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

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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

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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:

1 Assoc Prof Dr Luong Duc Long Chairman

2 Assoc Prof Dr Tran Duc Hoc Secretary, Member 3 Dr Nguyen Hoai Nghia Reviewer 1

4 Assoc Prof Dr Pham Vu Hong Son Reviewer 2

5 Dr Pham Hai Chien 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

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HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY Independence – Freedom - Happiness

THE TASK SHEET OF MASTER’S THESIS

Full name: Phan Thanh Thien Son Student code: 2192015

Date of birth: 22/09/1998 Place of birth: Ho Chi Minh city Major: Construction management 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 1 SUPERVISOR 2

Dr Bui Phuong Trinh Assoc Prof Dr Do Tien Sy

HEAD OF DEPARTMENT

Dr Le Hoai Long

DEAN OF FACULTY OF CIVIL ENGINEERING

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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

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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

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Tóm Tắt Luận Văn

Tỷ lệ cho vay trên chi phí (tỷ lệ LTC) của các khoản vay xây dựng là một cân nhắc quan trọng khi các chủ đầu tư muốn tối đa hóa lợi nhuận tài chính của các dự án phát triển mới Luận văn này đề xuất một mơ hình dùng để xác định tỷ lệ LTC tối ưu của các khoản vay xây dựng từ góc nhìn của các nhà phát triển đối với các dự án có vốn đầu tư tư nhân Mơ hình được đề xuất sử dụng Lý thuyết trị chơi làm nền tảng chính cùng với các thuật tốn tối ưu hóa phi tuyến tính để xác định tỷ lệ LTC tối ưu cho các khoản vay xây dựng nhằm tối đa hóa ba chỉ số tài chính khác nhau: giá trị hiện tại rịng kỳ vọng có địn bẩy (expected levered NPV), tỷ suất hồn vốn nội bộ kỳ vọng có địn bẩy (expected levered IRR), và tỷ suất sinh lợi kỳ vọng có địn bẩy trên vốn chủ sở hữu (expected levered ROE)

Mơ hình được phát triển trong luận án này (sau đây gọi là “mơ hình được đề xuất”) dựa trên các nghiên cứu và các mơ hình hiện hữu có xác định tỷ lệ LTC tối ưu cho các khoản vay xây dựng cũng như các nghiên cứu đã sử dụng Lý thuyết trò chơi trong các nghiên cứu khác liên quan đến các khoản vay xây dựng Ngoài các tỷ lệ LTC tối ưu, mơ hình cũng xác định được các chỉ số hữu ích khác như NPV kỳ vọng có địn bẩy, IRR kỳ vọng có địn bẩy, ROE kỳ vọng có địn bẩy được tối đa hóa, số tiền chi phí xây dựng khả thi tối đa, và lãi suất được đề xuất mà cả hai bên có thể chấp nhận được (mutually acceptable interest rate)

Do đó, luận án này đóng góp vào các tài liệu học thuật hiện có về chủ đề tối ưu hóa cấu trúc vốn (capital structure) bằng cách giới thiệu một công cụ mới trong số các mơ hình hiện hữu được dùng để giải quyết vấn đề tối ưu hóa này cũng như phát triển một công cụ hướng dẫn thực tế cho chủ đầu tư và các đơn vị phân tích tài chính trong việc thiết lập các điều khoản vay trước và trong khi đàm với bên cho vay

Từ khóa: khoản vay xây dựng, tỷ lệ vốn vay trên 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

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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

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Table of Contents

Abstract iii

Tóm Tắt Luận Văn iv

Table of Contents vi

List of Figures viii

1 Introduction 1

1.1 Research Problem 1

1.1.1 LTC Ratio, Feasibility Analyses, and Financial Analyses 1

1.1.2 Project Performance and Default 2

1.2 Thesis Objectives 4

1.3 Scope of Research 5

1.4 Significance of Research 8

1.4.1 Practical Significance 8

1.4.2 Academic Signficance 9

1.5 Research Process and Thesis Structure 10

2 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

3 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

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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

4 Case Studies and Model Comparisons 46

4.1 Case Study 1 47

4.2 Case Study 2 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

5 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

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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 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

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Abbreviations

The following abbreviations are used throughout this thesis: 1 $M – Millions of US Dollars

2 CAPM – Capital Asset Pricing Model 3 DSCR – Debt Service Coverage Ratio 4 IRR – Internal Rate of Return

5 LTC ratio – Loan-to-cost ratio 6 MC(S) – Monte Carlo (Simulation) 7 NPV – Net Present Value

8 PPP – Public Private Partnership (projects) 8.1 BOT – Build-Operate-Transfer (schemes) 8.2 BOO – Build-Operate-Own (schemes) 8.3 ROE – Return on Equity

