54 Journal of Marine Science and echnology, Vol. 13, No. 1, pp. 54-60 (2005) HE DEERMINAION OF POR FACILIIES MANAGEMEN FEE WIH GUARANEED VOLUME USING OPIONS PRICING MODEL Kee-Kuo Chen Key words: build-and-lease conrac, guaranee volume, real opions, blackscholes formula. ABSRAC his paper proposes a procedure o calculae he por faciliies managemen fee (MF) in a build-and-lease (BL) conrac wih guaraneed volume (GV). he MF pricing problem exiss in he conracs concluded by he lessees of por faciliies and harbor bureaus for a long ime. he problem is solved by analyzing he propery of GV firsly, and hen aking he real opion approach o find he MF in BL conracs wih GV. Finally, o demonsrae he mehod a real BL conrac wih GV is provided and is MF is calculaed in his paper. INRDUCION Build-and-Lease (BL) is one of he mos imporan approaches o operae he faciliies by harbor bureaus under he por privaizaion policy in aiwan. Under he radiional leasing conracs, harbor bureaus consruced he faciliies, and lessees pay annual renals and he managemen fees (MF). hese charges depend on wha kind of faciliies leased and how many volumes of raffic handled. In he BL conracs, he harbor bureau rens a zone o he lessee and also allows he lessee o build and use necessary faciliies in his zone during he conrac period. he ownership of faciliies consruced by he lessee, however, belongs o he harbor bureau. Insead of paying he consrucion cos of he faciliies on lump sum basis, he harbor bureau exemps he lessee from paying he annual renal in he BL conrac period. he lengh of such exempion is deermined in such a way ha all he consrucion coss be covered by he lessees. he lengh of an exempion usually is longer han 10 years because in he mos cases he consrucion coss of por faciliies are usually greaer han 10 imes of annual renals. Paper Submied 02/17/04, Acceped 01/13/05. Auhor for Correspondence: Kee-Kuo Chen. E-mail: kkchen@mail.nou.edu.w. *Associae Professor, Deparmen of Shipping and ransporaion Managemen, Naional aiwan Ocean Universiy. In order o mainain he sabiliy of annual revenue, harbor bureaus usually require ha a yearly guaranee volume (GV) be included in BL conracs. herefore, he minimum amoun of oal MF a lessee has o pay in a year is equal o he GV muliplied by per uni MF sipulaed in a paricular conrac in spie of he fac ha he lessee s annual operaion quaniy migh fall below he GV. In view of he GV, harbor bureaus usually give a MF discoun as a reward o he lessees as long as he annual volume handled exceeds GV. Bu here exiss no exac rule or formula o deermine he MF discoun. he MF discoun was deermined case by case in pracice. As a resul, here migh be cases in which similar conracs migh have significanly differen discouns. he range of MFs in BL conracs signed by Keelung, aichung and Kaohsiung harbor bureaus in he pas decades are summarized in able 1 [5, 6, 12]. If here were no GV, MF for he same goods in he same harbor should be much similar. Bu, as shown in able 1, here could be much differences beween MFs in differen BL conracs even hey were signed by he same harbor bureau for he same cargoes handled. For example, MFs of conainer could have a difference of 20% of oal annual lease paymen in BL conracs signed by Keelung harbor bureau. Ineviably, he differences have caused many dispues beween harbor bureaus and lessees who paid higher MFs han ohers. As a resul, he lessees used o call for a reasonable sandard mehod o compue MF discoun. Harbor bureaus are also eager o solve his problem. In his paper Black-Scholes call opion formula is applied o evaluae he value of GV engaged in a BL conrac and hen o derive a formula o deermine he MF discoun. his resul can provide a consisen sandard o calculae he MF discoun for any level of GV. his paper is organized ino six secions. he naure of BL conrac wih GV will be analyzed in he nex secion. he reason ha real opion analysis (ROA) mehod is an appropriae approach o evaluae he value
