Managing a Portfolio of Real Options: Sequential Exploration of Dependent Prospects. by James L. Smith and Rex Thompson

Transcription

1 Managng a Portfolo of Real Otons: Sequental Exloraton of Deendent Prosects by James L. Smth and Rex Thomson WP January 004

2 Managng a Portfolo of Real Otons: Sequental Exloraton of Deendent Prosects January 3, 004 James L. Smth and Rex Thomson Deartment of Fnance Edwn L. ox School of Busness Southern Methodst Unversty Dallas, TX 7575 Abstract We consder the mact of sequental nvestment and actve management on the value of a ortfolo of real otons. The otons are assumed to be nterdeendent, n that exercse of any one s assumed to roduce, n addton to some ntrnsc value based on an underlyng asset, further nformaton regardng the values of other otons based on related assets. We couch the roblem n terms of ol exloraton, where a dscrete number of related geologcal rosects are avalable for drllng, and management s obectve s to maxmze the exected value of the combned exloraton camagn. Management s task s comlex because the exected value of the nvestment sequence deends on the order n whch otons are exercsed. A basc conclusons s that, although deendence ncreases the varance of otental outcomes, t also ncreases the exected value of the embedded ortfolo of otons and magnfes the value of otmal management. Stochastc dynamc rogrammng technques may be used to establsh the otmal sequence. Gven certan restrctons on the rsk structure, however, we demonstrate that the otmal dynamc rogram can be mlemented by olces that are relatvely smle to execute. In other words, we rovde suffcent condtons for the otmalty of ntutve decson rules, lke bggest frst, most lkely frst, or greatest ntrnsc value frst, and we develo exact analytc exressons for the mled value of the ortfolo. Ths ermts the value of actve management to be assessed drectly. Fnally, the suffcent condtons we dentfy are shown to be consstent wth lausble exloraton rsk structures. Acknowledgement The authors thank John Semle for nsghts and gudance he rovded early n the research rocess, also Sten-Erk Fleten and Axel Perrru for several helful dscussons, and Jacquelne McLelland for assstance wth reference materals and lbrary research. Smth gratefully acknowledges research suort rovded by the MIT enter for Energy and Envronmental Polcy Research. one of these enttes are resonsble, however, for any mstakes the authors may have commtted.

3 Managng a Portfolo of Real Otons: Sequental Exloraton of Deendent Prosects. Introducton We consder the mact of sequental nvestment and actve management on the value of a ortfolo of real otons. The otons are assumed to be deendent, n that exercse of any one s assumed to roduce, n addton to some ntrnsc value based on ts underlyng asset, further nformaton regardng the values of other otons based on related assets. We take the values of the underlyng assets to be ostvely related; a hgh value on any one tends to ncrease the lkelhood of hgh values elsewhere. aluaton of such ortfolos s comlex n that the combned value of the entre ortfolo may deend on the order n whch otons are exercsed, and the otmal order s not always obvous (and sometmes counterntutve) when the number of otons exceeds two. As a frame of reference, we couch the roblem n terms of ol exloraton, where a dscrete number of related geologcal rosects are avalable for drllng and management s obectve s to maxmze the net resent value of the entre exloraton camagn. Such rosects tycally dffer n sze and robablty of success, and are sad to be deendent or assocated f success on one ncreases the condtonal robablty of success on others. Each rosect reresents a real oton, whch f successfully exercsed (va drllng) conveys the ntrnsc value of the underlyng ol, lus nformaton regardng the value of remanng rosects. How much should management be wllng to ay to acqure such a ortfolo? ertanly more than the sum of the ntrnsc values, because that measure gnores the value created by usng ntervenng nformaton to Tong (980, ) dscusses the assocaton of random varables (a roerty equvalent to ostve quadrant deendence n the bvarate case) and revews many of the relevant statstcal mlcatons.