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1 Introduction

Construction loans are a major source of financing for large construction projects, and their loan-to-cost ratios are a major consideration for developers when structuring and negotiating loan terms as different LTC ratios result in different financial performances A review of the current literature shows that existing models developed for this LTC ratio optimization problem can be further developed as they either omit certain aspects of the construction loans which may be useful or can be expanded to include additional financial goals that owners may be interested in This thesis aims to develop a model which employs a Game Theory approach to this optimization problem which would not only expand the existing set of tools available for this optimization problem but also address some of the areas which existing models do not cover that would be practically helpful if considered This section explores the research problem in greater depth, followed by an outline of the thesis’ scope and a discussion of the significance of the research This section concludes with an overview of how the different sections of this thesis are structured

1.1 Research Problem

1.1.1 LTC Ratio, Feasibility Analyses, and Financial Analyses

The main objective of an owner upon starting a construction project is to ultimately possess a structure which “looks and functions as intended and that provides an acceptable return for the money expended” [1] Therefore, the goal of a development project in most cases is to maximize financial returns as measured by metrics such as the Net Present Value (“N.P.V.” or “NPV”), Internal Rate of Return (“I.R.R.” or “IRR”), and Return on Equity (“ROE” or “R.O.E.”), see [2], [3] It is from these financial objectives that other construction-specific objectives, such as staying on budget and on schedule, are derived [1]

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directly represents the dollar amount which developers borrow and must repay in the form of cash inflows and outflows Therefore, the LTC ratio is a necessary input for any feasibility analysis that determines the degree to which developers use leverage in their project, which can theoretically range from 0% (full equity financing) to 100% (full debt financing); the capital structure of the project – the ratio of debt and equity that is employed – is therefore determined by the LTC ratio A project therefore cannot go past the pre-construction phase (where the feasibility and financial analyses take place) if the owner and lender cannot reach a sufficiently reasonable LTC ratio that would generate returns that match the owners’ objectives

The LTC ratio is also an important consideration from the lender’s perspective: Chan, Wang, and Yang [5] identifies the LTC ratio as one of three main factors which link construction loan risks to the pricing of construction loans [5] Without the LTC ratio as an input, loans cannot be priced (either by the owner or lender), which means projects cannot be financed and therefore makes them a lot less likely to take place Consequently, developers aren’t the only stakeholders who are affected by the finalized LTC ratio: the interest of lenders, contractors, and public agencies are also affected

Furthermore, Dias Jr and Ioannou [2] demonstrate mathematically that projects’ ROE and NPV are sensitive to changes in LTC Hence the determination of an optimal LTC ratio in order to maximize financial metrics such as ROE or NPV is not only necessary for a project to happen but also crucial for the owner that he may maximize his financial returns

1.1.2 Project Performance and Default

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through “high-quality, cost-effective, and timely construction”, while downside risk’s main concern “is the avoidance of delays and cost overruns.” Furthermore, it’s also argued that equity-holders “will be induced to control upside risk” while debtholders “will be induced to control downside risk.” As a result, “the relative importance of these two types of risk should be an important determinant of a project’s leverage”: projects with higher downside risk tend to be less levered, and vice versa [6]

Sundarajan [7] also identifies the importance of managing capital structures to ensure cost overruns and schedule delays (“project performance risks”) are mitigated As additional capital infusion may be required to manage and control such risks in the event they do materialize, capital structures may need to be altered in response in such a way that maximizes the project value Although Sundarajan [7] is innovative in this particular research through the use of dynamic capital structure to proactively manage project performance risks, the inherent relationship between such risks and their corresponding capital structure remains unchanged

Moreover, default on construction loans, which are made more likely from irresponsible lending and from developers borrowing more than what is economically optimal for them, can have significant impacts on the economy as it has the potential to create major losses for lenders (the degree of net loss would depend on the amount recovered from guaranteed assets used or recovered) For example, the S&P Global Market Intelligence [8] reported over $3Bn in delinquent construction loans across the U.S in Q1 of 2020 alone, representing 2.04% of all construction loans on banks’ portfolios In extreme situations such as in the aftermath of the financial crisis of 2008 up until 2013, the Federal Deposit Insurance Corporation (FDIC) resolved as many as $39Bn in delinquent construction loans across 144 U.S banks [9] While many factors contributed to the economic bubble and the crisis that ensued, ignorance of optimal loan ratios, whether willingly or not, prevented lenders and developers from making sound financing decisions

