Thema Working Paper n Université de Cergy Pontoise, France

Transcription

1 hema Working Paper n 11-1 Universié de Cergy Ponoise, France Real Esae Porfolio Managemen: Opimizaion under Risk Aversion Fabrice Barhelemy Jean-Luc Prigen June, 11

2 Real Esae Porfolio Managemen: Opimizaion under Risk Aversion Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France s: Absrac his paper deals wih real esae porfolio opimizaion when invesors are risk averse. In his framework, we deermine several ypes of opimal imes o sell a diversified real esae and analyze heir properies. he opimizaion problem corresponds o he maximizaion of a concave uiliy funcion defined on he erminal value of he porfolio. We exend previous resuls (Baroni e al., 7, and Barhélémy and Prigen, 9), esablished for he quasi linear uiliy case, where invesors are risk neural. We consider four cases. In he firs one, he invesor knows he probabiliy disribuion of he real esae index. In he second one, he invesor is perfecly informed abou he real esae marke dynamics. In he hird case, he invesor uses an ineremporal opimizaion approach which looks like an American opion problem. Finally, he buy-and-hold sraegy is considered. For hese four cases we analyze numerically he soluions ha we compare wih hose of he quasi linear case. We show ha he inroducion of risk aversion allows o beer ake accoun of he real esae marke volailiy. We also inroduce he noion of compensaing variaion o beer compare all hese soluions. Key Words Real esae porfolio, Opimal holding period, Risk aversion, Real esae marke volailiy. JEL Classificaion C61, G11, R1. 1

3 1. Inroducion Knowing he real esae holding period is imporan for invesmen in commercial real esae porfolios. However, resuls abou opimaliy of holding period are almos exclusively empirical. Hendersho and Ling (1984), Gau and Wang (1994) or Fisher and Young () show ha, for he US, he holding duraions depend mainly on ax laws. Brown and Geurs (5) deal wih small residenial invesmen. hrough a sample of aparmen buildings of beween 5 and unis over he period in he ciy of San Diego, hey find empirically how long an invesor mus own an aparmen building. he average holding period is around five years. For he UK marke, Rowley, Gibson and Ward (1996) prove he exisence of ex ane expecaions abou holding periods, relaed o depreciaion or obsolescence facors. As illusraed by Brown (4), he risk peculiar o real esae invesmens can jusify he behaviour of real esae invesors. However, Gelner and Miller (1) show ha CAPM is no a convenien ool o undersand porfolio managemen. Noe also ha values of ex pos holding periods are higher han hose usually claimed by invesors, as illusraed by Colle, Lizieri and Ward (3) 1. However, his kind of empirical sudy does no allow o precise conclusions abou he relaion beween asse volailiy and opimal holding period. Baroni e al. (7a, b) deermine he opimal holding period when i has o be chosen a iniial ime. hey suppose ha he markeing period risk corresponds o ime-on-he-marke ha deermines how long i akes o sell an asse once i is pu on he marke o be sold. I is an exogenous random variable. o beer model, he imes o sell are endogenously deermined from he opimizaion problems. Barhélémy and Prigen (9) examine also he deerminaion of he opimal holding period (opimal ime o sell in a real esae porfolio) by inroducing addiional crieria such he American opion approach. Neverheless, hey assume ha he invesor is risk-neural. Due o his laer hypohesis, he volailiy of he real esae asse eiher has no impac on he opimal holding period or is role is only implici. In his paper, our aim is o beer emphasize he impac of he volailiy. For his purpose, we consider ha he invesor maximizes his expeced uiliy a mauriy over a given ime period and is risk-averse. In his sandard approach, ime horizon and risk aversion are he key parameers. I is ypically assumed ha he individual preferences are described by uiliy funcions wih consan absolue risk aversion (CARA) and hyperbolic absolue risk aversion (HARA), which will be examined in his paper 3. Our resuls show in paricular ha he relaive risk aversion plays a key role o evaluae he moneary loss from no having access o he bes horizon. his feaure has o be relaed o previous works abou he influence of risk aversion as shown by Kallberg and Ziemba (1983). We also examine he robusness of our resuls wih respec o he uiliy specificaion. Our model provides soluions whose properies can poenially explain mos of he

4 previous empirical resuls. Firs, we deermine he opimal holding period when i has o be chosen a iniial dae, exending previous resuls of Baroni e al. (7a, b). he invesor is assumed o know probabiliy disribuion of real esae asse. We illusrae wha are he impacs of he risk aversion, he real asse value and he volailiy on he selling sraegies. Firs we deermine his laer one when he invesor is perfecly informed abou he growh rae dynamics bu mus choose his sraegy only a iniial ime. However, usually such a soluion is no ime consisen since he same deerminaion of opimal ime o sell a a fuure dae leads o a differen soluion. Second, we sudy he bes ideal case where he invesor knows exacly he price dynamics, as soon as a new period sars. In ha case, he can immediaely choose he bes ime o sell he asse. his approach provides he upper bound of he presen value of he porfolio as a funcion of holding period policy. Indeed, he presen value is maximized using perfec foresigh. We use his special framework as a benchmark. Finally, we deermine he opimal holding period according o he American opion approach. In his conex, a each ime during a given managemen period, he invesor compares he presen expeced uiliy of porfolio value wih he maximal expeced uiliy he could have if he would keep he asse. We show ha he invesor mus sell as soon as he presen uiliy is higher han is expecaion. Addiionally, we inroduce he noion of compensaing variaion o evaluae he moneary loss of no having he bes porfolio. As shown in de Palma and Prigen (8, 9), he compensaing variaion allows o measure he adequacy of a given porfolio o invesor s uiliy. he srucure of he paper is laid ou as follows: Secion provides a survey abou expeced uiliy heory, risk aversion and compensaing variaion. Secion 3 presens he coninuous-ime framework and he opimal ime o sell we ge in he neural risk invesor case. Resuls for he opimal holding period when he dae mus be chosen a iniial ime is developed in secion 4 for hree differen uiliy funcions. Secion 5 gives a heoreical framework for oher porfolio sraegies, as he perfecly informed invesor, he American opion soluion and he buy-and-hold sraegy. hese approaches are compared in Secion 6 by using compensaing variaions.. Uiliy funcions, risk aversion and compensaing variaion In his secion, firs we recall basic noions abou uiliy funcions and risk aversion. hen, we deail he concep of compensaing variaion..1. Expeced uiliy heory and measures of risk aversion he aim of expeced uiliy (EU) heory is o model problems of decision under uncerainy by means of a funcional represenaion of preferences over loeries. hese laer ones are composed of all possible evens and heir corresponding probabiliies. Preferences reflec he degree of saisfacion which resuls from he individual's choice. he objecive of raional individuals wih respec o his crierion is hus o maximize his funcion. Expeced uiliy heory assumes ha preference 3

