PROJECT RISK SIMULATION UNDER UNCERTAIN CONSTRUCTION DURATION. Kun-Jung Hsu

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1 Proceedigs of the 004 Witer Simulatio Coferece R.G. Igalls, M. D. Rossetti, J. S. Smith, ad B. A. Peters, eds. PROJECT RISK SIMULATION UNDER UNCERTAIN CONSTRUCTION DURATION Ku-Jug Departmet of Costructio Techology Leader Uiversity No.88 Sec. 5, A-Chug Rd. Taia, 709, TAIWAN ABSTRACT This paper establishes a two-phase model to explore the fiacial risk of costructio project uder ucertai costructio duratio. Treated as a radom variable i the model, the costructio duratio affects the radom preset worth of cash flow i both costructio ad operatio phases. A illustratio of BOT project uder ucertai costructio duratio was utilized for simulatio. The simulatio result was applied to reveal how ucertai costructio duratio affects project risk. The paper also shows how the stadard deviatio of the icreases wheever the variace of costructio duratio rises, thus accetuatig the cotigecy of the project. The results aalysis exteded to the discussio of its effect o the probability of egative ad its theoretical meaigs. INTRODUCTION Capital ivestmet ofte ivolves substatial costructio. Therefore, the fud eeded i the iitial stage is huge. Usually, after the costructio iitiated, i case of serious delay or work suspesio, the startig time for iitial busiess operatio is doomed to procrastiate. It implies detrimetal iflueces, particularly for a chartered cotract with fixed term. Usually, a costructio project is assumed to be completed o schedule withi predetermied budget, ad operates toward the ed of the project term uder the desiged capacity. So, it is commoly see i the busiess of capital ivestmet to iclude costructio project maagemet team credited with sizable resources. I such case, costructio work items are documeted i detail ad charted with accuracy. These maagemet activities are executed to make sure the works get completed o schedule. Most of the et preset value ( calculatios i the project evaluatio literatures assumed fixed costructio duratios. The risks geerated from this variatio that could affect the project evaluatio is still lack discourse i theory. As for risk aalysis, it is formulated by assessig the probability distributio of the factors affectig fiacial ecoomic criterio for the alterative selected (Hillier 963, Hertz 964, Hespos ad Strassma 965, Wagle 967, Hillier 97, Greeberg 98. The risk geerated from the variatio of costructio duratio could affect both the preset worth cash flow i costructio phase ad operatio phase. This correlated effect is ofte metioed i the realm of professioal practice, but ot thoroughly explored ad clarified. This paper first presets a two-phase cash flow model depictig the resulted from project fiace activities. The costructio duratio is treated as a radom variable i the formulatio of the model, which is used to derive the risk magitude affectig, ad the probability of resultig i egative. A illustrative example of BOT project uder ucertai costructio duratio was the utilized for simulatio. The coclusio remark is draw i the fial sectio. THE MODEL is the most popular idex i project evaluatio. I most BOT projects, the geeral cotractor faces a fixed period of implemetatio i accordace with the relevat cotract. I such cases, the period of the project is limited by N-year cocessios. If A j is the et cash flow arisig i year j coected with a capital ivestmet project. The of a project is A =, ( N j j= 0 ( + i where i is discrete discout rate. The of such project ca be preseted as cotiuous cash flow of period from 0 to N, ad ca be rewritte as defiite itegral form i equatio ( N 0 j j -rj = A e dj, (

