Developing a game theory approach to determine optimal loan to cost ratios for construction loans

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY PHAN THANH THIEN SON DEVELOPING A GAME THEORY APPROACH TO DETERMINE OPTIMAL LOAN-TO-COST RATIOS FOR CONSTRUCTION LOANS Major: Construction Management Major Code: 8580302 MASTER’S THESIS HO CHI MINH CITY, July 2023 THIS RESEARCH IS COMPLETED AT: HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY – VNU-HCM Supervisor 1: Dr Bui Phuong Trinh Supervisor 2: Assoc Prof Dr Do Tien Sy Examiner 1: Dr Nguyen Hoai Nghia Examiner 2: Assoc Prof Dr Pham Vu Hong Son This master’s thesis is defended at HCM City University of Technology, VNU- HCM City on 10/07/2023 Master’s Thesis Committee: Assoc Prof Dr Luong Duc Long Assoc Prof Dr Tran Duc Hoc Dr Nguyen Hoai Nghia Assoc Prof Dr Pham Vu Hong Son Dr Pham Hai Chien Chairman Secretary, Member Reviewer Reviewer Member Approval of the Chairman of Master’s Thesis Committee and Dean of Faculty of Civil Engineering after the thesis is corrected CHAIRMAN OF THE THESIS COMMITTEE DEAN OF FACULTY OF CIVIL ENGINEERING Assoc Prof Dr Luong Duc Long Assoc Prof Dr Le Anh Tuan i VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY SOCIALIST REPUBLIC OF VIETNAM Independence – Freedom - Happiness THE TASK SHEET OF MASTER’S THESIS Full name: Phan Thanh Thien Son Date of birth: 22/09/1998 Major: Construction management Student code: 2192015 Place of birth: Ho Chi Minh city Major code: 858032 I THESIS TOPIC: DEVELOPING A GAME THEORY APPROACH TO DETERMINE OPTIMAL LOAN-TO-COST RATIOS FOR CONSTRUCTION LOANS ĐỀ TÀI LUẬN VĂN: XÁC ĐỊNH TỶ LỆ KHOẢN VAY TRÊN CHI PHÍ TỐI ƯU CHO CÁC KHOẢN VAY XÂY DỰNG II TASKS AND CONTENTS  Conduct a literature review of existing models and studies pertaining to topic of optimization of LTC ratio for construction loans  Develop a Game theoretic model to determine the LTC ratio of construciton loans for different financial objectives based on literature review  Demonstrate model’s mechanics through 1) implementing it for two real-life case studies in Vietnam and 2) comparing its results to that from existing models  Evaluate model’s effectiveness and conclude thesis with a summary and suggestions for future research III TASKS STARTING DATE: December 2022 IV THESIS COMPLETION DATE: June 2023 V SUPERVISORS: Dr Bui Phuong Trinh, Assoc Prof Dr Do Tien Sy HCM City, date month year SUPERVISOR SUPERVISOR Dr Bui Phuong Trinh Assoc Prof Dr Do Tien Sy HEAD OF DEPARTMENT Dr Le Hoai Long DEAN OF FACULTY OF CIVIL ENGINEERING Assoc Prof Dr Le Anh Tuan ii Acknowledgements My family whose encouragement led me to pursue a master’s degree in the first place and whose support allowed me to persist through difficult times My professors, thesis instructors - Dr Bui Phuong Trinh and Assoc Prof Dr Do Tien Sy, and fellow master’s degree students at HCMC University of Technology whose guidance and assistance were essential to my academic progress throughout the course of the program A special thanks to Dr Nguyen Thanh Viet whose advice and counsel were tremendouly valuable to the completion of this thesis A special thanks to Professor Jang Ying at California State University, whose guidance was instrumeantal to the conception of this thesis iii Abstract The loan-to-cost ratio (LTC ratio) of construction loans are an important consideration for owners and developers embarking on new development projects who wish to maximize their financial returns This thesis proposes a model that solves for the optimal LTC ratio of construction loans from developers’ point of view for privately developed and privately financed development projects The proposed model uses Game Theory as its main framework along with non-linear optimization algorithms to determine optimal LTC ratios for construction loans that maximize three different financial metrics: the expected levered net present value (NPV) of project cash flows, the expected levered internal rate of return (IRR), and the expected levered return on equity (ROE) The model developed in this thesis (hereafter referred to as “the proposed model”) is based on research and existing models that solve for the optimal LTC ratios for construction loans as well as studies that incorporate Game Theory for various problems associated with construction loans In addition to optimal LTC ratios, the model also outputs other useful metrics such as the expected levered NPV, expected levered IRR, expected levered ROE values that it maximizes for, the maximum feasible construction cost, and the suggested mutually acceptable interest