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determined? A review of the existing literature shows that although previous studies, most notably that by Dias Jr and Ioannou [2] and Bakatjan et al [3], have developed models that solve for the optimal construction loan LTC ratio, they all have room for either improvement or modifications in some aspects, whether it’s the considerations and assumptions that are included to model the mechanics of construction loans or the perspectives which they consider For e.g., some models seek to maximize the ROE and IRR as the metric to be maximized, while others consider the IRR and the NPV; while some models consider the sale of the property at the end of the analysis period, others do not A more thorough review of these models can be found in section 2.3 of this proposal

Given the importance of the LTC ratio in the financial analyses of construction projects, it is worthwhile to enrich the current literature that pertains to modeling its optimization This thesis seeks to develop a model that supplements the existing host of models that deal with this particular optimization problem by proposing the use of Game Theory as an alternative approach which has yet to be considered from the perspective of the project owner for this particular optimization problem Game Theory as a framework offers several advantages that other approaches employed by previous models do not for this particular optimization problem, including accounting for decisions taken by more than one actor (in this case, the lender and borrower/developer) over the course of more than one time period as well as how decisions by different actors are affected as new information becomes available A more thorough analysis of the strengths of this framework is conducted in section 2.2.2

1.2 Thesis Objectives

This thesis sets out to achieve the following objectives:

1 Review and analyze the shortcomings and strengths of existing models that

determine the optimal LTC ratio; this is presented in the Literature Review

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2 Develop a model which determines the optimal LTC ratio using Game Theory;

this will be illustrated in the Methodology section

3 Demonstrate the mechanics and application of the developed model by determining the optimal LTC ratios and profitability levels for two case studies

taken from Vietnamese projects, as covered in sections Case Study 1 and Case Study 2 The model is also compared to two other existing models reviewed in the Literature Review section Both the results of the case studies and comparisons with existing models are detailed in the Case Study and Comparisons with Existing Models section

4 Evaluate strengths and limitations of model, suggest areas for its application,

and suggest areas for future research ; this is discussed in the Conclusion

section.

1.3 Scope of Research

The proposed model is intended to facilitate decision making and deal structuring between lenders, private owners of private projects, including private investors and private developers, and the specific personnel involved, which includes but is not limited to

 asset managers

 project managers

 CFOs, investment analysts, and financial analysts

 lenders, both local and institutional, and those who work for/under them

 loan originators

 loan servicers of different levels

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There are several reasons why this model does not assess construction loans in the case of public projects: firstly, the private sector in most countries is often involved in building a wider array of buildings, from commercial to industrial to residential, while public sector projects are largely focused on infrastructure As a result, the model when applied to private projects is able to expand its applicability range and thus theoretically should be more useful

Secondly, most countries’ private sectors spend a lot more on construction than their public sectors: for example, the U.S Census Bureau reported for calendar year 2021 that the private sector’s spending on construction was 3.91x larger than its public counterpart [10] The model therefore has greater potential to play a larger role for the macroeconomy if widely implemented and thus have a more notable impact when developed with private projects in mind

Thirdly, as the level of efficiency of public sectors across different countries is likely to vary due to public sector projects being more closely regulated and often subject to varying degrees of bureaucracy, red tape, and legislative barriers, it is difficult for a single model to account for all the differences in the ways which public sectors across the world approve and manage their construction projects More complex and country-specific models would need to be developed in order to realistically reflect the construction environment and loan practices for a given country’s public sector, which is different to the direction of general applicability that this model looks to achieve

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The model is constructed to account for the main working principles of a construction loan, which allows it to reasonably represent contemporary construction loan practices in most countries and hence ensuring that it achieves a fair level of applicability across the board with no major adjustments needed for any particular country Some of these main working principles include fund disbursement in phases, interest on loan amount, and the possiblity of the lender recovering some amount of the loan should the project be abandoned Furthermore, optimal LTC ratios and capital structures will be calculated in the context of project financing rather than corporate financing This is so that aspects of the model, which is built upon the original from [5], can be accommodated for, for e.g., the fact that lenders have limited recourse only to the unfinished construction project should the owner abandon it before its completion Secondly, limiting the research to only project financing removes the need to consider other aspects of corporate financing specific to the owner’s firm, such as the possibility of commingling of funds

