Timing Ventures: The Underinvestment Problem*

Transcription

1 Timing Venures: The Underinvesmen Problem by David C. Nachman J. Mack Robinson College of Business Georgia Sae Universiy lana, G dnachman@gsu.edu Tel: Fax: Sepember, 004

2 Timing Venures: The Underinvesmen Problem BSTRCT The incenives o exercise growh opions, referred o here as iming venures, are disored by he presence of a prior claim o venure value. This disorion is he underinvesmen problem of Myers 977). The soluion o he opimal iming problem does wo hings: i) i indicaes he ime o underake he venure when i is opimal o do so, or ii) i indicaes when he venure finishes ou of he money in he sense of having zero ime value. This soluion depends on whose value is being maximized. If here is a prior claim o venure value, he soluion ha maximizes he value of he residual claim exercises laer han he soluion ha maximizes he value of he venure. However, his levered venure finishes ou of he money before he value maximizing ime.

3 Myers 977), in a seminal conribuion o corporae finance, showed ha in he presence of a prior claim o value, owners of a value increasing invesmen opporuniy may choose o forego he opporuniy if hey have o conribue he capial o underake i. Myers coined he erms growh opion and real opion for such invesmen opporuniies and he disorion in invesmen incenives has come o be known as he underinvesmen problem. More specifically, Myers 977, p. 49) showed ha a he value maximizing ime, he incenive of he residual claiman was o... in some saes of naure, pass up valuable invesmen opporuniies. He did no characerize he opimal sraegy of he residual claiman, however, leaving open he quesion of when he residual claiman exercises he opion or when he opion of he residual claiman finishes ou of he money. The objecive of his paper is o answer his quesion wih an emphasis on he ime dimension of "pass up valuable invesmen opporuniies." I is no a far srech o hink ha he underinvesmen problem of Myers is an insance of he posponemen of a growh opion, referred o here as a iming venure, beyond is opimal exercise in he sense of value maximizaion. Properly framed and inerpreed, his phenomenon is demonsraed here in a fairly general discree ime seing. From he perspecive of value maximizaion, he iming venure is an opimal sopping problem ha has a well characerized soluion. The soluion is a sopping ime ha does wo hings: i) i indicaes he ime o exercise he opion, o underake he venure, when i is opimal o do so, or ii) i indicaes when he opion finishes ou of he money in he sense of having zero ime value.

4 This soluion depends on whose value is being maximized. If he venure is all equiy financed, he soluion is he one ha maximizes he ne presen) value of he venure. If here is a prior claim o venure value, he soluion ha maximizes he value of he residual claim, he owners claim, underakes he venure laer han he soluion ha maximizes he value of he venure. However, his levered venure finishes ou of he money before he value maximizing ime. Boh of hese feaures conribue o loss of value in he levered venure. So "passing up a valuable invesmen opporuniy" has a leas wo meanings. One meaning is foregoing he venure alogeher. This occurs when he ime value of he venure is zero. For a levered venure, his ime may come well before he value maximizing ime. The inuiion for his resul is sraigh forward. The presence of a prior claim raises he exercise price of he residual claiman s call opion. s a resul he opion may finish ou of he money before he value maximizing ime. In hose cases, however, i is possible o simply view he owner as waiing forever o exercise he opion. On he oher hand, if i is opimal for he residual claim in he levered venure o underake he venure exercise he opion), hen he ime a which his akes place is laer han he value maximizing ime, i. e., he unlevered venure would have been underaken earlier. Thus he oher meaning of "passing up a valuable invesmen opporuniy" is passing up in ime, delaying exercise. So owners of a levered venure pospone exercising heir growh opion beyond he value maximizing exercise ime. s a resul, hey inves subopimally relaive o owners in an unlevered venure. The problem ges worse he greaer he leverage. The greaer he leverage, he longer is he opimal exercise ime.

5 The resuls here clarify, a leas in he discree ime seing, he characer of he underinvesmen problem. The problem has been explored exensively in he conex of coninuous ime where value or some major componen of value follows a geomeric Brownian moion. Mauer and O 000, p. 53) review he lieraure ha examines he ineracion of financing and invesmen decisions in his coninuous ime conex. In a geomeric Brownian moion model of oupu price Sabarwal 003) shows, consisen wih he resuls here, ha he opimal oupu price ha riggers invesmen in a projec is higher for he levered firm han for an unlevered firm. In a similar seing, Mauer and O 000) solve for he opimal capial srucure and invesmen policy of a firm ha has a growh opion o expand he scale of operaions. In numerical simulaions hey show, consisen wih he resuls here, ha he opimal oupu price ha riggers exercise of he opion o expand is higher for levered equiy han for unlevered equiy, which is higher han he rigger price for he opimal capial srucure. In hese and oher coninuous ime models, he posiive par of ne presen value is a non-negaive diffusion ha is no absorbed a zero, and hence he opion never finishes ou of he money. I follows ha when he levered venure is underaken, i is underaken laer han he unlevered venure. In his paper he capial srucure of he firm, he presence of he prior claim o venure value, is given. The model is presened in Secion. Opimal iming sraegies are characerized in Secion. The opimal sraegies of levered owners and unlevered owners are compared in Secion 3. The effec of an increase in leverage on hese opimal exercise sraegies is derived in Secion 4. Secion 5 concludes he paper. Proofs and echnical resuls are gahered in he ppendix. 3