5 relaion is a linear relaion of he probabiliies 4. In oher words, here exiss a uiliy funcion u(.) defined (up o a posiive monoonic ransformaion) over he oucome a a a L = ( ω, p ),...,( ω, p ) and space Ω wih values in R such ha, for any loeries { 1 1 m m } b b b L = {( ω1, p1),...,( ωm, pm) }, he following equivalence holds: m m a b a b ( ωi) i ( ωi) i i= 1 i= 1 L L u p u p Risk aversion. he empirical observaion of individuals suggess ha usually invesors are risk-averse. herefore, invesmen in a risky asse implies ha he laer mus provide reurns ha are significanly higher han hose corresponding o risk-free invesmen. o illusrae he noion of risk-aversion, Friedman and Savage (1948) inroduce he following model: le X be a random variable wih only wo possible values x 1 and x. Le p be he probabiliy ha he value be x 1 and (1 p) he probabiliy ha he value be x. Le u (.) be a uiliy funcion defined over he possible oucomes. hen consider he wo following loeries,. a L and b L : he loery a L yields E[ X ] wih a probabiliy of 1; he loery L b pays x 1 wih probabiliy p and x wih probabiliy (1 p). hese wo loeries have he same expeced payoff. However, a risk-averse invesor prefers a L o b L. I means ha we have: a b UL [ ] = uex ( [ ]) UL [ ] = EuX [ ( )]. Previous propery is saisfied for all loeries if and only if he uiliy u (.) is concave. 5 Risk aversion is also linked o he concep of cerainy equivalen. Consider he safe loery, denoed by C[ X ]. I yields he same uiliy level as he loery L b. he invesor is indifferen beween receiving he cerain amoun C[ X ] and playing he b loery L, wih expeced payoff C[ X ]. he inequaliy CX [ ] < EX [ ] corresponds o risk aversion. he gap π ( X ) = EX [ ] CX [ ] has been inroduced by Pra (1964) as he risk premium. I is equal o he amoun ha he invesor would be willing o pay in order o benefi from a risk-free payoff. In he expeced uiliy framework, riskaversion is enirely characerized by assumpions abou he uiliy funcion u(.): 1) Invesors are risk-averse if CX [ ] < EX [ ], or equivalenly if π ( X ), for any random variable X. I is equivalen o concaviy of funcion u (.). ) Invesors are risk-neural if CX [ ] = EX [ ], or equivalenly if π ( X ) =, for any random variable X. I is equivalen o lineariy of funcion u (.). 3) Invesors are risk-loving if CX [ ] EX [ ], or equivalenly if π ( X ), for any random variable X. I is equivalen o convexiy of funcion u (.). he Arrow-Pra Measures of Risk-Aversion. he measure of individual s degree of risk-aversion is a horny problem. o solve i, one way is o analyze he risk premium 4

6 π and o examine is relaionship wih he concaviy of he uiliy funcion (see Pra, 1964; Arrow, 1965). he definiion of he noion of more risk aversion can be made as follows: Le u (.) and v (.) be wo uiliy funcions. he preference associaed wih u (.) is said o exhibi "more risk-aversion" han ha associaed wih v (.) if he risk π X π X for any random variable X. premia saisfy he following relaionship ( ) ( ) u hen, if u (.) and v (.) are coninuous, posiive, and wice-differeniable, he following properies allow us o define he noion of exhibiing more risk-aversion : 1) he derivaives of he wo uiliy funcions are such ha: u v > u v ( x) ( x),for all x in. ) here exiss a concave funcion, Φ, such ha: ( ) 3) he risk premia saisfy: ( X) ( X), π u v v ux ( ) =Φ vx ( ),for all xin. π for any random variable X. he raio u''( E[ X]) / u'( E[ X]) can be considered as a measure of risk-aversion 6. I is posiive as he uiliy funcion u (.) is increasing and concave. he erm A( x) = u''( x) / u'( x) is called he Arrow-Pra Absolue Risk Aversion (ARA). he Arrow-Pra Relaive Risk Aversion (RRA) is defined as R( x) = xu''( x)/ u'( x). I allows o ake he level of individual wealh ino accoun... Sandard uiliy funcions he specificaion of uiliy funcions remains a ough problem. Posiing a uiliy funcion is resricive, even if we esimae is associaed parameers, given ha differen uiliy funcions have differen behavioral implicaions (regarding he laer wih respec o invesmen and saving sraegies, see Gollier, 1, Gollier, Eeckhoud and Schlessinger, 5 and de Palma and Prigen, 9). he preceding definiions of risk-aversion allow us o characerize sandard uiliy funcions, and in paricular he HARA class of uiliy funcions. Hyperbolic Risk Aversion. A uiliy funcion u (.) is of ype HARA ( hyperbolic absolue risk aversion ) if he inverse of absolue risk-aversion is a linear funcion of 1 wealh. HARA uiliy funcions, u (.), are wrien as follows: ux ( ) = ab ( + ( x/ c)) c, where u (.) is defined over he domain b+ ( x/ c) >. he parameers a, b and c are consans such ha: a(1 c) / c >. he associaed ARA A( x ) is given by: ( ) 1 A( x) = b+ ( x/ c), he inverse of which is indeed a linear funcion of wealh, x. Noe ha he condiion a(1 c) / c > allows us o conclude ha u ' > and u '' <. hree sub-classes of funcions are ypically disinguished: 7 5