Mezzanine loans, however, have been omitted from this model because developers do not always use this type of quasi-debt to finance their projects, and so their omission would not significantly impact the model’s level of usefulness The model will also not analyze loans that also help in financing the acquisition phase of a lot, a piece of land, or a property in need of renovation, such as construction-to-permanent loans This is because such an analysis would have to extend beyond the construction phase into the operational phase of the project, which may dilute the focus of the model from assessing risks and rewards inherent to construction during the pre-construction and construction stages

The proposed model will consider three time periods in order to derive the optimal LTC ratio: at t=0 (during the pre-construction phase) when the developer performs his feasibility analysis upon being offered by the lender some interest rate

r; the developer chooses a leverage (LTC) ratio l and decides whether to finance his

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this point, the lender has released a portion of the full loan (known as the first phase loan amount), and the market price of the finished project as well as the final total construction cost is known (but not yet incurred/paid for) At this point the developer faces the decision to either abandon the incomplete project or continue with it, regardless of his initial financing decision At t=2, the owner sells the completed asset based on the known market price at t=2 and, depending on whether he used debt financing at the beginning, would have to repay the loan and the interest on it The development of this model, including how it is essentially founded on the model by Chan et al.’s 201 model [5], is thoroughly illustrated in Chapter 3

1.4 Significance of Research

1.4.1 Practical Significance

The proposed model could be an alternative or could be used alongside other models that also determine the optimal LTC ratio In the latter’s case, any major deviations across the models’ results could be the basis to evaluate not only the legitimacy of the inputs but also of the models themselves Both private projects and PPP projects under schemes such as BOO where the private sector is involved and does not have to transfer the asset back to its sponsor but instead owns the finished project and reserves the right to sell it could use the proposed model to determine the optimal LTC ratio for their feasibility and financial analyses

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1.4.2 Academic Signficance

In theory, it is possible to set up simple financial cash-flowing pro-forma whereby the user inputs some LTC ratio and then optimizes it through trial and error

or through some add-in function such as Microsoft (“MS”) Excel’s Solver or Goal Seek However, such a simple pro forma would not consider the effects of multi-stage

decision making that involves two parties – the lender and the borrower The reason why Chan et al [5] model is used as the basis for developing the proposed model, despite it being originally developed to find the mutually acceptable interest rate on a construction loan, is because it allows for the implementation of Game Theory, which is a particularly novel approach to this optimization problem While a two period Game Theory model with two player is able to realistically model the decision-making process of both developer and lender over the development life of the project, non-linear programming is used as an optimization tool that, given the assumptions and intermediary results (mutually acceptable interest rate and abandonment decision), calculates the optimal LTC ratio

The proposed model therefore contributes to expanding the toolset available to researchers and stakeholders who may need to tackle or would be interested in this subject Furthermore, the proposed model builds upon the foundation of other existing models by identifying both their strengths and what they omit in their modelling considerations that would’ve been useful or helpful if considered, hence it serves as a continuation of the existing scholarly conversations rather than as an addition of something entirely novel

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1.5 Research Process and Thesis Structure

This thesis is divided into seven chapters (including the Appendix and References), each with its own sections and subsections This thesis begins with an in-depth literature review of the existing studies that address the optimization of LTC ratios for construction loans in order to identify relevant models as well as to analyze their strengths and weaknesses The literature review section will also look into Game Theory as an approach to solving different types of problems The literature review section concludes with a summary of the findings from these existing studies as well as an assessment of the strengths and limitations of the developed models

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Figure 1.1 - Flowchart showing research process and thesis structure

2 Literature Review 2.1 Construction Loans

A review of construction loans in the existing literature is important as it provides insight into how they work and the different aspects that may be important for modelling them Zhang et al [11] categorizes construction project financing into three main sources: debt, equity, and quasi-equity (such as mezzanine loans) Construction loans are an important source of debt financing in many construction projects: Vu et al [12], for example, identifies construction loans as one of the main sources of debt financing in public infrastructure projects in Vietnam Such loans come in multiple different forms, including “ADF concessional loans, OCR ordinary loans and national debt securities.”

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disbursed in periods, unlike with mortgages when a lump sum is released to the borrower at loan closing Chan et al [5] and Ross et al [9] both point out that the project itself is often the collateral for the loan; as a result, the collateral is in essence being created as the loan matures and the project progresses, and is particularly significant to lenders as default or abandonment mid-project exposes the lender to a greater risk as incomplete projects are not only much more difficult to value but also do not have cash flows available to offset and debt obligations during the loan period

This raises the issue of structuring loan recourse from both the lender and borrower’s perspective, which is an important consideration for structuring loan terms and setting interest rates In the case of the degree of recourse on the loans, on one hand researchers such as Chan et al [5] and Ross et al [9] indicate that “some lenders request the developer to be personally liable for the construction loan or provide some other properties as additional collateral” so that borrowers “provide some “skin-in-the-game” on the part of the borrower, if the value of the raw land or partially built project that is pledged as collateral is not sufficient.”