6 The Model firm has he opporuniy o ime invesmen in a venure, a iming venure. Time is in discree daes = 0,, K, T, wih horizont, he expiraion of he venure. 3 Now is = 0. The venure can be underaken he growh opion can be exercised) a any dae = 0,, K, T. If he venure is underaken a dae, iniial capial K exercise price) is required o obain he venure worh Y. The value Y is he non-negaive change in he marke value of he firm ha owns he venure and Y K is he ne presen value. The value Y includes he change in marke value of asses in place, if here are any, as well as he marke value of any new asses ha are acquired by underaking he venure a dae. value The prior claim o venure value is D if he venure is underaken a dae. The D includes any explici paymens due he prior claiman and he change in marke value of he prior claim. The change in marke value of he residual claim a dae is herefore Y K D if he venure is underaken a dae. The naure of he prior claim is no imporan. Typically his would be a credior s claim. 4 I may however be a ax claim by some governmen. There is an ineresing case developed in he ppendix Theorem.3) where his could be a fracional share claim. Wha ever he naure of he prior claim, i consiues leverage in he venure, hence he erminology regarding he venure wih no prior claim as unlevered and regarding he venure wih a prior claim as levered. The quaniies Y, K, D are random variables whose values are known by dae. The same economic circumsances as Myers 977, Secion 4.) are assumed here for he exisence of hese values and for he imperfecions in he marke for real opions. 5 In 4

7 paricular, he exisence of valuable real opions presumes some marke imperfecions in real goods markes ha allow he possibiliy of earning economic rens, i. e., he possibiliy of Y K > 0. In addiion here mus be imperfecions in he marke for real opions so ha he firm ha owns he iming venure could no sell i for is value o his firm and hereby circumven he levered owners incenive o underake he venure subopimally. This could happen if he underlying, he Y, is firm specific or if he abiliy o acquire he underlying, he K, are firm specific, or boh. Le = Y K ) + and = ) Y K D +, denoing he ineress of limied liabiliy residual claims in he unlevered and levered venure, respecively. These and all oher random variables are defined on a probabiliy space Ω, F,P), and are adaped o he sequence { } F of σ-algebras, where F 0 F L F T F T = F. F are he evens knowable by dae, he firm s informaion a dae. Financial markes are such ha here is shor-erm riskless borrowing/lending. 6 The shor-rae for dae is r, a non-negaive F -measurable random variable, such ha one dollar invesed a reurns + risklessly a dae +. 7 For each pair of daes s, r wih s<, le R = + r ) + r ) L + r ) denoe he gross reurn on shor-erm s, s s+ riskless borrowing/lending a dae s for he period o dae. In general, R s, resuls from borrowing risklessly one dollar a dae s and rolling i over in shor-erm riskless borrowing repeaedly unil dae. Since r is known a dae is F -measurable) i follows ha R s, is known a dae is F -measurable) bu no before. For any, define R,. 5

8 For ease of inerpreaions i will be convenien o add an arificial dae T + o he problem wih Y D r 0. The reasons why will become clear in Secion 3. T+ T+ T The manager of he firm mus decide when o underake he venure, i. e., she mus choose a ime : Ω { 0,, K, TT, + } such ha { s} { ) s} = = F, : s s= 0,, K, T. Such a sraegy is a sopping ime. For such a sopping ime, he value of he unlevered) venure ne of invesed capial is where ) ) ) = s. The even { = } { > 0} underaking he venure. The even { = } { = 0} = s when I can be inerpreed as he even of exercise or I can be inerpreed as he even of no underaking he venure. These inerpreaions are made precise wih he characerizaion of opimal sraegies in he nex secion. I is assumed ha underaking or no underaking he venure will no change prices in financial markes and he values Y, K, D, r are predicaed on his assumpion. In his sense, we are alking abou small iming venures. 8 In he absence of arbirage here exiss a risk-neural measure P, equivalen o P, such ha for any securiy paying a dividend sream { δ }, he price of his securiy a dae, ex he dae dividend, is S δ S, ) T j T = E + j=+ R, j R, T where E denoes expecaion under expecaion under P, and likewise E x) E x ) = F, he condiional P given F. 9 lernaively, i could be assumed ha all he invesors, including shareholders, senior claimans, and managers in his world are risk neural. In 6

9 his case P = P and ) sill holds. The absolue values of all random variables in his paper are assumed o have finie expecaion wih respec o Opimal Sraegies For each dae le P. C denoe he se of sopping imes such ha, a. s., he se of sraegies ha delay underaking he venure unil a leas dae. 0 Denoe C 0 by jus C. value maximizing manager will choose C so as o maximize he change in marke value of he venure V E, where as he residual claimans in a levered R 0, venure would prefer he manager choose C o maximize he change in marke value of he residual claim V E. For ease of reference, we refer o V R as he marke 0, value of he unlevered venure and o V following sraegy. as he marke value of he levered venure, Each of hese problems is an opimal sopping problem. Soluions o hese problems are no unique in general, bu here are canonical soluions ha are easy o describe by dynamic programming argumens. The canonical soluion o he value maximizing problem is described here. Similar descripions obain for he sopping problem for levered equiy considered here and for oher problems considered in he res of he paper. These canonical soluions for levered and unlevered equiy are compared in Secion 3. Dynamic programming wih a finie horizon works backward. Define successively γ T+, γt, γt, K, γ0, by seing =, ) γ T + T + 7