7 1. he Quadraic Uiliy Funcion. If c = 1, he uiliy funcion u (.) is quadraic. For u (.) o be posiive, he domain here is resriced o he inerval ], b[. he ARA of a quadraic uiliy funcion is increasing in wealh ( increasing absolue risk aversion, IARA). his implies ha he risk premium π (.) is increasing, which is a fairly couner-inuiive propery, and which indicaes he limis of he applicaion of his funcion (despie he simpliciy of is use in he deerminaion of opimal porfolios, for example).. Consan absolue risk aversion (CARA). As he parameer c ends o infiniy, we obain A( x) = A, where A is a consan. In his case, he uiliy funcion u (.) is of he form: ux ( ) = (exp[ Ax]/ A). Noe ha he RRA here increases wih wealh. 3. Consan relaive risk aversion (CRRA). When b =, hen R( x) = c is 1c consan and u (.) is of he ype: ux ( ) = x / (1 c) if c> 1, ln[ x] if c= 1. his ype of funcion exhibis decreasing absolue risk-aversion (DARA)..3. Compensaing variaion he raio of expeced uiliies characerizes he invesor s choice behaviour bu i is only a qualiaive crierion since uiliies are defined up o affine ransformaions. In wha follows, we use insead a quaniaive index of invesor's saisfacion based on he sandard economic concep of compensaing variaion. As illusraed in de Palma and Prigen (8, 9), he noion of compensaing variaion is very useful o evaluae he moneary loss of no having he bes porfolio. he uiliy loss from no having access o a beer porfolio is provided by he compensaing variaion measure. If an invesor wih risk aversion γ and iniial invesmen V faces a choice (1) () beween wo (random) horizons and, he has o compare he wo expeced uiliies EU [ γ ( V ( i )); V ]. Assume ha horizon () provides higher uiliy han (1) (1) () mauriy. If he invesor selecs mauriy insead of, he will ge he same expeced uiliy provided ha he invess an iniial amoun V V such ha: E U ( V (1) ); V E U ( V () ); V γ = γ (1) herefore, his invesor requires (heoreically) a moneary compensaion ha can be evaluaed by means of he raio V / V. his amoun is in line wih he cerainy equivalen concep in expeced uiliy analysis. I can be viewed as an implici iniial 6

8 invesmen necessary o keep he same level of expeced uiliy. 3. Coninuous-ime model and risk neural invesor In his secion, he ime of sale is pre-se, commied irrevocably a ime, based on he expeced dynamics of he porfolio value and is cash flow. he real esae porfolio value is defined as he sum of he discouned free cash flows (FCF) and he discouned erminal value (he selling price). Denoe k as he weighed average cos of capial (WACC), which is used o discoun he differen free cash flows, and he erminal value. We assume ha he free cash flow grows a a consan rae g Coninuous-ime model As Baroni e al. (7a), we suppose ha he price dynamics, which corresponds o he erminal value of a diversified porfolio (for insance a real esae index), follows a geomeric Brownian moion: dp = μd + σdw, P where W is a sandard Brownian moion. We have: = +. () 1 P P exp μ σ σw his equaion assumes ha he real esae reurn can be modelled as a simple diffusion process where parameers μ and σ are respecively equal o he rend and o he volailiy. he expeced reurn of he asse a ime is given by: P E = exp( μ). (3) P hen he fuure real esae index value a ime, discouned a ime, can be expressed as: 1 P P = P exp μk σ + σw wih E exp( [ μ k] ) =. (4) P Denoe by FCF he iniial value of he free cash flow. he coninuous-ime version of he sum of he discouned free cash flows FCF s is equal o: which leads o ks [ kg] s s (5) C = FCFe ds = FCFe ds, FCF [ k g] C = 1 e k g. Inroduce he real esae porfolio value process V, which is he sum of he discouned free cash flows and he fuure real esae index value a ime, discouned a ime : (6) 7

9 1 ( μ k σ ) σw + V = C + P = c 1 e + P e. (7) a ( ) We deermine he porfolio value V for a given mauriy. his assumpion on he ime horizon allows o ake accoun of selling consrains before a limi dae. he higher, he less sringen his limi. Addiionally, his hypohesis allows he sudy of buy-and-hold sraegies (see secion 6). he fuure porfolio value a mauriy, discouned a ime, is given by: 1 FCF [ k g] ( μ k σ ) σw + V 1 e P e = +. k g he porfolio value V is he sum of a deerminisic componen and a Lognormal random variable. In wha follows, we deermine he opimal soluion a ime, for a given mauriy and for an invesor maximizing expeced uiliy. Firs recall ha he sum of he discouned free cash flows is always increasing due o he cash accumulaion over ime. Second, we have o analyze he expeced uiliy of he fuure real esae index value a ime, discouned a ime : if he price reurn μ is higher han he WACC k, hen, he opimal soluion for he linear uiliy case is simply equal o he mauriy. hus, in wha follows, we consider he case μ < k. Consequenly, no selling he asse implies a higher cumulaed cash bu a smaller discouned expeced erminal value e ( k ) P μ. Hence, he invesor has o choose beween more (discouned) flows and less expeced discouned index value. We also focus on he sub case g < μ, which corresponds o empirical daa. We invesigae wo main numerical cases: - Case 1: μ = 4. 4%, σ = 5 %, g = 3%, k = 8. 4%, P = 1, FCF = 1 /. I corresponds o an early selling, due in paricular o weak expeced reurn of he real esae asse. - Case : μ = 6%, σ = 5%, g = %, k = 9. 5%, P = 1, FCF = 1 / 15. I corresponds o a lae selling, due in paricular o higher expeced reurn of he real esae asse. (8) 3. Compuaion wih he linear uiliy funcion: (see Barhélémy and Prigen, 9) he opimizaion problem is: Max E V [ ],. (9) Since he expecaion of V is equal o: FCF [ kg] [ μk] EV = 1 e + Pe, k g (1) 8

10 we deduce: EV [ kg] ( ) [ μk ] = FCFe + P μ k e. hen, he opimal holding period is deermined as follows. (11) ( μ ) Case 1: he iniial price P is smaller han e FCF k k μ. hen, he opimal ime o sell corresponds o he mauriy. Since he Price P Earning Raio (PER) is oo small ( < ( kμ ) e FCF mauriy. k μ ), he sell is no relevan before FCF ( kμ ) FCF Case : he iniial price P lies beween he wo values μ e and μ hen, he opimal ime o sell is soluion of he following equaion: From Equaion (11), we deduce 9 : EV =. 1 FCF 1 μg P kμ = ln. k k. (1) (13) In paricular, noe ha is a decreasing funcion of he iniial price P and of he difference beween he index reurn μ and he growh rae g of he free cash flows. his laer propery was empirically observed by Brown and Geurs (5). I means ha invesors sell propery sooner when values rise faser han ren. FCF Case 3: he iniial price P is higher han k μ. hen, he opimal ime o sell corresponds o he iniial ime. Since he PER P is sufficienly large ( > 1 FCF k μ ), here is no reason o keep he asse P. As an illusraion, he cumulaive value C of he FCF values, of he expecaion of he index value EP [ ] and he expecaion of he porfolio value EV [ ] are displayed in Figure 1. We consider wo ses of parameer values for a year managemen period ( = ). We noe ha he discouned expeced value V of he porfolio is concave. he parameer values imply ha he opimal holding period,, is respecively equal o 913. years and years. For hese wo examples, he opimal ime o sell is smaller han he mauriy. In he second example, he discouned porfolio value varies up o % 1. Knowing he opimal ime o sell, which is deerminisic, he 9

