Optimal Capital Structure in Real Estate Investment: A Real Options Approach
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1 Optimal Capital Structur in Ral Estat Invstmnt INTERNATIONAL REAL ESTATE REVIEW Vol. 4 No. : pp. - 6 Optimal Capital Structur in Ral Estat Invstmnt: A Ral Options Approach Jh-Bang Jou School of Economics and Financ, Mass Univrsit (Alban), Privat Bag 94, North Shor Cit 745, Auckland, Nw Zaland; Graduat Institut of National Dvlopmnt, National Taiwan Univrsit, Taiwan; Tl: , Ext. 949, Fax: ; J.B.Jou@mass.ac.nz Tan (Charln) L Dpartmnt of Accounting and Financ, Univrsit of Auckland, Privat Bag 99; Own G Glnn Building, Grafton Road, Auckland, Nw Zaland; Tl: , Ext. 879, Fax: ; tan.l@auckland.ac.nz This articl mplos a ral options approach to invstigat th dtrminants of an optimal capital structur in ral stat invstmnt. An invstor has th option to dla th purchas of an incom-producing proprt bcaus th invstor incurs sunk transaction costs and rcivs stochastic rntal incom. At th dat of purchas, th invstor also chooss a loan-to-valu ratio, which balancs th tax shild bnfit against th cost of dbt financing rsulting from a highr borrowing rat and a lowr rntal incom. An incras in th sunk cost or th risk of invstmnt will not affct th financing dcision, but will dla invstmnt. An incras in th incom tax rat or a dcras in th dprciation allowanc will ncourag borrowing and dla invstmnt, whil an incras in th pnalt from borrowing, a dcras in th invstor s rquird rat of rturn, or wors ral stat prformanc through borrowing, will discourag borrowing and dla invstmnt. Kwords Optimal Capital Structur; Ral Estat Invstmnt; Ral Options; Transaction Costs Corrsponding author
2 Jou and L. Introduction This articl invstigats th invstmnt and financing dcisions of a ral stat invstor who considrs th acquiring of an incom-producing proprt through dbt financing. Th xisting litratur that thorticall invstigats this issu includs Cannada and Yang (995, 996), Gau and Wang (99), and McDonald (999). All of ths articls assum that th invstor must purchas th proprt now or nvr. Our articl significantl diffrs from thm bcaus w allow a proprt invstor to hav th option to dla th purchas. This articl, which blongs to th burgoning litratur that applis th ral options approach to invstmnt (Dixit and Pindck, 994), assums that an invstor chooss an optimal dat to maximiz th nt xpctd prsnt valu of an incom-gnrating proprt. Th invstor rcivs th stochastic incom gnratd from th srvic of this proprt, but incurs sunk costs such as statutor costs and third-part chargs (Bruggman and Fishr, 6). Th intraction of ths sunk costs and th stochastic cash flow confrs on th invstor an option valu to dla th purchas of proprt. Consquntl, th invstor will not purchas th proprt until s/h is sufficintl satisfid with th currnt incom gnratd b th srvic of th proprt. At th optimal dat of purchasing, th invstor also chooss a loan-to-valu ratio that involvs th tradoff as follows: th invstor njos tax dductibl bnfits from intrst pamnts and capital dprciation, but will b chargd a highr mortgag rat whn th loan-to-valu ratio incrass, and ma rciv a lowr incom bcaus th potntial tnants ma b willing to pa lss as th raliz that thir landlord is highl indbtd, and thus, highl suscptibl to bankruptc. Asid from allowing th invstor to dla th purchas of proprt, our articl also dparts from th xisting litratur in th following rspcts. First, w assum that proprt valu is ndognousl dtrmind, whil Cannada and Yang (995; 996), and McDonald (999) assum that th purchas pric and th nt slling pric of a proprt ar both xognousl dtrmind. Our assumption is mor plausibl bcaus th volution of th stochastic incom gnratd b th srvic of a proprt dtrmins th dnamic volution of th proprt valu. Scond, w assum that dbt financing ma advrsl affct ral stat prformanc, such that invstmnt and financing dcisions intract with ach othr. As such, factors that charactriz th volution of th proprt Evr sinc th sminal papr b Modigliani and Millr (958), th dtrminants of corporat borrowing hav bn a hatd topic in th corporat financ litratur. S, for xampl, th surv papr b Harris and Raviv (99), and Mrs (3). This topic has rcivd littl attntion, howvr, in th ral stat invstmnt litratur. S th discussions in Gau and Wang (99) and Claurti and Sirmans (6, Chaptr 5). This tradoff significantl diffrs from that addrssd in th financ litratur, which also allows th tax advantags of borrowing, but considrs th costs associatd with ithr financial distrss, or th conflict of intrst btwn quit and dbt holdrs. S, for xampl, Harris and Raviv (99) and Mrs (3).