2 where r is the omial cotiuous-compoud iterest rate, ad e is the base of atural logarithms. For practical purposes, we ca use r = l (+ i i order to relate i to r. Let represets the ed of the costructio phase at which the positive et cash-iflow of the operatio phase begis. Geerally speakig, project ivestmet is always plaed to geerate icome that covers the operatig cost ad the debt services resulted from project related fiacig activities. I the begiig of a costructio project, it requires a sizable fud to cover the acquisitio of lad, machiery, materials, egieerig ad costructio work. Meawhile, i most cases, there is o icome geerated from the projects. The cost of a project starts to accumulate as soo as the capital is ivested. The magitude of risk grows i proportio to the accumulated cost throughout the etire costructio duratio. I case of work suspesio durig costructio at ay time, the deferred cash icome for the project becomes uavoidable. So let j=0- represet period of the costructio phase, ad j= -N represet period of the operatio phase. The of such a BOT project thus ca be represeted as a cash flow model with a twophase framework, ad rewritte equatio ( as = -rj N -rj 0 A j e dj + A j e dj. (3 The first term represets the total preset value of the cash flow durig the costructio phase, ad the secod term represets the total preset value of the cash flow durig the operatio phase. I equatio (3, A j represets the et cash flow that occurs i year j. A j ca be treated as the summatio of cash iflows (icludig operatio icome, retal icome etc. ad cash outflows (icludig operatio expeditures, maiteace fees etc. I the cotext of project evaluatio, operatio expeditures ad maiteace fees are usually treated as a compoet of the operatio icome or the retal icome, respectively. A project evaluatio may also assig a costat growth rate for a reveue or expediture item. Also, the et cash flow of operatio expeditures ad maiteace fees from period to N i the operatio phase ca be calculated as the aual value i the evaluatio. I order to cotrol for other variables costat ad explore the effect of the ucertai costructio period o project risk both i aalytics ad simulatio model, we eed some reasoable simplificatio. For example, the risk durig the operatio phase is assumed to be egligible. This simplificatio results i a more parsimoious model without ay serious loss of realism. If the total costructio cost C is paid at the ed of costructio phase, the PV of the sigle paymet at is -rj C e. Let a represets a uiform cash flow i the time horizo of the operatio phase. The ca be represeted as - r N - rj e a e = C + dj. (4 Equatio (4 ca be simplified as -r a -r -r N = Ce + ( e - e. (5 r The first term i equatio (5 represets the preset value of the total costructio cost paid at the ed of the costructio phase (PV. The secod term represets the preset value of the cotiuous uiform cash flow of the operatio phase (PV. Mathematically, it ca be iterpreted as the preset value of the same perpetual auity, a (from time N oward beig subtracted from the preset value of a ifiite sequece of the aual cash flow a (from time oward. 3 THE MARGINAL EFFECT OF UNDER UNCERTAIN CONSTRUCTION DURATION I equatio (5, wheever costructio duratio postpoe, costat N-year cocessios will iduce a loss of cash iflow i operatio phase. But costructio duratio postpoe will also lower the preset worth of the total costructio cost because of delay paymet ad save the time cost of early paymet. So wheever costructio duratio postpoe, if loss of cash-iflow i operatio phase ca be compesated by iterest savig of total costructio cost paymet i costructio phase, the postpoe of costructio duratio will icrease the project value. But if the savig of total costructio cost i costructio phase caot compesate the losses of cash-iflow i operatio phase, the postpoe of costructio duratio will decrease the project value. But wheever is a radom variable ad with a give probability distributio. The expectatio ad variace of preset value of the cash outflow of C which depeds o the costructio duratio will be more complicate. Let expected preset value of C deoted as E [ PV ( ], ad is give by -r -r [ PV ( ] E[ C e ] = E[ C ] E[ e ] E =. (8 Sice the preset value factor is ot a liear fuctio -r -r [ ] of ; E[ e ] e E, we caot calculate E [ PV ( ] by isertig E( i i (8. The expec-