11 probabiliy disribuion of he discouned porfolio value V can be deermined. he value V is equal o: 1 ( kg) = FCF ( k g) + / +. ( kg) V 1 e P exp μ k 1 σ σw FCF ( kg) Denoe A= e he cumulaive discouned free cash flow value a. Since, from (13), he opimal ime o sell saisfies: 1 FCF = ln, μg P ( kμ) hen, we deduce: kg ( μg ) FCF FCF A = 1, ( kg) P ( kμ ) and he cdf FV of V is given by: if, v A FV () v = 1 v A N ln μ k 1 σ if v A /, > σ P (14) where N denoes he cdf of he sandard Gaussian disribuion. 4. Opimal ime o sell, chosen a ime 4.1 Compuaion wih he quadraic uiliy funcion Model wih he quadraic uiliy funcion he expeced uiliy of he porfolio value a ime in he case of he quadraic uiliy funcion is λ E U( V) = E( V) E( V ) (15) λ E U ( V) = C + E[ P] E( C + C E[ P] + E[ P] ) Using (8) we ge a a ( μ k 1 σ ) σw ( 1 μ k σ ) σw + + V = c 1 e + c 1 e P e + P e (16) and knowing ha ( ) ( ) 1 ( ) E + e = e A B BW A we deduce ha he expecaion of (16) is equal o:, where W N( ),, 1

12 1 a ( ) a E V = c ( 1 e ) + P e + c 1e Pe ( ) ( ) μ k σ μ k which leads o he following expeced value of he porfolio a ime described in (15): λ E U V c e P c e c e P P 1 a ( μk) ( ) ( ) ( ) ( 1 ) e a a μk ( 1 ) ( 1 ) e μk e σ = For insance, Figure 1 and illusrae he impac of λ on he porfolio uiliy funcion according o he selling ime. Noice ha λ = corresponds o he risk neural case sudied in Barhélémy and Prigen (9) and presened in secion 3.. In case 1, he opimal ime o sell is decreasing wih he level of he risk aversion λ (see Figure 1). In case, he effec of λ on he opimal ime o sell is opposie, even if he porfolio uiliies decrease (see Figure 1). In case 1, when λ is oo high (around.8), =, while in case, = (see Figure ). Case 1 represens a porfolio wih a high Price Earning Raio (PER). he imporance in he porfolio is given o he selling price whose value is sensiive o he risk aversion. A he conrary, case represens a siuaion wih a low PER. he imporance is hen aribued o he rens Limi of he quadraic uiliy funcion If risk aversion raises, he expeced uiliy can be no longer monoone wih respec o he selling ime. he quadraic uiliy funcion is no clearly defined for all he values of λ. If λ becomes high enough he uiliy funcion is hen a decreasing funcion for values higher han 1/λ. Indeed is derivaive funcion is equal o U( x) λ = x x = 1λx x x which is null for x = 1/ λ. For insance, wih U( x) = x.5 x,( λ =.5), x is consrained o be less han U + a < U, a> (as illusraed by Figure 3)., because ( ) ( ) λ λx In his case Ax ( ) = u''( x)/ u'( x) = and Rx ( ) = xu''( x) / u'( x) =. 1 λx 1 λx β 1 Seing Rx ( ) = β leads o x = 1 β. Usual values of relaive risk aversion lie + λ beween 1 and 1. For insance, in case, he funcion becomes convex wih λ =.15. his arises even wih σ = as observed on Figure 4. If λ =.9, he maximal value of he expeced uiliy is reached a x = 1/.9 = (see Figure 5a). able 1 shows ha for 5 years, he porfolio expeced value is greaer han he limi compued above. Hence, even if V 6 = is higher han V 4 = 19.97, he corresponding expeced value is lower, UV ( 6) < UV ( 4) as illusraed in Figure 5b. here are wo exrema, one maximum and one minimum, he laer being induced by he bias in he uiliy funcion Analysis of 11

13 In order o analyze he soluion expeced uiliy:, we have o compue he firs derivaive of he ( ) E U V ( ) ( μ μ ) a ac e P k e = + k here is no explici funcion for he soluion bu for he range where he uiliy funcion is concave, he derivaive funcion evaluaed a is null. Le E U( V ) d () =, hen d( ) =. Figure 6 illusraes his feaure for wo risk aversion levels and for each case (early or lae selling). More generally, he derivaive funcion may be expressed as a funcion of k, g, μ, FCF, m, σ and λ : dkg (,,, μ, FCF, m, σλ, ) We may sudy he implici funcions for he wo main parameers of ineres being σ and λ : d (, λ) = ( λ) which enables o sudy ( λ ). Figure 7 illusraes his funcion ( λ ) for case 1 and Figure 8 for case (he analysis is he one made a he end of secion abou he PER). We have seen ha for a given iniial endowmen V, he uiliy funcion is no more increasing for high risk aversion. In his case, he sudy of he implici funcion will no be relevan. Hence, we will no sudy he opimal ime o sell according o he differen parameers when considering he quadraic uiliy funcion. his will be done for he CARA and he CRRA funcions used o ake accoun of he risk aversion of he invesor. 4. Compuaion wih he CARA uiliy funcion 4..1 CARA uiliy maximizaion We have: 1 a ( ) ( μ k σ 1 ) σw 1 C c e P α + α α 1 α P e E U( V ) = α E e e = α e E e A ime, we ge: + 1 ( ) 1 e μ k σ σ x + αp αp x E e = e e / d. x π Le us denoe (, ) ( 1 ) hx = μk / σ + σ x, which gives: Is firs derivaive is equal o: + (, ) 1 e h x αp αp x / d π E e = e e x 1