10 For each, le be he firs dae s γ = max, E, = T, K,,0. 3) R + γ, + such ha s = γ s. By ) such a dae s exiss. Noe ha his does no depend on here being an arificial dae T +, because by ), 3) for dae T has γ T = T, which would be ) wih no arificial dae T +. The recursion relaion 3) has he usual dynamic programming inerpreaion. each dae = 0,, K, T, if he venure has no been underaken, he manager has wo choices. Underake he venure a ha dae and ge or wai one more dae. The value a dae of waiing one dae is he erm γ E +. This is he bes he manager can do R, + from following any coninuaion sraegy and hence he bes he manager can obain now is he maximum of hese wo alernaives. Tha is he essence of he following heorem. Theorem. For every dae, C, and σ E = γ E, σ C. 4). R, R, σ Since R 0, is known a dae and R0, σ = 0, ha for each, R, R σ for σ C, i follows from 4) σ E E, σ C, R 0, R 0, σ which implies he same hing for he uncondiional expecaions σ E E, σ C R 0, R 0, σ. 5) 8

11 The inequaliy in 5) says ha he marke value, as of oday, of he venure following he sraegy is greaer han he marke value of he venure following any oher sraegy ha pospones he underaking o a leas dae, i.e., maximizes he marke value of he venure among hose iming sraegies in C. I follows ha 0 is he sraegy ha maximizes he marke value of he venure. We call his sraegy. The sequence γ defined in ) and 3) is referred o commonly as he Snell envelope, for Snell s 95) original work on opimal sopping. We inerpre γ as he ime value of he venure a dae. The ime is he firs dae such ha he inrinsic value of he venure equals is ime value γ. 3 This is he canonical soluion o he problem of maximizing he marke value of he venure. The Snell envelope is useful in characerizing he opimal iming sraegy. I is useful as well in deermining when he venure finishes ou of he money, which we do below. The following resul records he opimaliy of and a necessary condiion of his opimaliy. Recall he noaion a he beginning of his secion ha he marke value of he sraegy C is V = E. R 0, Theorem. V = max C V and E R, on { } =, 6) for every C. If ˆ C and Vˆ = max C V, hen ˆ. Condiion 6) is he inuiive resul ha says when he marke value of he venure is a he opimum, he marke value of any sraegy ha coninues pas his ime is less. 9

12 The inequaliy is weak, however, and hence may no be he only opimal sraegy. The mulipliciy of opimal sraegies is illusraed in he Corollary 5 in he nex secion. The second saemen of Theorem assures us ha is he minimal opimal sraegy. I is emping o refer o he ime as he earlies ime o opimally exercise he venure. Bu i may be he earlies ime o decide o never underake he venure. This is he ime he venure finishes ou of he money. Define = { = } I { = 0}. I would be naural o inerpre his even as he even he venure finishes ou of he money a dae. The following resul provides moivaion for his inerpreaion. Theorem 3. On he even, = γ = 0, k = 0,, K, T +. + k + k Based on Theorem 3, on he value of he venure is zero in he mos meaningful sense of he erm value of he venure. I may be ha he inrinsic value of he venure a dae,, is zero in he sense ha he opion a dae is simply ou of he money. Bu of course he opion may sill have ime value, i. e., i may be ha γ > 0. The even is he even where a dae boh he inrinsic value of he venure and he ime value γ of he venure are zero, and his is he firs ime his happens. 4 Le T = 0 = U. Then is he even in which he venure finishes ou of he money, he even in which he venure is foregone. I follows ha c, he complemen of his even, is he even in which i is opimal o underake he venure. Indeed, c = T = 0 { = } I{ > 0} U. Theorems and and 3 apply as well o he problem wih replaced by he value of he residual claim in he levered venure. This resuls in variables,, γ, 0

13 where = is he canonical soluion o he problem of maximizing he marke value of 0 = = I = 0, he residual owners ) claim in he levered venure. The evens { } { } = T U = 0 and c have he same inerpreaions as above bu for he levered venure. The even in which he levered venure finishes ou of he money is and he even in which he levered venure is underaken is canonical soluions o he levered and he unlevered venures. 3 Comparing Opimal Sraegies 3. Framing he underinvesmen problem. c. In he nex secion we will compare he Inuiion, based on Myers 977) insighs and resuls such as hose in Sabarwal 003) and Mauer and O 000), suggess ha. Bu his is no he case in general, because hese iming sraegies have wo componens, he opimal ime o exercise and he ime he venure finishes ou of he money. s saed earlier, his is jus a maer of inerpreaion. The saemen is rue of appropriae modificaions of hese iming sraegies. This is done below in Corollary 5. Firs, he following assumpion on he behavior of he value of he prior claim o venure value is needed o rule ou possibiliies ha here is no underinvesmen. The prior claim o venure value is significan if and here is no significan value leakage if Recalling ha D 0, = 0,, K, T, 7a) ) DR E D K T. 7b), 0,,,, + + = D includes he change in marke value of he prior claim, 7a) rules ou pure risk shifs where he only effec of underaking he venure is a decrease