4 actvely manage the exloraton sequence. Holdng all else equal, the exstence of deendence among rosects adds value to the ortfolo of otons. Our results hghlght an mortant dfference between real oton alcatons and the standard fnancal oton aradgm: n many alcatons of real otons, the value of the underlyng asset wll not be revealed untl after the oton has been exercsed. The true value of the asset wll often deend on addtonal sources of uncertanty that can be resolved only through nvestment and exlotaton of the asset. Paddock, Segel, and Smth (988) demonstrated n the case of etroleum exloraton (where the number of shares to be acqured va drllng an exloratory well s uncertan) that, where only a sngle asset s nvolved, the basc analogy to fnancal otons s reserved and standard technques based on the rsk-neutral valuaton rncle may be aled. When several deendent assets are nvolved, however, the valuaton roblem changes n a fundamental way. The flow of avalable nformaton s endogenzed subect to managment s decson to exercse one oton that could reveal nformaton regardng the values of others. The flow of nformaton has to be managed n concert wth nvestment n the underlyng assets, and the value of the ortofolo as a whole wll reflect managment s skll n combnng these two functons. A smle llustraton shows the mortance sequencng deendent nvestments otmally. onsder three rosects, each requres the exendture of $80 mllon to test, and returns a gross value of $00 mllon f successful. Jont and margnal robabltes of success for the three rosects are shown below:

5 Prosect s the most lkely, and rosect 3 the least lkely to succeed. It s nonetheless otmal to test the thrd rosect frst because t generates valuable nformaton that more than comensates for ts lesser ntrnsc value. The value of the ortfolo f rosect 3 s tested frst amounts to $5. mllon. In contrast, the value of the ortfolo f all three rosects are tested smultaneously, or wthout regard for ntervenng outcomes, amounts to only $3.3 mllon. Thus, deendence among ndvdual rosects quadrules the value of ths ortfolo, but only f the nvestments are made n roer sequence and the resultng nformaton acted uon n an otmal manner. Subect to certan regularty condtons, stochastc dynamc rogrammng technques may be aled to dentfy the otmal order of trals n roblems of ths tye and to ascertan ortfolo value. That aroach reles heavly on comutatonal ower but does not contrbute much economc nsght regardng the elements of a successful sequental nvestment strategy. Of course, as the sze of the ortfolo grows, dynamc rogrammng sollutons mose ever larger comutatonal demands and nformaton requrements, as well. 3 We show that, gven certan lausble constrants on the structure of deendent rsks, the soluton to ths ortfolo management roblem reduces to a form that s much smler and easer for management to execute. In the extreme, for examle, s the case where the value of each asset s ndeendent of the others, whch mles that the value of Ths value s obtaned by solvng a smle decson tree. 3 For examle, see Haugen (996) and Jorgenson (999). 3

6 the ortfolo s ndeendent of the order n whch the otons are exercsed. But that examle throws out the baby wth the bath water. Our goal s to dentfy, where ossble, smle rules for managng a ortfolo of deendent otons. We also show that the rsk structure most commonly used to descrbe deendent etroleum exloraton outcomes s suffcent for the otmalty of smle decson rules, lke bggest frst, most lkely frst, or greatest ntrnsc value frst ; and we develo exact analytc exressons for the value of the ortfolo n such cases. Ths ermts the ncremental value of actve management to be assessed smly and drectly. We are ust as much nterested n establshng the lmts beyond whch smlfed decson rules would fal to otmze the value of the ortfolo. Postve assocaton s not, by tself, suffcent for our results as the recedng examle already demonstrated. A secal form of assocaton among the underlyng assets s requred to acheve much n the way of smlfcaton.. Related Lterature Relatvely few aers have consdered the mact of sequental nvestment and roect nterdeendence on the value of a ortfolo of real otons. Trgeorgs (993) was among the frst to consder the mlcatons of nterdeendence and establsh the nonaddtvty of real oton values. Hs analyss, however, ertans to a collecton of otons all wrtten on the same underlyng asset, whereas we have n mnd alcatons where dstnct assets underlay each oton n the ortfolo. In addton, the sequence n whch otons mght be exercsed s redetermned n hs analyss, whereas flexblty n determnng ths sequence s aramount n our study. 4