14 h x α ( ) + (, ) α P 1 P e x / π E e = e e dx which corresponds o + (, ) e h x α P α P h 1 (, x ) 1 1 x / E e P e ( k x ) e dx π α + μ σ σ = + For he case 1, he uiliy funcion for a risk neural invesor leads logically o he same opimal soluion as found in secion 3: α =, = his is of course he same resul for case, where for α =, = In he case of early selling (case 1), he opimal ime o sell ( α ) decreases when he risk aversion increases. In his case, he rens are relaively low comparaively o he asse price. Hence, an increase in he risk aversion makes he lae selling more risky from he poin of view of he invesor, as he porfolio benefis come more from he erminal value. In order o reduce his risk, he asse has o be sold earlier as illusraed in Figure 9. his is he same analysis as he one made for he quadraic uiliy funcion represened on Figure 1. In he conrary, when considering he case of lae selling (case ), he opimal ime o sell ( α ) increases wih he risk aversion. he cash flows induce by he rens are relaively high comparaively o he one of he erminal value. In he arbirage beween ren cash flows and selling price, he cash flows become more and imporan as he risk aversion increases (remember he rens are deerminisic). In order o ge a higher porfolio value, he asse has o be sold laer as illusraed in Figure 1. We find he same qualiaive analysis as he one made for he quadraic uiliy funcion illusraed on Figure. 4.. Volailiy and risk aversion We analyze he evoluion of opimal ime o sell ( α, σ ) when he level of risk aversion and he volailiy may change. Figure 11 (for case 1) and Figure 1 (for case ), clearly show how hese wo parameers modify he opimal ime o sell in he same way. In he case of early selling (case 1), he opimal ime o sell decreases wih he risk aversion and he volailiy. he volailiy reinforces he risky effec on he selling price (as seen previously on Figure 9). In he case of lae selling (case ), he opimal ime o sell increases wih hese wo parameers. As in case 1, more volailiy implies a more risky erminal value. Bu, as he sraegy focuses on he rens componen of he porfolio, he asse has o be sold laer. Funcions ( α, σ ) may be analyzed by considering fixed one of hese wo parameers. Figure 13 examines he impac of he volailiy level on he relaion ( α ). For case 1, he decreasing shape is more and more pronounced as he volailiy increases. When he volailiy level is quie small ( σ =.1), he risky effec on he selling price is low and he opimal ime o sell is relaively consan whaever he risk aversion level. Wih higher volailiies, he opimaliy ends o be a sell a ime (in order o avoid a poenial loss in he erminal value). For case, he increasing of ( α ) is more and more imporan as he volailiy increases. As for case 1, a low volailiy ( σ =.1) implies a relaively consan 13

15 opimal ime o sell (as in he risk neural case). Wih higher volailiies, he opimaliy ends o be a sell a ime. Figure 14 illusraes he evoluion of ( σ ) for a given risk aversion level Risk aversion and iniial price P Whaever he cases (1 or, see Figure 15), ( P ) is a decreasing funcion of he iniial price P. Considering a higher iniial price P for a given level of rens, implies P an increase of he Price Earning Raio (PER) FCF. hen, as more imporance is given o he erminal value, he opimal ime o sell decreases in order o ake accoun of he increase of he risk in he selling price. P A he conrary, when decreases, he rens have a higher weigh in he porfolio FCF value, and he opimal ime o sell increases. A he limi, if he PER is oo small, he sell is no relevan before mauriy. his analysis will be exended wih he CRRA uiliy funcion. 4.3 Compuaion wih he CRRA uiliy funcion CRRA uiliy maximizaion he expeced uiliy is given by: 1 ( ) = ( + ) 1γ E U V 1 E C P γ 1 1γ 1 a ( μk ) ( ) σ + σw = 1γ E c 1 e + P e and he firs derivaive is equal o: + 1 hx (, ) γ a hx (, ) 1 1 x/ E [ UV ( ) ] = ( C + Pe ) ace + Pe ( μk σ + σ x ) e dx π Previous formula does no allow o ge explici relaions beween he opimal selling ime and various parameers such as he relaive risk aversionγ, he volailiy σ and he iniial real esae asse value P. In wha follows, we begin by examining he shape of he uiliy funcion according o he relaive risk aversion γ. hen, we sudy he impac of he volailiy. Figure 16 for case 1, and Figure 17 for case lead o he same qualiaive resuls as he ones described wih he use of he quadraic or he CARA uiliy funcions. 14

16 Volailiy and risk aversion Figures 18 o lead o he same qualiaive resuls obained wih he CARA uiliy funcion Risk aversion (γ ) and iniial price P Wih his funcion, we have he same resuls as hose described in secion We develop hese resuls analyzing he variaion in he iniial price P differenly. According o he iniial price (he rens remaining consan), we may change he siuaion from an early selling o a lae selling and vice versa. For insance, in case 1, wih an iniial price P equal o 15, he opimal ime o sell is a decreasing funcion of he risk aversion γ (wih γ =.5, = 9, wih γ =, = 8 and wih γ = 1, = ). his is rue for each value of P equal or greaer han 1 (he reference value), as his case illusraes an early selling siuaion. Le us noice, he higher P, he more decreasing ( γ ). here exiss a value of P below which he effec of he risk aversion is reversed: we have hen a lae selling siuaion. For insance, wih an iniial price P equal o 95, he opimal ime o sell is an increasing funcion of he risk aversion γ (wih γ =, = 13.1 and wih γ = 1, = 16.5 ). γ =.5, = 1.9, wih he same analysis may be made wih case. For values of P less han 1, he opimal ime o sell is a increasing funcion of he risk aversion γ. For insance an iniial price of 9 leads o wih γ =.5, = 18.6, wihγ =, = 18.8 and wih γ = 1, = ). here exiss a pivoal iniial price poin from which he impac of he risk aversion on he opimal o sell is changing. his is illusraed on Figure. 5. Oher opimal imes o sell for a risk averse invesor 5.1 Perfecly informed invesor In his secion, he invesor is supposed o have a perfec foresigh abou he enire fuure price pah. rajecories are random (he invesor does no choose he realized pah) bu, a ime, he whole pah is known. herefore, he invesor can maximize wih respec o his rajecory. hus, he opimal soluion is deerminisic condiionally o his informaion. Neverheless, he pah is unknown jus before ime. Consequenly, he opimal ime o sell is a random variable. his ideal framework is no realisic bu provides an upward benchmark. Noe ha, since he invesor is raional, his uiliy funcion is increasing. herefore, since he pah is known, he 15

17 maximizaion of he uiliy of his porfolio value is equivalen o he maximizaion of a linear uiliy. his means ha we recover previous soluion provided in Barhélémy and Prigen (9). In wha follows, we recall he disribuions of he opimal holding period and of he opimal value V. Barhélémy and Prigen (9) provide an explici formula by means of a mild approximaion. Inroduce he funcion G defined by: 1 y 1 my y G( m, y, ) = 1 Erfc m e Erfc m +, where he funcion Erfc is given by: Denoe also u Erfc( x) = e du. π FCF v Av () = + μk 1 / σ, and Bv () = ln. v P hen, he approximaed cdf of V is given explicily by: x for, v< P, PV [ v] = Av ( ) Bv ( ). (15) G,, for v > P σ σ he probabiliy ha he real esae porfolio value is higher han P is equal o 1. hus, whaever he pah, he invesor receives a leas P. Indeed, if all he fuure discouned porfolio values are lower han he iniial price, he knows he has o sell a ime and hen receives exacly P. 5. American opimal selling ime In his hird case, we allow ha he invesor may choose he opimal ime o sell, according o marke flucuaions and informaion from pas observaions. In his case, he faces an American opion problem. Recall ha he invesor preferences are modelled by means of uiliy funcion. A any ime before selling, he compares he uiliy of he presen value P wih he maximum of he fuure uiliy value he expecs given he available informaion a ime (mahemaically speaking he compues he maximum expeced uiliy of his porfolio on all J -measurable sopping imes τ )., I means ha he decides o sell a ime only if he uiliy of his porfolio value a his ime is higher han he maximal expeced uiliy ha he can expec o reach if he does no sell a his ime. hus, he has o compare Uγ ( C + P) wih sup τ J EU γ( Cτ + Pτ) J, where C s denoes he FCF value a ime s., 16