On the other hand, researchers such as Hainz and Kleimeier [12] and Esty [13] point out that in the context of infrastructure project financing (rather than corporate financing) through independent legal project entities, “liability is limited such that the lenders have no or only limited recourse to the sponsoring firms” From this we see that since there is in practice room for the lender to recover some portion of the loan in the form of limited recourse (rather than pure non-recourse or pure recourse agreements) through either project financing, corporate financing, or from the value of the incomplete project, this proposed model will also account for such limited recourse in the form of some level of recovery of the construction from the persepective of the lender and the borrower, which will be reflected in their cash flows, most notably that of the developer/borrower

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construction loan taken out to finance the Son La hydropower plant in Vietnam An earlier and more distantly related study by Dang and Huynh [15] uses statistical methods to analyze the effect of portfolio diversification on Vietnamese banks’ financial returns Topics centering or surrounding construction loans in Vietnam, therefore, seem to receive very little attention, particularly in the commercial context (rather than in residential projects)

2.2 Game Theory

2.2.1 Theoretical Applicability

In order to develop a model for any type of optimization problem (in this case, the optimization of the LTC ratio for a construction loan), it is essential that a foundational framework is first established or identified The choice of a suitable framework is based on the evaluation of its “appropriateness, ease of application, and explanatory power” for the intended research purpose, which in this case is to model and find construction loans’ optimal LTC ratios [16] It is therefore necessary to analyze the mechanics and environment in which such loans and their corresponding development projects take place in order to select a suitable framework to approach the optimization problem

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the model assumptions, and his preference would be how much he prefers one level of return to another

Although useful, decision problems are ineffective as a framework for certain scenarios where more than one entity can potentially have an influence on the outcome of another; Tadelis [17] illustrates this limitation through an example where a student’s overall “curved” test score depends not only on his own performance on the test but also on that of other students who also take that test It is therefore necessary to modify the decision problem framework to better model and analyze strategic situations “in which players who interact understand their environment, how their actions affect the outcomes that they and their counterparts will face, and how these outcomes are assessed by the other players.” Game Theory becomes suitable in these circumstances as it “provides a framework based on the construction of rigorous models that describe situations of conflict and cooperation between rational decision makers [17].”

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not affect the use of backwards induction that the original authors employ to determine the M.A.I.R, nor does it make any major changes to any existing aspects of the game [5] The proposed model is built upon the model in [5] but modified and improved for the objectives of this thesis is fully presented in Section 3

2.2.2 Advantages Offered by Game Theory

While the optimal LTC ratio can simply be solved for using spreadsheet software such as MS Excel after cash flow forecasts have been projected, Game Theory offers a number of advantages for this particular task that traditional pro-forma cash flow modelling may lack To evaluate these advantages, it is helpful to first examine other methodologies which existing models use to approach this particular topic Research that considers the optimal LTC ratio with respect to indicators other than financial returns (such as budget and schedule adherence) are not considered here (for e.g., see [7]) Moreover, research that focuses on debt not specific to construction projects are also omitted as such loans and forms of debt may possess characteristics that differ from construction loans The following assessment compares the merits of Game Theory with the approaches used by other existing models built for this purpose and not necessarily to evaluate these models as a whole A more in-depth evaluation of these models can be found in section 2.3

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and subsequently any analysis of how the interest rate could be arrived at in an environment where both parties consider each other’s interests is not reflected

Although not the most flawless and insightful of all existing models, Chan et al.’s model [5] is unique in that it utilizes Game Theory to considers the issue of two parties that engage in setting the interest rate given their opposing interests The model does this by implementing backwards induction to solve for the interest rate that makes a borrower indifferent between 0% leverage and a given 𝑙% leverage Although its goal is not to determine an optimal LTC ratio that maximizes a financial metric, this model does calculate an expected profit function given its modelling inputs The calculated expected profit is hence a reflection of the strategic nature which the developer is in, meaning it can be contextualized as the realistic expected profit given an interest rate which is likely reached by both parties in reality This contextualization is significant because researchers such as Vu et al [12] and Sundararajan [7] have identified interest rate payment as burdens that contribute to risk categories that in turn have a notable effect on construction cost overruns and schedule delays