3 -r tatio of e ca be writte as the Laplace Trasform of the probability desity fuctio E -r ( e = L ( r -r The variace of e ca be writte as ( r L ( r -r Var( e L - Equatio (8 ca be rewritte as. (9 [ ] =. (0 [ PV ( ] L ( r E µ c =. ( Var[EXP(-r] Ucertai Costructio Duratio: Var[EXP(-r] vs. var(exp(-r 5 9 The variace of the preset value of such a sigle paymet is c c c [ [ L ( r ( r ] [ L ( r ] r r [ PV ( ] = ( µ + Var( e + E( e ] Var = ( µ c + c (L + c -. ( The Laplace Trasform of differet probability desity fuctios has bee explored by Youg ad Cotreras (975 ad Rosethal (978. Give a probability distributio fuctio of -r, ( -r E e ad Var ( e ca be derived usig the Laplace Trasform. Let µ represets mea value of costructio duratio, oe ca simulates Var( e vs. µ -r ad -r -r Var( e vs. r. The results of simulatio Var( e vs. µ represet i Figure, which show a decreasig curve. But the o-liear relatioship betwee -r Var( e vs. r will be more complicated. The paper -r simulates Var( e vs. r ad shows results the oliear relatioship i Figure. The o-liear relatioship -r shows the directio of margial effect o Var( e vs. r will be o-defiite. The variace of the of the total project will be affected by margial chage or the variace of. But, the compoet of the effect will be more complicatedly. I such situatio, the paper give a illustrative example to simulate the two-phase model i Sectio 4, the discusses the simulatio results i details thereafter. 4 SIMULATION ON ILLUSTRATIVE EXAMPLE I most BOT projects, the geeral cotractor faces a fixed period of implemetatio i accordace with the relevat Var[EXP(-r] Figure : Ucertai Costructio Duratio: -r Var( e vs. µ Ucertai Costructio Duratio: Var[EXP(-r] vs. r % 5% 9% 3% 7% % 5% 9% r var(exp(-r Figure : Ucertai Costructio Duratio: -r Var( e vs. r cotract. The period of the project is limited by N-year cocessios. The fiacial idex of a BOT-Project ca be represeted as a cash flow model with a two-phase form, which has bee discussed i Sectio 3. I order to reveal how costructio duratio ucertaity affects project risk, the simulatio case i this paper was largely simplified from the case study of Taiwa High Speed Rail (THSR. So that result ad meaig exhibited here is applicable i similar case. Bureau of Taiwa High Speed Rail (BOTHSR implemeted BOT policy ad ivited private sectors to fiace ad costruct the HSR Project i 996. THSRC ad the ROC Miistry of Trasportatio ad Commuicatios siged the BOT Agreemet. Govermet cocessios awarded to THSRC the costructio ad operatio of the 345 km HSR Project, the maagemet of subsidiary commercial eterprises, ad the developmet of the five statios

areas (iclude 35 year cocessios of Costructio ad Operatio, the Statio Zoe Developmet Agreemet, 50 years cocessios of Busiess Developmet Lad Agreemet, the Memoradum o Govermet Commitmets, ad the

4 areas (iclude 35 year cocessios of Costructio ad Operatio, the Statio Zoe Developmet Agreemet, 50 years cocessios of Busiess Developmet Lad Agreemet, the Memoradum o Govermet Commitmets, ad the Cotract Executio Memoradum. By the year 000, THSRC etered ito all major civil works cotracts. The costructio commeced soo after, ad scheduled to operate at the ed of 005. Total costructio budget for HSR project approximates 44.9 billio NTD ( com.tw/emai/hsr/hsr.asp. I HSR Project, the operatio expediture is represeted as the fuctio of ridership ad the growth rate of ridership. Let C be the operatio expediture of the first year operatio phase, g is the growth rate of ridership, the preset value of total operatio expediture (PV ca be writte as PV + ( + g ( + r = C * ( + r r g. (3 Let R represets operatio icome of the first year operatio phase, g is the growth rate of average ridership price, the preset value of total operatio icome (PV ca be writte as PV ( + g ( + r r g * N N R * ( + r * =, (4 where g* = g + g +g g. For simplificatio, let average icome of the total the statio zoe developmet ad busiess developmet lad be costat (R. The preset value of total developmet (PV 3 ca be writte as effect o totalig the statio zoe developmet ad busiess developmet lad. Table : Risk Profile of the Project uder Ucertai Costructio Duratio (Uit: 00 Millio NTD CL(% Value 5% % % % % % % % % % % % %.54 70% % % % % % Miimum Maximum Mea Std Dev 8.44 Variace 33,83.86 PV ( + i i( + i 3 = R * ( + i. (5 So oe ca decompose the ad rewrite equatio (5 as = VP + PV + PV + PV 3. (6 The risk profile of THSR Project uder ucertai costructio duratio is show i Table, Figure 3, Figure 4. I which Mote Carlo Simulatio was applied usig ad ru 0000 iteratios. The simulatio also shows the compoet of project risk uder ucertai duratio. I Table, the compoet shares of the project risk show that the mai cost risk derive from the effect o costructio cost. The mai reveue risk derive from the (Uit: 00 millio NTD Figure 3: PDF of the Project Risk uder Ucertai Costructio Duratio