14 in he value of a credior s claim because of an increase in risk of he asses backing ha claim. I is clear ha in such insances here may be no underinvesmen, and may even be overinvesmen. See Myers 977, Secion 4..3) for a discussion of his case. There also may be no underinvesmen if here is value leakage sufficien o induce early exercise of he levered venure. Increase in he exercise price of an merican call opion is a form of value leakage. If he exercise price increases enough over he life of he opion, early exercise may be opimal. Under 7a), relaive o an unlevered venure he value of D is an increase in he exercise price. I urns ou ha if he rae of increase in he exercise price is no larger han he riskless rae, hen early exercise will no be opimal. 5 The inequaliy in 7b) is his needed resricion on he rae of increase of he exercise price. 6 In hese respecs, 7a) and 7b) consiue a weak regulariy requiremen, bu one sufficien o frame he underinvesmen problem. I should be noed ha hese condiions are saisfied in he simple bu imporan case where he D arise from a fixed posiive promised paymen o a credior and he discouned value process Y is a maringale under P. 7 numerical example wih his feaure is given below in Secion Comparison The following heorem gives he comparisons of he wo componens of he opimal sraegies of equiy in an unlevered venure and an equivalen bu levered venure. Here and elsewhere I is he indicaor of he even. Theorem 4. ssume 7a) and 7b). Then, I I, and I c I c. s a consequence, E I c V V. R 0,

15 The firs par of Theorem 4 saes ha if he unlevered venure finishes ou of he money, hen so does he levered venure. The second saemen says ha in he even ha he unlevered venure finishes ou of he money, he levered venure finished ou of he money earlier no laer). By aking complemens in he firs par of he heorem we ge ha c c ; if levered equiy underakes he venure, hen so does unlevered equiy. The hird saemen of he heorem says ha in he even ha levered equiy underakes he venure, unlevered equiy underakes i sooner. The las saemen of Theorem 4 says ha he value of he unlevered) venure following he opimal sraegy of levered equiy is less han he value of he venure following he sopping ime which of course is less han he value of he venure following he value maximizing sraegy. In his compuaion, we had o muliply he payoff R0, by he indicaor of he even in which levered equiy acually underakes c he venure, because he ime does indicae ha he levered venure finishes ou of he money on and here = 0, bu he unlevered venure finishes ou of he money only on he subse of his even. On he even I c, he levered venure finishes ou of he money, bu he unlevered venure does no, i. e., > 0. s a consequence E I c V R 0,. In valuing he unlevered venure simply following would lead o more value on he even, when levered equiy finishes ou of he money. The even I c is one inerpreaion of he even ha Myers 977, p. 49) refers o in he saemen The firm financed wih risky deb will, in some saes of naure, pass up valuable invesmen opporuniies... Of course on c, levered equiy 3

16 underakes a valuable invesmen opporuniy, bu does so laer han unlevered equiy, and in his sense passes up, in ime, he valuable invesmen opporuniy. These are he wo meanings of "passing up a valuable invesmen opporuniy." Boh inerpreaions of Myers saemen are correc. When a venure finishes ou of he money can be inerpreed as he invesor or firm never exercising he opion o inves. Indeed, we can redefine he opimal sraegy o be beyond he horizon T in ha even. For example, le ) ν = ) I T I c Then ν is a sopping ime and i equals he opimal exercise ime whenever i is opimal o exercise; oherwise i wais ill he expiraion of he opion and allows i o expire unexercised. Similarly we can le ν = I T + + I. 9) ) c This sopping ime is he opimal exercise ime for he levered venure when i is opimal o underake his venure; oherwise i wais ill he expiraion of he opion and les i expire unexercised. The following is a corollary of Theorem 4. Corollary 5. V = V, and V = V. Under 7a) and 7b), ν ν. ν ν The firs wo equaliies indicae he mulipliciy of soluions o he sopping problems characerized in Theorems and. Clearly ν and ν, bu ν and ν are opimal for heir respecive problems. I is in he sense of he las par of Corollary 5 ha he incenives of equiy in a levered venure is o wai pas he value maximizing ime o underake he venure, someimes forever. 4

17 3.3 numerical example. simple numerical example will help illusrae he resuls here. 8 There are wo periods and hree daes. Consider a firm whose only asse is a iming venure whose values are described in he following value ree. Y 0 Y Y The values in he ree are he gross values if he venure is underaken a hose poins in ime. For example, if he venure is underaken a dae = 0, he venure is worh Y 0 = 00 gross of required invesmen. If i is underaken a dae = following an iniial up move, he venure is worh Y = 80, ec. Here one can ake {,,, } Ω= 3 4 and in he familiar language of binomial rees, denoe by he pah in he ree of up followed by up, is he pah in he ree of up followed by down, 3 is he pah down followed by up, and 4 is down followed by down. The following able summarizes he gross values. 5

18 Pah 3 4 Values of Y Y Y ssume ha ineres raes are deerminisic and equal o 8% per period. So ) R 0, =.08, = 0,,. These values are hen consisen wih a risk-neural probabiliy of.4 of an up move anywhere in he ree. So P { } =.4.4) =.6, { } =.4, P { } =.6.4) =.4, and { } 3 process Y is a maringale under 4 P =.4.6) P =.6.6) =.36. Then he discouned value P in ha Y + Y = E, = 0,. R, + Suppose ha o underake he venure requires an iniial invesmen. Of course if he invesmen is fixed, he venure is an merican call opion on an asse ha has no value leakage and hence he opimal exercise sraegy is o wai ill dae = and exercise he opion when i is in he money. The above reasoning only applies o he exercise componen of he opimal iming sraegy. I may well be ha even for merican call opion, i finishes ou of he money before he erminal dae. 9 To illusrae all aspecs of Theorems -4 and Corollary 5, i is useful o induce earlier exercise by assuming ha he required invesmen K for underaking he venure a dae is as follows. To underake he venure a dae = 0 or a = he required invesmen is 00, regardless of he sae. To underake a = he required invesmen is 00 in saes, 3, and 4, bu i jumps up o 45 in sae. So K0 K 00 and K ) = i =, bu ) i 00,,3,4 K = 45. 6

