ side are consans. herefore, problem (P') can be wrien as problem (P''), Min x s.. y d y d= (P ) he soluion can be wrien by inspecion. he objecive funcion is a sum of quadraic erms. he minimum value occurs a y =. From he definiion of y and equaion (.) we have he proof of he proposiion.( 4

28 able 1 Mean-variance model and opimal porfolio selecion wih coninuous sochasic rebalancing on he average price pah. Assumed parameers: r= 5%; µ=1%; σ = %; α = 6.5%; = 3 Calculaed parameers: λ = 31.8%; β = 1.45; k =.13 Year B E(S ) E(X ) SD(X ) x() on E(S ) X on E(S ) $1. $1. $1. % 86.34% $1. 1 $1.5 $1.13 $ % 75.% $1.1 5 $1.8 $1.8 $ % 45.65% $ $1.65 $3.3 $.8 47% 6.33% $.3 15 $.1 $6.5 $3.1 5% 15.86% $3.5 $.7 $11. $ % 9.8% $4.7 5 $3.49 $.9 $ % 6.14% $ $4.48 $36.6 $ % 3.88% $6.97 5

29 able Iniial sock allocaion and sandard deviaion of final wealh for differen drif and volailiy values. Assumed parameers: r= 5%; α = 6.5%; = 3 σ / µ 9% 1% 15% x SD(X ) x SD(X ) x SD(X ) 18% 91% 138% 14% 6% 175% % % 75% 196% 86% 57% 118% 11% 6% 66% 5% 66% 91% 85% 8% 6

30 able 3 Mean-variance model and opimal porfolio selecion in he deerminisic case. Assumed parameers: r = 5%; µ=1%; σ = %; α = 6.5%; = 3 Year B E(S ) E(X ) SD(X ) x() on E(S ) X on E(S ) $1. $1. $1. % 1.43% $1. 1 $1.5 $1.13 $1.7 5% 1.43% $1.7 5 $1.8 $1.8 $ % 1.43% $ $1.65 $3.3 $1.9 9% 1.43% $. 15 $.1 $6.5 $.65 49% 1.43% $.8 $.7 $11. $ % 1.43% $ $3.49 $.9 $5.8 11% 1.43% $5.6 3 $4.48 $36.6 $ % 1.43% $7.94 7