Secondly, as Game Theory is essentially built on the framework of decision problems, the advantages of decision problems are also inherited by Game Theory, specifically its “precise, well structured, and generally applicable, and most importantly it lends itself to systematic and consistent analysis” [17] Precision and sound structure lies in the presentation of a decision tree (which “evolves” into a game tree when used in Game Theory) where each player’s set of available actions and their corresponding payoffs are clearly delineated in a traceable manner at the outset; such action-outcome pairing from different decision scenarios is not as readily illustrated with spreadsheet cash flow pro-formas, especially when more than one player is involved

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in response to changes in the knowledge made available to each player in each time period This allows for additional dynamism when modeling the interaction between the entities involved as well as how circumstances may change halfway through the construction of the project

Chan et al.’s model [5], for example, models the loan life with two time periods, marked by three instances of time: t=0, t=1, and t=2 At each time instance a major decision by one or more player and/or an additional piece of information is presented, and decisions and information at each time instance is made in light at what happened before (e.g., previous moves made by other players, events that occur at previous time instances) and what is given or made known at that time instance; particularly notable is the decision at t=1 by the developer who has the choice to either abandon or continue with his project, see [5] This sequential analysis characteristic is powerful as it allows for a more realistic modelling of how certain events may unfold over time and how decision making is influenced as new information is made known As actions and outcomes now occur sequentially rather than simultaneously, backwards induction is also possible as beginning at the terminal node each preceding node’s optimal move can be deduced

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useful in more complex models that deal with optimization and other aspects of construction management

2.3 Existing Models

One of the earliest models that deal with optimal capital structures in construction projects is the research by Shah and Thakor [18] which is in effect a one time period constrained maximization program based on the concept of Reactive Capital Structure Equilibrium by Riley [19] Although research by Riley [19] contains concepts related to game theoretic models, including Nash Equilibrium, neither it or Shah and Thakor’s [18] explicitly structures its model in the form of a game with essential game components (further discussed in Section 3) specifically intended for finding the optimal capital structure While one of its strengths is that it considers the perspective of both the lender and borrower, the model developed by Shah and Thakor [18] only looks at net expected discount profit (NPV) as the maximization objective, and considers only one time period (demarked by two points-in-time)

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Zhang [11] develops a Monte Carlo simulation-based model which simulates construction costs in order to find an optimal LTC ratio that maximizes project IRR while meeting some minimum IRR requirement The model simulates data for two stochastic variables: construction cost and construction duration, based on some selected probability distribution for each of the variables The optimal capital structure’s bankruptcy risk is safeguarded by a given minimum debt service coverage ratio The model does not consider ROE maximization or NPV maximization as the optimization function for any set of inputs The solution algorithm iterates through loops of different equity levels with increments of 1% in each loop and calculates the IRR level corresponding to that loop All IRR levels are gathered together after all iterations have been completed are gathered together and evaluated based on some viability criteria, such as maintaining minimum DSCR level If viable, the capital structure that corresponds to the highest IRR is the optimal capital structure [11]

As with the models by Dias Jr and Ioannou [2] and Bakatjan et al [3], this model is static in nature, is not decision and strategy-based in structure, and does not consider more than one time period Jian-Cheng et al [20] develops a similar optimal capital structure model for PPP projects based on the framework of utility functions of the IRR metric from the perspective of the owner and the utility function of the debt service coverage ratio (DSCR) from the perspective of the lender; probability distributions of variables such as inflation, construction cost, etc are given as inputs and the IRR is maximized subject to meeting a number of constraints regarding the project’s IRR, NPV, and DSCR [20]

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on the optimal capital structure for either PPP projects or projects that are in some way funded or guaranteed by the government; no research has been conducted which focuses exclusively on projects financed and developed by private entities with no government involvement in any form

Other researches that discuss optimal capital structures for projects that are loosely but not specifically relevant to construction loans with the objective of financial return maximization are important to note but are not foundational in formulating the proposed model; the only exception being Chan et al.’s model [5] which is the only model that actually clearly utilizes a game theory framework Research by Moszoro and Gasiorowski [21] looks at the optimal public to private funding ratio on a project; Sundararajan and Tseng [7] develop a dynamic capital structure model which is optimized based on project performance indicators (cost overruns and schedule delays) Kong et al [22] develops a “BOT credit risk model”” which calculates the ““default riskiness” of a project, which essentially comprises maximum default losses over a range of confidence levels.” Other relevant but not directly related research surrounding the topic of optimal capital structure is compiled in Sundararajan and Tseng research [7] No relevant studies exist in the Vietnamese literature pertaining to construction loans’ LTC ratio optimization No study pertaining to optimization of LTC ratio for private commercial loans at the project level after 2020