19 Then he values of ineres here are he ne presen values ), 0,, + = Y K =. These values are summarized in he following able. Pah 3 4 Values of Given hese values, i is sraigh forward o deermine he value maximizing ime. Since 0 0, > 0 since here is value in waiing, i. e., γ 0 > 0. Following an iniial down move, i is clearly in he ineres of unlevered equiy o wai ill = since γ.48) +.60) = 0 and waiing has he value γ = E = = R, This is also an even when he venure is ou of he money bu has ime value. The venure is hen underaken in a subsequen up move bu i finishes ou of he money in a subsequen down move. So ) ) = =. 3 4 To deermine he value of in he oher insance, noe ha following an iniial up move, he value of waiing one more period is ) γ ) E = R,.08 = < 80 =. Thus following an iniial up move, γ = and his is he firs ime ha happens, so ) ) I follows ha = =. The venure is underaken since > 0. V = E = E I I + { = } { = R } = R 0, 0, R 0, ) +.08).08) ).360) + =

20 Noice ha he venure is underaken in every sae i has posiive NPV, he even {,, }. The venure finishes ou of he money in { } 3 = =. 4 c = Suppose he venure is levered wih a deb claim ha has a promised paymen of 50, so [ ] D min 50, Y, = 0,,. This is he case described in Lemma. in he ppendix and hence he D saisfy 7a) and 7b). The values of levered equiy = Y K D + ) are given in he following able. Pah 3 4 Values of Obviously > 0, here is some value o waiing. However, following an iniial down move, he value of waiing is zero and so is. Indeed, = γ = 0 in his even, and hence = γ = 0, and his is he firs ime his happens, i. e., ) = ) 3 4 =. Following an iniial up move, he value of waiing is γ.4 9 ) +.60) E = R,.08 = > 30 = γ. s a consequence γ = E > R, and ) = ) =. Levered equiy will wai ill dae = o underake his venure. Following a second up move, he venure will be underaken, bu i finishes ou of he money in a subsequen down move. The even in which he levered venure is no underaken is he even in which his venure finishes ou of he money, he even = {,, } = { } s indicaed 8

21 in Theorem 4, he levered venure finishes ou of he money before he unlevered venure in ha ) = < = ). Noe also ha ) = < = ) , so levered equiy passes up a valuable invesmen opporuniy before he value maximizing ime. 0 The c even in which he levered venure is underaken is { } = ) = < = c ), as indicaed in Theorem 4. The even = {, } even in which levered equiy foregoes a posiive NPV venure. Indeed, here is value loss here for 3 c bu I is he E I c = R 0, ).6 79 E I c I { } + I = { = } = R0, R 0,.08) = 4.55 < V. Here V = E 0, R E I + I R = { = } { = } R = 0, 0, ) + ).08) = 6.0, oversaing he value of following he opimal sraegy of levered equiy. The value of he deb claim in his example, as raionally anicipaed by crediors,.650) would be ).08 = 6.86, since levered equiy only underakes he venure when i makes enough o pay crediors in full. This calculaion indicaes how lile deb financing can be raised agains such growh opions, a major poin of Myers 977) The modified sraegies of 8) and 9) are compared in he following able. 9

22 Values of Pah ν ν = = ) ) ) = 3 ) = Increase in Leverage The incenives of owners of a levered iming venure o wai beyond he value maximizing ime o underake he venure ges worse as leverage increases. 4. Generalizaion of Theorem 4 and Corollary 5 To make hings precise, le D, D denoe he prior claim o venure value arising i i from wo differen levels of leverage. Le = ) Y K D +, he value of he residual i claim in he levered venure wih leverage D, i =,. For he sake of economy in describing hese we may jus refer o hem as venure i =,. i i i Similarly, he relevan variables of Theorems and are denoed,, γ, where i i = 0 is he canonical soluion o he problem of maximizing he marke value of he residual owners ) claim in venure, I and i i i i =. The evens = { = } { = 0} i = T i U = 0 and ic have he same inerpreaions as above bu for he levered venure i =,. Theorem 6. ssume ha D D, = 0,, K, T, saisfy 7a) and 7b). Then, I I, I c I c, and E I c E I c V R R 0, 0,. 0

23 The assympion of Theorem 6 ha he difference D D saisfies 7a) is a sense in which he leverage in venure is greaer han he leverage in venure. Theorem 6 is really a general resaemen of Theorem 4, for he case of D 0, = 0,, K, T is Theorem 4. The firs par of Theorem 6 saes ha if he venure wih lower leverage finishes ou of he money, hen so does venure wih he higher leverage. The second saemen says ha in he even ha he venure wih low leverage finishes ou of he money, he venure wih higher leverage finished ou of he money earlier no laer). The firs par also can be read as, if he venure wih high leverage underakes he venure, hen so does he venure wih lower leverage. The hird saemen of heorem says ha in he even ha he venure wih high leverage underakes he venure, he venure wih low leverage underakes i sooner. The las par of Theorem 6 saes ha here is loss no gain) of value using deb and his loss weakly) increases wih leverage. Now le ν and ν be defined as in 9) of he previous secion. Then as in he case of Corollary 5, we have ha Corollary 7. V = V, V = V, and if D D, = 0,, K, T, saisfy 7a) and 7b), hen ν ν. ν ν 4. The numerical example again. Consider again he numerical example of Secion 3.3. In ha example, he firm s only asse is a iming venure and he prior claim o value was a promised paymen of 50. = = We had ) ) 3 4 = = and ) ) and here was loss of value wih ha much leverage. Bu if leverage is sufficienly low, somehing beer can obain. s a funcion of he promised paymen D, afer an iniial up move he value of waiing ill