3 Optimal Capital Structur in Ral Estat Invstmnt 3 valu will also affct th optimal lvl of dbt. In contrast, Cannada and Yang (995; 996), and McDonald (999) abstract from this advrs ffct, and thus, th invstmnt and financing dcisions ar indpndnt. 3 Th rmaining sctions ar organizd as follows. W first prsnt th basic assumption of th modl, and thn driv th conditions for th invstmnt timing and th loan-to-valu ratio dcidd b an invstor who indfinitl holds th proprt. W furthr considr th polar cas whr dbt financing dos not affct ral stat prformanc, in which w driv som tstabl implications with rgards to th dtrminants of dbt financing. W thn mov to a mor gnral cas, in which dbt financing advrsl affcts ral stat prformanc, but find that most of our thortical prdictions bcom indfinit. Consquntl, w mplo plausibl paramtrs in ordr to carr out som numrical comparativ-statics tsting in th following sction. Th last sction concluds and offrs suggstions for futur rsarch.. Th Modl Th modl prsntd in this sction xtnds that of McDonald (999), which in turn, rsmbls that of Cannada and Yang (995, 996). W dpart from ths studis b allowing non-ngligibl transaction costs, uncrtaint in dmand, as wll as ndognousl dtrmind proprt valus. Considr an invstor who chooss an optimal dat to purchas a commrcial proprt, as wll as th prcntag of dbt to financ th purchas. For as of xposition, w considr th intrst onl mortgag loan. That is, w assum that th invstor pas onl intrst in th holding priod, and rpas th principal whn slling th proprt. Suppos that w start at tim t. Thn, th xpctd nt prsnt valu of this invstmnt is givn b: W ( P( t ), T, M ) Et [ T + t T ATCF ( s) ρ( s t ) ds + ATER ( T + t ) ρ( T + t t ) ρ( T t ) ( EI ( T ) + f ) ], () whr T is th dat on which th proprt is purchasd; ATCF(s) is th aftr-tax cash flow from th nt oprating incom at tim t; ATER ( T + t ) is th aftr-tax quit rvrsion from slling th proprt at tim ( T + t ), whr t is th holding priod of th ral stat invstmnt; ρ is th quit invstor s rquird rat of rturn; EI (T) is th initial quit invstmnt; and f is th transaction cost. 3 Our articl also diffrs from Gau and Wang (99) and McDonald (999), as ths two studis allow for th cost associatd with bankruptc (Stiglitz, 97) whn th invstor fails to pa off dbt obligations. Our articl, howvr, abstracts from this bankruptc cost.
4 4 Jou and L Each of th four trms in Equation () is dfind as follows. Th aftr-tax cash flow for th invstor is writtn as: ATCF ( s) ( τ) P( s) + τδh ( T ) / n ( τ ) MH ( T ) r( M ), () whr T < s < T + t. Th trm τ is th (constan incom tax rat, δ is th proportion of th proprt that is dprciabl capital (that is, not land), M is th loan-to-valu ratio, n is th lngth of th dprciation priod (39 ars for commrcial ral stat in th U.S.) 4 r (M) is th borrowing rat (whr r (M) > ), H (T) is th initial housing pric at tim T, and P(s) is th nt oprating incom gnratd from th proprt invstmnt at tim s, which follows th gomtric Brownian motion as givn b: dp ( s) µ ( M ) P ( s) ds + σp ( s) dz ( s), (3) whr µ (M) is th xpctd growth rat of P (s), xprssd as a non-positiv function of M, σ is th instantanous volatilit of th growth rat, and dz (s) is an incrmnt to a standard Winr procss. Th housing pric at tim s, H (s), is qual to th xpctd discountd prsnt valu of th nt oprating incom, and is thus givn b: P( s) H ( s). (4) ρ µ( M ) Not that both th intrst pamnts, MH (T) r (M), and straight-lin dprciation prmittd undr th tax cod, δh (T)/n, ar tax dductibl. Upon invstmnt, th proprt invstor trads th tax shild bnfits with two tps of costs associatd with dbt financing whn choosing a loan-to-valu ratio. Th first on, which is alrad addrssd in Cannada and Yang (995, 996), and McDonald (999), indicats that th borrowing rat incrass with th loan-to-valu ratio, givn that th invstor is mor likl to dfault whn borrowing mor. This positiv rlation is supportd b th mpirical stud of Maris and Elaan (99). Th scond on, which is novl to th litratur, indicats that th xpctd growth rat of th nt oprating incom is non-incrasing with th loan-to-valu ratio. This non-positiv rlation indicats that thos who intnd to rnt commrcial proprt ma b willing to pa lss whn th raliz that thir landlord bars mor dbt and is thus, mor suscptibl to bankruptc. This is plausibl bcaus thos who rnt in a commrcial proprt, such as a shopping mall, would tpicall rathr sta at th sam plac for a long priod of tim so that th can attract loal customrs. 5 4 Not that dprciation is onl allowd for th priod of n vn if th holding priod t is longr than n. 5 This assumption is also plausibl for a comptitiv commrcial proprt markt whr landlords who substantiall borrow ma nd to lowr th rnt to attract potntial tnants.
5 Optimal Capital Structur in Ral Estat Invstmnt 5 Th aftr-tax quit rvrsion for th invstor at tim T + t is givn b: 6 ATER ( T + t ) H( T + t ) MH( T) τ[ H( T + t ) H( T) + (δh ( T) t / n)], (5) whr H ( T + t ) is th slling pric on dat T + t at which th invstor rcivs th pamnt. On this dat, howvr, th invstor must also pa off th loan balanc, MH (T ), and pa taxs on th capital gain of H ( T + t ) H ( T ) + ( δh ( T ) t / n) In addition, th amount of quit invstmnt at tim T is simpl: EI( T) ( M ) H ( T), (6) Finall, th transaction cost f is also novl to th litratur. As Bruggman and Fishr (6, Chaptr 4) suggst, a mortgag loan borrowr, who is also th bur of a proprt in our framwork, incurs statutor costs and thirdpart chargs. Th formr includs crtain chargs for lgal rquirmnts that prtain to th titl transfr, rcording of th dd, and othr fs rquird b stat and local law. Th lattr includs chargs for srvics, such as lgal fs, appraisals, survs, past inspction, and titl insuranc. All of ths changs, howvr, ar unrcovrabl aftr th proprt is purchasd. 7 Givn that th invstor incurs sunk costs in purchasing a proprt and that th proprt offrs a stochastic cash flow in th futur, th invstor must thus wait for a sufficintl good stat of natur to purchas th proprt, as th ral options litratur suggsts (Dixit and Pindck, 994). Spcificall, th invstor simultanousl chooss a dat T and a loan-to-valu ratio M, so as to maximiz th xpctd nt prsnt valu of th invstmnt. This problm is dfind as: W ( P( t ), t ) Max E t W ( P ( t ), t, T, M ). (7) T, M As indicatd b Dixit and Pindck (994, p.39), whn th nt oprating incom is stochastic, w ar unabl to find a non-stochastic timing of invstmnt. Instad, th invstmnt rul taks th form whr th invstor will not purchas th proprt until th nt oprating incom P(t ) rachs a critical lvl, dnotd b P. At that instant, th invstor will choos a loan-to-valu ratio, dnotd b M. Consquntl, th initial purchas pric of th proprt, P / (ρ µ (M )) as givn b Equation (4), is ndognousl dtrmind. Our modl thus significantl dparts from that in th litratur as w ndogniz th valu of th proprt. V (P (, is dnotd as th gross valu of invstmnt, i.., 6 Not that Equation (5) applis to th cas in which t n. Whn n > t, w nd to impos n t. 7 Broadl spaking, th proprt bur also incurs th transaction sunk cost such as opportunit cost in th form of tim spnt on ngotiating with both th proprt sllr and th mortgag loan providr.