18 Inuiively, he opimal ime mus be he firs ime a which he uiliy Uγ ( C + P) is sufficienly high. A his price level, he fuure free cash flows (received in case of no sell) will no be high enough o balance an index value lower han he price P a ime (he expeced index value decreases wih ime as he discouned rend μ k is negaive). he opimal ime corresponds exacly o he firs ime a which he asse price P is higher han a deerminisic level (see Appendix C). his resul generalizes he case considered in Barhélémy and Prigen (9) where he invesor has a linear uiliy. In ha case, he sells direcly he asse if he FCF price P is higher han. hen, since he reurn of he discouned free cash flows is equal o k μ ( k g) e, he price ( ) P has o be compared wih he value e FCF k g k μ he American opion problem Denoe by V ( x, ) he following value funcion: ( ) V ( x, ) = sup E Uγ Cτ + P P = x. τ τ F, V (, ) γ +, since τ = Noe ha we always have x U ( C x) ( ) V ( x, ) = Uγ C + x. As usual for American opions 11, wo regions have o be considered: he coninuiy region: J and, in ha case,, ( ) + C = ( x, ) R [, ] V ( x, ) > Uγ C + x he sopping region: ( ) + S = ( x, ) R [, ] V ( x, ) = Uγ C + x he firs opimal sopping ime afer ime is given by hen: ( ) = inf u [, ] V ( P, u) = U C + P. u γ u u { } = inf u [, ] u C. 5.. Compuaion of he value funcion V o deermine, we have o calculae V ( x, ). We have o compue: 17

19 a aτ ( μ σ ( τ ) σ τ ) sup E Uγ C + c e e + P exp k 1/ + ( W W) P = x. τ J, In paricular, we have o search for he value τ for which he maximum a aτ ( μ σ ( τ ) σ τ ) sup E Uγ C + c e e + xexp k 1/ + ( W W). τ J, is achieved. his problem is he dynamic version of he deerminaion of presened in Secion 4. Inroduce he funcion f x, defined by: a a ( + θ ) γ ( ) fxz,, γ( θ) = U C + c e e + xexp μk 1/ σ θ + σ θz., his funcion is sricly increasing wih respec o x. We have o solve: sup E fxz,,, γ ( θ ). θ J, Case 1. he opimal soluion is equal o he mauriy. hen: a a V ( x, ) = E Uγ ( C + ce e + xexp μk 1/ σ ( ) + σ( W W ) ). Using Jensen inequaliy, we deduce in ha case ha V ( x, ) > Uγ ( C + x), for all hree sandard uiliy funcions (since hey are concave and sricly increasing). Case : he opimal soluion lies sricly beween and. hen, he opimal ime τ is equal o ( + θ ), where θ is he soluion of he following equaion: E fxz,,, λ ( θ ) =. θ a aτ ( μ σ ( τ ) σ τ ) V ( x, ) = E Uγ C+ ce e + xexp k 1/ + ( W W). Case 3: he opimal ime τ corresponds o he presen ime, and ( ) V ( x, ) = Uγ C + x. (5) Consequenly, from he hree previous cases, we deduce he value of V ( x, ). Finally, he American opimal ime is deermined by: 18

20 ( ) = inf [, ] V ( P, ) = U C + P. herefore, we can check ha ( P, ) = U ( C+ P) hus, we have: where (,,,, ) γ γ V if and only if: ( θ) <, θ,. E f P,, z, λ θ ( ) = inf [, ] P l, FCF, k, g, μ, l FCF k g μ is deermined from opimaliy condiion of being opimal for he firs problem (see secion 4). 6. Comparison of he hree opimal sraegies and he buy-and hold one We examine boh he probabiliy disribuions of he opimal imes o sell, and he corresponding discouned porfolio values V, and V and V. We inroduce also he comparison wih he buy-and hold porfolio V. he sensiiviy analysis is done wih respec o he volailiy σ and o he mauriy. Noe ha, since he porfolio value V dominaes he oher ones, is cdf is always below he cdf of he ohers ( firs order sochasic dominance ). We also examine compensaing variaions Cdf of V according o Figures 3 and 4 provide respecively cdf and pdf of porfolio values according o selling ime. For numerical case 1, we noe if he invesor sells oo quickly (for insance =3), he probabiliy o ge high reurns is smaller han for opimal sraegy. Addiionally, he opimal sraegy dominaes sochasically he buy and hold sraegy a he firs order. For numerical case, he invesor mus sell laely. he opimal sraegy clearly dominaes a he firs-order he oo early selling sraegy (=3). 6.. Compensaing variaions: compuaions Recall ha, if an invesor wih risk aversion γ and iniial amoun V mus selec one of he wo (random) horizons (1) and (), he has o compare he wo expeced uiliies EU [ γ ( V ( i) ); V ]. In wha follows, we suppose ha horizon () provides higher uiliy han mauriy (1). If he invesor selecs mauriy (1) insead of (), he will ge he same expeced uiliy provided ha he invess an iniial amoun V V such ha: E [ U ( V ); V ] = E [ U ( V ); V ] γ (1) () γ 19