24 = is D) and he payoff o underaking he venure a dae = is 80 D. Seing hese equal and solving for D gives a breakeven value of.76. So ake and D = min D, Y, = 0,,. Then ) = ) = = = ) ) 3 4, bu = {, } 3 4 c and = {, }. Evaluaing he venure value wih deb D = D, 0. lso E I c = R 0, E I = c R 0,.4 80).08 = Leing ) ) = = D = 50 and, and we have ha D = min D, Y, = 0,,, ) 3 = 4) = and E I c R 0, = 4.55 < E I c R 0, = 9.63 < V = 3.8. Check ha D D saisfies Hypohesis D. The values are presened in he following able Pah 3 4 Values of D D D D D D Noice ha incenives o no underinves are improved wih less leverage. s he example illusraes, however, hese incenives are no resored o full efficiency unil he level of he promised paymen ges below 8. Indeed = {, } finishes ou of he money, for all levels of D, ) = D, where he venure <.76. So here he

25 riskiness of he deb has o be judged relaive o he lowes ne venure value in which he venure would be underaken by he unlevered firm. Wih a promised paymen of 0, he value of he deb claim, as raionally anicipaed by crediors, would be ).4 0 = 7.4 > This indicaes ha crediors.08 would be willing o pay more for a lower promised paymen of 0 han for a higher promised paymen of 50, anoher indicaion of he perversion of invesmen incenives caused by greaer leverage. 5 Conclusions Owners of levered iming venures have incenives o pospone implemenaion of hose venures beyond he value maximizing ime. he exreme hey may forego such venures as indicaed in Myers 977). Viewing growh opporuniies as iming venures, owners of firms wih such opporuniies or managers of firms wih such opporuniies acing in shareholders ineres will no exercise hese opions opimally in erms of value maximizaion. The reason for hese pervered incenives is he convex form of he residual claim. Owners of levered venures are risk loving relaive o owners of equivalen bu unlevered venures. Their incenives o wai longer may resul in loss of value. These incenives o wai ge worse he greaer he leverage. We would expec ha firms wih subsanial growh opporuniies would use lile risky deb o avoid he non-opimal exercise of hese opions. Indeed all he cauions abou poenial soluions o he underinvesmen problem and heir coss examined in deail in Myers 977) sill apply here. 3

26 The implicaion for opimal capial srucure for iming venures is clear. Levered iming venures are less valuable han unlevered ones, so he opimal capial srucure is no leverage. ppendix Proofs and references for proofs of resuls in he ex are presened in his appendix. ll properies of condiional expecaions used here and in he ex can be found in Williams 99, Secion 9.7). Proof of Theorem. Fix,< T, ake any C ' C and since R, =, and le ' max [,] =. Then R, +,.) ' = I{ } I = { } R, ' where I is he indicaor of he even. Wih.) he inducion argumen in Chow, Robbins, and Siegmund 99, Theorem 3.) goes righ hrough, o esablish 4).[] Proof of Theorem. The firs par follows from 5). The second and hird pars follow from Chow, Robbins, and Siegmund 99, Lemma 3.), since here sup C V <.[] Proof of Theorem 3. By definiion of, γ = 0 on F. I follows by 3) ha 0 = γ γ + + I γ I E E I = R, + R, +. Then 0 = γ + γ+ E I dp = I dp R, + R, + implies ha I γ + = 0. Using 3) again for each laer dae, i hen follows ha I γ + k= 0, k = 0,, K, T, and hence ha I + k= 0, k = 0,, K, T, from ) and 3).[] The following resul is needed for he proof of Theorem 4. 4

27 Lemma.. Under 7b), D D E, for all C R, = 0,, K, T., Proof: Given and C, E D R, T D j T D j D T E I E j I I E = j T R =, j R, j R, T R T, T = { } = = j { j} + = { = T} D E I + I D T j T j= { = j} { T } R, j R, T D E I I E T j T j T = R, j R, T RT, T = { } + = j { T } D E I + I D T j T j= { = j} { T } R, j R, T D... E I{ } D =. R, The equaliies follow from he law of ieraed condiional expecaion and he fac ha { j} j F. The inequaliies follow from 7b).[] pplying his resul for = 0 i follows ha D D D0 E R 0,, for all C. So if prior claimans were given he righ o decide when o underake he venure using shareholders capial and hey decided by maximizing he marke value of heir claim o venure value, heir opimal choice would be o underake he venure a dae 0. Of course when D 0 = 0, prior claimans would forego he venure, and obviously do so before he value maximizing ime. 5