rate This thesis therefore contributes to the existing academic literature on the topic of optimal capital structure by introducing a new tool alongside other existing models that also address this optimization problem as well as serving as a practical guide for owners and analysts in structuring their loan terms for negotiations Key words: construction loans, loan-to-cost ratio, LTC ratio, Game Theory, non-linear programming, optimization, expected levered, NPV, ROE, IRR, feasibility analysis iv Tóm Tắt Luận Văn Tỷ lệ cho vay chi phí (tỷ lệ LTC) khoản vay xây dựng cân nhắc quan trọng chủ đầu tư muốn tối đa hóa lợi nhuận tài dự án phát triển Luận văn đề xuất mơ hình dùng để xác định tỷ lệ LTC tối ưu khoản vay xây dựng từ góc nhìn nhà phát triển dự án có vốn đầu tư tư nhân Mơ hình đề xuất sử dụng Lý thuyết trị chơi làm tảng với thuật tốn tối ưu hóa phi tuyến tính để xác định tỷ lệ LTC tối ưu cho khoản vay xây dựng nhằm tối đa hóa ba số tài khác nhau: giá trị rịng kỳ vọng có địn bẩy (expected levered NPV), tỷ suất hồn vốn nội kỳ vọng có địn bẩy (expected levered IRR), tỷ suất sinh lợi kỳ vọng có địn bẩy vốn chủ sở hữu (expected levered ROE) Mơ hình phát triển luận án (sau gọi “mơ hình đề xuất”) dựa nghiên cứu mơ hình hữu có xác định tỷ lệ LTC tối ưu cho khoản vay xây dựng nghiên cứu sử dụng Lý thuyết trò chơi nghiên cứu khác liên quan đến khoản vay xây dựng Ngoài tỷ lệ LTC tối ưu, mơ hình xác định số hữu ích khác NPV kỳ vọng có địn bẩy, IRR kỳ vọng có địn bẩy, ROE kỳ vọng có địn bẩy tối đa hóa, số tiền chi phí xây dựng khả thi tối đa, lãi suất đề xuất mà hai bên chấp nhận (mutually acceptable interest rate) Do đó, luận án đóng góp vào tài liệu học thuật có chủ đề tối ưu hóa cấu trúc vốn (capital structure) cách giới thiệu công cụ số mơ hình hữu dùng để giải vấn đề tối ưu hóa phát triển công cụ hướng dẫn thực tế cho chủ đầu tư đơn vị phân tích tài việc thiết lập điều khoản vay trước đàm với bên cho vay Từ khóa: khoản vay xây dựng, tỷ lệ vốn vay chi phí, tỷ lệ LTC, Lý thuyết trị chơi, lập trình phi tuyến tính, tối ưu hóa, địn bẩy kỳ vọng, NPV, ROE, IRR, nghiên tích khả thi v AUTHOR’S COMMITMENT The undersigned below: Name : PHAN THANH THIEN SON Student ID : 2192015 Place and Date of Birth : Ho Chi Minh city, 22nd September 1998 Address : Ho Chi Minh City, Vietnam declares that the master thesis titled “Developing A Game Theory Approach to Determine Optimal Loan-To-Cost Ratios for Construction Loans” is completed by the author under supervision of the supervisors All works, idea, and material that gained from other references have been cited in the correct way Ho Chi Minh City, June 10th, 2023 PHAN THANH THIEN SON vi Table of Contents Abstract iii Tóm Tắt Luận Văn iv Table of Contents vi List of Figures viii Introduction 1.1 Research Problem 1.1.1 LTC Ratio, Feasibility Analyses, and Financial Analyses 1.1.2 Project Performance and Default 1.2 Thesis Objectives 1.3 Scope of Research 1.4 Significance of Research 1.4.1 Practical Significance 1.4.2 Academic Signficance 1.5 Research Process and Thesis Structure 10 Literature Review 11 2.1 Construction Loans 11 2.2 Game Theory 13 2.2.1 Theoretical Applicability 13 2.2.2 Advantages Offered by Game Theory 15 2.3 Existing Models 18 Methodology 24 3.1 The Model 24 3.2 Modelling Assumptions 26 3.3 The Game 27 3.4 Solving for The Optimal LTC ratio 32 3.4.1 Backwards Induction 32 3.4.2 Expected Levered NPV 32 vii 3.4.3 Mutually-Acceptable Interest Rate 34 3.4.4 Abandonment Decision 35 3.4.5 Solving for the Optimal LTC Ratio 36 3.5 Flowchart 43 3.6 Cash Flow Schedule 45 Case Studies and Model Comparisons 46 4.1 Case Study 47 4.2 Case Study 52 4.3 Comparison with Other Models 57 4.3.1 Dias Jr and Ioannou (1995) 57 4.3.2 Bakatjan et al (2003) 66 4.4 Discussion and Evaluation 72 4.5 Interpreting and Applying The Results 74 Conclusion 75 References 78 Appendix 81 A Variables Chart 81 B Abbreviations 81 C Evaluation of Integrals and Sample Excel VBA Code 82 Expected Levered NPV Evaluation 82 Expected Levered ROE Evaluation 83 Sample VBA Code 84 D Equations 85 viii List of Figures Figure 1.1 - Flowchart showing research process and thesis structure 11 Figure 3.1 - Game Tree for model 31 Figure 3.2 - Relationship between 𝑐̅ ± 𝜃, 𝑐′, and 𝑐̂ ′ 31 Figure 3.3 - Flow chart of game 44 Figure 4.1 - Relationship between expected levered NPV and LTC ratio 51 Figure 4.2 - Relationship between expected levered IRR and LTC ratio 51 Figure 4.3 - Relationship between expected levered ROE and LTC ratio 52 Figure 4.4 - Relationship between expected levered NPV and LTC ratio 55 Figure 4.5 - Relationship between