6 6 Jou and L V t + t ρ( s t ) ρ( T + t t ) ( P(, Et ATCF ( s) ds + ATER( T + t ), (8) t whr t T, and V ( P ( ) is dnotd as th invstor s option valu from waiting in th rgion whr P ( t ) < P. Th invstor s option valu is tim-indpndnt, i.., V ( ) / t, bcaus th invstor has som lwa in choosing th timing of invstmnt rathr than bing forcd to purchas th proprt during a finit priod of tim. B appling Ito s lmma, V (P() satisfis th ordinar diffrntial quation givn b: d V P t σ ( ( )) dv ( P( ) P( + µ ( M ) P( ρ V ( P( ), (9) dp( dp( B contrast, if P( t ) P and t t, thn th invstmnt is mad, and thus, V (P (, satisfis th partial diffrntial quation givn b: σ P( V ( P(, V ( P(, V ( P(, + µ( M ) P( + + P( P( t τδ P P ( τ) P( + ( τ) M r( M ) ρv n ( P(, () Th boundar condition is givn b: ( P( T + t ), T + t ) ATER( T t ) V +. () Equation () has an intuitiv intrprtation. If w trat V (P (, as an asst valu, thn th xpctd capital gain of th invstmnt (th sum of th first thr trms on th lft-hand sid) plus th dividnd (th sum of th last thr trms on th lft-hand sid) must b qual to th rturn rquird b th invstor (th trm on th right-hand sid). Equation () simpl sas that whn th invstor slls th proprt, th valu of th proprt must b qual to th aftr-tax quit rvrsion for th invstor. Appndix A shows that whn an invstor holds a proprt for an infinit tim horizon, thn th invstmnt and financing dcisions for th invstor rspctivl satisf th two quations givn b: and whr P D ( P, M ) ( ) A + f, () β H ( P, M µ' ( M ) A ( τ) ) + [ ( r( M ) + M r' ( M ρ ))], (3)
7 and A Optimal Capital Structur in Ral Estat Invstmnt 7 ρn τδ ( τ) τ + ( ) M r( ), (4) nρ ρ M M β ) ) + ) > µ( M σ µ( M ( σ ρ + σ. (5) Equation () is drivd basd on th condition that an invstor balancs th immdiat bnfit from purchasing a proprt against th bnfit from waiting for a mor favorabl stat of natur. Equation (3) is drivd basd on th condition that an invstor trads off th bnfit from th tax advantags of dbt financing against th advrs ffct of dbt financing that rsults from a highr borrowing rat and a possibl lowr xpctd growth rat of th nt opration incom. W can simultanousl us Equations () and (3) to driv th solution for th choic of th loan-to-valu ratio, M, and that for th critical lvl of th nt oprating incom that triggrs invstmnt, P. To compar our modl with thos in th xisting litratur, w first invstigat th polar cas whr dbt financing dos not affct ral stat prformanc at all, i.., µ (M). From Equation (3), this condition implis that: ρ ( τ)( r ( M ) + Mr '(. (6) Equation (6), which is xactl th sam as that in McDonald (999), suggsts that an invstor will choos a highr loan-to-valu ratio, if th invstor ithr rquirs a highr rat of rturn, facs a lowr incom tax rat, or is pnalizd lss whn borrowing mor. Lt us switch to th cas whr dbt financing advrsl affcts ral stat prformanc, i.., µ (M ) <. Givn this prmis and th rquirmnt that H ( P, M ) / M <, it follows that M < M a, whr M a is dfind as th M that satisfis Equation (6). In othr words, whn dbt financing advrsl affcts ral stat prformanc, thn th loan-to-valu ratio chosn b th invstor will b lowr than its countrpart whn dbt dos not affct ral stat prformanc at all. W assum that an invstor simultanousl maks th invstmnt and th financing dcision. In ordr to mak comparisons with th rsults of th litratur, w will first sparatl invstigat ach dcision, assuming that th othr dcision is xognousl givn. Diffrntiating H (P, M ) in Equation (3) with rspct to its various undrling paramtrs ilds th following rsults. Proposition Givn th timing in th purchas of a proprt, th invstor will tak on mor dbt (M incrass) if: (i) th invstor is allowd to dprciat capital lss rapidl (n incrass); (ii) th invstor is pnalizd lss through dbt financing (r (M) dcrass); (iii) th invstor xpcts borrowing to xhibit
8 8 Jou and L a lss advrs impact on ral stat prformanc (th absolut valu of µ (M) is smallr); and (iv) th invstor has lss dprciabl capital (δ dcrass). 8 Proof: S Appndix B. Th rsult of Proposition (ii) is th sam as that in th litratur such as McDonald (999), and th rason for Proposition (iii) is obvious. Th rsult for Propositions (i) and (iv) ma sm to countr intuition at first sight bcaus tax dductions from dprciation allowanc will b lowr as th invstor is ithr allowd to dprciat capital lss rapidl (n incrass) or has lss dprciabl capital (δ dcrass). Howvr, it is th intraction ffct btwn µ (M) and δ or n that mattrs for th financing dcision. As suggstd b Equation (3), an incras in n or a dcras in δ will mitigat th ngativ impact on ral stat prformanc which rsults from an incras in th loan-to-valu ratio, thus ncouraging th invstor to borrow mor. Diffrntiating D (P, M ) in Equation () with rspct to its various undrling paramtrs ilds th following rsults. Proposition Givn an invstor s loan-to-valu ratio, th invstor will dla th purchas of a proprt (P incrass) if: (i) th invstor incurs a largr transaction cost ( f incrass ); (ii) th invstor is allowd to dprciat capital lss rapidl (n incrass); (iii) th invstor is pnalizd mor through dbt financing (r (M ) incrass); (iv) th invstor xpcts to rciv lss rturn through dbt financing (th absolut valu of µ (M ) is largr); (v) th invstor has lss dprciabl capital (δ dcrass); and (vi) th invstor facs a highr risk in purchasing th proprt (σ incrass); and (vii) th invstor facs a highr incom tax rat (τ incrass). 9 Proof: S Appndix C. Propositions (i) and (vi) ar th standard rsults of th ral options litratur (s, for xampl, Dixit and Pindck, 994), which indicat that gratr uncrtaint and/or irrvrsibilit will dla invstmnt. Th othr scnarios statd in Proposition follow bcaus an invstor will rciv lss rturn from invsting immdiatl. Propositions and hlp us invstigat how th various forcs affct th invstmnt and financing dcisions for th cas whr ths two dcisions ar intracting with ach othr. W, howvr, can onl rach dfinit comparativ -statics rsults for th two xognous forcs, naml, th sunk costs and th risk of invstmnt, as statd blow in Proposition 3. 8 Furthrmor, w find that an invstor s incntiv to borrow is ambiguousl affctd if th invstor ithr facs a highr incom tax rat (τ is highr) or rquirs a highr rat of rturn (ρ is highr). S Equations (B5) and (B6), rspctivl. 9 Furthrmor, w find that an invstor s incntiv to purchas proprt is ambiguousl affctd if th invstor rquirs a highr rat of rturn, as suggstd b Equation (C7).