7 shwanath (99) derves, as we do, suffcent condtons for the alcaton of relatvely smle rules to solve the roblem of otmal sequental nvestment n a collecton of roects. She also shares our vew regardng the nherent ractcal lmtatons of the dynamc rogrammng aroach, whch n her words s lkely to shed lttle economc nsght (and) would be a comlex brute force task. Unlke us, however, she confnes her analyss to roects whose ayoffs are mutually ndeendent, all of whch must be exercsed. 4 Thus, two crucal asects of our framework are mssng n her work. ortazar, Schwartz, and asassus (00) nvestgate the mact of geologc and rce rsk on the value of a collecton of nterrelated natural resource otons. As n Trgeorgs (993), however, the analyss ertans to multle otons wrtten on the same underlyng real asset, and the nvestment sequence s redetermned. hlds, Ott, and Trants (998) undertake what s erhas the most comrehensve study of the mact of nterdeendence on real oton valuaton and nvestment sequence. They descrbe roblems wheren the form of nterdeendence ranges from mutual exclusvty to erfect comlmentarty. Ther analyss s lmted to the two-rosect case, however, and only the mutually exclusve case s dscussed n the text. Ther results antcate one of our man observatons: that t s not always advsable to exercse the most valuable oton frst; and they descrbe the condtons that work for and aganst such an outcome. Our work ertans to otons whose values tend to be ostvely correlated, rather than mutually exclusve, and we have found that some results obtaned easly for the two-rosect case fal to generalze even to the three- 4 The sequence of nvestments matters n that framework, but for reasons that relate to rsk references. 5

8 rosect case the dfference comng from the extra degrees of freedom that are nherent n multvarate dstrbutons. 5 Several aers n the racttoners arena are also ertnent to our work. Murtha (996) has nvestgated the mact of deendence among etroleum rosects, but wthn a rgd nvestment structure that would requre all rosects to be drlled. He fnds (correctly wthn that framework) that exected reserve volumes are unaffected by deendence among rosects, although the varance of reserves (ayoffs) must ncrease. Hs concluson, that deendence ncreases the rskness of exloraton, overlooks the value of flexble management a value that we conclude can be very hgh. Delfner (000) adheres closely to Murtha s aroach to ortfolo analyss, to the ont of reeatng Murtha s otentally msleadng concluson, that deendences ncrease the exloraton rsk. What the real otons aroach ncororates s what both Murtha and Delfner leave out: the ablty of manageral flexblty to turn nflated varance nto enhanced return. Wang, et. al. (000) do recognze that deendence creates manageral otons to sequence etroleum exloraton rosects otmally based on udated nformaton, but they rovde no analyss of the value of such otons. 3. Prelmnares: The ase: We start by assumng there are two rosects, wth ntrnsc values: and, where reresents the robablty of success on the th rosect, s the exected value of that rosect condtonal on success, and s the cost of erformng the tral (we assume dentcal drllng costs over all rosects). 6 5 It s mossble wth only two rosects, for examle, to construct the tye of llustraton we resented n the ntroducton;.e., where the lower-robablty rosect should go frst. 6

9 Wthout loss of generalty, we order rosects n terms of decreasng ntrnsc value, thus: >. To smlfy the resentaton, we wll assume that all rosects are ntally n the money, thus: >0. 7 As noted already, assocaton of outcomes mles:. and. ; that s, success on ether rosect ncreases the chance of success on the other. Due to the mact of the secfc nformaton generated by the frst tral, the value of the ortfolo deends on whch rosect goes frst. The exected value of startng wth rosect s gven by: ( ) (. ) m(. ) Π, where for convenence we defne: m(x) max(x,0),, and /,. m embodes the value of the oton not to test the second etc. The term ( ). rosect after falng on the frst. Lkewse, the exected value of startng wth rosect s gven by: ( ) (. ) m( ). Π. The remum earned by startng wth rosect one s gven by the dfference: ( ) ( ) (. ) (. ) m ( ) m( ).. ( ) ( ) ( ) m( ) m( )... 6 In the vernacular of the etroleum ndustry, s sometmes referred to as the unrsked value of a rosect, whereas s the rsked value. 7 The value of the ortfolo may be ostve even f < 0, and we beleve t s ossble to derve smlar results for such cases, but that goes beyond the scoe of the resent work. 7