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Table 2.1 - Research from the current literature relevant to the optimization of construction loans’ LTC ratio

Research Relevant Assumptions and Methods included Aspects omitted

1 Shah and Thakor, Optimal Capital Structure and Project Financing, (1987) [18]

 Reactive Capital structure equilibrium

 Maximized return as the objective function

 Pareto optimality

 Project duration modeled as one time period (demarked by two points in time)

Does not consider

 Two or more time periods during which abandonment can happen

2 Dias Jr and Ioannou, Debt Capacity and Optimal Capital

Structure for Privately-Financed Infrastructure Projects (1995) [2]

 Uses CAPM model and algebra to solve for optimal LTC ratio for a PPP BOT project by considering

 Bankruptcy possibility and associated costs

 Time Value of Money

 Debt Capacity and relationship with LTC ratio

 Effect of taxes on cash flow

 NPV and ROE as return metrics

Does not consider

 Sale of asset

3 Chan, Wang, Yang, The Pricing of Construction Loans (2016) [5]

 Uses a two-period Game Theory model to solve for mutually acceptable Interest Rate for a private project by considering

 Abandonment decision of developer

 Mutually acceptable fixed interest rate

 Lender’s loss ratio

 Sale of asset

Does not consider

 Time value of money

 Bankruptcy possibility and associated costs

 Debt capacity of project

 IRR or ROE as measures of financial return

 Operation period post-construction

 LTC ratio is an exogenous input

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Table 2.1 – Research from the current literature relevant to the optimization of construction loans’ LTC ratio (continued)

4 Bakatjan et al., Optimal Capital Structure Model for BOT Power Projects in Turkey (2003) [3]

 Uses Linear Programming and algebra to determine optimal capital structure for PPP BOT projects in Turkey by considering:

 IRR and NPV as measures of return

 Site acquisition costs

 Minimum DSCR as a linear programming constraint



Does not consider

 Debt capacity of project

 Value of asset 5 Zhang X., Financial

Viability Analysis and Capital Structure Optimization in Privatized Public Infrastructure Projects (2005) [11] Considers

 Weighted average cost of debt

 Some operating period post-construction

 Construction base cost, construction escalation cost, and Construction Duration based on variable distributions and simulated using Monte Carlo simulation

 DSCR and Loan Life Coverage Ratio

 NPV and IRR as measures of return

Does not consider

 Sale of asset

 No numerical examples or case studies included

6 Jian-Cheng et al., Determine Optimal Capital Structure for Metro PPP Projects to Reduce Financial Risks: Theory and Empirical Analysis, (2013) [20]

Considers

 Utility of functions and of debt service coverage ratio

 NPV and IRR as measures of return

 Operation cost and revenue

 Income tax

 Probability distribution specified for different types of variables

Does not consider

 Sale of asset (although taxes on NOI are modeled to be paid)

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Table 2.1 – Research from the current literature relevant to the optimization of construction loans’ LTC ratio (continued)

7 Proposed Model  Uses a two period Game Theory model to solve for mutually acceptable Interest Rate by considering

 Mutually acceptable fixed interest rate

 Lender’s loss ratio

 Expected levered IRR, NPV, and ROE as measures of financial return

 Sale of asset

Does not consider:

 Bankruptcy costs

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3 Methodology

This chapter proceeds to develop a game theoretic approach to the research problem and objectives as described in sections 1.1 and 1.2 The areas which this Game Theory approach will need to focus on are laid out in Table 2.1 in the Literature Review chapter, which provides a succint overview of the areas that have not been addressed by existing models pertaining to the subject Chan et al.’s model [5] – the foundation which the proposed model is based on - will be described in greater detail in this chapter, followed by explanations for the modifications made to it in order to arrive at the proposed model, including modifications to the game tree, the equations for the expected net profit amounts (i.e., the expected levered NPV), the objective functions for additional financial objectives, and additional constraints not originally considered

3.1 The Model

Games in Game Theory can be classified as either static or dynamic, the latter being a class of games that unfold over time as players make moves sequentially Tadelis [17] defines an extensive form game as “one of the most common representation of games that unfold over time in which players move after they learn the actions of other players” Tadelis identifies seven elements that characterizes such games:

1 Set of players, N

2 Players’ payoffs as a functions of the outcomes from each move 3 Moves made in some particular order

4 Actions that can be made when a player is able to move 5 Knowledge that players have when they’re able to move 6 Probability distribution over exogenous events