K.K. Chen: he Deerminaion of Por Faciliies Managemen Fee wih Guaraneed Volume Using Opions Pricing Model 55 able 1. A summary of he MFs range in aiwan harbor bureaus Por Cargoes Min. MF Max. MF Keelung cemen 29 N/on 31 N/on liquid 26 N/on 30 N/on general cargo 15 N/on 40 N/on conainer 10% of oal annual 30% of oal annual lease paymen lease paymen consrucion and 10% of oal annual 30% of oal annual building lease paymen lease paymen oil 24N/on aichung Cemen 20N/on 23 N/on liquid 21 N/on 23 N/on general cargo 15.7% of handling 18.2% of handling charge charge conainer fron yard G13%~20% of sum of handling charge and equipmen expense back yard: 140 N/conainer grain 12% of operaion 14% of operaion revenue revenue coal (general) 15 N/on coal (EC) 26.16 N/on oil 17.66~30N/m 2 or 10N/on Kaohsiung cemen 22 N/on 35 N/on liquid 2.1 N/on 61 N/on general cargo 2%~4% of operaion revenue conainer 10% of oal annual lease paymen grain 10% of oal annual lease paymen consrucion and 10% of oal annual building lease paymen oil 11 N/on iron 4.3 N/on Source: Saisical Absracs 2003 of Keelung, aichung, and Kaohsiung [5, 6, 12]. of GV will be illusraed in Secion 3. In Secion 4, a formula o deermine he MF discoun in a BL conrac wih a paricular level of GV will be derived. Finally, a real case will be sudied using he formula derived from his paper. he las secion gives a brief conclusion of he presen sudy. HE NAURE OF BL CONRAC WIH GV Under he radiional conrac, harbor bureau migh inves by iself, say, C 0, o build he necessary faciliies for leasing. Assuming ha here is no echnology advanage o lessee o build he faciliies, he lessee also has o spend C 0 o build hem. Hence, we can assume ha, in a BL conrac, he presen value of he sum of annual renals paid by he lessee o he harbor bureau mus also be C 0. Oherwise he conrac should no be concluded. Le R be he oal renal of year under a BL conrac. I is also assumed ha R is also he harbor bureau s annual amorizaion of faciliies consrucion cos. Moreover, le r be he cos of capial of he lessee. Under he assumpion of value maximizaion objecive of harbor bureau [9], he BL conrac period,, can be deermined by he following equaion: R C 0 = Σ (1 + r) (1) Based on he heory of capial budgeing, he value of a projec can be represened by he ne presen
56 Journal of Marine Science and echnology, Vol. 13, No. 1 (2005) value (NPV) of he incremenal free cashes creaed by his projec. he free cash flow is defined as [3]: Free cash flow = Earning before ineres and axes (EBI) Cash axes on EBI + Incremenal accrued axes + Depreciaion Capial expendiures Incremenal operaing working capial (2) Wihou loss of generaliy, i is assumed ha is an ineger and MF = MF 0 + P Q (3) MF 0 is a consan, and P is he MF per handling uni of year. If MF does no depend on he volume of cargo, is equal o 0. o simplify, i is also assumed ha here are no side effecs on he oher revenues of he harbor bureau and no accrued axes. Because harbor bureau does no have o pay income ax, and because here is no reason o believe ha depreciaions, incremenal working capials and addiional capial expendiures are differen in differen BL conracs and in radiional conracs, wihin hese periods, he annual free cash flow coming from a BL conrac can be expressed as follows: Free cash flow = Renal Amorizaion of faciliies consrucion cos + MF = R R + MF = MF = MF 0 + P Q (4) for all = 1, 2,...,, (renal amorizaion of faciliies consrucion cos ) is considered as he EBI of his conrac. Hence, he value of his conrac can be deermined by calculaing he Ne Presen value (NPV) of MF. ha