28 The claim abou he simple bu imporan case when 7a) and 7b) hold is he following. Lemma.. Suppose ha for some 0 D >, D min [ D, Y ] = and suppose ha he discouned value process is a maringale in ha Y + Y = E, = 0,, K, T. Then D R, + saisfies 7a) and 7b). Proof: Clearly 7a) is saisfied since he value process is non-negaive and D > 0. For given ) ) ) D E D Y D E Y E Y + D Y E = E = D = D Y = D, R, + R, + R, + R, + R, + where he firs equaliy follows by definiion of D +, he second because R =, + + r is known a dae, he firs inequaliy from Jensen s inequaliy for condiional expecaions, he nex inequaliy follows from he fac + r. Finally, he las equaliy follows from he maringale assumpion on he value process Y. This esablishes 7b). [] This resul is of course he case of a concave ransformaion of a maringale being a supermaringale. The maringale hypohesis of Lemma. would per force hold by ) if he value process was he value of a raded asse, bu need no hold if he underlying is no a raded asse. Convex ransformaions of maringales of course are submaringales. I is his propery ha gives rise o he disored incenives of levered equiy. fer he echniques for he proofs of Theorems 4 and 6 are shown, an ineresing case of he prior claim being a proporional share claim is reaed. Under he maringale hypohesis, he resuls of Theorem 4 obain as well. 6

29 + + = Y K D = D. By Theorem Proof of Theorem 4. Under 7a), ) ) 3, I + = 0, k = 0,, K, T + and hence I + = 0, k k k = 0,, K, T +. I follows from ) and 3) ha I γ + = 0, k k = 0,, K, T +. Hence I I =, esablishing he second par of he heorem. T c I and noe ha = B. Then Le B = { = } { > 0} I = I D) = I γ I E, C B B B B,.) R, where he firs equaliy comes from he fac > 0, he second from he definiion of, and he inequaliy from 4) for he problem wih oher side of his inequaliy, gives ha U = 0 replaced by + D I B I E I D I E D B + + B B R, R,. Taking D o he D I B E + D R, D = I B E + D E R, R, I E, C B..3) R, where he firs inequaliy follows from.), he second from Jensen s inequaliy for condiional expecaions, he hird inequaliy follows from he posiive par, and he las inequaliy follows from 7b) and Lemma.. I hen follows from.3) ha 7

30 I = I γ C. Thus B B, by 4) because I = I B B, since is he firs ime ha s = γ, proving he hird par of he heorem. s The firs par of he heorem follows from he inequaliies in he second and hird pars. The firs inequaliy in he las par of he heorem follows from he fac ha I R 0, 0, and he second inequaliy follows from Theorem and he fac ha C.[] Proof of Corollary 5. The firs wo equaliies follow from he fac ha = 0 = T on + and = 0 = T + on. From 8), ν ) = I T + + I c ) = I T + + I + I c c I I T + + I T + + I ) c ) c I = ν, by 9), where he decomposiion in he second equaliy and he inequaliy follow from Theorem 4.[] Proof of Theorem 6. The proof of Theorem 6 follows jus as in he proof of Theorem 4. By 7a), D + D implies = D D ). The inequaliy.) is for he ) higher deb level D and only D D is moved o he oher side o produce inequaliy.3) for he lower deb level D. Then use Lemma. o conclude ha D D D D E 0.[] R, Proof of Corollary 7. Follows from Theorem 6 as Corollary 5 does from Theorem 4.[] 8

31 Consider now he case when D = αy for some fracion α, 0 < α <. This migh arise in he following seing. universiy professor has developed scienific resuls ha may have commercial applicaion. The professor may rade he applicaion, inellecual capial, o a commercial developer wih financial capial for a slice of he pie when developed. There is no claim ha a fracional share is an opimal conrac here, bu i is no difficul o imagine he professor aking a slice of he pie. In his case, he desired resuls obain when he value of he underlying saisfies he maringale hypohesis. gain, we noe ha his would be he case if he underlying were a raded asse, e. g., if here was a secondary marke for he underlying asse. The case here exends he resuls of Theorem 4 o a prior claim ha is a proporional claim. The noaion of Secion 4 applies here. Theorem.3. ssume ha D αy =, = α ) Y K) + and ha Y + Y = E, R, + = 0,, K, T. Then, I I, and I c I c. s a consequence, E I c V V R 0, Proof: The non-negaiviy of Y and he maringale hypohesis ensures ha he 7a) and 7b). In fac, by Lemma., i follows ha = α + follows ha ) 4.[] D saisfy D D E = 0, for all C R. I, Y. The proof proceeds jus as in he proof of Theorem 9

32 References rnold, T., and R. L. Shockley, 003, Real Opions, Corporae Finance, and The Foundaions of Value Maximizaion, Journal of pplied Corporae Finance, 5, Bodie, Z.,. Kane, and. J. Marcus, 00, Essenials of Invesmens 4 h ed.), McGraw- Hill/Irwin, New York. Chow, Y. S., H. Robbins, and D. Siegmund, 99, The Theory of Opimal Sopping, Dover, New York. Dalang, R.,. Moron, and W. Willinger, 990, Equivalen Maringale Measures and No-rbirage in Sochasic Securiies Marke Models, Sochasics and Sochasic Repors, 9, Duffie, D., 00, Dynamic sse Pricing Theory 3 rd ed.), Princeon Universiy Press, Princeon, N. J. Grenadier, S. R., 00, Opion Exercise Games: n pplicaion o he Equilibrium Invesmen Sraegies of Firms, The Review of Financial Sudies, 5, Mauer, D. C., and S. H. O, 000, gency Coss, Underinvesmen, and Opimal Capial Srucure: The Effec of Growh Opions o Expand, in M. J. Brennan and L. Trigeorgis eds.), 000, Projec Flexibiliy, gency, and Compeiion, Oxford Universiy Press, New York. Myers, S. C., 977, Deerminans of Corporae Borrowing, Journal of financial Economics, 5, Nachman, D. C., 975, Risk version, Impaience, and Opimal Timing Decisions, Journal of Economic Theory,,