10 8 We evaluate ths exresson, dstngushng three cases: ase A: ether rosect s condemned by falure of the other In ths case, both exressons of the form ( ) m. are ostve and we have: ( ) ( ) ( ) ) A ( ) ( ) ( ) ( ) ( ) ( ) 0. () Thus, order doesn t matter n ase A snce management would test both rosects regardless of ntervenng outcomes. ase B: Ether rosect s condemned by falure of the other In ths case, both exressons of the form ( ) m. are zero and we have: ( ) ( ) ( ) B. ( ) ( ) ( ) () But, < / (by assumton,) so: ( ) ( ) ( ) B / ( ) ( )( )., whch takes the sgn of ( ) snce. > 0 (recall that. ). Thus, n ase B t s otmal to test frst the rosect wth hgher ntrnsc value f that rosect also has the hgher robablty of success. If the two rosects have equal values condtonal on success (.e., ), you would always test the more lkely rosect frst. If they have

11 equal robabltes of success, then the one wth the greater condtonal value must go frst. On the other hand, you would test the nd rosect (lower ntrnsc value) frst f ts falure conveys enough nformaton to comensate for ts lower ntrnsc value. Secfcally, the condton for testng the second rosect frst s (from Eq. ): Equvalently: ( ) ( ) ( ) 0 < ( ) < ( ) ; whch mles that you would test the lower ntrnsc value frst f and only f: <. (3) In terms of the rmtve arameters, lower values of the rato / make t more lkely that the lower ntrnsc value rosect should go frst. Intutvely, low values of means that the odds are aganst rosect two generatng many false negatves, / at least relatve to rosect one, whch enhances the value of nformaton gleaned from t. ase : Only one rosect s condemned by falure of the other It s easy to show and B have the same sgn. To see ths, examne the dfference: ( ) ( ) ( ) ( ) ( ) B.. where the frst equalty s mled by Eq., and the second by Eq.. Thus: > ( ) ( ) 0 > 0 B <.. ; < whch, snce and are non-negatve, s equvalent to: 9

12 > < ( ) 0 and ( ) 0 > B 0 <. <., > where we have used the fact that n ase these two exressons must dffer n sgn. Therefore, f B > 0, we know: ( ) ( ) ( ) ( ). ( ) 0 Lkewse, for B < 0, we have: A. >. ( ) 0 A. <. Imlcatons: Some relmnary results can now be summarzed. If nether rosect has the ower to condemn the other, both may be tested smultaneously. 8 Otherwse, the necessary and suffcent condton for testng the th frst s gven by (cf. Eq. 3): Test the th rosect frst f and only f: >. For, ths condton s assured f, n addton to >, we have >. On the other hand, f we have, the condton mles testng rosect one frst snce we have assumed >. Our conclusons regardng the otmal nvestment sequence s summarzed n the followng dagram. 8 Throughout ths aer, we neglect the tme value of money n order to emhasze the oton-value comonent of ortfolo value. 0

13 Otmal Sequence wth ommon Rsk Structure: If Prosect Falls In Shaded Regon, Test Prosect Frst; In Unshaded Regon, Order Deends on Degree of Deendence < & > Order deends on ont robabltes < & < Prosect Frst E E Oton alue We defne the statc value of the ortfolo (Π 0 ) to be the sum of ntrnsc values;.e., the exected value of the ortfolo f management gnores the nformaton content of revous outcomes: ( ) ( ) Π 0. We may then defne the oton value of the th rosect (O ) as the addtonal value that comes by testng t frst and usng the resultng nformaton to make subsequent nvestment decsons: O ( ) ( ) ( ) Π Π 0. m.. ow, f falure on the th rosect does not condemn the th, we have: O ( ) ( ) ( ).. 0..