21 Recall also ha, a any ime of he managemen period,, he porfolio value is given by: FCF C = 1e k g Inroduce he reurns [ kg] and V = C + P, wih exp 1 P = P μk / σ + σw R = V / V and R = V / V. (1) (1) () () 6..1 he compensaing variaion for he quadraic case Suppose ha he invesor s uiliy U is of quadraic. Funcion U is equal o: v Uv () = v λ, wih λ>. If we fix he level of risk aversion λ, hen Relaion E [ U ( V (1) ); V ] E [ U ( V () ); V λ = ] λ is equivalen o: λ λ V E (1) (1) = () () R V E R V E R V E R. he previous relaion provides he expression of he compensaing variaion for he quadraic case, hrough he resoluion of he following polynomial equaion: λv λv x E R (1) - xe R (1) + E R () E R (), = where x denoes he possible values of he compensaing variaion V / V. Se: ( ) λv Δ= E (1) - (1) () () R λv E R E R E R. hen, we deduce: V E R (1) + Δ =. V λve R (1) Since he relaive risk aversion is increasing for he quadraic case, i is no surprising ha he compensaing variaion depends on he wealh level V. 6.. he compensaing variaion for he CARA case Suppose ha he invesor s uiliy U is of CARA ype. Funcion U is equal o: av e Uv ( ) =, wih a>. a hen, if a is fixed, Relaion E [ U ( V ); V ] = E [ U ( V ); V ] is equivalen o: a (1) () a

22 av R (1) av R () E e = E e. which yields o an implici relaion beween V and V he compensaing variaion for he CRRA case Suppose ha he invesor s uiliy U is of CRRA ype. Funcion U is equal o: hen, Relaion which yields o: 1γ v Uv () =, wih γ >. 1 γ E [ U ( V ); V ] = E [ U ( V ); V ] is equivalen o: γ (1) () γ V 1γ E R 1γ 1 1 (1) = V γ E R γ (), 1 1γ 1 V E R () 1γ V E R (1) γ =. he previous relaion provides he expression of he compensaing variaion for he CRRA case he hree compensaing variaions beween and. Recall ha is necessarily beer han since i is opimal among all deerminisic daes. herefore, we have o search V such ha he following equaliy holds: Recall ha (, ) ( 1 ) = γ E[ U ( V ); V ] E[ U ( V ); V ] γ hx = μk / σ + σ x. hus, we ge: = + 1 π + 1 π = γ E[ U ( V ); V ] E[ U ( V ); V ] γ hx (, ) x / a U ((1 c e ) + P e ; V ) e dx γ h (, x) x / V a U ((1 c e ) + P e ; ) e dx γ his laer equaion can be numerically solved for he hree ypes of uiliy funcions. 1

23 and he hree compensaing variaions beween respecively and and Recall ha is necessarily beer han boh and since i is opimal among all possible daes. herefore, we have o search V such ha he following equaliies hold: i) Compensaion beween and : his is equivalen o: = γ E[ U ( V ); V ] E[ U ( V ); V ] γ π γ γ P a h (, x) x / U ((1 c e ) + P e ; V ) e dx= U (; v V ) f dv ii) Compensaion beween and : his is equivalen o: = γ E[ U ( V ); V ] E[ U ( V ); V ] γ a hx (, ) x / ((1 ) Pe ; ) e x ( v; π γ + = γ V U c e V d U V ) f dv 6.3 Compensaing variaions: comparisons for he CARA case V In wha follows, we illusrae he compensaing variaions in he CRRA case (he mos usual case). We compare he uiliy of he porfolio opimal selling ime corresponding o ime wih all hose corresponding o he oher possible selling imes, including of course he mauriy iself. We invesigae he wo numerical base cases as previously (see Figures 5 and 6, respecively for case 1 and case ). We analyze on he same graphic four compensaing variaion curves, for four differen risk aversion levels ( γ =.5, γ =, γ = 5 and γ = 1). As well, for each case, four volailiy values are considered ( σ =.1, σ =.5, σ =.1 and σ =., named respecively Fig a, Fig b, Fig c and Fig d). For numerical case 1, by considering a low volailiy level (Figure 5a), he impac of a wrong selling dae is underlined. he opimal ime o sell is around 9.16, whaever he risk aversion level (because of he low volailiy value). he four curves (according o he risk aversion level) are nearly he same. Selling a ime could be seen as a cos of % (managemen cos for insance). Addiionally, he cos increases wih he gap beween he selling dae and he opimal ime o sell. Wih a volailiy of 5% (Figure 5b), he opimal imes o sell are significanly differen according o he risk aversion level (=9. wih γ =.5 while = wih γ = 5 or γ = 1). he heoreical managemen cos could reach 15% wih a volailiy of 1% (Figure 5c)

24 and even 6% (Figure 5d) wih a volailiy of %. he laer can be seen as 3% per year heoreical managemen cos during years. For numerical case, heoreical managemen coss are higher for he small volailiy levels (Figure 6). 7. Conclusion his paper emphasizes he impac of he real esae marke volailiy on opimal holding period. For his purpose, he invesor is assumed o be risk-averse, which is an usual assumpion when dealing wih porfolio opimizaion. hree kinds of opimal imes are considered. he firs one supposes ha he invesor can only choose he opimal ime o sell a he iniial dae. he second one assumes ha he invesor is perfecly informed. his is no oo realisic bu provides an upward benchmark. Finally, we consider he American opion framework, which is more raional since he invesor is allowed o ake accoun of ineremporal managemen and cumulaive informaion. For each of hese models, he opimal imes o sell and porfolio values are analyzed and compared, using various parameer values of he real esae markes, in paricular he volailiy index and he porfolio mauriy. We show ha risk aversion allows acually o beer ake accoun of he volailiy, according o he raio of cash flows upon real esae price. he higher he cash flows, he longer he opimal imes. However, he resul wih respec o he volailiy is more mixed. According o specific assumpions on oher parameers such as he risk aversion and he real asse value, he opimal ime o sell can be increasing or decreasing w.r.. he volailiy. We illusrae also he comparison by means of (moneary) compensaing variaion. Furher research would inroduce oher funcional represenaion of uiliy funcion such as an addiive represenaion w.r.. curren ime. his would allow separaion beween uiliy and free cash flows (for example, if hey are used o consume) and uiliy on real asse value a he selling ime. Oher models such as recursive uiliy or uiliy wih loss aversion and probabiliy deformaion may be also examined. 3