expected levered IRR and LTC ratio 56 Figure 4.6 - Relationship between expected levered ROE and LTC ratio 56 Figure 4.7 - Relationship between expected levered NPV and LTC ratio 63 Figure 4.8 - Relationship between expected levered IRR and LTC ratio 63 Figure 4.9 - Relationship between expected levered ROE and LTC ratio 64 Figure 4.10 - Relationship between expected levered NPV and LTC ratio 70 Figure 4.11 - Relationship between expected levered IRR and LTC ratio 71 Figure 4.12 - Relationship between expected levered ROE and LTC ratio 71 Figure C.1 - Sample code written (MS Excel VBA) to determine optimal LTC ratio for expected levered IRR 84 List of Tables Table 2.1 - Research from the current literature relevant to the optimization of construction loans’ LTC ratio 21 Table 3.1 - Discounted Cash Flow Schedule For Each Scenario At Each Time Instance 46 74 4.5 Interpreting and Applying The Results The results achieved hitherto in this section satisfies the list of objectives identified in section 1.2 Further interpretation of the results produced so far in this section is needed to concretely illustrate the significance of the thesis The first thing to bear in mind when interpreting the results is that the calculated optimal LTC ratios should be viewed as ideal outcomes (assuming mutually acceptable interest rates are achieved) that should be strived for when negotiating the terms of the loan with the lender, and so in terms of applicability the calculated optimal LTC ratios can be interpreted and strategically used in this way For example, suppose the owner in the first case study chooses to maximize his ROE; in this case, he would approach the lender knowing the optimal LTC ratio is 60% and that higher LTC ratios are more preferable, assuming him and the owner can agree on a M.A.I.R The owner could also use the optimal LTC ratio of 60% as a baseline to create a cash flow pro-forma that can be dynamically analyzed with a sensitivity analysis in a similar fashion to that found in the model comparisons in section 4.3.1 and section 4.3.2 Alternatively, if the owner in the above example seeks to equally balance the rewards of optimal expected levered ROE and expected levered NPV, he could split right down the middle, i.e., select the arithmetic average of the optimal LTC ratio for the expected levered NPV (15.1%) and the optimal LTC ratio for the expected levered ROE (60%), which would be 37.5% This value (and its corresponding M.A.I.R.) would be the targeted value for negotiation with the lender as well as being the base case for the cash flow pro-forma Should the owner want more or less of either financial metric (e.g., a higher expected levered NPV which would mean giving up some expected levered ROE), he would pick a value closer to 15.1% than to 60%, and vice versa In summary, whether the owner chooses to maximize a financial metric or some balance/trade-off between two or three objectives, the calculated optimal LTC ratios are useful as end 75 ranges for the overarching objective The selected optimal LTC ratio can then be used for negotiating loan terms and as the base case for cash-flow pro-formas Conclusion The importance of the LTC ratio in financial and feasibility analysis for construction projects lies not only in how it affects the bottom-line cash flows of the project, but also how it changes the level of risk exposure that the project is subject to Finding the optimal LTC ratio for different financial objectives not only ensures that developers are able maximize the financial performance at the end of the project’s life, but also acts as a planning guide during the preconstruction phase of the project The significance of an optimal LTC ratio in making a project a financial success motivated further research into the topic of LTC ratio optimization The review of the existing literature carried out in this thesis shows that existing models that specifically address the determination of optimal LTC ratios, while practical, all have room for improvement or expansion in some way: for example, research by Bakatjan et al [3] as well as Dias Jr and Ioannou [2] not consider more than one time period from project conception to the end of the construction phase of a project, which in turn fails to capture the possibility of developers’ abandonment of the project midway through as well as any changes in the availability of information pertinent to the project Chan et al ’s model [5] on using Game Theory to find mutually acceptable interest rates for construction loans opened up the possibility of using a similar