14 Thus, f the th rosect has no ower to condemn the other, t has no oton value. Alternatvely, f falure on the th rosect does condemn the th, we have: O ( ) ( ) ( ) ( ). > 0. (4) Ths oton value has the natural nterretaton of beng the exected cost savngs (n terms of deferred testng cost) less the foregone revenue due to the occurrence of a false negatve (.e., the th rosect wrongly deferrng the th ). Partal dfferentaton of Eq. (4) gves: O < 0 O > 0 O. Thus, oton value falls as the ntrnsc value of the other rosect rses snce there s less chance that falure wll defer t. Oton value rses as the cost of trals rses snce the otental cost savngs s larger. Fnally, oton value falls as the robablty of false negatves rses. 4. The General ase: Before these results can be generalzed to the case of multle rosects, t s necessary to lace further restrctons (beyond assocaton) on the rsk structure. In ths secton, we restrct attenton to what s referred to n etroluem exloraton as the shared rsk nformaton structure. 9 Shared Rsk Informaton Structures We let: (F ) q ; and ( F ) -q ; for 0,,..., ; where the F reresent ndeendent events. Then, defne: 9 The name comes from Stabell (000), although alcatons of ths tye have a much longer hstory n the etroleum ndustry. See Megll (979) for examle.

15 S F 0 F ;,,...,. Intutvely, F 0 denotes the resence of a common factor that s necessary for success on each of the rosects (e.g., the orgnal deoston of carbonferous sedments n a rosectve etroleum basn). For,...,, each of the F reresents the resence of an addtonal rosect-secfc factor that s necessary for success (e.g., a local trang mechansm) on that secfc rosect. The rosect-secfc factors are assumed to be ndeendent of each other and ndeendent of the common factor. Thus: (S ) q 0 q ;,,...,.; and:. (S S ) (S S )/(S ) (F 0 F F )/(F 0 F ) q 0 q q /q 0 q q Also: q k 0qq q / q q k 0 q. k.,. k q0qq qk / q0q qk., etc. In the shared rsk structure, relatve robabltes of success among remanng rosects are not affected by revous outcomes, snce:.{} k.{} k q q, where the set {k} reresents any set of outcomes on other rosects. From ths, t also follows: > > < <.{} k. {} k I.e., the rankng of remanng common-rsk rosects by ntrnsc value s not affected by the outcomes of revous trals. Wthn ths framework, we can now rove: 3

16 Theorem : Gven rosects such that and, for all and such that <, then at each stage n the nvestment sequence t s otmal to test next the rosect wth hghest ntrnsc value. (Proof see Aendx) Oton alues We now extend the defnton of oton value to the -rosect case. The secfc results to follow are based on Theorem, and therefore resume that, for all and such that <. Moreover, for the tme beng, we wll assume that falure on rosect one would condemn rosect two. The oton value of the st rosect as t affects the th can be defned as n the case: O, for,...,. The value of the ortfolo can then be comuted as the sum of these elementary oton values (cf. Eq. 4): Theorem : If falure on rosect one would condemn rosect two, then the value of the ortfolo s gven by: Π* Π Π 0 O O3... O. ( ). Π 0. (5) (Proof see Aendx) The value of actvely managng the ortfolo s therefore: Π ( ). Π 0. 4