9 Optimal Capital Structur in Ral Estat Invstmnt 9 Proposition 3 An invstor who incurs a largr sunk cost of invstmnt or facs a highr risk of invstmnt will not altr th loan-to-valu ratio, but will dla invstmnt and rciv a highr nt invstmnt valu. Proof: S Appndix D. W us Figur to xplain th rsults of Proposition 3. Suppos that w start from an initial point E, which is th intrsction of lin I I and lin F F. In th figur, lin I I charactrizs th optimal condition for th choic of invstmnt timing as shown b Equation (). Not that w assum that M xhibits a ngativ ffct on P in this figur (our rsult will b qualitativl th sam vn if M xhibits a non-ngativ ffct on P ). Furthrmor, lin F F, which charactrizs th optimal condition for th financing dcision as shown b Equation (3), is vrtical bcaus P will not affct M at all. Proposition indicats that an invstor who incurs a highr transaction cost or facs a highr risk of invstmnt will dla th purchas of a proprt. This is shown in Figur, whr th optimal timing dcision charactrizd b lin I I will shift upward to lin I I, whil th optimal dbt financing dcision charactrizd b lin F F will rmain unchangd. Thus, th invstor will wait for a bttr stat to invst, but will not altr th loan-to-valu ratio. Th nt valu of invstmnt will incras, givn that th invstor purchass th proprt at a bttr stat of natur. Th rsults of Proposition 3 impl that nithr irrvrsibilit nor uncrtaint will affct an invstor s choic of th loan-to-valu ratio. This coms from our assumption that an invstor has th option to dla th purchas of a proprt, but not th option to dfault th loan. As a rsult, th invstor will choos th sam loan-to-valu ratio rgardlss of th stat of natur at which th invstor purchass th proprt. If w allow th invstor to hav th dfault option (s.g., Kau t al., 993), thn ths two xognous forcs will also affct th dbt financing dcision of th invstor bcaus diffrnt stats of natur will ntail diffrnt liklihoods of dfault. W will us plausibl paramtrs to mplo a numrical analsis to mak our thortical prdictions statd in Propositions -3 mor vivid. W considr both cass, that is, whr th holding priod is infinit and finit. Appndix E shows th procdurs to find th solutions for th lattr cas. S Equation (C8) which indicats that M xhibits an ambiguous ffct on P. If w allow th option to dfault, thn an invstor will both purchas a proprt at an arlir dat and borrow mor bcaus th invstor will rciv th (pu option valu to dfault, which also incrass th bnfit from borrowing.
10 Jou and L Figur Th Effct of an Incras in Eithr th Sunk Cost or th Risk of Invstmnt. This graph shows that ithr chang will mov th quilibrium point from E, th intrsction of I I (th lin that rprsnts th optimal condition of th invstmnt dcision) and F F (th lin that rprsnts th optimal condition of th financing dcision), to E. As a rsult, choics of th loan-to-valu ratio will rmain unchangd at M ; whil th critical lvl of th nt oprating incom that triggrs invstmnt will incras from P to P. P F P I E P I E I F I M M 3. Numrical Analsis W assum that r( M) r + λm, and µ ( M) µ + λm, such that r '( M) λ (> ) and µ ' ( M ) λ ( < ). Our chosn bnchmark cas is as follows: sunk cost f ; incom tax rat τ %; rquird rat of rturn on quit ρ% pr ar; th numbr of ars allowd for dprciation for tax purposs n 39 ars; proportion of dprciabl capital δ.5; minimum borrowing rat r 7% pr ar; as an invstor incrass th loan-to-valu ratio b %, thn th mortgag rat that th invstor facs will b incrasd b.5%, i.., λ.5; th nt oprating incom is xpctd to grow at most %, i.., µ % pr ar; an invstor xpcts th growth rat of th nt oprating incom to dclin b.% if th invstor incrass loan-to-valu incrass b %, i.., λ.;
11 Optimal Capital Structur in Ral Estat Invstmnt th instantanous volatilit of that growth rat is qual to 5% pr ar, i.., σ 5% pr ar; and th holding priod is infinit, i.., t. Tabl Dtrminants of th Invstmnt Timing and Loan-to-Valu Ratio. Bnchmark cas: f, τ %, ρ % pr ar, n 39 ars, δ.5, r 7% pr ar, λ.5, µ % pr ar, λ.,σ 5% pr ar, t, M 79.5%, P 4.534, and W Variation in f M P W Variation in τ % 5% % 5% 3% M P W Variation in ρ.5%.75% %.5%.5% M P W Variation in n M P W Variation in δ M P W Variation in λ M P W (Continud ) According to Gotzmann and Ibbotson (99), during th priod of 969 to 989, th annual standard dviation for REITs on commrcial proprt was qual to 5.4%. W us this as a prox for th volatilit of th growth rat of th nt oprating incom.
12 Jou and L (Tabl continud) Variation in λ M P W Variation in σ %.5% 5% 7.5% % M P W Variation in t M P W Not: M : th optimal loan-to-valu ratio; P : th critical lvl of th nt oprating incom that triggrs invstmnt; W : th nt valu of invstmnt; f : th sunk cost of invstmnt; τ: th incom tax rat; ρ: an invstor s rquird rat of rturn; n: th numbr of ars allowd for dprciation for tax purposs; δ: th proportion of dprciabl capital; r : th minimum borrowing rat; λ : th siz of th ffct of dbt financing on th borrowing rat; µ : th maximum xpctd growth rat of th nt oprating incom; λ : th siz of th ffct of dbt financing on that xpctd growth rat; σ: th instantanous volatilit of that xpctd growth rat; and t : th holding priod. Givn this st of bnchmark paramtr valus, w find that th invstor will not purchas a proprt until th nt oprating incom rachs (P 4.534). At that instant, th invstor will us 79.5% dbt to financ this purchas (M 79.5%), and will rciv a nt valu qual to.4483 (W W also find that th P and M dfind in Equation () ar ngativl corrlatd, as shown b lin I I in Figurs,, and 3. Tabl shows th rsults for f changs in th rgion (.5,.5), τ in th rgion (%, 3%), ρ in th rgion (.5%,.5%), n in th rgion (3, 47), δ in th rgion (.4,.6), λ in th rgion (.45,.55), λ in th rgion (,.), σ in th rgion (%, %), and t in th rgion of (, ), holding all th othr paramtrs at thir bnchmark valus. 3 Th ratio of th sunk cost, f, to th housing pric, P / (ρ-µ (M )), is qual to.38%, which is a littl lowr than th avrag lvl (s,.g., 5-6% stimatd b Stok, 9, p.8). Eithr a lowr tax rat or lowr dgr of uncrtaint will driv this ratio clos to th avrag lvl (S Tabl ).
13 Optimal Capital Structur in Ral Estat Invstmnt 3 Figur Th Effct of an Incras in Eithr th Tax Rat or th Lngth of Dprciation for Tax Purposs, or A Dcras in Dprciabl Capital. This graph shows that ach chang will mov th quilibrium point from E to E, such that choics of th loan-to-valu ratio will incras from M to M, and th critical lvl of th nt oprating incom that triggrs invstmnt will incras from P to P. P F F I P P I E E I F F M M I M Tabl indicats th following rsults. First, (a) an invstor will wait for a bttr stat to purchas a proprt and rciv a highr nt valu (both P and W incras), but will choos th sam lvl of dbt (M rmains unchangd) if th invstor incurs a highr transaction cost (f incrass) or facs a highr risk (σ incrass). Ths rsults conform to thos statd in Proposition 3. Scond, (b) an invstor will wait for a bttr stat to purchas a proprt and us mor dbt, but rciv a lowr nt valu (both P and M incras, but W dcrass), if th invstor facs a highr incom tax rat (τ incrass), is allowd to dprciat capital lss rapidl (n incrass), or has lss dprciabl capital (δ dcrass). Third, (c) an invstor will wait for a bttr stat to purchas th proprt and rciv a highr nt valu, but us lss dbt (both P and W incras, and M dcrass), if th invstor ithr rquirs a lowr rat of rturn (ρ dcrass) or is pnalizd mor through dbt financing (λ incrass). Fourth, (d) an invstor will wait for a bttr stat to purchas th proprt, but will us lss dbt and rciv a lowr nt invstmnt valu (P incrass, and both M and W dcras), if borrowing xhibits a mor advrs impact on ral stat prformanc (λ incrass). Finall, () an invstor will choos almost th sam dbt-to-loan valu ratio for all holding priods. Howvr, this is not th cas for th choic of invstmnt timing. Whn th holding priod is
14 4 Jou and L shortr than thirt ars, th invstor will wait for a bttr stat to purchas a proprt and rciv a highr nt valu (both P and W incras) if th invstor holds th proprt longr ( t incrass). Howvr, for holding priods longr than thirt ars, both P and W will thn dclin toward thir rspctiv stad-stat lvls. Figur 3 Th Effct of an Incras in Pnalt Through Borrowing, A Dcras in th Invstor s Rquird Rat of Rturn, or a Mor Advrs Effct of Dbt Financing on Ral Estat Prformanc. Th graph shows that ach chang will mov th quilibrium point from E to E, such that choics of th loan-to-valu ratio will dcras from M to M, and th critical lvl of th nt oprating incom that triggrs invstmnt will incras from P to P. P F F I P E I P E I F M F I M M Th rason for Rsult (b) is as follows. Considr an invstor who is allowd to dprciat capital lss rapidl (n incrass) or has lss dprciabl capital (δ dcrass). Each lads to a dirct ffct that forcs th invstor to purchas th proprt latr, givn th dbt lvl, as suggstd b Propositions (ii) and (v), rspctivl. This is shown in Figur whr lin I I shifts upward to I I. Each chang also lads th invstor to us mor dbt as shown b Propositions (i) and (iv), rspctivl, such that th invstor will b inducd to purchas at an arlir dat. This is shown in Figur whr lin F F shifts rightward to lin F F. Th quilibrium point thus shifts from E to E, which indicats that th invstor dlas th purchas and borrows mor. An incras in th
15 Optimal Capital Structur in Ral Estat Invstmnt 5 loan-to-valu ratio, in turn, dcrass th nt valu through lowring ral stat prformanc. Similar argumnts as th abov also appl to th cas whr an invstor facs a highr incom tax rat (τ incrass). Th rason for Rsults (c) and (d) is as follows. Suppos that an invstor is pnalizd mor through dbt financing (λ incrass). Proposition (iii) indicats that an invstor will dla purchasing, givn dbt lvls. This is shown b a shift from lin I I upward to lin I I in Figur 3. Proposition (iii), on th othr hand, indicats that th invstor will borrow lss, givn th invstmnt timing. This is shown b a shift of lin F F lftward to lin F F in Figur 3. Th quilibrium point shifts from E to E, thus suggsting that th invstor will dla th purchas and also borrow lss. Similar argumnts as abov can appl to th cas whr th invstor rquirs a lowr rat of rturn (ρ dcrass) or dbt xhibits a mor advrs ffct on ral stat prformanc (λ incrass). Th nt invstmnt valu will incras whn ithr λ incrass or ρ dcrass bcaus th invstor invsts at a bttr stat of natur. B contrast, th advrs ffct of an incras in λ will outwigh th positiv ffct that rsults from invsting at a bttr stat of natur such that th nt invstmnt valu will dcras as a rsult. Finall, th rason for Rsult () is as follows. Considr that an invstor incrass th holding priod in th rgion cappd b thirt ars. Th valu of th option to wait thus bcoms mor valuabl as th holding priod incrass. As a rsult, th nt invstmnt valu also incrass. Nonthlss, th abov pattrn will vntuall rvrs whn th holding priod is longr than thirt ars. Th rason is obvious. Givn that an invstor njos tax dduction bnfits from dprciation allowanc for at most thirt nin ars, th invstor is unabl to continuousl rciv a highr nt valu from th invstmnt ovr an infinit horizon. 4. Conclusion This articl mplos a ral options approach to invstigat th dtrminants of an optimal capital structur in ral stat invstmnt. W hav assumd that an invstor incurs transaction costs whn purchasing an incom-producing proprt that ilds a stochastic nt oprating incom. W find svral tstabl implications as follows. First, an invstor who incurs a largr sunk cost or facs a highr risk of invstmnt will not altr th loan-to-valu ratio, but will dla invstmnt. Scond, an invstor who ithr facs a highr incom tax rat or rcivs lowr dprciation allowanc for tax purposs will borrow mor and dla invstmnt. Finall, an invstor who ithr pas mor pnaltis from borrowing, rquirs lss rturn for quit invstmnt, or has wors ral stat prformanc through borrowing will borrow lss and dla invstmnt.
16 6 Jou and L This articl builds a simplifid modl, and thus, can b xtndd in th following was. First, this articl implicitl assums that an invstor has a monopolizd right to purchas a crtain incom-producing proprt (s, for xampl, Dixit and Pindck, 994). A mor sophisticatd modl ma allow diffrnt invstors to compt for a crtain proprt, or allow th sllr of th proprt to pla a mor activ rol. Scond, this articl abstracts from svral aspcts of ral stat financing, such as variabl mortgag rats and prpamnt pnaltis. It dsrvs furthr invstigation of whthr ths factors also mattr for th dtrminants of optimal capital structurs. Acknowldgmnts W would lik to thank th ditors (Ko Wang and Ros Lai), two anonmous rviwrs, Jungwon Suh, Siu Kai Cho, and participants at th 7 th SFM confrnc, and th 3 rd ACE confrnc for thir hlpful commnts on arlir vrsions of this manuscript. W also thank Hsin-Jung (Gar) Chung for rsarch assistanc. Appndix A If Th Cas for t t, thn V ( P(, / t, and ρ( T + t t ) E ATER ( T + t ) t. For this cas, suppos that V ( P( )) and F ( P( )) dnot th option valu of waiting t t in th rgion whr P ( t) < P and th proprt valu in th rgion whr P( t ) P, rspctivl. Substituting V (P() P ( β into Equation (8) ilds th quadratic quation for solving β: σ φ (β) β(β ) µ ( M )β + ρ. (A) Consquntl, th solution for V (P () in Equation (9) is givn b: β β ( P( ) A P( A P( V +, (A) whr β and β ar, rspctivl, th largr and smallr roots of β in Equation (A), and A and A ar constants to b dtrmind. Similarl, if P (t ) P such that invstmnt is mad, thn w can rwrit Equation () as:
17 Optimal Capital Structur in Ral Estat Invstmnt 7 σ P( F ( P( ) F ( P( ) + µ( M ) P( + ( τ) P( + P( P( τδ P ( τ) M n P r( M ) ρf Th solution for F (P) in Equation (A3) is givn b: F ( P( ) B P( + ( ρn β + B P( β τδ P ( τ) MP r( M ) ), nρ ρ ( P( ). P( + ( τ) (A3) (A4) whr B and B ar constants to b dtrmind. Th trms A, A, B, B, and th critical lvl of th nt oprating incom that triggrs invstmnt, P, ar simultanousl solvd from th boundar conditions as follows: lim V ( P( ), (A5) P( β β lim B P( + B P( P( β β lim B P( + B P( P(, (A6), (A7) and P V ( P ) F ( P ) ( M) f, (A8) M)) V ( P ) P( t t F ( P ) P( t t ( M ). (A9) Equation (A5) is th limit condition, which stats that th invstor s option valu from dlaing th purchas is worthlss as th nt oprating incom approachs its minimum prmissibl valu of zro. This condition rquirs that A. Equations (A6) and (A7) ar two othr limit conditions, which rspctivl stat that aftr an invstor purchass a proprt, th invstor s option valu from abandoning th proprt is worthlss, whn th nt oprating incom is xtrml bad and xtrml good. Ths two conditions rquir that B B. Equation (A8) is th valu-matching condition, which stats that at th optimal timing of purchasing (t T in our cas), th invstor is indiffrnt btwn xrcising and not xrcising th invstmnt. Equation (A9) is th smooth-pasting condition, which guarants that th invstor will not driv an arbitrag profits b dviating th optimal xrcis stratg.