Game Theory framework for this particular optimization problem Further review of the existing literature also shows that no concrete effort had been made to incorporate Game Theory into determining the optimal LTC ratio for construction loans Given the unique strengths and applicability of Game Theory as well as the possibility of expanding and developing aspects of existing models pertaining to this topic, this thesis sets out to develop a model 76 based on a Game Theory framework which determines the optimal LTC ratio for construction loans that addresses some of the shortcomings of existing models This thesis therefore not only adds to Game Theory the existing toolbox of available models and frameworks that address this optimization problem, but also The proposed model is based on that built by Chan et al [5] and contains modifications and features that makes it in some ways more complete than some existing models, including allowing for two stages of construction, availability of additional information mid-way through the project, and evaluation of LTC ratio for three different financial objectives Its effectiveness is demonstrated in section in two different case studies from real life projects in Vietnam as well as through comparisons of its results with that produced by other relevant existing models using their input data The case studies show that the model is able to produce optimal LTC ratios for different financial objectives along with additional data that would be useful to a developer planning out a project’s capital structure (e.g., the M.A.I.R, the maximized financial metric for the objective in question, such as the expected levered NPV) Despite the relative strengths which the proposed model exhibits, the comparisons with other existing models in section 4.3 show that the proposed model may produce similar and also significantly different optimal LTC ratios and optimized financial metrics from those which these models suggest (for example, the proposed model produced results that aligned much more closely to those of Dias Jr and Ioannou [2] than it did with Bakatjan et al [3] This is due to the fact that the proposed model is quite input sensitive, making the quality of the assumptions important for the model to accurately depict and solve for each project’s optimal LTC ratio, a problem which financial modelling in general inherently faces; this is why it is strongly encouraged that sensitivitiy analyses are also performed on the variables d and k (the variable h is omitted in such analysis as d and h are dependent on one another), as conducted in sections 4.3.1 and 4.3.2 77 Other limitations that the proposed model exhibits include not having a hold period during which the property may generate additional cash flows and requiring that the probability distribution for the price and construction cost variables must be symmetrical, such as in the case of the normal distribution or triangular distribution These limitations, however, are in theory not hugely difficult to overcome and even when they are left unaddressed not diminish the value that Game Theory brings as a novel and usable approach for this particular optimization problem The intended objectives and significance of the thesis outlined in section 1.2 and 1.4 are achieved: the proposed model can be practically used alongside other existing models (such as those by Dias Jr [2] and Bakatjan et al [3], to which the proposed model was compared in section 4.3) or as a standalone model, depending on factors such as availability of inputs and exit strategy, to guide developers on how to optimize their capital structures to fit their financial objectives Section 4.5 illustrates how the results from chapter can be practically applied to an actual case study from the perspective of an owner during the pre-construction phase who would, if he chooses to It also contributes to the existing pool of academic research pertaining to the topic of capital structure, construction loans, and Game Theory Further research into this topic could explore certain modifications to the model, including increasing the number of time periods from loan origination to project completion, accommodating asymmetrical probability distribution for the random variables 𝑝̃ and 𝑐̃ , and including an operation period before the asset is disposed The model could also be expanded to account for other financial instruments as well as forms of financing, such as mezzanine loans and real options Research into the use of Game Theory in other aspects of construction loans such as in negotiating