is, NPVMF NPVMF = Σ MF 0 + Σ (1 + r f ) E (P Q ) (1 + r p ) = MF 0 r 1 1 1 + f Σ (1 + r f ) E (P Q ) (1 + r p ) (5) r f is he risk-free ineres rae wih consan MF 0, is an appropriae risky ineres rae depending on he inrinsic risk of P Q [9], and E( ) is he noaion of expecaion operaion. In a BL conrac wih GV, le be he GV of he year, hen he free cash flow of he year becomes: Free cash flow = max{mf, GV P } = max{mf 0 + P Q, GV P } = MF 0 + max{0, (GV Q ) P MF 0 } (6) which is differen from equaion (4). REAL OPIONS Since GV in a BL conrac reduces he conrac risk involved by harbor bureau, he radiional discouned cash flow (DCF) approaches o he appraisal of capial invesmen projec, such as equaion (4), can no properly capure he characerisics of his problem. GV gives harbor bureau he righ o sell he annual MF and receive he guaraneed revenue of GV P. If MF is greaer han GV P, he value of GV would be worhless and harbor bureau has he same annual paymens wheher hey have BL conrac wih or wihou GV. Oherwise i would be worhy of GV P MF which is greaer han 0. So he value provided by GV would be posiive. Hence, he value of BL conrac wih GV = Value of BL conrac wihou GV + Value of GV (7) Several echniques are available for evaluaing he value of GV. hey are, for example, NPV, Decision ree Analysis (DA), Real Opions Analysis (ROA), ec. However, Copeland and Anikarov [4] considered ha NPV can no capure he value of flexibiliy in managemen. McDonald and Siegel [8] and rigeorgis [13] showed heir examples ha NPV rule always underesimaes he value of invesmen projecs when hey involve managerial flexibiliy. rigeorgis also menioned ha DA rule could no adjus he discoun rae
K.K. Chen: he Deerminaion of Por Faciliies Managemen Fee wih Guaraneed Volume Using Opions Pricing Model 57 o reflec he change of risk in projecs. Unlike NPV and DA, real opions analysis (ROA) can be saed ha he value of he projec resuling from ROA already includes he value of opion due o uncerainy and flexibiliy in managemen. ROA is a sysemaic and inegraed decision analysis process used o evaluae he invesmen projec wih managerial flexibiliy. I is he echnique ha exends from he financial opion heory, which is adoped in he sock marke, o be applied in real invesmen. Currenly, ROA is already acceped as an evaluaion process for projec under uncerainy in various fields. For example, Pichayapan, Kishi, Hino and Saoh [11] used ROA o evaluae he expressway projecs in Hokkaido, Japan. McCormeck and Sick [7] adoped ROA for valuing undeveloped reserves in oil and gas indusry. Yamagushi, akezawa and Sumia [14] used ROA o analyze he land developmen in okyo. Concas, Glaesel, Reich and Yelds [2] valued he economic impac of ransporaion research aciviies using ROA approach. Brand, Mehndiraa and Parody [1] used ROA o analyze he risk in ransporaion planning. However, here has been a lack of sudy applying ROA in he por planning and managemen field ye. On he oher hand, researches on he problem of pricing MF or relaed por faciliies have no found eiher excep he fac ha he formula of DCF has always been used by harbor bureaus o evaluae various projecs in pracice. Because ha when GV exiss in a BL conrac, he annual cash flows and is risk, and herefore is discoun rae of his conrac will be differen from ha of a radiional conrac. hese changes can no be refleced in he radiional DCF and DA mehods. Real opion mehod considers he changes of annual cash flows in is cash flow equaion (6), and uses he risk-free rae of reurns as is discoun rae o solve he problem of changing discoun raes. By means of real opions, a value is assigned o he opions a he managemen s disposal, GV. his GV value can be deermined in a manner ha is similar o he valuaion echniques for financial opions. A summary of he models