18 8 Jou and L ρn Dfin A M τ + ( )τδ/ np ( τ) M r( M ) / ρ. Multipling Equation (A9) b P /β, and thn subtracting Equation (A8) from it ilds: P D ( P, M ) ( ) A + f, (A) β β A A P, (A) β and A. Furthrmor, th choic of M is found b stting th drivativ of V (P ) in Equation (A), or quivalntl, P F ( P ) ( M) f, with M)) rspct to M quals to zro. This ilds: H ( P, M µ' ( M ) A ( τ) ) + [ ( r( M ) + M r( M ρ Th scond-ordr conditions rquir that: ))]. (A) and D ( P, M ) / P <, (A3) H ( P, M ) / M <, (A4) D( P D( P, M, M ) / P ) / M H ( P H ( P, M, M ) / M ) / P (A5) >. Appndix B Proof of Proposition Totall diffrntiating H( P, M ) in Equation (3) with rspct to n, r (M ), µ (M ), δ, τ, and ρ ilds: M > n M < r ( M ), (B), (B) M 3 >, (B3) µ ( M ) M 4 <, (B4) δ M 5 >, (B5) τ <
19 Optimal Capital Structur in Ral Estat Invstmnt 9 M ρ 6 > <, (B6) whr H ( P, M ) / M <, H ( P, M ) µ' ( M )τδ ρn ρn ( ρn ), (B7) n n ρ > < H ( P, M ) ( τ) M, (B8) r' ( M ) ρ H ( P, M ) A, (B9) µ' ( M ) 3 > 4 < H ( P, M ) µ' ( M )τ ρn ( ), (B) δ nρ and 6 5 H ( P, M ) µ' ( M ) M r( M ) ρn δ [ + + ( ) ] τ ρ nρ + ( r( M ρ τδ + ( + nρ) nρ ) + M r' ( M )) H ( P, M ) -µ' ( M ) ρ ρn > <, ( τ) ] + ( r( M ρ µ' ( M ) ( τ) A + [ M ρ ) + M r' ( M )) > <. r( M ) (B) (B) Q. E. D. Appndix C Proof of Proposition Equation () implis that: P f ρ ( ), (C) A β whr w hav usd th rlationship β β ( ρ µ ( ( β )( β ) ρ. Diffrntiating P with rspct to f, n, δ, σ, τ, ρ and M ilds:
20 Jou and L P f P f >, (C) and P fρ τδ ρn ρn ( ) ( ρn )] >, (C3) n A β n ρ P fρ ρn τ ( )( ) ] <, (C4) δ A β nρ P fρ β >, (C5) δ Aβ σ P fρ M r( M ) ρn δ ( )[ + + ( ) ] >, (C6) τ A β ρ nρ P ρ τδ + nρ f β f ( τ) M r( M ) ( )[ ρa β ρ ρa β ρ (C7) ( + nρ) ρn )] + f A ( P fρ β fρ > ( )[ ( τ)( r( M ) + M r'( ]. (C8) M Aβ M A β < Proposition (iii) follows bcaus as r (M ) incrass, thn r (M ) in Equation (C) also incrass. Proposition (iv) follows bcaus as th absolut valu of µ (M ) incrass, thn µ (M ) in Equation (C) will dcras. Q. E. D. β ) > <, Appndix D Proof of Proposition 3 Equation (3) indicats that H( P, M ) is indpndnt of f and σ, thus suggsting that th optimal lvl of M is nithr rlatd to f nor σ. Equations (C) and (C5) thn suggst that P / f > and P / σ >. Substituting A in Equation (A) and P in Equation (C) into th lft-hand sid of Equation (A8) ilds th nt valu of invstmnt qual to f / ( β ). Diffrncing this valu with rspct to f and σ ilds and ( f /(β ) >, (D) f (β )
21 Optimal Capital Structur in Ral Estat Invstmnt ( f /(β ) f β >. (D) σ (β ) σ Q. E. D. Appndix E Th Cas for Finit t W follow Brnnan and Schwartz (978), and Hull and Whit (99) to find P and M. Lt ( ln P ( such that ln P, U( ( ) V ( P( ) for ( < and Z( (, V ( P(, for ( t) and t t. As a rsult, Equation (9) can b rwrittn as: σ U ( ( ) σ U ( ( ) + (µ( M ) ) ρu ( ( ), if ( ( ( <. (E) Furthrmor, Equation () can b rwrittn as: Z( (, σ Z ( (, Z( ( ) σ + (µ( M ) ) + + ( τ) ( ( t if ( t ), and t t. τδ + ( τ) M r( M ) ρz( (,, n ( (E) Equation () can also b rwrittn as: ( T+ t ) ( τ) Z( ( T +, T + M)) τδt ( M τ + ). (E3) n M)) Finall, Equation (6) can b rwrittn as: EI ( T ) ( M ) ( ρ µ ( M. (E4) )) Th choic of M is drivd b stting th drivativ of W ( ) in Equation () with rspct to M quals to zro. LtG( P(, V ( P(, / M. As a rsult, W ( ) ( M ) µ ( M ) G( P( T ), T ) + E [ ]. (E5) M ( ρ µ ( ( ρ µ ( ρ( T t ) t
22 Jou and L Lt g( (, Z ( (, / M. Diffrntiating Equation (E) trm b trm with rspct to M ilds: σ g( (, ( τ) t M τδ [ ( Mr( M )] n g( (, σ g( (, Z ( (, + (µ( M ) ) + µ' ( M ) + ( ( ( µ' ( M ) [ r( M ) + Mr' ( M )] + )) ρg( (,. (E6) Evaluating Equation (E5) at T t and M M ilds: ( M ) µ ( M ) t g(, ) + [ ]. (E7) ( ρ µ ( ( ρ µ ( Th boundar condition for g( (, t ) is drivd b diffrntiating Equation (E3) with rspct to M, which ilds: g ( ( T + t ), T ( T + t ) ( τ) µ' ( M ) + t ) (E8) ( M τ + τδt µ' ( M ) ) n. W implmnt th xplicit finit diffrnc mthod (Hull and Whit, 99) to solv for M and P. W bgin b choosing a small tim intrval, t, and a small chang in ( t ),. A grid is thn constructd to considr th valus of Z ( whn ( is qual to and tim is qual to, +,..., max, t, t + t,..., t + t, Th paramtrs and max ar th smallst and th largst valus of (, and t is th currnt tim. Lt us dnot + i b i, t + j t b t j, and th valu of th drivativ scurit at th (i, j) point on th grid b Z i,j. Th partial drivativs of Z ( with rspct to ( at nod (i, j) ar approximatl as follows: Z ( Zi+, j Zi, j, (E9) ( and
23 Optimal Capital Structur in Ral Estat Invstmnt 3 Z( Z + Z Z i+, j i, j i, j ( ( ), (E) Th tim drivativ for Z ( is approximatl: Z ( Zi j Zi j t t,,. (E) Substituting Equations (E9) to (E) into Equation (E) ilds: Z whr i, j az i, j + bz i, j + cz i+, j t + ( τ) ( + ρ t ) t τδ + ( ( τ) Mr( M)), ( + ρ t ) n M)) + i (E) σ t σ t a [ (µ ) ], (E3) ( + ρ ( ) and σ t b [ ], (E4) ( + ρ ( ) σ t σ t c [ + (µ ) ]. (E5) ( + ρ ( ) Similarl, Equation (E6) can b writtn as: g i, j agi, j + bgi, j + cgi+, j + ( Zi+, j Zi, j t ( τ) [ r( M ) + Mr' ( M )] ( + ρ t τδ µ' ( M ) + [ ( τ) Mr( M )] ( + ρ n µ' ( M ) t ) ( +. (E6) W also nd to impos th optimal condition for th timing of invstmnt. Th solution to U ( ( t )) in Equation (E) is givn b: β( β( A A, (E7) U ( ( ) +
24 4 Jou and L whr A and A ar constants to b dtrmind, and β and β ar dfind in Appndix A. Th optimal timing is dtrmind b th following boundar conditions: lim U ( ( ), (E8) ( and ( ) (, ) ( ) U Z T M f, (E9) ( ρ µ ( U ( ) Z(, ( M ). (E) ( ( ( ρ µ ( ( ( Solving Equations (E8)-(E) simultanousl ilds: and A. (E) β A [ Z(,) ( M ) f ] ( ρ µ ( β Aβ Z (, ( ( t ) ( M ), (E). (E3) Th law of motion for Z((, shown in Equation (E) and that for g((, shown in Equation (E6) ar subjct to two optimal conditions shown in Equations (E7) and (E3), rspctivl, and two boundar conditions shown in Equations (E3) and (E8), rspctivl. Solving ths four conditions simultanousl ilds th solutions for A, M, g i,, and Z i,, whr i Z is th gross valu of invstmnt. W can furthr us th rlation P to find th critical lvl of th nt oprating incom that triggrs invstmnt, as wll as th nt valu of invstmnt, P Z i ( M ) f., i,
25 Optimal Capital Structur in Ral Estat Invstmnt 5 Rfrncs Brnnan, M.J., and E.S. Schwartz. (978). Finit Diffrnc Mthod and Jump Procsss Arising in th Pricing of Contingnt Claims, Journal of Financial and Quantitativ Analsis, 3, Bruggman, W.B., and J.D. Fishr. (6). Ral Estat Financ and Invstmnts (3th dition), McGraw-Hill Cannada, R.E., and T. Yang. (995). Optimal Intrst Rat-Discount Points Combination: Stratg for Mortgag Contract Trms, Ral Estat Economics, 3, Cannada, R.E., and T. Yang. (996). Optimal Lvrag Stratg: Capital Structur in Ral Estat Invstmnts, Journal of Ral Estat Financ and Economics, 3, Claurti, T., and G.S. Sirmans. (6). Ral Estat Financ (5th dition). Uppr Saddl Rivr, NJ: Prntic-Hall. Dixit, A.K., and R.S. Pindck. (994). Invstmnt undr Uncrtaint, Princton, NJ: Princton Univrsit Prss. Gau, G., and K. Wang. (99). Capital Structur Dcisions in Ral Estat Invstmnt, AREUEA Journal, 8, 5-5. Gotzmann, W.N., and R.G. Ibbotson. (99). Th Prformanc of Ral Estat as an Asst Class, Journal of Applid Corporat Financ, 3,, Harris, M., and A. Raviv. (99). Th Thor of Capital Structur, Journal of Financ, 46, Hull, J., and A. Whit. (99). Valuing Drivativ Scuritis Using th Explicit Finit Diffrnc Mthod, Journal of Financial and Quantitativ Analsis, 5,, 99, 87-. Kau, J.B., D.C. Knan, W.J. Mullr III, and J.F. Epprson. (993). Option Thor and Floating - Rat Scuritis with a Comparison of Adjustabl - and Fixd - Rat Mortgags, Journal of Businss, 66, Maris, B.A., and F.A. Elaan. (99). Capital Structur and th Cost of Capital for Untaxd Firms: Th Cas of REITs, AREUEA Journal, 8, -39. McDonald, J.E. (999). Optimal Lvrag in Ral Estat Invstmnt, Journal of Ral Estat Financ and Economics, 8,, 39-5.
26 6 Jou and L Modigliani, F., and M. Millr. (958). Th Cost of Capital, Corporation Financ, and Th Thor of Invstmnt, Amrican Economic Rviw, 48, Mrs, S. (3). Financing of Corporations in: Handbook of th Economics of Financ, Volum A: Corporat Financ, Constantinids, G.M., Harris, M., and R.M. Stulz (d.), Elsvir, North Holland, Stiglitz, J. (97). Som Aspcts of th Pur Thor of Corporat Financ: Bankruptcis and Tak-Ovrs, Bll Journal of Economics, 3, Stok, N.L. (9). Th Economics of Inaction, Princton Univrsit Prss.
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