17 Portfolo alue omaratve Statcs We dfferentate Eq. (5) to observe the mact of arameter changes on the ortfolo value, and on the value of actve management. Frst, wth resect to the cost of testng the rosects: Π ( ) < 0 ; ( Π Π ) 0 ( ) > 0 ; whch means that a hgher testng cost decreases the value of the ortfolo, but ncreases the value of actve management. Wth resect to the condtonal value of each rosect: Π > 0 ; ( Π Π ) 0 ( ) < 0 ; whch means that hgher rosect value (condtonal on success) ncreases the value of the ortfolo, but decreases the value of actve management. Fnally, wth resect to the robablty of obtanng a false negatve from each rosect (whle holdng constant the margnal robabltes of success): Π. ( Π Π ). 0 < 0. But, note that ; thus:.. Π. ( Π Π ). 0 > 0 ; 5

18 whch means that, holdng other thngs equal, greater deendence ncreases the term, and therefore ncreases both the value of the ortfolo and the value of actve. managment; whereas greater robablty of a false negatve regardng any rosect decreases both the value of the ortfolo and the value of actve management. ext we account for the case where falure on the st rosect may not condemn the nd. To make an nterestng roblem, some rosect must be condemned by one or more ror falures, else all rosects would be tested and the value of the ortfolo would be gven smly by the statc value, Π 0. We let rosect m (where m<) reresent the most valuable condemnable rosect (.e., the rosect of lowest ndex that could ossbly be condemned by ror falures). We can then establsh: Theorem 3: If rosect m s the condemnable rosect of hghest ntrnsc value, then the value of the ortfolo s gven by: ( m) Π * Π Π (6) 0 0( m) m 0( m) where ( ) s the robablty of no success among the frst m trals, and ( ) s the 0 m robablty that the th rosect ( m) succeeds and there are no successes among the frst m trals. Proof: (see aendx). Our revous eq. (5) reresents the secal case of (6) obtaned by settng m. The same natural nterretaton of oton values ales here as n that case, but where the decson to test the frst m rosects smultaneously s treated as a sngle act. The form of the exresson s otherwse entrely analogous. Fnally, the value of actvely managng the ortfolo s gven by: 0 m 6

19 Π Π0 0( m) ( m) 0( m), m from whch comaratve statc roertes can be derved smlar to those gven above. 5. Summary At ths uncture we are able to organze the followng smlfcatons to the general roblem of managng a ortfolo of deendent otons.. When choosng between two rosects, t s otmal to test both smultaneously f nether has the ower to condemn the other.. When choosng between two rosects, t s otmal to test frst the rosect wth larger ntrnsc value f that rosect also has the larger robablty of success. However, t s otmal n some cases to test frst the rosect wth smaller ntrnsc value f t has the larger robablty of success. 3. When choosng between two rosects, knowledge of the rato of success robabltes condtonal on falure of the other rosect s suffcent to order the rosects as a functon of unrsked valuatons. 4. If rosect deendence conforms to the common-rsk rsk structure, then these results generalze to comarsons among rosects: a. When choosng among rosects, t s otmal to test frst the rosect wth the largest ntrnsc value f t also has the largest robablty of success. b. When choosng among rosects, t s also otmal to smultaneously test any rosects that would not be deferred (condemned) by falure on 7

20 the rosect dentfed n art a, regardless of ts ntrnsc value and/or rsk. c. If all rosects are of the same sze and conform to the commonrsk structure, then t s otmal to test frst the one wth the largest robablty of success. d. If all rosects have the same robablty of success and conform to the common-rsk structure, then t s otmal to test frst the one wth the largest sze. 5. The oton value of a rosect measures the extent to whch nformaton revealed va a test of that rosect enhances the value of the rest of the ortfolo of rosects. 6. The oton value of a rosect ncreases drectly wth that rosect s degree of afflaton wth other rosects. 7. The oton value of any rosect vares drectly wth the cost of testng, but nversely wth the ntrnsc value of other rosects. In these two resects, crcumstances that are assocated wth a decrease n the statc value of the ortfolo are assocated wth an ncrease n the value of managng the ortfolo actvely. Our relmnary nqury encourages us nto further contemlaton of how the structure underlyng a ortfolo of real otons nterlays wth the otmal oton exercse. We suggest broadenng the search for robablty saces wheren smle decson rules are otmal and characterzng these rules n the vernacular of real otons. Wll t be ossble to state suffcent and necessary condtons under whch secfc smle rules are otmal? What wll these condtons look lke and how closely wll they conform to meanngful alcatons? How much value s contrbuted by the otmal 8

21 exercse of mbedded otons? If research outcomes are ndeed assocated, our results to date leave us otmstc about obtanng useful answers to these and smlar questons. 9

22 Lst of References hlds, P., S. H. Ott, and A. J. Trants, atal Budgetng for Interrelated Proects: A Real Otons Aroach, Journal of Fnancal and Quanttatve Analyss, 33:3(998), ortazar, G., E. S. Schwartz, and J. asassus, Otmal Exloraton Investments Under Prce and Geologcal-Techncal Uncertanty: A Real Otons Model, R&D Management, 3:(00), Delfner, P., Modelng Deendences Between Geologc Rsks n Multle Targets, Socety of Petroleum Engneers, SPE Paer 6300, 000. Haugen, K. K., A Stochastc Dynamc Programmng Model for Schedulng of OffShore Petroleum Felds wth Resource Uncertanty, Euroean Journal of Oeratonal Research, 88(996), Jorgenson, T., Proect Schedulng as a Stochastc Dynamc Decson Problem, doctoral dssertaton, Det. of Industral Economcs and Technology Management, orwegan Unversty of Scence and Technology, Trondhem, 999. Megll, R. E., An Introducton to Rsk Analyss, Tulsa: PennWell Publshng (979). Murtha, J. A., Estmatng Reserves and Success for a Prosect wth Geologcally Deendent Layers, SPE Reservor Engneerng, February, 996. Paddock, J. L., D. R. Segel, and J. L. Smth, Oton aluaton of lams on Real Assets: The ase of Offshore Petroleum Leases, Quarterly Journal of Economcs, August 988. Stabell,. B., Two Alternatve Aroaches to Modelng Rsks n Prosects wth Deendent Layers, Socety of Petroleum Engneers, SPE Paer #6304, 000. Tong, Y. L., Probablty Inequaltes n Multvarate Dstrbutons, ew York: Academc Press, 980. Trgeorgs, L., The ature of Oton Interactons and the aluaton of Investments wth Multle Real Otons, Journal of Fnancal and Quanttatve Analyss, 8(993), -0. shwanath, T., Otmal Orderngs for Parallel Proect Selecton, Internatonal Economc Revew, February, 99. Wang, B., et. al., Deendent Rsk alculatons n Multle-Prosect Exloraton Evaluatons, Socety of Petroleum Engneers, SPE Paer 6398,

23 Aendx Proof of Theorem : The roof s by nducton. The result has already been establshed for the case of, so begn now wth 3. Pck any rosect other than the frst ( ) to test frst. Among the - rosects that reman, we have already shown that t s otmal to test the hghest ntrnsc value frst. Snce the order s reserved, the rosect wth hghest ntrnsc value (and hghest robablty of success) after has been tested s the same as before was tested. There s no ambguty therefore n the labelng of rosects. The maxmal exected value of all rosects, gven that you start wth, may then be wrtten: Π ( ) (. ) (. E E ). [( ) E E ] m... ; (A) where:. condtonal robablty of success on rosect after falng on. E exected value of remanng rosects after success on and falure on. m [] x maxmum of (0,x). If a negatve value aears n the square bracket, then you would choose to not test after falng on. But, n that case you would test no further (else rosect would not have been the otmal choce to follow ) and the seres ends. We clam that a value not less than Π could be obtaned by startng wth the frst nstead of the th. The maxmal exected value, gven that you start wth rosect, can be wrtten as:

24 Π ( ) (. ) (. E. E ) m[ ( ) ]. E E E E.. (A) where: E exected value of remanng rosects after falure on and not ermttng to go next. Equaton () dffers n form from () only because there s no assurance that (whch was chosen arbtrarly) should otmally follow. If the value n square brackets s non-negatve, t should follow ; otherwse not. To rove our clam, we must show Π -Π 0. Usng (A) and (A), we have: Π Π ( ) (. ) ( ) (. ) (T ) (. E E ) ( E E )... (T ) m[ d ] E m [ d ].. ; (T 3 ) where: d ( ).. E E E.. and: d. E E. ( )... The terms d and d.. show the mact on ortfolo value f each rosect s tested, rather than deferred, after the falure of the other. A negatve value ndcates that deferral s otmal. We evaluate each of these three comonents searately, then combne results. It s straghtforward to show (cf. the case): ( ) ( ) T > 0. We roceed to T, where due to the common-rsk structure, we have:

25 E E E E 0. I.e., confrmaton of any rosect confrms the common factor on all remanng rosects. After substtutng these nto T, we get: snce > by assumton. T ( )E 0, Regardng T 3, there are three ossble cases to consder. ase A: ether d nor d.. s negatve (nether s deferred by falure of the other). T 3 then takes the form: E E E E T 3 ( ) E E. But, the common-rsk structure mles: E E E > 0. After makng ths substtuton and cancellng lke terms, T 3 reduces to: ( ) ( )( E ) T3 (T T ) < 0 Thus, n ase A: A T T T3 0 Thus, f falure of nether rosect would cause the other to be deferred, the order s of no consequence; they could be tested smultaneously. ase B: Both d and d.. are negatve (ether s deferred by falure of the other). T 3 then takes the smle form: T 3 E 0 (snce E cannot be negatve), whch when combned wth T and T gves: 3

26 B T T E 0. > Thus, f falure on each rosect would cause the other to be deferred, the hghest exected value (and most lkely) should be tested frst. ase : Only d. s negatve (only one s deferred by falure of the other). The fact that t s the st rosect that would defer the th can be deduced from the ase A result, where we showed: T whch mles: ( T ) 0 3 d E d.. T <, d d. < E 0.. ow, f d and d.. are to dffer n sgn (as ase requres), then t must be that d <. 0 whle d. > 0. Thus, t must be the st rosect that has the ower to defer the th. We can now easly evaluate T 3 n ase by reference to ase A: what entered there nto T 3 (and therefore ) as d enters here as 0. All else remans the same.. Thus, we can smly subtract ths term from the ase A result to obtan: A 0. d >. Thus, f falure on only one of the rosects s nformatve enough to cause deferral of the other, the most lkely (and hghest ntrnsc value) rosect would be the nformatve one, and t should be tested. Proof of Theorem : Theorem establshed that the value of the ortfolo s gven by Π, whch can be comuted drectly usng the decson tree aroach. We kee n mnd that f the st rosect succeeds, then all rosects wll be tested, and f the st rosect fals, no more 4

27 5 wll be tested. (We assumed that the st would condemn the nd, but the nd would otmally follow the st under Theorem, thus no other rosect could follow the st but the nd. In other words, f the st has the ower to condemn the nd, then t has the ower to condemn them all. Thus, we can comute the value of the entre ortfolo as follows: ( ) Π. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Π O 0. Proof of Theorem 3: The frst m rosects wll be tested smultaneously, en block. Prosect m (and all remanng rosects) would be condemned unless at least one success occurs among the frst m trals n whch case rosect m (and all remanng rosects) would be tested. Thus, we can wrte the value of the ortofolo as: ( ) ( ) Π Π m m m m ).0( ) 0( *,

28 where s the robablty of at least one success among the frst m trals, and ( ). 0( m) 0 m reresents the robablty that the th rosect (>m) succeeds gven that there was at least one success among the frst m trals. Ths exresson can be smlfed as follows: Π m ( ) ( 0( m) ( ) ) 0( m) m ( ) ( 0( m) 0( m) ) Π m 0 0( m) ( m) 0( m). m 6