for opion valuaion is described by Mun [10]. he real opion problem can be solved by solving he parial differenial equaion (for example, he Black and Scholes model), by dynamic programming (for example, he binomial opion model) or by simulaion (for example, Mone Carlo simulaion). As a general rule, binomial rees are frequenly applied in real opion valuaion, as hey allow simulaneous valuaion of various opions and pu less resricions on he disribuion of he underlying value [6]. DEERMINAION OF MF IN A BL CONRAC WIH GV In he problem of GV evaluaion, usually he annual uni MF, P, is aken as a consan in he whole conrac period, ha is, P = P 0, for all in he conrac period. Because GV in BL conrac is irrecoverable and is known a he beginning of he conrac, and he exercise price of he pu harbor bureau obained is fixed, he value of GV can be evaluaed by European pu formula [13]: p(s, ) = Xe r f N( d 2 ) SN( d 1 ) (8) ln X S + r + σ 2 d 1 =, σ d 2 = d 1 σ p = price of he pu S = price of underlying asse X = exercise price r f = risk-free ineres rae = ime o mauriy of he opion in years σ = sandard deviaion of he annualized coninuously compounded rae of reurn on he underlying asse ln = naural logarihm e = he base of he naural log funcion N (d) = he probabiliy ha a value draw randomly from a sandard normal disribuion will less han d In he case of GV valuaion problem, p = value of GV S = Q 0 P 0 = he faciliies ren a he poin of ime when he conrac is arranged X = GV P 0 r f = ln (1 + he average annual ineres raed of bank loan) = ime o he evaluaed years σ = sandard deviaion of he annualized coninuously compounded rae of reurn on ha can be calculaed by: σ = ln(u) (9) Le sp be he NPV of GV, i is he value ha he lessee offers o harbor bureau and is fair o be paid o he lessee by harbor bureau. Hence he NPV of MF o harbor bureau in a BL conrac wih GV should be equal o NPVMF sp, say ν. Le p * be he uni MF in a BL conrac wih GV, he following wo formulas can be
58 Journal of Marine Science and echnology, Vol. 13, No. 1 (2005) obained: Formula 1. he uni MF in a BL conrac wih GV P * = A = v A + B C Σ GV (1 + r f ), B = Σ Q 0 N (d 1 ), (10) S = Q 0 P * X = GV P * d 1 = ln Q 0 + r'f + σ 2 GV 2 d 2 = d 1 σ σ σ = sandard deviaion of he annualized coninuously compounded rae of reurn on Q P * ha can be calculaed by σ = ln(u) C = Σ GV e r'f N (d 2 ).and, r' f = ln(1 + r f ), makes he value of MF in he BL conrac equal o ν wih annual GV and a geomeric operaion quaniy Q having a value Q 0, a he beginning of conrac period wih an up movemen facor u > 1. Proof. Because of he requiremen of annual GV, he annual MF can be wrien as Max{GV P *, Q P * } = GV P * + Max{0, Q P * GV P * } = 1, 2,...,. he firs erm in he righ-hand side of he above equaion sipulaed a he beginning of he conrac period should be reaed as a riskless asse. Is NPV can be compued as follows: NPV (GV P * ) = GV P * (1 + r f ). he second erm can be reaed as a call opion wih underlying asse Q P * and exercise price GV P *, under he assumpions of Q having a value Q 0 a he beginning of conrac period wih an up movemen facor u > 1. his erm can be evaluaed by Black- Scholes European call opion formula as follows: [13] C = value of Max{0, Q P * GV P * } = SN (d 1 ) Xe r'f N (d 2 ) (2) herefore, r' f = ln(1 + r f ) N (d) = he probabiliy ha a random draw from a sandard normal disribuion will less han d ν = Σ Max{GV P *, Q P * } A = = Σ GV P * + Σ (1 + r f ) (Q 0 P * ) N (d 1 ) Σ GV P * e r'f N (d 2 ) = P* (A + B C) Σ GV (1 + r f ), B = Σ Q 0 N (d 1 ), and C = Σ GV e r'f N (d 2 ). his implies ha P * = ν A + B C Q.E.D. Formula 2. Under he same assumpions made in Formula 1, he following resul can be derived:
K.K. Chen: he Deerminaion of Por Faciliies Managemen Fee wih Guaraneed Volume Using Opions Pricing Model 59 MFD = P * P 0. represened by A CASE SUDY o demonsrae he mehod proposed in he above secion, a real case of conrac concluded by a shipping company, say Company A, and aichung Harbor Bureau on July 1, 2000 is provided below. [12] Currenly, he faciliies charge calculaion mehod used by aichung Harbor Bureau is arbirary. ha is, here exiss no rule o deermine how large he MFD should be offered o he faciliies lessee when a conrac includes a GV agreemen. Hence, insead of presening he complee conens of he conrac, only he basic informaion relaed o he evaluaion of he value of GV is described. In his conrac, i was arranged ha Company A was responsible for building faciliies, composed of a wharf, hree silos, road, and digging a waer way. heir coss were N$ 300,000 housands, N$ 941,170 housands, N$ 13,667 housands, and N$ 84,000 housands, respecively. he oal consrucion cos is N$ 1,338,837 housands. he iems of annual lease paymens are lised in able 2. he oal annual lease paymen was N$ 197,919 housands. I was esimaed ha he firs year operaion quaniy would be 1,000,000 ons and GV was also 1,000,000 ons. he uni price of MF was N$ 37 per on in he conracs wihou GV. Boh paries agreed ha he discouned rae of harbor bureau was 8%, he cos of capial for Company A was 10% and he uni price was fixed in he conrac period. Based on he procedure proposed in previous secion, i is necessary o calculae he conrac period,, by equaion (1) and esimae he volailiy of annual operaion quaniy, Q. In his case, C 0 = N$ 1,338,837 housands, R = N$ 197,919 housands, and r = 10%. By solving equaion (1), = 11.8 years is obained. Suppose ha Company A and aichung Harbor Bureau agreed ha he operaion quaniy model can be able 2. Annual lease paymens included in he case sudy BL conrac Ren of wharf N$ 30,000 (housands) Ren of silos 94,117 Ren of road buil 1,367 Ren of land 8,495 Mainenance expenses 35,211 Insurance expenses 13,209 Oher expenses 15,520 oal N$ 197,919 (housands) Source: A BL conrac concluded by Company A and aichung Harbor Bureau, 2000. E(Q ) = Q 1 e rq, (12) and also suppose ha boh paries agreed ha he operaion quaniies are expeced o grow on average a a consan rae of 6% and wih 95% confidence, he acual operaion quaniy would no be below he curren level for he nex 11.8 years. Based on hese esimaes he value of he operaion quaniy volailiy can be derived by he following equaion: Σ ρ i ln Q lower i Q 0 σ = (13) 2 ρ i, i = 1, 2,...,, are he expeced growh raes, lower and Q is he lower 95 h percenile value of Q. Subsiuing he above esimaes ino equaion (13), we have σ = 10.8 0.06 ln 1000000 1000000 = 0.099 2 10.8 If he average risk-free ineres rae is 5%, hen r'f = 0.049. Subsiuing hese parameers o equaion (8), we obain he annual presen values of GV for he nex 11.8 years lised in able 3. he oal presen value of GV in his conrac, sp, is N$ 6,838 housands. I is noed ha he value almos comes from he firs year. his fac can be easily realized because, when he ime passes, he operaion quaniy is expeced o increase so ha he probabiliy ha operaion quaniy less han he level of GV becomes very small. On he oher hand, he value, c, is increasing when he ime inerval becomes longer and longer. he oal amoun B C is N$ 3,410 housands. he value of A is equal o N$ 8,729 housands. he NPVMF of his conrac is N$ 347,390 housands calculaed by equaion (5) if GV agreemen was no conained in his conrac. Subracing sp from his value, ν = N$ 340,550 housands is obained. Hence, and P * = ν A + B C = 28.1 (N$) MFD = 28.1/37 = 0.76. CONCLUSION he dispues beween por faciliies lessees and four harbor bureaus in aiwan wih respec o MF in he