25 References [1] Anderson, S., de Palma, A., hisse, J.-F. (199): Discree Choice heory of Produc Differeniaion, MI Press, Cambridge, MA. [] Arrow, K.J., (1965): Aspecs of he heory of Risk-Bearing, Helsinki: Yrj Hahnsson Foundaion. [3] Barhélémy, F., and J.-L. Prigen, (9): Opimal ime o Sell in Real Esae Porfolio Managemen, Journal of Real Esae Finance and Economics, 38, [4] Baroni, M., Barhélémy, F., and M. Mokrane, (7a): Mone Carlo Simulaions versus DCF in Real Esae Porfolio Valuaion, Propery Managemen, 5(5), [5] Baroni, M., Barhélémy, F., and M. Mokrane, (7b): Opimal Holding Period for a Real Esae Porfolio, Journal of Propery Invesmen and Finance, 5(6), [6] Bond, S.A., Hwang, S., Lin, Z. and K.D. Vandell, (7): Markeing Period Risk in a Porfolio Conex: heory and Empirical Esimaes from he UK Commercial Real Esae Marke, Journal of Real Esae Finance and Economics, Vol. 34, [7] Borodin, A. N., and P. Salminen, (): Handbook of Brownian Moion (Facs and Formulae), Birkhäuser. [8] Brown, R.J., (4): Risk and privae Real Esae Invesmens, Journal of Real Esae Porfolio Managemen, Vol. 1, 113. [9] Brown, R.J., and. Geurs, (5): Privae Invesor Holding Period, Journal of Real Esae Porfolio Managemen, Vol. 11, [1] Colle, D, Lizieri, C., and C.W.R. Ward, (3): iming and he Holding Periods of Insiuional Real Esae, Real Esae Economics, Vol. 31, 5-. [11] de Palma, A. and J.-L. Prigen, (8): Uiliarianism and fairness in porfolio posiioning, Journal of Banking and Finance, Vol. 3, [1] de Palma, A. J.-L. Prigen, (9): Sandardized versus cusomized porfolio: a compensaing variaion approach, Annals of Operaions Research, 165, [13] Dexer, A.S., Yu, J.N.W. and Ziemba, W.., (198): Porfolio selecion in a lognormal marke when he invesor has a power uiliy: compuaional resuls, in Sochasic Programming, M.A.H. Dempser (Ed.), Academic Press, New-York, [14] Ellio, R.J., and Kopp, P.E., (1999): Mahemaics of Financial Markes, Springer Finance series, Springer. 4

26 [15] Elon, E., Gruber, M. (1995): Modern Porfolio heory and Invesmen Analysis, J. Wiley, New York. [16] Fisher, J., and M. Young, (): Insiuional Propery enure: evidence from he NCREIF daabase, Journal of Real Esae Porfolio Managemen, Vol. 6, [17] Friedman, M., Savage, L. (1948): Uiliy analysis of choices involving risk, Journal of Poliical Economics, 56, [18] Gau, G., and K. Wang, (1994): he ax-induced Holding Periods and Real Esae Invesors: heory and Empirical Evidence, Journal of Real Esae Finance and Economics, Vol. 8, [19] Gollier : Gollier, C, (1): he Economics of Risk and ime. he MI Press, Cambridge, MA. [] Gelner, D. M., and N.G. Miller, (7): Commercial Real Esae Analysis and Invesmens, Cengage/Souh-Wesern, Cincinnai. [1] Hendersho, P., and D. Ling, (1984): Prospecive Changes in ax Law and he Value of Depreciable Real Esae, AREUEA Journal, Vol. 1, [] Kallberg, J.G. and W.. Ziemba, (1983): Comparison of alernaive uiliy funcions in porfolio selecion problems, Managemen Science, 9 (11), [3] Karazas, I., and S. Shreve, (1): Mehods of Mahemaical Finance, Springer. [4] McFadden, D. (1): Economic choices, American Economic Review, 91(3), [5] Oksendal, B., (7): Sochasic Differenial Equaions: An inroducion wih applicaions, 6h ed. 3. (Corr. 4h), Springer-Verlag. [6] Pra, J.W., (1964): Risk aversion in he small and in he large, Economerica, 3, [7] Prigen, J-L., (7): Porfolio Opimizaion and Performance Analysis, Chapman & Hall, Florida. [8] Rohschild, M., Sigliz, J. E. (197): Increasing risk: I. A definiion, Journal of Economic heory,, [9] Rohschild, M., Sigliz, J. E. (1971): Increasing risk: II. Is economic consequences, Journal of Economic heory, 3, [3] Rowley, A., V. Gibson, and C. Ward, (1996): Qualiy of Urban Design: a Sudy of he Involvemen of Privae Propery Decision-Makers in Urban Design, Royal Insiuion of Charered Insiuion of Charered Surveyors, London. [31] von Neumann, J., Morgensern, O. (1944): heory of Games and Economic Behavior, Princeon Universiy Press. 5

27 1 Using he daabase of properies provided by IPD in he UK over an 18-year period, heir empirical analysis shows ha he median holding period is abou seven years. hese opimizaion problems are specific o real esae invesmens and differ from sandard financial porfolio managemen problems (see Karazas and Shreve, 1, or Prigen, 7). Firs, he asse is no liquid (no divisible). Second, he conrol variable is he ime o sell and no he usual financial porfolio weighs (see Oksendal 7, for a relaed problem abou opimal ime o inves in a projec wih an infinie horizon). 3 See Elon and Gruber (1995) for a discussion abou he esimaion of risk aversion parameers. 4 he independence axiom has been implicily inroduced by von Neumann and Morgensern (1944). his crierion allows he characerizaion of he expeced uiliy. his axiom implies he uiliy funcion U defined over he loeries is linear wih respec o he probabiliies ha he differen evens occur. 5 he concaviy of he uiliy funcion yields he following inequaliy, known as Jensen's inequaliy: λ [,1], a, b, λ ua ( ) + (1 λ) ub ( ) u( λa+ (1 λ) b). 6 Consider he variance σ X = E[( X E[ X]) ]. By using a aylor s expansion, we deduce: Eu [ ( X)] uex ( [ ]) + (1/ ) u''( E[( X)]) E[( XEX [ ]) ) We also have: uex ( [ ] π [ X]) uex ( [ ]) u'( E[( X)]) π [( X)], By definiion, uex ( [ ] π [ X]) EuX [ ( )]) Finally, we have: π[ X] ( u''( E[ X])/ u'( E[ X])) σ and π[x] -((u (E[X]))/(u (E[X])))σ X ². 7 See Gollier (1) for main definiions and properies of uiliy funcions. 8 his assumpion allows explici soluions for he probabiliy disribuions of he opimal imes o sell and of he opimal porfolio values. he inroducion of sochasic raes would lead o only simulaed soluions. 9 his is he coninuous-ime version of he soluion of Baroni e al. (7b). 1 We can also examine how he soluion depends on he index value P. For example, proporional ransacion coss imply a reducion of P. For insance, for he case, a ax of 5% leads o an opimal ime o sell equal o years, insead of years when here is no ransacion cos. Wih a 1% ax, he soluion becomes years. his is in line wih he empirical resuls showing ha high ransacion coss imply longer holding periods (see for example Colle e al., 3). 11 See Ellio and Kopp (1999, p. 193). 1 his equaion can be analyzed hrough Laplace ransforms of Lognormal disribuions. 6