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The proposed model (“the model”) is built on the same basic extensive form game framework as outlined above which Chan et al [5] developed in order to calculate the mutually acceptable interest rate The game in the model is marked by three time periods: t=0, t=1, and t=2; the game could terminate at any time period depending on the moves that the players make The proposed model uses the basic framework of Chan et al.’s model [5] and adds the following modifications to the original model to calculate the optimal LTC ratio for a given construction loan:

1 Developer’s decision for the LTC ratio (l) at t=0 added before the decision to finance the model either with full equity or with leverage (see Figure 1.1)

2 Implementation of time value of money to discount cash flows from each period

under some discount factor j in order to arrive at the expected levered NPV value,

the expected levered IRR, and the levered Return on Equity (ROE) for the project (rather than using only the expected undiscounted profit amount as in the original model)

3 Length of the two time periods may vary (rather than strictly equal each other, as in the original model)

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3.2 Modelling Assumptions

The following assumptions, most of which come from the original model in [5], are made about the factors and circumstances of the loan and the game which the lender and borrower (developer) enter into:

 Limited recourse construction loan (either with land/project as collateral or personal guarantee) with fixed interest rate, as argued in section 2.1

 Two period game marked by three points in time; a portion of the loan disbursed for the first construction phase if decision to lever up is made; loan interest and principal repaid in full at end of game (as in [5])

 Income tax on net earnings after interest and debt obligations

 No operation period, only sale at end of game (as in [5])

 Two players who act rationally at every stage of the game (as in [5])

 Complete information at every stage of the game

 Exogenous variables: k, d, h, j, q, m, n, upper bounds for the LT ratio (see section 3.4.5d (as in [5], modified to exclude the variable l here)

 Objective functions: maximize either expected levered NPV, expected levered IRR or expected levered ROE, each of which corresponds to a different optimal LTC ratio

 Time between construction completion and sale is negligible (both take place at t=2) (as implied from [5])

 Constraints: see section 3.4.5 d)

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milestone in the construction process, for e.g., completion of concrete framing, is reached Estimations of the cost up until such a milestone can be garnered from sources such as cost breakdowns in bid submissions or historical data which may be available in databases such as RSMeans by Gordian©

An example of estimating the value for h can be seen from Bakatjan et al.’s model [3] contains a case study where a project in the negotiation (presumably pre-construction) stage between a contractor and owner (in this case, the Turkish government) is estimated to take four years to complete, with 40% of total construction works completed in the first two years In this example the value of m would be 2, and the value of h would be within the realm of 40%, contingent on detailed estimations which the contractors may put forward in their bid submissions

The value of k is much less obvious to estimate, as the recovery amount a lender can realistically expect depends on many factors, including the specific recourse structure of the loan as well as the amount of the remaining project can be salvaged at abandonment As a result the use of sensitivity analysis is encouraged for this model, particularly for the value of d As for the specific distributions of the market price p at completion, sources such as market reports, such as monthly or quarterly market reports for location and sector-specific markets produced by real estate services companies such as CBRE Group Inc © and Savills Inc ©, are often the go-to starting point for deriving expected market rents Further adjustments and valuation reports may be necessary for historical distributions and trends for specific projects

3.3 The Game

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Consequently, he faces two choices with respect to its capital structure: either have an equity-only structure or one with some combination of equity and debt

Tadelis [17] defines a game of perfect information as one which every single information set is a singleton; the implication of this is that 1) every player makes their move knowing their position in the game tree, and 2) every player upon making a move knows every move of every player that has been made hitherto [10] Identifying a game as having perfect information is important as it determines the method to be used to solve for the loan’s LTC ratio, which is the overarching goal of this model

The rest of this section illustrates how the game unfolds from the owner’s selection of the LTC ratio at the beginning to when the project is completed and sold, as detailed in [5]: at t=0, the modelling assumption stipulates that the future sale price of the finished project 𝑝 and its total cost 𝑐 are both unknown to both players until t=1; their values, however, belong to random variables that follow their respective probability distributions and are known to both players at t=0 This particular game is one of perfect information because the model assumes that the decision to enter into the construction project (rather than just selling/operating it as-is) has been determined prior to the game (an exogenous factor to the model) regardless of both players’ knowledge at t=0 of 𝑝 and 𝑐 Therefore, players are always aware of where they are in the game tree and all the moves that have been made by all players at any given time

The game begins at stage t=0 where the developer selects some LTC ratio l in the

event he does decide to borrow to finance his project He then makes the decision to either finance the project entirely with equity or with some combination of equity and debt, in which case he would approach the lender and request a to take out a construction loan

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