loan terms and loan structures could also be a fruitful investigation path 78 References [1] S C Ward, B Curtis, and C B Chapman, “Objectives And Performance in Construction Projects,”, Constr Mgt and Econ., vol 9, no 4, pp 343-353, 1991, doi: 10.1080/01446199100000027 [2] A Dias Jr and P G Ioannou P G., “Debt Capacity and Optimal Capital Structure for Privately-Financed Infrastructure Projects,” Journal of Constr Eng and Mgt., vol 121, no.4, pp 404-414, Dec 1995, doi: https://doi.org/10.1061/(ASCE)07339364(1995)121:4(404) [3] S Bakatjan, M Arikan, and R L K Tiong, “Optimal Capital 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Hoboken, NJ, USA: John Wiley and Sons, Inc., 2011 81 Appendix A Variables Chart Variable Total Cost Average Total Cost Total Cost Dispersion value Sale Price Average Sale Price ($ million) Sale Price Dispersion value 1st phase loan ratio Loss Ratio 1st Phase Construction Cost % Loan-to-cost ratio 1st phase duration (years) Total duration (years) Discount rate Tax rate Interest rate (mutually acceptable) Lender’s loan limit Data Source Model’s Random variable Forecasted/Budgeted Modelling assumption Model’s Random variable Forecasted Modelling assumption Modelling assumption Modelling assumption Modelling assumption Calculated Forecasted/scheduled Forecasted/scheduled Modelling assumption Modelling assumption Calculated Modelling assumption Symbol 𝑐̃ 𝑐̅ 𝜃 𝑝̃ 𝑝̅ 𝛿 d k h l m n j q r (r*) 𝑙𝑈3 B Abbreviations $M – Millions of US Dollars 10 CAPM – Capital Asset Pricing Model 11 DSCR – Debt Service Coverage Ratio 12 IRR – Internal Rate of Return 13 LTC ratio – Loan-to-cost ratio 14 MC(S) – Monte Carlo (Simulation) 15 NPV – Net Present Value 16 PPP – Public Private Partnership (projects) 16.1 BOT – Build-Operate-Transfer (schemes) 16.2 BOO – Build-Operate-Own (schemes) 17 ROE – Return on Equity 82 C Evaluation of Integrals and Sample Excel VBA Code This section contains evaluation of the integrals in section 3.4.5 which then allows the calculation in the case studies and model comparisons in Section 4.3 It also contains a sample of Excel VBA code that could be used to determine the optimal LTC ratio that maximizes the expected levered IRR values Firstly, the following shorthand and variable changes are made and will be used throughout this section to make evaluating equations more readily legible: Expected Levered NPV Evaluation The first term of Equation 3.5 can be re-written as: C.1 C.2 83 Next, the second and third term of Equation 3.5 are evaluate to be as follows: C.3 C.4 The values of Ψ(𝑎), Ψ(𝑏) and Ψ(𝑘) are definite integrals of the cumulative normal distribution function Φ(𝑐̃ ) with respect to 𝑐̃ (as defined in their respective case studies/model comparisons sections) and can be estimated using numerical methods; in this thesis we use Simpson’s rule with five partitions to estimate them Substitute the values for d, h, u, k, q, j, m, n, 𝑎, 𝑏, 𝑡, 𝑤, 𝑐̅, 𝑝̅, into equation C.2, Equation C.3, and Equation C.4, followed by the summation of these three equations to acquire an expression for the expected levered NPV in terms of 𝑙 that can be optimized for Expected Levered ROE Evaluation As defined in section 3.4.5d), we have: 𝐸 [𝑅𝑂𝐸 ′ ] = 𝐸[𝜋 ′ ]/𝐸[𝑆 ′ ] The value of 𝐸[𝜋 ′ ] is essentially the undiscounted version of the expected levered NPV, hence it is the sum of Equation C.2, Equation C.3, and Equation C.4 without their respective discounting factors 𝐸[𝑆 ′ ] is given by equation 3.12.2, and its first and second terms are evaluated to be C.5 84 C.6 Substitute the values for the corresponding variables in Equation C.5 and Equation C.6 followed by the summation of these equations to acquire an expected levered ROE expression in terms of 𝑙 that can be optimized for Sample VBA Code Figure C.1 below is a sample code created on MS Excel’s VBA platform to determine the optimal LTC ratio that corresponds to the highest expected levered IRR value within the specified LTC ratio range Figure C.1 - Sample code written (MS Excel VBA) to determine optimal LTC ratio for expected levered IRR 85 D Equations Listed below are the equations referenced in this thesis 3.1 3.2 3.3 3.4 3.5 3.6 where 3.7 3.8 3.9 3.10.1 86 3.10.2 3.11 3.12.1 3.12.2 3.13 3.14 3.15 3.16 3.17 3.18 C.1 C.2 where 87 C.3 C.4 C.5 C.6 88 PROFILE Full name: Phan Thanh Thien Son Date of birth: 22 September 1998, Place of birth: HCMC, Vietnam Address: 77 Cao Thang, District Ward 3, HCMC Vietnam Email: 2192015@hcmut.edu.vn , Phone number: 0907428899 TRAINING PROCESS 2021 B.Sc Real Estate, New York University 2021-present: Thien Hung R.E Co., Ltd