33 Sabarwal, T., 003, The Non-Neuraliy of Deb in Invesmen Timing: New NPV Rule, Deparmen of Economics, BRB.6, The Universiy of Texas a usin, July. Snell, J. L., 95, pplicaions of Maringale Sysems Theorems, Transacions of he merican Mahemaical Sociey, 73, Trigeorgis, L., 996, Real Opions: Managerial Flexibiliy and Sraegy in Resource llocaion, The MIT Press, Cambridge, Massachuses. Williams, D., 99, Probabiliy and Maringales, Cambridge Universiy Press, Cambridge. Endnoes None of he resuls of his paper are peculiar o discree ime. ny discree ime model can be imbedded in a coninuous ime seing where nohing happens beween he discree daes. ll he resuls here obain in ha coninuous ime seing. In a paper wih an economic seing as we posi here, bu where values follow geomeric Brownian moion, McDonald and Siegel 986) emphasize he imporance of allowing he opion o have zero ime value as his is a feaure of invesmen opporuniies where compeiion may erode economic rens. They do no, however, address he underinvesmen problem. 3 The horizon is he opion expiraion. The fixed finie horizon is no essenial bu is convenien in describing opimal sraegies. The resuls esablished here hold if he horizon or opion expiraion is an arbirary sopping ime ha is finie almos surely. See Nachman 975) for his general case. More realisic economic circumsances, such as compeiion, ha deermine expiraion are beyond he scope of his paper. Grenadier 00) shows ha compeiion among homogeneous firms in a coninuous-ime Courno-Nash framework erodes he value of he opion o wai o inves. 4 For simpliciy we ignore axes. To he exen ha any par of he prior claim o value is ax deducible so he afer-ax claim o value is less, he leverage is less and he underinvesmen problem is less. See Secion 4 for he effec of changes in leverage. 5 See also McDonald and Siegel 986). 6 See Duffie 00, Secion.G). 7 Riskless borrowing here plays he role of impaience in Nachman 975), bu no resul here requires i, i. e., every resul in his paper is valid even if r 0 for all. 3

34 8 The venures need no be small if financial markes are complee. See rnold and Shockley 003). 9 See Duffie 00, Theorem.G) for he finie sae space case. For more general sae spaces, see Dalang, Moron, and Willinger 990). These resuls apply for an economy wih a finie number of daes. If he horizon for his economy coincides wih he horizon T for he venure, hen he absence of arbirage enails ha S T = 0, almos surely. If he horizon for he economy is longer han ha for he venure, hen in general S T is he expeced value a dae T, under P, of he discouned dividends beyond dae T. 0 Unless explicily saed o he conrary, all relaions among random variables in his paper hold almos surely, abbreviaed a. s. Wih his undersood, we henceforh drop his qualifier in he ex. See for example Duffie 00, pp. 33,84). Only in he finie horizon case can his envelope be defined consrucively as in ) and 3). See Chow, Robbins, and Siegmund 99, Chaper 4) for he nonconsrucive definiions for he infinie horizon case. The resuls of his paper go hrough for he case. s a resul here is no more finance jus more complicaed mahemaics. The erms inrinsic value and ime value used here are he familiar erms used in describing opion value. See for example Bodie, Cane, and Marcus 00, pp. 54,54). 3 I is never he case ha T + is he firs ime his happens. 4 The inuiion for his resul is ha he process γ is non-negaive and expecaion decreasing a nonnegaive supermaringale) in ha γ E. Once his process reaches zero i mus say here. By he R + γ, + recursion ), 3), he ime value γ is he smalles supermaringale ha is larger han 5 This holds when he discouned value process is a maringale under P see Lemma. and he commens ha follow i in he ppendix). The ac of early exercise saves he increase in he exercise price, bu if wha you save is less han wha you can earn on he curren exercise price, i would no pay o exercise early. 6 In mahemaical erms, 7a) and 7b) is he assumpion ha he discouned sequence of prior claim values. in he venure is a non-negaive supermaringale. See Lemma. in he ppendix and he commens ha follow i for one implicaion of his hypohesis. 3

35 7 This is shown in Lemma. of he ppendix. The discouned value process would per force be a maringale by ) if he value process was he value of a raded asse, e. g., if here was a secondary marke for he asse. 8 This is he generic example of Trigeorgis 996, Chaper 5). 9 For any consan exercise price greaer han 08 bu less han 34 in his example, he venure finishes ou of he money a dae = following an iniial down move. 0 This can be made more sark. Suppose K 4) = 30 insead of 00, bu all else is he same. Then he payoffs o levered and unlevered equiy remain he same excep ha 4) = 6. The opimal imes and remain unchanged, bu now = and he unlevered) venure is posiive ne presen value in boh 3 and 4. Ye in boh hese saes levered equiy passes up he valuable invesmen before he value maximizing ime. See Figure 3, Myers 977, p. 54). This example is one where he promised paymen of D = 50 is o he righ of he promised paymen ha gives he maximum amoun of deb. 33