60 Journal of Marine Science and echnology, Vol. 13, No. 1 (2005) able 3. Annual NPV of GV uni: N$ 1,000 1 2 3 4 5 6 7 8 9 10 11 11.8 oal d 1 0.50 1.14 1.56 1.90 2.19 2.45 2.69 2.90 3.10 3.29 3.46 3.61 d 2 0.00 1.00 1.39 1.70 1.97 2.21 2.43 2.62 2.8 2.97 3.13 3.27 p 6197.5 333 151.7 81.4 33.3 33.3 2.96 1.48 1.48 1.48 0.37 0.37 6838 c 215 112 151 189 226 262 297 331 363 392 422 450 3410 BL conracs wih GV have been prevailing for a long ime. Alhough his issue has been discussed for long ime, he resoluion o he problem has no been found ye. In his paper, real opions approach was used o analyze he problem and a procedure was proposed o evaluae MF in BL conracs wih GV. he firs sep in he procedure is o evaluae he value of MF and NPVMF in he conrac by he radiional discouned cash flow mehod as if i was a BL conrac wihou GV. Afer ha, he propery of GV was analyzed and found ha GV offered by he lessee o harbor bureau resembles he fac ha harbor bureau ges a pu opion from faciliies lessee. Hence, i is suiable o use Black-Scholes pu formula o evaluae he value of GV, sp. Moreover, he value of a BL conrac wih GV should be equal o NPVMF sp. Nex, i was poined ou ha he cash flows in a BL conrac wih GV is he same as he cash flows ha could be goen from buying a consan annuiies and a call opion. In his poin of view, Black-Scholes call formula was applied, and he calculaion formulas of P * and MFD were derived. Finally, a real case of conrac was invesigaed and is MF was calculaed using he proposed mehod. I was found ha he final MF calculaed by Formula 1 was N$ 28.1 per on, which is only 76 percen of he original MF of N$ 37 per on. he real opions mehod applies financial opions heory o quanify he value of managemen flexibiliy under he condiion of uncerainy. his mehod was applied in various fields successfully. In his paper an academic reasoning why his mehod can be applied o pricing MF for a BL conrac wih GV was explained and he pricing formula based on his mehod was successfully derived. REFERENCES 1. Brand, F., Mehndiraa, S.R., and Parody,.E., Opions Approach o Risk Analysis in ransporaion Planning (ransporaion Research Record 1706), BR, Naional Research Council, Washingon, DC (2000). 2. Concas, S., Glaesel,., Reich, S.L., and Yelds, A.., Valuing he Economic Impac of ransporaion Research Aciviies Using a Real Opions Approach (RB 2003 Annual Meeing CD-ROM), RB, Naional Research Council, Washingon, DC (2003). 3. Copeland,., Valuaion in Pracice, Recen rends in Valuaion, John Wiley and Sons, New York, pp. 35-81 (2003). 4. Copeland,. and Anikarov, V., Real Opions, exere Publishing, New York (2001). 5. Kaohsiung Harbor Bureau, 2003 Saisical Absrac, Kaohsiung, aiwan (2003). 6. Keelung Harbor Bureau, 2003 Saisical Absrac, Keelung, aiwan (2003). 7. McCormack, J. and Sick, G., Valuing PUD Reserves: A Pracical Applicaion of Real Opion echnique, J. Appl. Corpor. Financ., Vol. 13, No. 4, pp. 8-13 (2001). 8. McDonald, R. and Siegel, D., Opion Pricing when he Underlying Asse Earns a Below Equilibrium Rae of Reurn: A Noe, J. Financ., Vol. 39, No. 1, pp. 261-265 (1984). 9. Megginson, W.L., Corporae Finance heory, Addison- Wesley Publishing, MS (1997). 10. Mun, J., Real Opions Analysis: ools and echniques for Valuing Sraegic Invesmens and Decisions, John Wiley and Sons, Hoboken, NJ (2002). 11. Pichayapan, P., Hino, S., Kishi, K., and Saoh, K., Real Opion Analysis (ROA) in Evaluaion of Expressway Projecs Under Uncerainies, J. Eas. Asia Soc. ransp. Sud., Vol. 5, No. 2, pp. 3015-3030 (2003). 12. aichung Harbor Bureau, 2003 Saisical Absrac, aichung, aiwan (2003). 13. rigeorgis, L., Real Opions: Managerial Flexibiliy and Sraegy in Resource Allocaion, he MI Press, Cambridge (2002). 14. Yamagushi, H., akezawa, N., and Sumia, U., he Real Opion Premium in Japanese Land Prices, Proceeding 4 h Annual Real Opions Conference, Universiy of Cambridge (2000).