5 which will icrease the probability of a egative project. This will icrease cotigecy of the BOT-Project. 5 CONCLUSION REMARK (Uit: 00 millio NTD Figure 4: CDF of the Project Risk uder Ucertai Costructio Duratio Followig the same process, it s easy to simulate how the variace of costructio duratio will icrease the variace project. So the stadard deviatio of the project icreases as well. Thus affect the probability areas of a egative of the project. Give the expectatio ad the variace of project, the probability of a egative of the project ivestmet ca be derived as - µ 0 - µ µ Pr( 0 = Pr ( - = Pr ( Z, (7 where - µ Z =. I equatio (7, the term o the right had side of the bracket icludes a egative sig, so that wheever the stadard deviatio of ( icreases, the critical - value of Z-Score ( µ for a 0 will also i- crease. Hece, the critical value of Z-Score icreases ad moves toward the right had side alog the axis of the probability distributio fuctio, the icreasig the area uder the probability distributio fuctio from - Z, Capital ivestmet ofte ivolves substatial costructio. Costructio project maagemet is executed to cotrol ad isure the work to be completed o schedule. Most of the et preset value calculatios i the project evaluatio literatures assumed fixed costructio duratios. But this paper allows the costructio duratio to be radom, which will be more realistically applicable for project maagemet i large capital ivestmet projects. This relaxatio provides profoud meaigs. A illustrative example of BOT-Project uder ucertai costructio duratio was the simulated. I the ed, the paper shows that wheever the variace of the costructio duratio icreases, the stadard deviatio of the project icreases as well. Hece, icrease the probability of a egative project. ACKNOWLEDGMENTS The author would like to thak Professor David Emauel Aderso for his Eglish grammar review of the paper. REFERENCES Bureau of Taiwa High Speed Rail Project Overview of the high speed railroad system i Taiwa. Available olie via < hsr.asp> [accessed Jauary 5, 004]. Greeberg, Joel S. 98. Ivestmet Decisios: The Ifluece of Risk ad other factors, New York: Research ad Iformatio Service, America Maagemet Associatios. Hertz, D.B Risk Aalysis i Capital Ivestmet, Harvard Busiess Review 4: Hespos, R. G. ad P. A. Strassma Stochastic Decisio Trees for the Aalysis of Ivestmet Decisios, Maagemet Sciece : Hillier, Frederick S The Derivatio of Probabilistic Iformatio for the Evaluatio of Risk Ivestmet, Maagemet sciece 9: Hillier, Frederick S. 97. The evaluatio of risky iterrelated ivestmets, Amsterdam: North-Hollad. Rosethal, Richard E The Variace of Preset Worth of Cash Flows Uder Ucertai Timig, The Egieerig Ecoomist 3: Wagle, B A Statistical aalysis of risk i ivestmet projects, Operatioal research quarterly 8: Youg, D. ad L. E. Cotreras Expected Preset Worths of Cash Flows Uder Ucertai Timig, The Egieerig Ecoomist 0:

6 Table : Compoets of the Project Risk Simulatio uder Ucertai Costructio Duratio (Uit: 00 millio NTD Cost Reveue Reveue from the statio zoe Name Costructio Operatio Operatio Costructio developmet Cost Expediture Icome Duratio ad busiess PV PV PV developmet lad PV 3 Mea -3, , , , Source 00% 0% % 99% Std Deviatio Source -75% -0% % 73% % 0% % 99% Source of Variace 8, , Variace -*4.94 -*4, *0.03 -*3.8 AUTHOR BIOGRAPHY KUN-JUNG HSU is Assistat Professor ad Director of the Departmet of Costructio Techology, Leader Uiversity. He received his Ph.D. degree i Civil Egieerig from Natioal Taiwa Uiversity. He is also a seior lecturer of the Graduate Istitute of Buildig ad Plaig, Taiwa Uiversity. Dr. had twety years experiece i professioal practice as a seior architect ad costructio maager. He teaches i Statistics, Regressio Aalysis, Ecoomics, Costructio Ecoomics, Quatitative Method, ad Costructio System for Architectural Desig, Architectural Desig, etc. His curret research iterests iclude housig ecoomics, costructio ecoomics ad special topics o project evaluatio. His address is <hsu@mail.leader.edu.tw>. Total Variace = 33,

Subject CT1 Financial Mathematics Core Technical Syllabus

Subject CT1 Financial Mathematics Core Technical Syllabus Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig