The publication co-financed by the European Union from the European Social Fund

Economics of Finance Agency Problem and Risk in Corporate Finance

Ireneusz Dąbrowski

Warsaw School of Economics JUNE 2015

This publication is the result of the project Młodzi projektują zarządzanie co-financed by the European Social Fund within Human Capital Operational Programme, Priority IV “Higher education and science,” Measure 4.1 “Strengthening and development of didactic potential of universities and increasing the number of graduates from faculties of key importance for knowledge-based economy,” Sub-measure 4.1.1 “Strengthening and development of didactic potential of universities.” Publisher: Szkoła Główna Handlowa w Warszawie (SGH) Reviewer: Tomasz Dębski First Edition

© Copyright by Szkoła Główna Handlowa w Warszawie (SGH) al. Niepodległości 162, 02-554 Warszawa, Polska Free copy ISBN: 978-83-65416-15-5 Circulation: 200 copies Cover design: Monika Trypuz Typesetting, printing: Agencja Reklamowa TOP 87-800 Włocławek, ul. Toruńska 148 tel. 54 423 20 40, fax 54 423 20 80 e-mail: [email protected]

to my Mother and in memory of my Father

Contents 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 Corporate Finance and The Financial Manager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Long-term Financial Decisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Capital Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Capital Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. The Agency Problem in Corporations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Principal-Agent Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 The Objectives of Financial Management. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Managers and Stockholders’ in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. Time Value of Money. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 3.1 Future Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 Investing for a Single Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.2 Investing for More Than One Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Present Value and Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 The Single-Period Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.2 Present Values for Multiple Periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Present versus Future Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4. Capital Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.1 Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Typical Cash Outflows and Inflows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Net Present Value (NPV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3.1 Estimating Net Present Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 Internal Rate of Return (IRR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4.1 Estimating Internal Rate of Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4.2 Net Present Value and Internal Rate of Return. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5 Payback Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.5.1 Evaluation of the Payback Period Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.5.2 Criticism of Payback Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5. Risk and Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.1 Rates of Return. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Variance and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 Diversification and Risk Reducing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3.1 Portfolio Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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CONTENTS

5.3.2 Portfolio Expected Returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Portfolio Variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Portfolio Risk and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Diversification, Systematic and Unsystematic risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Limits of Diversification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7 Beta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 32 33 36 39 40

6. Cost of Capital. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.1 Required Return, Cost of Capital and Financial Policy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 The Cost of Equity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.2.1 The Security Market Line and Cost of Capital. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.2.2 Advantages and Disadvantages of the Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.3 The Cost of Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.4 The Weighted Average Cost of Capital. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.4.1 The Capital Structure Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.4.2 Taxes and the Weighted Average Cost of Capital. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.4.3 The Security Market Line and the Weighted Average Cost of Capital. . . . . . . . . . . . . . 49 7. Capital Structure and Financial Leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.1 Capital Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.2 The Effect of Financial Leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.2.1 Financial Leverage and ROE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.2.2 Financial Leverage and EBIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2.4 Business Risk and Financial Risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.3 Capital Structure and the Cost of Equity Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.3.1 Traditional View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.3.2 The Miller & Modigliani Proposition I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.3.3 Cost of Equity and Financial Leverage: M&M Proposition II. . . . . . . . . . . . . . . . . . . . . 60 7.4 Taxation, Financial Distress and Bankruptcy Cost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.4.1 The Interest Tax Shield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.4.2 Taxes and M&M Proposition I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.4.3 Taxes, the WACC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.5 Financial Distress and Bankruptcy Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.5.1 Cost of Capital in Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.5.2 The Static Theory of Capital Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

1. Introduction Companies finance their business activities from both short-term and long-term sources. Even if short-term financial decisions constantly involve short-lived assets, there is a link between short-term and long-term financing decisions arising from a firm’s cumulative capital requirements. If we have a surplus of long-term financing, we would need less short-term funds. If the company wants to run a new project, regardless of what type it has started, it must answer two long-term questions. Long-term corporate finance, generally speaking, is the study of following issues: 1. Long-term investments that company should undertake. 2. Sources of long-term financing of projects. Short-term financial decisions differ from long-term ones in two important ways. First, they may be quickly reversed in most cases. Second, there is far less risk involved in the decision parameters as we are concerned with the horizon period counted in months rather than years. This does not mean that short-term financial decisions are any less important. Shortterm financial decisions ensure the company’s liquidity which is essential for the short-term operating capacity of the company.

1.1 Corporate Finance and The Financial Manager Corporate Finance means any monetary or financial action concerning a company itself and its cash. It is the set of actions and policies that decides how capital is employed in a company. An important problem of large corporations is that the owners are usually not directly involved in making company decisions, especially on a  day-to-day basis. The corporation usually hires managers to represent the owners’ interests and make decisions on their behalf. In a large corporation, the financial manager would be in charge of resolving the problems outlined in the preceding section. The financial management task is typically associated with a top officer of the company, such as a vice president for finance or some other chief financial officer (CFO). Figure 1 is a basic organizational chart that shows the finance position in a typical corporation. Normally the vice president of finance manages the activities of the controller and the treasurer. The controller’s office is responsible for financial accounting, tax payments, and management information systems. The treasurer’s office is responsible for managing the company’s credit and cash, its financial planning, and its capital expenditures. Generally, financial managers try to match the maturity of capital sources with the life of the assets funded by them. For example, some minimum level of working capital is needed permanently in the company and is financed from permanent sources, but the seasonal increase in working capital is characteristically financed from short-term sources.

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1. INTRODUCTION

Figure 1 Organization chart

1.2 Long-term Financial Decisions As the preceding discussion suggests, the financial manager must be concerned with two basic problems. We discuss these issues in greater detail next. 1.2.1 Capital Budgeting The first long-term problem concerns the company’s new investment. The process of planning and managing a company’s long-term investment is called capital budgeting. In capital budgeting, the financial manager tries to recognize investment opportunities that are worth more to the company than they cost to obtain. In other words, we may say that the value of the cash flow generated by an asset exceeds the cost of that asset. The types of investment opportunities that would naturally be considered depend in part on the type of the company’s industry. In spite of the specific nature of an opportunity under consideration, financial managers must be concerned not only with an amount of money they expect to collect, but also when they anticipate to receive it and how likely they are to receive it. Evaluating the size, timing, and risk of future cash flows is the essence of capital budgeting. In fact, whenever we evaluate a business decision, the size, timing, and risk of the cash flows will be, by far, the most important things we will consider. 1.2.2 Capital Structure The second long-tem problem for the financial manager concerns ways in which the company acquires and manages the long-term financing to sustain its long-term investment. A company’s capital structure (or financial structure) is the specific combination of long-term

9 debt and equity the company employs to finance its activity. The financial manager has two concerns in this area. Firstly, how much debt relatively to equity should the company employ, i.e. what combination of debt and equity is optimal. The chosen capital mix will influence both the risk and the value of the company. Secondly, what are the least expensive sources of capital for the company? Company’s capital structure determines the fraction of the company’s cash flow that goes to debtholders and the one that goes to shareholders. Companies have a great deal of elasticity in selecting their financial structure. The problem of whether one structure is better than any other for a particular company is in the core of the capital structure problem. Additionally, deciding on the financing mix, the financial manager has to choose how exactly and where to raise the cash. The operating costs associated with raising long-term financing can be substantial, so different possibilities must be carefully estimated. Moreover, companies borrow money from a variety of investors in a lot of different, and sometimes peculiar, ways. Selecting bankers and debt types is another task handled by the financial manager.

2. The Agency Problem in Corporations In traditional (neoclassical) approach a corporation is treated as a single entity, it is often called a holistic approach. That is one of the characteristics of a sole proprietorship. There is no conflict of interest between owners and managers. In large corporations we almost always have the separation of owners and managers. Financial manager is supposed to work in the best interest of the owners by taking actions that increase the value of the company. However, we have also witnessed that in large corporations ownership can be spread over a huge number of owners (shareholders). If we assume that shareholders buy shares as they seek financial gain, then the answer is obvious: good choices increase the value of the shares, and poor choices decrease the value of the shares. Specified our clarification, it follows that the financial manager acts in the shareholders’ best interest by making decisions that increase the current value of the shares. On the other hand the separation of management and owners has a few advantages. It allows owners to change management without interfering the business operation. It allows the company to employ professional managers. This dispersion of ownership means that managers, not owners, can control the company, and it brings problems if the managers’ and owners’ objectives are not the same and if management do not really acts in the best interest of the owners. We explicitly mean that management goal is to maximize the current share value. By this we mean that owners are only entitled to what is left after suppliers, staff, debtholders and anyone else with a legitimate claim to paid their due. If any of these groups go unpaid, the owners get nothing. Because the goal of financial management is to maximize the value of the stock, we need to learn how to recognize those investments and financial planning that positively impact the value of the company. We can say that corporate finance is the study of the relationship between business decisions and the value of the shares. 2.1.1 Principal-Agent Problem The relationship between managers and owners is called principal-agent problem (or agency relationship). Such a  connection exists whenever someone (the principal) employs another (the agent) to represent his interests. Shareholders are the principals and the managers are their agents. Shareholders want the management to raise the value of the company, but managers may have their own aims. Agency costs are incurred when (1) managers do not attempt to maximize company’s value and (2) shareholders incur expenses to monitor the managers and influence their actions. Usually, the expression agency costs refers to the costs of the conflict of interest between owners and management. Naturally, there are no costs when the owners are also the managers.

11 Agency costs can be indirect or direct. An indirect agency cost is an opportunity cost. Direct ones come in two types. The first type is corporate spending that benefits management but constitute excessive costs for the owners (i.e. luxurious and unnecessary corporate cars or even jets). The second type of direct agency cost is an expense that arises from the need to control management actions. Hiring external auditors to assess the correctness of financial statement information is a typical example. 2.1.2 The Objectives of Financial Management Under the neoclassical profit maximization assumption, we expect the financial management to make profits or add value for the owners. This aim is however a  little unclear, so we observe somewhat different ways of formulating it while trying to come up with a more precise definition. Such a description is important because it provides an objective basis for making and evaluating financial decisions. If we consider possible financial goals, we may list some ideas such as the following: • Survive, • Avoid financial distress and bankruptcy, • Beat the competition, • Maximize sales or market share, • Minimize costs, • Maximize profits, • Maintain steady earnings growth. There are economic theories which assume that management may tend to overemphasize organizational survival to protect their job security. Management may have an aversion to outside interference, so independence and corporate self-sufficiency may be the important goals for them. Baumol presented a  managerial theory of the firm based on the sales maximization. According to the managerial theory, if the management have no control at all it seeks prestige and higher salaries by expanding the company’s sales even at the expense of its profits. It is the idea which comes from the outside mainstream economy that, with the separation of ownership and control in modern companies, managers prefer the company to be rather bigger than more profitable. Therefore, managers left on their own would tend to maximize the amount of resources over which they have control or, more generally, corporate power or wealth. This goal could lead to an overemphasis on corporate size or growth. 2.1.3 Managers and Stockholders in Practice Agency problems would be easier to solve if owners and managers have the complete information. That is however rarely the case in finance. Managers, shareholders, and debtholders may all have different information about the value of a financial asset, and it may be many years before all the information is revealed. Financial managers have to recognize these information asymmetries and find ways to convince investors that there are costly surprises on the way ahead.

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2. THE AGENCY PROBLEM IN CORPORATIONS

Two basic aspects decide if managers act in the best interest of owners: how closely are the goals of management aligned with those of stockholder and, if management can be replaced when they do not follow owners’ goals. The first aspect relates to the way managers are compensated. The second aspect relates to control of the company. Usually, there are some reasons to believe that, even in the corporations with complex structures, management has a significant incentive to perform in the interests of owners. 2.1.3.1 Managerial Compensation Managerial compensation is usually analyzed by means of four categories: • Wage Level • Base Pay • Variable Pay • Benefits Management will normally have a significant economic incentive to increase share value for two reasons. First, managerial compensation, particularly at the top, usually depends on the financial performance in general and to share value in particular. Managers are regularly given the option to buy shares at an individual price. The more the shares are worth, the more valuable this option is. Actually, options are increasingly being used to motivate managers of all types, not only the top management. The second incentive for managers refers to their career prospects. Better performers within the company will tend to be promoted. More generally, those managers who are successful in pursuing stockholder’s goals will be in greater demand on the labor market and thus they will command higher salaries. In fact, managers who are successful in pursuing stockholder’s goals can reap enormous rewards. 2.1.3.2 Control of the Company The control of the company depends ultimately on stockholders. They vote for the board of directors which employ management. A key mechanism that allows dissatisfied stockholders to change the current management is called a proxy fight. A proxy is the authority to vote someone else’s stock. A proxy fight occurs when a group solicit proxies in order to replace the elected board, and thus replace the incumbent management. Another way that management can be replaced is by takeover. Those companies that are badly managed are more attractive as acquisition targets than well-managed companies because a greater possible gain exists. Therefore, avoiding a hostile takeover by another company provides the management another incentive to take action in the owners’ interest. Management and owners are not the only players that may have an interest in the company’s decisions. Customers, employees, suppliers, and even the state authority all have a financial interest in the company. These different groups are called stakeholders in the company. Normally, a stakeholder is someone other than a stockholder or creditor who potentially has a claim on the cash flows of the company. Such groups will also try to exert control over the company, possibly to the loss of the owners.

3. Time Value of Money Time value of money refers to the fact that the cash today is worth more than the cash promised at some time in the future. One explanation for this is that one could receive interest while waited, so the money today would grow over the time. The trade-off between cash now and cash later thus depends on, among other things, the rate of return we can receive from investing.

3.1 Future Value Future value (FV) is the amount of money that an investment will grow to over some period of time at some given interest rate. Put differently, future value is the cash value of an investment at some time in the future. Let us consider now a single-period investment first. 3.1.1 Investing for a Single Period Suppose we invest $1 in a savings account that pays 100% interest per year. How much will we get after one year? We will get $2. This $2 is equal to our original principal of $1 plus $1 in interest that we earn. We say that $2 is the future value of $1 invested for one year at 100%, and we simply mean that $1 today is worth $2 in one year, given that 100% is the interest rate. Generally, if we invest funds for one period at an interest rate of r, our investment will grow to (1 + r) per dollar invested. In our example, r is 100%, so our investment grows to 1 + 1 = 2 dollars per dollar invested. We invested $1 in this case, so we end with $1*(1+1)= $2. 3.1.2 Investing for More Than One Period Going back to our $1 investment example, what will we get after two years, assuming the interest rate doesn’t change? If we deposit the entire sum of $2 in the bank, we will earn $2* 1 = 2 in interest during the second year, so we will have a total of $2* (1+1)=$4. This $4 is the future value of $1 in two years at 100%. Another way of looking at it is that one year from now we are effectively investing $2 at 100% for a year. This is a single-period problem, so we will end up with $2 for every dollar invested, or $2 (1+1) = $4 total. This $4 contains four elements. The first element is the $1 original principal. The second element is the $1 in interest we earned in the first year, and the third element is another $1 we earn in the second year, for a total of $3. The last $1 we end up with (the fourth part) is interest we earn in the second year on the interest paid in the first year. This method of putting our money and any accumulated interest in an investment for more than one period, thereby reinvesting the interest, is called compounding. Compounding the interest means receiving interest on interest, so we call the result compound interest. With simple interest, the interest is not reinvested, so interest is earned each period only on the original principal.

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3. TIME VALUE OF MONEY

Now we discuss how we calculated the $4 future value. We multiplied $2 by 2 to get $4. The $2, however, was $1 also multiplied by 2. In other words: $4 = $2*2= ($1*2)*2= $1*(2*2)= $1 * 22 Now let’s ask: How much would our $1 grow to after three years? Once again, in two years, we will be investing $4 for one period at 100%. We will end up with $2 for every dollar we invest, or $4 *2 = $8 total. This $8 is thus: $8 = $4*2= ($2*2)*2= [($1*2)*2] *2= $1*(2*2 *2)= $1 * 23 After becoming aware of a pattern to these calculations, we can go ahead now and state the general result. As our examples suggest, the future value of $1 invested for t periods at a rate of r per period is: Future value = $1 * (1 + r)t The expression (1 + r)t is called the future value interest factor (or just future value factor) for $1 invested at r% for t periods and can be abbreviated as FVIF(r, t). Let us consider what would your 1$ be worth after five years with interest rate 10%? We can first calculate the relevant future value factor as: (1 + r)t = (1 + 0.10)5 = 1,15 = 1,6105 Your $1 will thus grow to: $1 * 1.6105 = $1.6105

3.2 Present Value and Discounting After we discuss future value, now we have a similar issue that is even more present in financial management and is clearly related to future value. Assume we need to have $10000 in 5 years, and we can earn 10% on our money. How much do we have to invest today to reach our objective? 3.2.1 The Single-Period Case Consequently, now we want to find out how much do we have to invest today at 10% to get $1 in one year? To make things simpler, we have to calculate the present value (PV) if we know that the future value is $1. The solution is not too complicated to find. The amount we invest today will be 1,1 times bigger at the end of the year. Since we need $1 at the end of the year: Present value * 1,1 = $1

15 Or, solving for the present value: Present value = $1/1,1 = $0,909 In this example, we must find the amount (the present value), invested today, that will grow to $1 after one year if the interest rate is 10%. Present value is therefore just the opposite of future value. Instead of compounding the money forward into the future, we discount it back to the present. From our examples, the present value of $1 to be obtained in one period is basically given as:

3.2.2 Present Values for Multiple Periods Assume we want to have $100 in two years and if we can earn 10% per year. How much do we have to invest now to make sure that we have the $100 when we want it. So now the issue is, what is the present value of $100 in two years if the relevant rate is 10%? Based on our understanding of future values, we know the amount invested must grow to $100 over the two years. In other words, it must be the situation that: $100 = PV * 1,1 * 1,1 = PV * 1,12 = PV * 1,21 Given this, we can easily find the solution for the present value:

Consequently, $82,65 is the amount we must invest in order to achieve our goal. As we have known by now, calculating present values is quite similar to calculating future values, and the general result looks much the same. The present value of $1 to be collected t periods into the future at a discount rate of r is:

Because the measure

1 is used to discount a future cash flow, it is called a discount (1 + r) t

factor and the rate r used in the calculation is named the discount rate. We will tend to call it in talking about present values. The measure

1 is also called the present value interest (1 + r) t

factor (or just present value factor) for $1 at r% for t periods and is sometimes abbreviated as PVIF(r, t). Calculating the present value of a future cash flow to determine its value today is usually called discounted cash flow (DCF) valuation.

16

3. TIME VALUE OF MONEY

3.3 Present versus Future Value If we look back at the formulas we provided for present and future values, we will see there is an straightforward connection between the two. We investigate this relationship and some related issues. What we called the present value factor is just the reciprocal of (that is, 1 divided by) the future value factor: Future value factor = (1 + r)t Present value factor =

1 (1 + r) t

Certainly, the simple way to calculate a present value factor on many calculators is to calculate the future value factor first and then press the “1/x” key to flip it over. If we let FVt stand for the future value after t periods, then the relationship between future value and present value can be formulated very simply as one of the following: PV (1 + r)t = FVt

This last formula we will name the central present value equation. We will use it throughout the text. There are a number of differences that will occur, but this simple equation lies beneath many of the most important ideas in corporate finance.

4. Capital Budgeting Today, the professional allocation of capital resources is a most essential function of the financial management. This purpose involves organization’s decision to invest its resources in long-term assets such as a new production facility or investing in machinery and equipment, land, vehicles, etc. All these assets are very important to the company because, usually all the profits are obtained from the use of its capital in investment in assets which correspond to a  very large commitment of financial resources, and these funds typically remain invested over a long period of time. The future expansion of a company is contingent on the capital investment projects, the replacement of existing capital assets, and the decision to abandon previously accepted activities which appear as less attractive to the company than was originally assumed, and redeploying the resources, new concepts and planning. In an ideal world, companies should reject all projects and opportunities that harm shareholder’s value. The process during which a  company determines whether particular endeavor such as building a new factory or investing in a long-term project is worth investing is called capital budgeting (or investment appraisal). Generally, a prospective project’s lifetime cash outflows and inflows are evaluated in order to determine whether the returns generated meet a target baseline. In real economy the total available capital at every given time for new businesses is strictly limited, executives need to use capital budgeting methods to determine which businesses will yield the highest return over an applicable period of time. Consequently, capital budgeting can be defined as long-term planning for proposed capital outlays and their financing. Therefore, it includes both raising long-term funds and their use. To be more specific, capital budgeting decision may be defined as long-term investments in which the assets involved have useful lifes of multiple years. Capital budgeting is a method of estimating the financial viability of a capital investment over the life of the investment. The capital budgeting question is probably the most important issue in corporate finance. How a company chooses to finance its operations (the capital structure question) and how a company manages its short-term operating activities (the working capital question) are certainly issues of concern, but it is the fixed assets that define the business of the company. Capital budgeting, unlike other types of investment analysis, concentrates on cash flows rather than profits. It involves identifying the cash inflows and cash outflows rather than accounting revenues and expenses flowing from the investment, such non-expense items as debt principal payments are included in capital budgeting because they are cash flow transactions. Over the long run, capital budgeting and traditional performance analysis will yield analogous net values. However, capital budgeting methods include the time value modifications.

18

4. CAPITAL BUDGETING

There is a huge number of viable investment available for each company. Each viable investment is an option available to the company. Some alternatives are valuable, others are not. The essence of successful financial management, of course, is learning to recognize which are which. Capital investments create cash flows that are often spread over several years into the future. To accurately assess the value of a capital investment, the timing of the future cash flows is taken into account and converted to the current time period (present value). There are two approaches to making capital budgeting decisions using Discounted Cash Flow model (DCF). One is the net present value method (NPV), and other is the internal rate of return method (also called the time adjusted rate of return method). Another popular method of capital budgeting is payback period.

4.1 Cash Flows In capital budgeting issues, the most important are cash flows (not accounting net income). The idea is that accounting net income is based on results that ignore the time value of cash flows into and out of a company. From a capital budgeting perspective, the timing of cash flows is essential, because money received today is more valuable than money received in the future. Consequently, even though accounting net income is useful for many things, it is not generally used in discounted cash flow analysis. Instead of determining accounting net income, the manager concentrates on recognizing the specific cash flows of the investment project.

4.2 Typical Cash Outflows and Inflows Typical businesses will have an immediate cash outflows in the form of an initial outlay or other assets. Any estimated resale value of an asset at the end of its useful life (salvage value) can be recognized as a cash inflow or as a reduction in the required investment. Some projects require that a company increase its working capital as well. When a company decides to do a new project, the balances in the current assets will often raise. Opening a new Orlen gasoline station would require additional cash, increased accounts receivable for new customers, and more inventory. These additional working capital requirements should be treated as a part of the initial investment in a business. These all should be treated as cash outflows for capital budgeting purposes. On the cash inflow side, a project will usually either increase incomes or reduce expenditures. Either way, the amount involved should be treated as a cash inflow for capital budgeting purposes. We can observe that so far as cash flows are concerned, a reduction in costs is equivalent to an increase in revenues. Cash inflows are also regularly realized from salvage of equipment when a project ends, although the company may actually have to pay to dispose of some low - value or hazardous items. Furthermore, any working capital that was used in the project can be available for use somewhere else at the end of the project and should be treated as a cash inflow at that time. Working capital is released, for example, when a company sells off its stock or collects its receivables.

19

4.3 Net Present Value (NPV) A project is worth undertaking if it generates value for its owners. In the common sense, we create value by recognizing a project worth more in the marketplace than it costs us to obtain. It is a case of the whole being worth more than the cost of the parts. The real challenge is to somehow recognize ahead of time whether or not a particular investment is a good idea in the first place. This is what capital budgeting is all about, specifically, trying to determine whether a proposed investment or project will be worth more, once it is in place, than it costs. The difference between an investment’s market value and its cost is called the net present value of the investment (NPV). Thus, the net present value is a measure of how much value is added or created today by undertaking a project. Given our aim of creating value for the owners, the capital budgeting procedure can be viewed as a search for investments with positive net present values. We can imagine how we can get on with making the capital budgeting decision of new technology. First we look at what price a similar technology is selling for in the market. Then we must estimate the cost of creating a new technology and bringing it to market. Therefore, we have an estimated total cost and an estimated market value. If the difference is positive, then this investment is worth undertaking because it has a  positive estimated net present value. Naturally, there is uncertainty, because there is no guarantee that our estimates will turn out to be correct. Investment decisions are significantly simplified when there is a market for assets comparable to the investment we are taking into consideration. Capital budgeting becomes much more complicated when we cannot monitor the market price for almost comparable investments since we are then faced with the problem of estimating the value of an investment using only indirect market information. 4.3.1 Estimating Net Present Value We can assume that we are considering starting a new project. First, we can estimate the initial costs with reasonable accuracy because we know what we will need to buy to begin business. We will try to estimate the future cash flows we expect the new project to produce. Then we will apply our basic discounted cash flow procedure to estimate the present value of those cash flows. Once we had this estimate, we will then estimate NPV as the difference between the present value of the future cash flows and the initial cost of the project. This method is often called a discounted cash flow (DCF) valuation. In order to see how we might estimate NPV, suppose we believe the cash revenues from our project will be $30 per year, assuming everything goes as expected. Financial costs will be $10 per year. We will end the project in 6 years. The property and equipment will be worth $50 as salvage at that time. The project costs $100 to launch. We use a 10% discount rate on new projects such as this one.

20

4. CAPITAL BUDGETING

Table 1 Discount rate=

10%

Year Initial cost

0

1

2

3

4

5

-100

Salvage value Cash infow

6

50 30

30

30

30

30

30

30

Cash outflow

-10

-10

-10

-10

-10

-10

-10

Net cash flow

20

20

20

20

20

20

20

1,00000

0,90909

0,82645

0,75131

0,68301

0,62092

0,56447

18,18

16,53

15,03

13,66

12,42

11,29

Present value factor Present value cash flow

20,00

Total PV of cash flow

107,11

+PV of salvage value

28,22

28,2237

135,33 -Initial cost Net present value

- 100,00 35,33

From a simply mechanical perspective, we need to calculate the present value of the future cash flows at 10%. The net cash inflow will be $30 cash income less $10 in costs per year for six years. These cash flows are illustrated in Table 1. As Table 1 suggests, we effectively have a six-year annuity of $30 - $10 = $20 per year, along with a single lump-sum inflow of $50 in six years. The total present value is: Present value = When we compare this to the $100 estimated cost, we see that the NPV is: NPV = 135,33 – 100 = 35,33 Consequently, this is a good investment. Based on our estimates, undertaking it would increase the total value of the company by $35,33.

21 Our example illustrates how NPV estimates can be used to determine whether or not a particular project is desirable. Thus we can observe that if the NPV is negative, the effect on company’s value will be unfavorable. If the NPV were positive, the effect would be favorable. As a result, all we need to know about a particular proposal for the purpose of making an accept-reject decision is whether the NPV is positive or negative. There are some advantages and disadvantages of the net present value method. On one hand, NPV accounts for time value of money. Thus it is more reliable than other investment appraisal methods such as the payback period or the internal rate of return. NPV is quite straightforward to calculate, as well. On the other hand, NPV is based on estimated future cash flows of the project and estimates may be far from real results.

4.4 Internal Rate of Return (IRR) This method seeks to identify the rate of return that an investment project yields based on the amount of the original project remaining exceptional during any period, compounding interest annually. 4.4.1 Estimating Internal Rate of Return The Internal Rate Of Return (sometimes referred to as an economic rate of return ERR) is closely connected to the Net Present Value discount rate. To calculate IRR using the formula, we must set NPV equal to zero. Thus IRR of a project is the discount rate which if applied to the project yields a zero NPV, so it is the solution, for i, to the following simple equation:

which could be solved using the standard solution to a quadratic equation. The answer incidentally is i = 0,099 or 9,9%.) We can now look at a little bit more complicated example of the equation from which we must solve for i:

300 300 300 300 − 1000 300 + + + + + =0 0 1 2 3 4 (1 + i ) (1 + i ) (1 + i ) (1 + i ) (1 + i ) (1 + i ) 5 Solving for i here is not that simple and because of the nature of the formula, IRR cannot be calculated analytically, and must instead be calculated either through trial-and-error or using software program. In practice, we usually solve for i by trying various values of i until we find one which satisfies or almost satisfies the equation.

22

4. CAPITAL BUDGETING In Table 2 we can calculate that the NPV is $137 when discounted at 10%.

Table 2 i= 10,00% CF 0 - 1 000,00

PV - 1 000

1

300,00

273

2

300,00

248

3

300,00

225

4

300,00

205

5

300,00

186 NPV= 137

Now we know that we must have an IRR of above 10%, because the higher the discount rate, the lower the present value of each cash flow. How much above 10% is the IRR we actually do not know, so some higher discount rate needs to be tried, say 15 and 16%. In Table 3 referring to the 15 and 16% we can calculate NPV. Table 3 i= 15,00% CF 0 - 1 000,00

i= 16,00% CF

PV - 1 000

0 - 1 000,00

PV - 1 000

1

300,00

261

1

300,00

259

2

300,00

227

2

300,00

223

3

300,00

197

3

300,00

192

4

300,00

172

4

300,00

166

5

300,00

149

5

300,00

143

NPV= 6

NPV= -18

Since the NPV when the cash flows are discounted at 16% is negative and at 15% is positive, we know that the discount rate which gives this project a zero NPV lies below 16% and apparently close to the midpoint between 15 and 16%. We can prove this in Table 4 by discounting at 15,5%. As the NPV when the cash flows are discounted at 15,5% is negative and at 15% is positive, we know that the discount rate which gives this project a zero NPV lies between 15,0% and 15,5%. In Table 4 we also can try discounting at 15,25%.

23 Table 4 i= 15,50% CF 0 - 1 000,00

i= 15,25% CF

PV

0 - 1 000,00

- 1 000

PV - 1 000

1

300,00

260

1

300,00

260

2

300,00

225

2

300,00

226

3

300,00

195

3

300,00

196

4

300,00

169

4

300,00

170

5

300,00

146

5

300,00

148

NPV= -6

NPV= 0

As NPV is now zero, we know that the IRR is about 15,25%. Generally, the higher a project’s Internal Rate of Return, the more likely it is to be undertaken. IRR is standardized for investments of varying types and, as such, IRR can be used to sort out multiple prospective projects a  company is considering on a  quite even basis. Assuming projects are competing and the costs of investment are equal among the various ones, the project with the highest IRR would probably be considered the best one and undertaken first. 4.4.2 Net Present Value and Internal Rate of Return As we can see, Internal Rate of Return calculations rely on the same formula as net present value. The NPV method is a direct measure of the cashflow contribution to the owners. The IRR method shows the return on the original money invested. NPV method has several important advantages over the IRR method. The net present value method is often easier to use. The internal rate of return method may require looking for the discount rate that results in a net present value of zero. This can be a very laborious trial-and-error process, although it can be programmed to some degree using a computer software. The key assumption made by the internal rate of return method is sometimes doubtful. Both methods assume that cash flows generated by a project during its useful life are immediately reinvested in another project. In short, when the net present value method and the internal rate of return method are not coherent in terms of the attractiveness of a project, it is best to go with the net present value method. Of the two methods, it makes the more realistic assumption about the rate of return that can be earned on cash flows from the project. The NPV method does not measure the project size. The IRR method can, at times, give us inconsistent results when compared to NPV for mutually exclusive projects (multiple IRR problem). A multiple IRR problem occurs when cash flows during the project lifetime are negative (i.e. the project operates at a loss or the company needs to contribute more capital). This is known as a “non-normal cash flow,” and such cash flows will give multiple IRRs.

24

4. CAPITAL BUDGETING

When a project is an independent project, meaning the decision to invest in a project is independent of any other projects, both the NPV and IRR will always give the same result, either rejecting or accepting a project. While NPV and IRR are useful metrics for analyzing mutually exclusive projects - that is, when the decision must be one project or another - these metrics do not always point the same direction. This is a result of the timing of cash flows for each project. In addition, conflicting results may simply occur because of the project sizes.

4.5 Payback Period Payback period in capital budgeting refers to the period of time required for the return on an investment to “repay” the sum of the original investment. This period is sometimes referred to as “the time that it takes for an investment to pay for itself.” The basic premise of the payback method is that the more quickly the cost of an investment can be recovered, the more desirable the investment is. The payback period is expressed in years. When the net annual cash inflow is the same every year, the following formula can be used to calculate the payback period. 4.5.1 Evaluation of the Payback Period Method The formula or equation for the calculation of payback period is as follows: Payback period = Investment required / Net annual cash inflow To illustrate the payback method, we can consider the following example. The company needs a new device. The company is considering two devices. Device A and device B. Device A costs $15,000 and will reduce operating cost by $5,000 per year. Device B costs only $12,000 but will also reduce operating costs by $5,000 per year. Device A payback period = $15,000 / $5,000 = 3.0 years Device B payback period = $12,000 / $5,000 = 2.4 years According to payback estimations, the company should acquire device B, since it has a shorter payback period than device A. Payback period, as a tool of analysis, is often used because it is easy to apply and easy to interpret for most individuals, but it is not a true measure of the profitability of an investment. When used carefully or in order to compare similar investments however, it can be quite useful. All else being equal, shorter payback periods are preferable to the longer ones. As a stand-alone tool to compare an investment to “doing nothing,” payback period has no explicit criteria for decision-making (except, perhaps, that the payback period should be less than infinity).

25 Payback period simply tells the manager how many years it will take to recover the original investment. Unfortunately, a shorter payback period does not always mean that one investment is more desirable than another. 4.5.2 Criticism of Payback Method The payback period is an effective measure of investment risk. The project with a shorter payback period has less risk than those with a longer payback period. The payback period is often used when liquidity is an important criteria to choose a project. Payback period method is suitable for projects of small investments. It is not worth spending much time and effort on sophisticated economic analysis in such projects. Criticism of payback method is that it does not consider the time value of money. A cash inflow to be received several years in the future is weighed equally with a cash inflow to be received right now. On the other hand, under certain conditions the payback method can be very useful. In addition, the payback period is often of great importance to new companies with cash shortage. In such companies, a project with a short payback period but with a low rate of return might be preferred to another project with a high rate of return but a long payback period. The reason is that the company may simply need a faster return of its cash investment. And finally, the payback method is sometimes used in industries where products become obsolete very rapidly (i.e. laptops or mobile phones). As products may last only a year or two, the payback period on investments must be very short.

5. Risk and Portfolio The first and basic thing we should know: an economic risk is the risk that happens then and only then when we expect something in the future. The risk is the possibility of difference between agent’s expectations and reality. If one is not concerned about future, (s)he has no risk (i.e. each state of future has identical utility). Risk is everywhere and affects every consumer and company, because every agent expects something, even if one is not aware of this. We remember the discount rate for safe projects, and an estimate of the rate for averagerisk projects. However we do not know so far how to find discount rates for assets that do not fit these simple cases. Therefore, we have to learn (1) how to measure risk and (2) the relationship between risks borne and required risk premiums.

5.1 Rates of Return To make things less complicated, we assume no inflation in economy thus all rates are nominal. If investor buys an asset of any kind, one’s gain (or loss) from that investment is named the return on investment. This return typically has two components. First, receive some cash directly while one’s own the investment. This is called the income component of the return. Second, the change in value of the purchased asset and it’s named capital gain (or loss) on investment. On the other hand, the term is also used in reference to percentage return, which is e.g. stock’s total return - dividend and change in value - divided by the investment amount (Figure 2).

Figure 2 Components of the return on investment.

Suppose investor bought the stock at the beginning of 2010 when its price was $10.00 a share. By the end of the year the value of that investment rose to $15.00 yielding a capital gain of $5: $15- $10 = $5.

27 Additionally, in 2010 the company paid a dividend of $1 a share. The return on investment is therefore: Nominal return = capital gain + dividend = (15-10) + 1 = 6 Percentage rate of return is income on an investment expressed as a percentage of the investment’s purchase price. With a common stock, the rate of return is dividend yield, or annual dividend divided by the price paid for the stock, thus the percentage return on investment was:

The percentage return can also be formulated as the sum of the dividend yield and percentage capital gain. The dividend yield is the dividend formulated as a  percentage of the stock price at the beginning of the year:

In the same way, the percentage capital gain is Percentage=capital gain =

capital gain (15 10 ) 5 = 50 % = = initial share pnce 10 10

Thus the total return is the sum of 10% + 50% = 60%.

5.2 Variance and Standard Deviation Even if there can be different definitions of risk, the standard statistical measures of risk are variance and standard deviation. Variance measures the variability of realized returns around an average level. The larger the variance, the higher the risk in the portfolio. The variance of the market return is the expected average value of squared deviations from mean squared deviation from the expected return. In other words:

where r is the actual return and E(r) is the expected return. The standard deviation is typically used by investors to measure the risk of a  stock or a stock portfolio. The fundamental idea is that the standard deviation is a measure of spread: the more stock’s returns vary from the stock’s average return, the more volatile is the stock. The standard deviation is simply the square root of the variance:

28

5. RISK AND PORTFOLIO

Think of the following two stock portfolios and their individual returns over the last six months. Both portfolios end up increasing in value from $1,000 to $1,058 but they clearly differ in volatility. Portfolio A’s monthly returns range from -1.5% to 3% whereas Portfolio B’s range from -9% to 12%. The standard deviation of the returns is a better measure of volatility than the range because it takes all the values into account. The standard deviation of the six returns for Portfolio A is 1,52 while for Portfolio B it is 7,24. Another uncomplicated example demonstrates how variance and standard deviation are calculated. Suppose that you have the chance to play the following financial game. You start by investing $1000 and return on this investment depends on combination of two different states of economy: • Changing interest rates by FED • Changing interest rates by EBC For each interest decreasing that occurs you get back your initial capital plus 10%, and for each interest increase you get back your starting capital less 5%. There are obviously four equally expected outcomes determined by possible states of economy: • FED-decreasing, EBC-decreasing: profit 10+10=20%. • FED-decreasing, EBC-increasing: profit 10-5=5%. • FED-increasing, EBC-decreasing: profit -5+10=5%. • FED-increasing, EBC-increasing: profit -5+-5 =-10%. There is 25% probability (1 in 4) that you will make 20%; 50% probability (2 in 4) that you will make 5%; and 25% probability (1 in 4) that you will lose 10%. The gamble’s expected return is, as a result, a weighted average of the possible profits: Table 5 illustrates that the variance of the percentage returns is 0,0113. Standard deviation is the square root of variance. This figure is in the same units as the rate of return, so we can say that the game’s variability is 10,61%. One way of defining uncertainty is to say that more things can occur than will occur. The risk of an asset can be thoroughly expressed by writing all possible result and the probability of each. In practice this is awkward and often impossible. Thus we use variance or standard deviation to summarize the spread of possible profits. These measures are natural indexes of risk. If the outcome of the game had been certain, the standard deviation would have been zero. The actual standard deviation is positive because we do not know what is going to happen.

29 Table 5. FED

EBC

return ri

probability p

ri*p

E(rp)- ri

[E(rp)- ri]2

p [E(rp)- ri]2

Decreasing

decreasing

20%

25%

5,00%

-15%

0,0225

0,0056

Decreasing

increasing

5%

25%

1,25%

0%

0,0000

0,0000

Increasing

decreasing

5%

25%

1,25%

0%

0,0000

0,0000

Increasing

increasing

-10%

25%

-2,50%

15%

0,0225

0,0056

σ2=

0,0113

σ=

10,61%

E(rp)=

5%

Table 6 shows a second game, again, there are four equally likely profits: Table 6 FED

EBC

return ri

probability p

ri*p

decreasing

decreasing

40%

25%

10%

-35%

0,1225

0,0306

decreasing

increasing

5%

25%

1%

0%

0,0000

0,0000

increasing

decreasing

5%

25%

1%

0%

0,0000

0,0000

increasing

increasing

-30%

25%

-8%

35%

0,1225

0,0306

σ2=

0,0613

σ=

24,75%

E(rp)=

5%

E(rp)- ri [E(rp)- ri]2 p [E(rp)- ri]2

In this game the expected return is 5%, the same as that of the first game. However its standard deviation is more than double that of the first game: 24,75% versus 10,61%. By this measure the second game is more than twice as risky as the first one.

30

5. RISK AND PORTFOLIO

5.3 Diversification and Risk Reducing By now we have concentrated on single assets considered separately, but most investors hold a portfolio of assets. Investors usually own more than just a single bond, stock, or other asset. Therefore, portfolio return and portfolio risk are of obvious significance. Consequently, we now explain portfolio expected returns and variances. 5.3.1 Portfolio Weights Portfolio can be described in many equivalent ways. The most popular approach is to list the percentage of the total portfolio’s value that is invested in each portfolio asset. We call these percentage the portfolio weights. Table 7 USD Asset A

$ 200,00

Asset B

$ 300,00

Asset C

$ 500,00

Total

$1 000,00

portfolio weights (Asset A)/ (Asset A+Asset B+Asset C) (Asset B)/ (Asset A+Asset B+Asset C) (Asset C)/ (Asset A+Asset B+Asset C)

portfolio weights 20% 30% 50% 100%

Table 7 shows simple illustration of a porfolio. An investor has $200 in asset A, $300 in asset B and $500 in asset C, then the portfolio is worth $1000. The percentage of the portfolio is summarized in the last column: the asset A is $200/$1000, or 20%, the asset B is $300/$1000, or 30%, the asset C is $500/$1000, or 50%. The weights of the portfolio are thus 0,20; 0,30 and 0,50. It’s easy to see that the weights have to add up to 100% because the entire amount that investor holds is somewhere invested. 5.3.2 Portfolio Expected Returns Let’s now consider the portfolio of the same three assets, but the portfolio weights are different: Asset A 40%, Asset B 30% and Asset C 30%. The rate of return depends on two different states of economy (decreasing or increasing interest rates by FED), each with the same probability 50%. First in Table 8 we calculate expected returns, variance and standard deviations for each asset.

31 Table 8 Return FED

Asset A

Probability

Asset B

Asset C

decreasing

20%

30%

10%

50%

increasing

-10%

0%

-10%

50%

5,00%

15,00%

0,00%

Expected return σ=

0,0450

2

σ= portfolio weights

0,0450

0,0200

21,21%

21,21%

14,14%

40%

30%

30%

Now we calculate the rates of expected return on this portfolio. To answer these questions, suppose the FED increased interest rates and economy actually declines. In this case, 40% of invested money loses 10%, 30% of invested money does not change its value and the other 30% also loses 10%. The portfolio return, rP, in a depression is thus: rP = 0,40 * (-10%) + 0,30 * 0 + 0,30 * (-10%) = -7% Table 9 summarizes the remaining calculations. When FED decreases interests, during boom the portfolio will return 20%: rP = 0,40 * 20% + 0,30 * 30% + 0,30 * 10% = 20% Table 9 FED

Return Asset A Asset B Asset C

Probability p

rp

rp*p

decreasing

20%

30%

10%

50%

20,00%

10,0%

increasing

-10%

0%

-10%

50%

-7,00%

-3,5%

portfolio weights

E(rP) = 40%

30%

6,50%

30%

As specified in Table 9, the expected return on portfolio, E(rP), is 6,5%. We can also calculate the expected return more directly. Having these portfolio weights, we could have reasoned that we expect 40% of our money to earn 5%, 30% to earn 15% and 30% to earn 0%. Our portfolio expected return is as a result: E(rP) = 0,40 * rA + 0,30 * rB + 0,30 * rC = 0,40 * 5% + 0,30 * 15% + 0,30 * 0% = 6,5%

32

5. RISK AND PORTFOLIO

This is exactly the same portfolio expected return as we calculated previously. This method of calculating the expected return on a portfolio is successful, regardless of how many assets there are in the portfolio. Assume we had n assets in our portfolio, where n is any number. If we let wi stand for the percentage of our money in Asset i, then the expected return would be: where: wi – weight of i-th asset, ri – return of i-th asset. This implies that the expected return on a portfolio is a simple combination of the expected returns on the assets in that portfolio. This seems rather obvious, but, as we will examine it next, the obvious approach is not always the correct one. 5.3.3 Portfolio Variance To understand the effect of diversification, we must find out to what extent the risk of a portfolio depends on the risk of the individual shares. We must calculate the risk of portfolio knowing the expected return on a portfolio that contains investment in Assets A, B and C is 6,5%. At the beginning we may tend to wrongly assume that the standard deviation of the portfolio is a weighted average of the standard deviations of the two stocks. Simple intuition suggests that by analogy the risk of portfolio, measured by standard deviation is (Table 10):

Unfortunately this method is completely wrong as a rule (sic!). That would be correct only in an extremely exceptional case if the prices of the two stocks were moving in ideal lockstep, with which we will deal later on. Table 10 Return Asset A

Asset B

Asset C

Portfolio

Expected return

5,00%

Variance

0,0450

15,00%

0,00%

6,50%

0,0450

0,0200

0,04

Standard deviation

21,21%

21,21%

14,14%

19,09%

Portfolio weights

40%

30%

30%

Wrong!

Table 11 illustrates what the portfolio’s standard deviation really is. After summarizing the relevant calculations, we can see that the portfolio’s variance is about 0,018 and its standard

33 deviation is much less than we tried to predict - it’s only 13,5%. It implies that the variance of a portfolio is basically not a direct combination of the variances of the assets in the portfolio. Table 11 FED

Probability pj

Return

rj

pj rj

[E(rp)- rj]2

pj [E(rp)- rj]2

Asset A

Asset B

Asset C

decresing

20%

30%

10%

50%

20,00% 10,0%

0,018225

0,009113

incressing

-10%

0%

-10%

50%

-7,00%

0,018225

0,009113

-3,5%

σ Portfolio weights

40%

30%

30%

E(rp)=

6,50%

2 P

=

σP =

0,01823 13,50%

First basic formula for variance is:

where: pj – probability of j-th state of economy, rj – return of portfolio in j-th state of economy E(rp) – expected return of portfolio 5.3.4 Portfolio Risk and Correlation As we noticed, calculating the expected portfolio return is straightforward. The more challenging part is to find the risk of the portfolio. Let us assume that in the past the standard deviation of returns was σA for Asset A and σB for Asset B. We assume that these figures are a good estimate of the spread of possible future outcomes. At the moment we must ask the most important question: how strong and in what way these two assets are “linked”? The portfolio variance is dependent on the way in which individual securities are correlated with each other. In order to calculate the portfolio variance of a two-stock we can use the following formula:

where: ρ12 – correlation coefficient σAB – covariance The portfolio standard deviation is, of course, the square root of the variance.

34

5. RISK AND PORTFOLIO

σ P = σ P2 Let us examine this formula more thoroughly. At first, we must weigh the variance of the returns on Asset A by the square of the proportion invested in it. In the same way, we weigh the variance of the returns on Asset B by the square of the proportion invested in Asset B. The first two parts of this formula depend on the variances of Asset A and B whereas the last part of this formula depend on their covariance. Just as we weighted the variances by the square of the proportion invested, so we must weigh the covariance by the product of the two proportionate holdings wA and wB. The covariance essentially tells us whether or not two securities returns are correlated. The covariance can be expressed as the product of the correlation coefficient and the two standard deviations: , Covariance measures themselves do not provide any measurement of the degree of correlation between two securities. Thus, correlation is standardized by dividing covariance by the product of the standard deviation of two individual securities. The correlation coefficient is therefore a standardized measure of correlation:

The correlation coefficient is a pure measure of the co-movements between the two securities and it ranges from –1 to +1. For the most part stocks are likely to move together. In this case the correlation coefficient is positive, and therefore the covariance is also positive. Portfolio risk can be effectively reduced (diversified) by combining securities with returns that do not move in the same direction. A correlation of +1 means that the returns of the two securities always move in the exactly same direction; they are perfectly positively correlated.

ρ12=1

generally:

Therefore, we get an analogical result as the return on portfolio E(rP ). In any other case, diversification reduces the risk below this level.

35 A correlation in interval 0 < ρ12 < 1 means that the returns on the two securities always move in the same direction. Positive correlation coefficient is typical in real economy and it means that we have limited possibility of risk reduction. A correlation of –1 means the returns are negatively correlated and always move in the exactly opposite direction.

ρ12= -1

If the stocks just tend to move in opposite directions, the correlation coefficient and the covariance would be negative.

-1 < ρ12 < 0 The last interesting example is when the correlation coefficient is zero, which means that the two securities are independent and unrelated to each other. If the prospects of the stocks were totally unrelated, both the correlation coefficient and the covariance would be zero; Therefore, assume that assets are independent, ρ12=0, but additionally they are equally weighted

and equally risky

σ1= σ2 = σn but with different rates of return the portfolio variance is:

Consequently, standard deviation is:

We can see that in this case we can significantly reduce the risk by adding very large amount of assets to our portfolio.

36

5. RISK AND PORTFOLIO In Table 12 and Table 13 we have two examples of a portfolio comprised of two Assets.

Table 12 FED

Return

Probability pj

rj

pj rj

[E(rp)- rj]2 pj [E(rp)- rj]2

Asset A

Asset B

decresing

24%

3,0%

50%

13,50%

6,8%

0,000506

0,000253

incressing

12%

6,0%

50%

9,00%

4,5%

0,000506

0,000253

E(rp)=

12,25%

Expected return

σ

2

σ

18,00%

4,50%

=

0,0072

0,0005

=

8,49%

2,12%

50%

50%

σ = σP = 2 P

ρ= portfolio weights

0,00051 2,25%

- 1,00

Table 13 FED

Return

Probability pj

rj

pj rj

[E(rp)- rj]2 pj [E(rp)- rj]2

Asset A

Asset B

decreasing

24%

3,0%

50%

7,20%

3,6%

0,000000

0,000000

increasing

12%

6,0%

50%

7,20%

3,6%

0,000000

0,000000

E(rp)= Expected return

18,00%

4,50%

σ2

=

0,0072

0,0005

σ

=

8,49%

2,12%

7,2%

σ σP = 2 P

=

ρ= portfolio weights

20%

0,000000 0,00%

- 1,00

80%

5.3.5 Diversification, Systematic and Unsystematic Risk. We can observe a  significant differences among various sources of risk. Some fraction of risk is specifically attached to particular assets, and some part of risk is more universal. Specific risk stems from the fact that many of the threats that surround a specific company are peculiar to that company and perhaps its direct competitors.

37 Diversifiable risk (unsystematic, specific or residual risk) represents the fraction of an asset’s risk and risk aspects affecting only that particular company. It’s typical for company-specific events, such a lawsuit, strikes, regulatory actions, and the loss of a key account. Unsystematic risk is due to factors specific to a company or an industry such as product category, labor unions, research and development, marketing strategy, pricing etc. The market risk (un-diversifiable or systematic risk) refers to economy-wide (macroeconomic) factors which affect all companies. This is the relevant fraction of an asset’s risk typical for market factors that affect all companies such as inflation, war, international incidents, and political events. So systematic risk is inherent to the entire market or entire market segment and stems from the fact that there are other economy wide perils that threaten all businesses. Figure 3 Systematic and Unsystematic Risk

Unsystematic risk can be reduced through diversification because it is associated with random causes. Company or industry specific risk is inherent in each investment. The fraction of unsystematic risk can be reduced through appropriate diversification. On the other hand, systematic risk comes from economy-wide sources of risk that generally affect the entire capital market. Sources of systematic risk affect the entire market and cannot be avoided through diversification. Systematic risk can be mitigated only by being hedged. The combination of a security’s non-diversifiable risk and diversifiable risk is called the total risk. Total Risk = Systematic Risk + Non-Systematic Risk. Systematic risk is due to risk factors that affect the entire market such as investment policy modifications, a change in socio-economic or in taxation clauses, foreign investment policy, international security threats and measures etc. Systematic risk underlies all other investment risks is beyond the control of investors and cannot be mitigated to a large extent. In contrast to this, the unsystematic risk can be mitigated through portfolio diversification. It is a risk that can be avoided and the market does not compensate for taking such risks. Specific risk disappears in the portfolio construction process when you diversify among assets that are not correlated. Diversification is succesful because prices of different assets do not move exactly together. If we randomly choose a certain number of marketable assets and form them into portfolios of varying sizes, measure the expected returns and standard devia-

38

5. RISK AND PORTFOLIO

tions of returns from each of our various-sized portfolios and then plot standard deviation against the size of portfolios, we should obtain a picture similar to that in Figure 4. The market portfolio is a combination of individual stocks, but its variability does not reflect the average variability of its components. It means that diversification decreases the overall variability. Even a small diversification can provide an important reduction in variability. Market risk is the risk that remains after creating the market portfolio, which apparently contains all risky assets. These are the risks that cannot be diversified away. The varioussized portfolios are constructed randomly. Various portfolios of each size are created, and the standard deviation of returns shown in Figure 4 is the average of the standard deviations of all of the portfolios for each size. As the standard deviation of returns around the mean is a reasonable measure of risk, Figure 4 suggests that large reduction in risk can be achieved simply by randomly combining assets in portfolios. Figure 4 Market risk and the portfolio size

These two characteristics might be a surprise: •

•

Very limited diversification yields significant reductions in the level of risk. Even splitting the investment funds available into a couple of different assets effectively eliminates a large portion of risk. Each additional and different asset added to the portfolio produces less marginal risk reduction. If the portfolio contains about 15 to 20 assets there is little benefit to be gained by way of risk reduction from additional increases in its size. Part of the risk seems to be immune to attempts to reduce it through diversification.

39 There are two basic implications of the phenomenon depicted in Figure 4. First, the investors should hold assets in portfolios as, by doing so, risk can be reduced at little cost and second, that there are limits in diversifying into many more than 15 to 20 different assets as nearly all the benefits of diversification have been exhausted at that size of portfolio. Additional diversification implies that the investor will have to pay higher transaction costs (e.g. dealing commissions) to establish the portfolio and then will have more expenses and more work in managing it. Market risk must be taken into consideration by investors because it is caused by marketwide factors so it seems possible that some securities are more vulnerable to these factors than others. We will now examine if the level of risk attached to the returns from all securities is the same, even though no one is forced to bear any specific risk. In Figure 4 we have split the risk into its two parts – specific risk and systematic risk. If we have only a single asset, specific risk is very important; but once we have a portfolio of 20 or more assets, diversification has done its job. For a rationally well-diversified portfolio, only market risk is important. Consequently, the major source of uncertainty for a diversified investor is that the market will expand or contract, moving the investor’s portfolio value with itself. 5.3.6 Limits of Diversification A level of risk can possibly be reduced by diversification, but there is also some risk that we cannot avoid, regardless of how much we diversify. Normally market risk affects each asset and that is why assets have a propensity to move together. So that is why investors are exposed to market risk, no matter how many assets they hold. If the expected returns on the assets were completely unrelated, both the correlation coefficient and the covariance would_ be zero. We assume that all _ assets in portfolio are equally weighted, so w1= w2 = wn =1/n, σ is average variance and σ ij is average covariance. To begin with, we start our discussion with a portfolio of two assets:

In case of an n-asset portfolio:

Consequently, with n goes to infinity we have:

We can observe that the portfolio variance goes to average covariance with number of assets goes to infinity.

40

5. RISK AND PORTFOLIO

5.3.7 Beta Beta is a measure of the systematic risk, or volatility, of assets or a portfolio in comparison to the entire market. Therefore, beta gives a sense of a stock’s market risk compared to the greater market. Beta is also used to compare a stock’s market risk to that of other stocks. In compare to standard deviation, beta measures volatility relative to a relevant baseline rather than to the mean of the asset that is being evaluated. Beta is the appropriate measure of an asset’s input to our portfolio’s risk, as it measures only the market risk. Beta is estimated using a regression analysis, and you can think of beta as the tendency of a security’s returns to respond to swings in the market. We should recall that Y = a + bX is the standard form of the equation of the regression line. When we regress one asset on another asset or a benchmark index, the slope of the regression line, b, is referred to as the dependent variable’s beta and it describes the movement of the asset (the dependent variable) relative to its benchmark (the independent variable). For example, if we assume, based on the daily data, that the pension fund beta is 1,5 it informs us that for every 1% move in the market the pension fund can be expected to move around 1,5% in the same direction. The market’s beta is by definition 1,0 and it is the baseline market risk. The risk-free asset’s beta is obviously 0. In our example, there is very little dispersion of the data points with respect to the regression line. This is predictable, as the dependent variable, being a “large market capitalization” pension fund is supposed to be extremely correlated with the market and it is a well-diversified portfolio. If we have the same exercise with an individual “large market capitalization” asset, the data points would probably be much more dispersed. In addition, the “large market capitalization” asset’s market risk, beta, could be significantly higher or lower than 1,0, considerably depending on the character of the company. Beta is a frequently published value that we know how to use to estimate the market risk of assets that we are considering adding to our portfolio. On the other hand, betas are usually obtained by using the WIG 20 as an initial value that can be used to compare past, current and projected future values (called benchmark). It is fine if we are evaluating “large market capitalization” domestic stocks or if we want to see how any specific asset moves relative to the WIG 20. We must remember that betas thus obtained are relative to the WIG 20 and that they simply correspond to the residual volatility. Portfolio betas are estimated as the weighted average of the betas of the assets that compose the portfolios. If we enlarges economy well beyond the market, to get a really relevant portfolio beta, the individual asset’s betas would have to be obtained using a benchmark that is model of the assets in the portfolio. The market risk of our economy is still 1.0 by assumption but it may be more or less volatile than the market. For a largely diversified portfolio, this would involve developing a weighted average index of the appropriate indexes. We do not say that investors have to do this, we only state an item of importance. The portfolio standard deviation is all we need to estimate portfolio’s possible variability. Since beta determines the risk, it can be associated with the standard deviation. Certainly, an asset’s beta is equal to the product of its correlation coefficient and its standard devia-

41 tion divided by the market’s standard deviation. Mathematically, the correlation coefficient divided by the market’s standard deviation factor the specific risk out of the asset’s standard deviation leaving only the systematic risk, which is the market risk of the asset. We can recapitulate that: if β = 0 asset is risk free, if β = 1 asset return is equal to market return, if β > 1 asset is riskier than market risk, if β < 1 asset is less risky than market risk. An asset’s market risk, relative to the average, can be measured by its beta coefficient. The risk premium on an asset is then given by its beta coefficient multiplied by the market risk premium, [E(rM) - rf] *β The expected return on an asset, E(ri), is equal to the risk-free rate, rf, plus the risk premium: E(ri) = rf + [E(rM) - rf] *β As a result risk free rate is a type of reference point adjusted by the risk which has qualitative and quantitative part. This is the equation of the Security Market Line (SML), and it is often called the capital asset pricing model (CAPM). We tried to complete our arguments of risk and return as we have a better understanding of what decides of a company’s cost of capital for an investment. Now we can move on to examine more closely how companies analyze the cost of capital in practice.

6. Cost of Capital The required return (or an appropriate discount rate) is essential in making a capital budgeting of new businesses or projects (such as building a new plant), worthwhile is often called the cost of capital associated with the investment. The cost of capital decides how a company can raise money (through equity, debt, or a mix of the two). This is the rate of return that a company would receive if it invested in a different projects with similar risk. It is referred to as the required return is what the company must earn on its capital investment in a project just to break even. From an economic point of view it can be interpreted as the opportunity cost linked with the company’s capital investment. Observe that when it is said that an investment is lucrative if its expected return exceeds what is offered in capital markets for investments of the identical risk, we are effectively using the internal rate of return (IRR). The only difference is that now we have a much better concept of what determines the required return on an investment. This will be crucial when we discuss the capital structure and the cost of capital. As we identify the relation of risk and return, we must observe that the correct discount rate depends on the riskiness of the project. The new business will have a positive NPV only if its return exceeds what the financial markets offer on investments of similar risk. Consequently, this minimum required return is the cost of capital linked with the business. One of the most important idea we develop is that of the weighted average cost of capital (WACC)1. WACC is an estimate of a company’s cost of capital, or the minimum that a company must earn to gratify all debts and support all assets. The calculation includes the company’s debt and equity ratios, as well as all long-term debt. Companies usually do an internal WACC estimation to assess overall company. The more complex and larger a company is, the more complicated it is to estimate WACC. Tax treatments are an important reflection in determining the required return on an investment, because we are always interested in valuing the after tax cash flows from a business. Unfortunately, only some of the information needed to calculate WACC can be found in a statement of financial positions. The weighted average cost of capital shows how much it costs a company to raise money for a business. WACC is important when a company needs to raise money to expand. Beta is the systematic risk – risk that is intrinsic with the market – that a company has comparing to the market. The security market line (SML) can be used to explore the relationship between the expected return on a  security and its systematic risk. We concentrated on how the risky returns from buying securities looked from the viewpoint of a  shareholder in the company. This helped us understand more about the alternatives available to an investor in the capital Sometimes also called the marginal cost of capital

1

43 markets. Now, we look more closely at the other side of the issue, which is how these returns and securities look from the point of view of the companies that issue them. The important fact to observe is that the return on a security an investor receives is the cost of that security to the company that issued it.

6.1 Required Return, Cost of Capital and Financial Policy Required return is the minimum annual percentage earned by an investment that will encourage companies or individuals to put money into a particular project or security. The required rate of return is used in both corporate finance and in equity valuation. Stockholders use the required rate of return to decide where to put their funds. They evaluate the return of an investment to all other available alternatives, taking the risk-free rate of return, liquidity and inflation into consideration in their estimations. For stockholders using the dividend discount method to choose stocks, the required rate of return influences the maximum price they are willing to pay for a stock. The required rate of return is also used in calculations of net present value in discounted cash flow study. Companies use the required rate of return to choose if they should start a new business or a project . In corporate finance, the required rate of return is equal to the weighted average cost of capital (WACC). When we state that the required return on an investment is 10%, we generally mean that the investment will have a positive NPV simply if its return exceeds 10%. A different way of understanding the required return is to observe that the company must earn 10% on the investment just to compensate its investors for the use of the capital needed to finance the business. Thus we could also say that 10% is the cost of capital associated with the investment. Assuming that the other information is unchanged, required return is obviously higher if a project is risky. In other words, if the project is risky the appropriate discount rate would exceed the risk-free rate and the cost of capital for this project is greater than the risk-free rate. From now on we will use the terms cost of capital, required return and appropriate discount rate more or less interchangeably because they all mean basically the same thing. The key fact to grasp is that the cost of capital associated with an investment depends on the risk of that investment. The cost of capital depends primarily on the use of the funds, not the source. It is a common mistake to forget this critical point and fall into the trap of thinking that the cost of capital for an investment depends primarily on how and where the capital is increased. The particular mixture of debt and equity a company chooses to employ its capital structure is a managerial variable. We presume the company’s financial policy as given. In particular it means that the company has a given debt-equity ratio that it maintains. This ratio reflects the company’s target capital structure. We know that a company’s overall cost of capital will reflect the required return on the company’s assets as a whole. If a company uses both equity and debt so overall cost of capital will be a combination of the returns needed to compensate its debtholders and those needed to compensate its stockholders. In other words, a company’s cost of capital includes the cost of debt and the cost of equity.

44

6. COST OF CAPITAL

6.2 The Cost of Equity A company’s cost of equity represents the compensation that the market requires in exchange for possessing the asset and bearing the risk of ownership. Put it in different words: this is the return that stockholders demand from a company. The capital asset pricing model (CAPM) or security market line (SML) is the method used to find the cost of equity. 6.2.1 The Security Market Line and Cost of Capital Company’s cost of capital is based on the average beta of the assets. The average beta of the assets is based on the proportion of funds in each asset. From security market line we know that the expected or required return on a risky asset depends on three things: 1. The expected risk-free return in that market (e.g. treasury bond yield), rf 2. The risk premium of market assets over risk free assets, E(rM) - rf 3. The sensitivity to market risk for the security, beta coefficient βE, With the SML, we can reformulate the expected return on the company’s equity, E(rE), as: E(rE) = rf +βE [E(rM)- rf ]

or just:

rE = rf +βE(rM- rf ) rf

3%

rM

9%

rM-rf

6% rE

Beta Equity value

1,2

10,20%

To apply Security Market Line approach, we must have a  risk-free rate, rf, an estimate of the market risk premium, rM - rf, and an estimate of the appropriate beta, βE. We assume estimated market risk premium (based on large common stocks) is 9%. Treasury bills yield around 3,0%, so we will use this as our risk-free rate. Beta coefficients for publicly traded companies are usually available. We could thus estimate the cost of equity as: rE = rf + βE *(rM - rf)= 3,0% + 1,2*(9-3)% = 10,2% As a result, using the SML approach, we estimate that cost of equity is about 10.2%. 6.2.2 Advantages and Disadvantages of the Approach The Security Market Line approach has two most important advantages. First, it explicitly adjusts for systematic risk. Second, it is applicable to all companies. As long as we com-

45 pute beta it is applicable to companies others than just those with steady dividend growth. Therefore, it may be useful in a wider variety of situations. There are however some disadvantages as well. The SML approach requires two things to be estimated: the market risk premium and the beta coefficient. We have to estimate the expected market risk premium, which differs over time: to some degree that the estimates are poor, the resulting cost of equity will be imprecise. Using different stocks or different time periods could result in very different estimates. In SML approach we have to estimate beta, which also fluctuates over time. We trust the past statistics in order to predict the future, which is not always reliable. Economic circumstances can change very quickly, thus, the past may not be a good guide to the future. If the dividend growth model and the SML happen yield similar answers, we might have some confidence in our estimates.

6.3 The Cost of Debt Calculating the cost of capital associated with debt is much easier than calculating the cost of equity. The cost of debt is calculated by taking the rate on a risk free bond whose duration matches the term structure of the corporate debt, then adding a default premium. This default premium will rise as the amount of debt increases (as all other things being equal, the risk goes up as the amount of debt increases). Because in most cases the cost of debt is a deductible expense, the cost of debt is calculated as an after-tax cost to make it comparable with the cost of equity (incomes are after-tax as well). Consequently, for profitable companies, debt is discounted by the tax rate. For the sake of consistency with our other notation, we will use the symbol rD for the cost of debt and the formula can be written as: rD=(rf + default premium)(1-T), where T is the corporate tax rate and rf is the risk free rate. The yield to maturity can be used as an estimation of the cost of debt.

The cost of debt is the return that the company’s debtholders require for new debt. A company will use various loans, bonds and other forms of debt, so this measure is useful for giving a concept as to the overall rate being paid by the company to use debt financing. In general, we could determine the beta for the company’s debt and then use the SML to estimate the required return on debt just as we estimated the required return on equity. This is in fact not necessary. The measure can also give investors an idea as to the riskiness of the company compared to others, because riskier companies generally have a higher cost of debt. Company’s cost of debt, unlike its cost of equity, can usually be observed either directly or indirectly, because the cost of debt is basically the interest rate the company must pay on new debt, and we can monitor interest rates in the financial markets. Thus company’s cost of debt is the effective rate that a company pays on its current debt. If the company already has bonds outstanding, then the

46

6. COST OF CAPITAL

yield to maturity on those bonds is the market-required rate on the company’s debt. On the other hand, if we know that the company’s bonds are rated A, then we can simply find out what the interest rate on newly issued A-rated bonds is. Consequently, there is no need to estimate beta for the debt because we can directly observe the rate we want to know. However, there is one thing to be accurate about. The coupon rate on the company’s outstanding debt is irrelevant here. That rate simply tells us approximately what the company’s cost of debt was back when the bonds were issued, not what the cost of debt is today. This is why we have to monitor the yield on the debt in the present day’s market.

6.4 The Weighted Average Cost of Capital When we have the costs linked with the main sources of capital the company employs, we will take this mix, which is the company’s capital structure, as given for now. We will focus mostly on debt and ordinary equity. As we mentioned earlier investors frequently focus on a company’s total capitalization, which is the total of its long-term equity and debt. This is mostly true in determining cost of capital; short-term liabilities are often ignored in the process. The general approach of explicitly not distinguishing between total value and total capitalization is basically applicable. 6.4.1 The Capital Structure Weights We calculate market value of the company’s equity by taking the number of shares currently held by all its shareholders and multiplying it by the price per share. We will use the symbol E (for equity). In the same way, we will use the symbol D (for debt) to stand for the market value of the company’s debt. For long-term debt, we calculate this by multiplying the current price of a single bond by the number of bonds currently held by all its debtholders. If there are multiple bond issues (as there normally would be), we repeat this calculation of D for each and then add together the results. If there is debt that is not currently traded we must monitor the yield on similar, currently traded debt and then estimate the current market value of the privately held debt using this yield as the discount rate. For short-term debt, the book (accounting) values and market values should be to some extent comparable, thus we can use the book values as approximation of the market values. Figure 5 Capital structure.

47 As a final point, we will use the symbol V (for Value) to stand for the combined market value of the equity and debt: V=D+E We can calculate the fractions of the total capital represented by the debt and equity if we divide both sides by V:

These fractions can be interpreted identically to portfolio weights, and they are often called the capital structure weights. For example, if the total market value of a company’s stock was calculated at $50 million and the total market value of the company’s debt were calculated at $50 million, then the combined asset value would be $100 million:

so 50% of the company’s financing would be equity and the remaining 50% would be debt.

We highlight here that the proper way to proceed is to use the market values of the debt and equity. Under certain circumstances, such as when calculating figures for a privately held company, it may not be possible to get reliable estimates of these quantities.

Debt value (D) Equity value (E) Asset value (V)

USD 50 50 100

Weights 50% 50% 100%

6.4.2 Taxes and the Weighted Average Cost of Capital Since we are always interested in after-tax cash flows, there is one final problem we need to discuss. If we estimate the discount rate appropriate to after-tax cash flows, then the discount rate needs to be expressed on an after-tax basis as well. The interest paid by a company is deductible for tax purposes, but cash flows to equityholders, such as dividends, are not. In effect, the government pays some of the interest. Consequently, in determining an after tax discount rate, we need to distinguish between the pretax and the after tax cost of debt. In Table 14 we presume a company borrows 50 at 4,8% interest. The corporate tax rate is 35%. The total interest payments are tax deductible, thus the interest reduces the company’s tax bill by: 0,35 * 0,048 * 50 = 0,84

48

6. COST OF CAPITAL

The after-tax interest bill is thus The after-tax interest rate is thus

2,4 - 0,84 = 1,56 1,56/50 = 3,12%

The after-tax interest rate is basically equal to the pretax rate multiplied by 1 minus the corporate tax rate. For example, using the previous numbers, we notice that the after-tax interest rate is: 4,8%*(1-0,35) = 3,12%. Table 14 rf

3%

rM

9%

rM-rf

6%

T

35% no tax

Debt value

WACC

r

beta 50

0,3

4,80%

Equity value

50

1,2

10,20%

Asset value

100

0,024 0,051 0,075

tax WACC

r

beta Debt value

50

0,3

4,80%

0,0156

Equity value

50

1,2

10,20%

0,051

Asset value

100

0,067

We have the capital structure weights along with the cost of equity and the after-tax cost of debt. To find the company’s total cost of capital, we multiply the capital structure weights by the associated costs and add them up. The total is the weighted average cost of capital (WACC). WACC = (E/V) * rE + (D/V) * rD * (1 - T) This WACC has a very simple explanation. It is the total return the company must earn on its existing assets to keep the value of its stock. Moreover, the required return on any investment in the company has essentially the same risks as existing operations. Thus, if we were estimating the cash flows from a recommended expansion of our existing operations, this is the discount rate we would use. If a  company uses preferred stock in its capital structure, then our expression for the WACC needs a small extension. If we define P/V as the fraction of the company’s financing that comes from preferred stock, then the weighted average cost of capital is:

49 WACC = (E/V) * rE + (P/V) * rP + (D/V) * rD * (1 - T) where rP is the cost of preferred stock. 6.4.3 The Security Market Line and the Weighted Average Cost of Capital When we consider investments with risks that are significantly different from those of the company taken as a whole, the use of the weighted average cost of capital will possibly lead to ineffective decisions. In Figure 6, we have plotted a Security Market Line corresponding to a risk-free rate of x% and a market risk premium of y%. To keep the discussion simple, we assume an all-equity company with a beta of 1. As we have indicated, the weighted average cost of capital and the cost of equity exactly equal (x+y)% for this company because there is no debt. Figure 6 Security Market Line (SML) and Weighted Average Cost of Capital WACC

We can assume that our company uses its weighted average cost of capital to evaluate all investments. This means that any investment with a return of greater than (x+y)% will be accepted and any investment with a return of less than (x+y)% will be rejected. From risk and return discussion we know, however, that an attractive investment is the one that plots above the Security Market Line. As Figure 6 illustrates, using the weighted average cost of capital for all types of projects can prompt the company to incorrectly accept relatively risky projects and incorrectly reject relatively safe ones.

7. Capital Structure and Financial Leverage Capital structure refers to the way a company finances its assets through some combination of debt and equity. A company’s capital structure is then the composition (structure) of its liabilities. To this point, we assumed the company’s capital structure as fixed. In real world debt-equity ratios are the result of some kind of a dynamic optimization. We want to find out where the real capital structure comes from. Decisions about a company’s debt-equity ratio are referred to as capital structure decisions. Company can select any capital structure that it wants and such activities that change the company’s existing capital structure are recognized as capital restructurings. Company could either issue some bonds and use the proceeds to buy back some stock (increasing the debtequity ratio) or issue stock and use the money to pay off some debt (reducing the debt-equity ratio). Restructurings take place when the company substitutes one capital structure with another leaving the company’s total assets unchanged. Capital restructuring does not directly influence assets of a company, thus we can look at the capital structure changing separately from its business activities. Thus, capital budgeting decisions are not dependent on capital structure and therefore restructuring decisions. Consequently, we can assume investment decisions as given and focus on capital structure matters (long-term financing). Subject to several assumptions, financing by debt rather than equity does not seem to make any difference to shareholders’ value. Let us now review the traditional view, that the capital structure does have an effect on the wealth of the shareholders.

7.1 Capital Structure Weighted average cost of capital demonstrates the company’s overall cost of capital and should reflect the mixed returns expected by the different suppliers of capital. A company’s WACC accounts for both the company’s cost of equity and its cost of debt, weighted according to the proportions of equity and debt in the company’s capital structure. When we defined the WACC, we assumed the company’s capital structure as fixed. Now we need to find what happens to the cost of capital if a company increases the amount of debt financing, or the debt-equity ratio. Most important reason for studying WACC is that WACC is the rate investors use to discount cash flows. A company’s WACC is in general a required return on the company as a whole and, as such, it is often used internally by company directors to determine the economic feasibility of expansionary opportunities and mergers. It is the appropriate discount rate to use for cash flows with risk that is similar to that of the overall company.

51 Hence, the value of the company is altering when the WACC is changing. Values and discount rates alter in opposite directions, so minimizing the WACC will maximize the value of the company’s cash flows. This optimal capital structure is also called the long-term targeted capital structure.

7.2 The Effect of Financial Leverage We know why the capital structure that produces the highest company value (or the lowest cost of capital) is the most beneficial to stockholders. Now, we examine the impact of financial leverage (gearing, solvency) on the cash flows to stockholders. Financial leverage can be appropriately described as the extent to which a business or investor uses debt. A  lot of companies meet part of their long-term financing requirements through debt, often by means of the loan stocks issuance. This gives debtholders contractual rights to receive interest, typically at a predetermined rate and on specified dates. Debt finance could also be provided by a bank or a similar institution, which would acquire similar contractual rights. The important thing about debt finance lies in the fact that neither interest nor redemption payments are matters of the borrowing company’s discretion. Interest on such loans amounts to an annual charge on profits. This must be satisfied before the equity shareholders (owners), who in the typical company provide the larger part of the finance, may participate. As we observed, under several assumptions, financing through debt rather than equity does not make any change in a company’s value. We can now review the traditional point of view, that capital leverage does have an effect on a company’s value. Then we can try to make some conclusions on the matter. Any additional debt financing in a company’s capital structure raises its financial leverage. As we explained, financial leverage can significantly modify the payoffs to shareholders in the company. On the other hand, financial leverage may not affect the total cost of capital. Consequently, a company’s capital structure is irrelevant because changes in capital structure will not have an effect on the value of the company. 7.2.1 Financial Leverage and ROE Let us begin by demonstrating how financial leverage works. For the moment, we take no notice of the impact of taxes. Moreover, for ease of presentation, we explain the impact of leverage in terms of its effect on earnings per share (EPS) and return on equity (ROE). We use cash flows instead of accounting numbers, which would lead us to exactly the same conclusions, but a little more work would be needed. We show the impact on market values in a following part.

52

7. CAPITAL STRUCTURE AND FINANCIAL LEVERAGE

Table 15 Capital structure A 1000 10 10000 0

Shares Price per share Equity value Debt value Asset value

rD=10%

Capital structure B 500 10 5000 5000

10000

10000

Table 15 shows two capital structures. As predented, the company’s assets have a market value of $10 000, and there are 1000 shares outstanding. Since it is an all-equity company the price per share is $10. The planned debt issue would raise to $5000; the interest rate would be 10%. As the stock sells for $10 per share, the $5000 in new debt would be used to purchase $5000/10=500 shares, leaving 500. After the restructuring, this company would have a capital structure that was 50% debt, thus the debt-equity ratio would be 1. Observe that, for now, we assume that the stock price will stay at $10. Table 16 compares the company’s capital structure A and capital structure B under three scenarios. The scenarios mirror different assumptions about the company’s EBIT. Under the expected scenario (unchanged i), the EBIT is $1500. Table 16 Shares Price per share Equity value Debt value Asset value

Capital structure A

Capital structure B

1000 10 10000 0

500 10 5000 5000

10000 scenario

10000 scenario

Decrease i Operating results: Increase i Unchanged i Net income before 500 1500 2000 interest (EBIT) Interest expense 0 0 0 Net income 500 1500 2000 Earnings per share 0,5 1,5 2

Return on equity

5%

15%

20%

Increase i Unchanged i

rD=10%

Decrease i

500

1500

2000

500 0 0

500 1000 2

500 1500 3

0%

20%

30%

In the recession scenario, EBIT falls to $500. In the expansion scenario, it rises to $2000. In order to demonstrate some of the calculations behind the figures in Table 16, consider the expansion case. EBIT is $2000. With no debt (capital structure A) and no taxes, net in-

53 come is also $2000. In this case, there are 1000 shares worth $10 000 total. EPS is therefore $2000/1000=2. Also, because accounting return on equity, ROE, is net income divided by total equity, ROE is $2000/10000=20%. With $5000 in debt (capital structure B), things are fairly different. Since the interest rate is 10%, the interest bill is $500. With EBIT of $2000, interest of $500, and no taxes, net income is $1500. Now there are only 500 shares worth $5000 total. EPS is therefore $1500/500 =3, versus the $2 that we calculated in the previous scenario. Additionally, ROE is $1500/5000=30%. This is well above the 20% we calculated for the current capital structure. 7.2.2 Financial Leverage and EBIT The impact of leverage is clear when the effect of the restructuring on EPS and ROE is examined. Particularly, the variability in both EPS and ROE is much bigger under the proposed capital structure. This demonstrates how financial leverage works to amplify profits and losses to shareholders. The key thing to observe in Figure 7 is that the slope of the line in this second case is steeper. Actually, for every increase in EBIT, EPS rises faster, so the line is twice as steep. This shows us that EPS is twice as sensitive to changes in EBIT because of the financial leverage employed. Let us take a look at the effect of the capital restructuring. This figure plots earnings per share (EPS) against earnings before interest and taxes (EBIT) for both capital structures. The first line, labeled “No debt,” represents the case of no leverage. This line begins at the origin, indicating that EPS would be zero if EBIT were zero. From there, every increase in EBIT increases EPS. The second line represents the capital structure B. At this point, EPS is negative if EBIT is zero. It happens so because interest must be paid apart from the company’s profits, the EPS is negative as illustrated. In the same way, if EBIT were $500, EPS would be exactly zero.

54

7. CAPITAL STRUCTURE AND FINANCIAL LEVERAGE

Figure 7 The impact of leverage. EPS versus EBIT

7.2.3 Homemade Leverage At that time we can formulate a few conclusions: • The effect of financial leverage depends on the company’s EBIT. When EBIT is relatively high, leverage is favorable.

55 • Under the expected scenario, leverage increases the returns to shareholders, as measured by both ROE and EPS. • Stakeholders are exposed to more risk under the proposed capital structure because the EPS and ROE are much more sensitive to changes in EBIT in this case. • Since the impact that financial leverage has on both the expected return to stockholders and the riskiness of the stock, capital structure is an important consideration. The first three of these conclusions are undoubtedly correct. However the last conclusion does not follow. As we discuss next, the explanation is that shareholders can adjust the amount of financial leverage by borrowing and lending on their own. This use of personal borrowing to modify the degree of financial leverage is called homemade leverage. Given that the ordinary shares are contain priced when assumed returns are expected we may presume that investors regard 10% as the proper return for such an investment. If the expected returns increase as debt increases, this seems likely to push up the price of an ordinary share. This means that, were the assets to be financed by an issue of equity, the value of each ordinary share would be higher, but if the assets were to be financed by debt, the company’s ordinary shares would be worth more. In short, the shareholders’ wealth would be increased by debt instead of issuing equity. Finally, let us look at the situation from the point of view of the suppliers of debt. If there are higher returns to be made from direct investing in assets, they are prepared to do so for less return from debt. They just should buy ordinary shares in this company. The difference between direct investment in a company by buying the shares, and lending money on a fixed rate of interest is the different risk level. From expected profits from the new projects, only a specific part would be paid to debtholders (providers of the new finance). The other part goes to the stockholders, but with it goes all of the risk. As with all real businesses, returns are uncertain. Assume that there were to be a recession in the industry, so that the profits would fell from new businesses. This would mean that stockholders can get nothing. From this it seems clear that debt provide an apparently cheap source of finance, but it has an unseen cost to stockholders. 7.2.4 Business Risk and Financial Risk The inclusion of debt enhances returns on equity over those which could be earned in the all-equity company. Where profit falls below the breakeven point, however, the existence of debt finance weakens the ordinary shareholder’s position. In effect, below certain profit there would be insufficient profit to cover debt interest payments. At almost all levels of profit, the debtholders could view the situation philosophically. Finally, not only do they have the legal right to enforce payment of their interest and repayment of their capital, they even have the company’s assets as collateral. Only if major losses to the market value were to occur, would the debtholders’ position seriously be in danger. Wherever leverage occurs, the risk to which stockholders are exposed is clearly higher then the risk they would bear in the all-equity company. To business risk, the normal risk at-

56

7. CAPITAL STRUCTURE AND FINANCIAL LEVERAGE

tached to investing in the real world, is added financial risk, the risk caused by being burdened with the obligation to meet fixed charges. Figure 8 illustrates the relationship between business and financial risk where operating returns fluctuate. The amount of business risk depends on the area of activity in which the company is involved; financial risk depends on how the company is financed. Modern portfolio theory, as well as intuition, tells us that risk and return are correlated. If investors recognize high risk they immediately require high returns. Consequently, while leverage will lift expected dividends to ordinary shareholders, this will not necessarily increase the share price and as a result the wealth of the shareholder. Accompanying the increased expected dividend is a wider range of possible outcomes. Figure 8 ROE for levered company with fluctuations

The new opportunities which the leverage brings to stockholders are not always bad. For any profit over certain expected earnings, share dividends would be enhanced. The majority of investors are risk-averse, which means that the possibility that profit could be below expected earnings tends to be more important to them than the possibility that earnings could be greater than expected. Ordinary shareholders will only be wealthier through the introduction of leverage if the capital market (after it reassesses the situation following the debt issue) values the ordinary shares less than their expected yield. Prior to the company’s expansion these ordinary shares had a certain expected yield, but as a result of the higher risk level, which occured after the

57 expansion, the market expects a higher rate of return and therefore the price per share would increase. If alternatively capital market requirements for the level of risk are lower, the price per share would become lower. We must ask if the introduction of leverage lowers the weighted average cost of capital (WACC) and therefore makes more valuable the investments in which the company is involved. Given the company’s general objective of maximization shareholder’s wealth, the capital market responds, in terms of required returns, to the introduction of leverage are very important issue. Leverage is presumably only undertaken with the purpose of increasing stockholders’ wealth.

7.3 Capital Structure and the Cost of Capital There is nothing special about corporate borrowing because investors can borrow or lend on their own. As a  consequence, whichever capital structure company chooses, the stock price will be the same. Capital structure is thus irrelevant. 7.3.1 Traditional View The traditional view seems to be that the expected rate of return from equity investment will not be significantly affected by the introduction of capital leverage, not at any rate up to reasonable levels. This view of the effect of leverage on capital market expectations of returns from equities, debt and WACC is shown in Figure 9 and illustrates that as leverage increases, both shareholders and debtholders recognize additional risk and require higher returns. At lower levels of leverage neither group wants significantly increased returns to compensate for this risk, and WACC reduces. When leverage reaches higher levels, the risk problem becomes increasingly important to both groups, so required returns start to increase radically and WACC starts to rise sharply. According to traditionalists even if the point at which each group’s required return starts going up sharply is not necessarily the same, there is a point (or a range of points) where WACC is at a minimum. This is the optimal level of leverage. At this point the shareholders’ wealth is maximized, i.e. the price per share would reach a peak. The underlying principle for the traditional view is that debtholders would recognize that at high levels of leverage their security is significantly lost and would start to demand successively higher levels of interest as compensation. Moreover, there is a common belief that, up to a certain level of leverage, equity shareholders would not see the increased risk to their returns as a particularly important issue.

58

7. CAPITAL STRUCTURE AND FINANCIAL LEVERAGE

Figure 9 Cost of capital for different levels of leverage (gearing) - traditional view Figure 9 Cost of capital for different levels of leverage (gearing) - traditional view Cost of capital (%) Cost of Equity

WACC

Cost of Debt

Debt (D) Optimal debt (D*)

After that point, however, they would start to demand substantial increases in returns for further increases in leverage. A traditionalist would disagree that a company 50% loan financing still has significant security in the value of the assets for the interest and the capital repayment, After point,dropped however,significantly. they wouldNevertheless, start to demand main increases returns increased for furtherto even that if market if the level of loan in financing 90%, a fallinof leverage. more thanA 10% in the marketwould value would erode thewith security of debtholders. increases traditionalist disagree that, company at 50% Equity loan holders would not consider their situation as badly threatened by the existence of moderate financing there is still plenty of security in the value of the assets for the interest and the levels of loan finance and would not require returns great enough to negate the advantage of capital repayment, even if market dropped significantly. Nevertheless, the that levelleverage of loanis the appraise entry cheap loan finance. Generally, the traditional conclusionif was a good thing, in terms of shareholder’s wealth maximization, at least up to a certain level past financing increased to 90%, a fall of more than 10% in the market value would erode the which it would start to have an adverse effect on WACC and therefore on shareholders’ wealth. security of debtholders. Equity holders would not notice their situation too badly

threatened by the&existence of Proposition moderate levels of loan finance and would not require 7.3.2 The Miller Modigliani I returns great enough to negate the advantage of the appraise entry cheap loan finance. We know from previous arguments on weighted average cost of capital (WACC) that the Generally, the structure traditional was that the leverage good thing, terms ofin optimal capital forconclusion a corporation is when WACCisis aminimized. Thisinis basically line with thewealth Miller maximization, & Modigliani Propositions I and II aboutlevel Capital Corporations, shareholder at least up to a certain pastStructure which itof would start to proposed by Franco Modigliani and Merton Miller, two famous Nobel prize winners. have an adverse effect on WACC and therefore on shareholder wealth. The simplest way to show Miller & Modigliani Proposition I is to imagine two companies that have exactly the same assets and that they have exactly the same business operations. Consequently, the left side of their balance sheet is exactly the same. The only difference between the two companies is the right-hand side of the balance sheet (liabilities), that is to say how they finance their business performance.

59 In the first company (Figure 10 right), stocks (equity) make up 70% of the capital structure while bonds (debt) account for 30%. In the second diagram (Figure 10 left), it is the exact opposite, stocks (equity) make up 30% of the capital structure while bonds (debt) make up 70%. It happens so because the assets of both capital structures are exactly the same. Any financial assets are valued by reference only to their expected return and risk. Both structures offer equal risk/return expectations. Figure 10 Capital structure

Miller & Modigliani Proposition I states that the value of a company is NOT dependent on its capital structure. A change from the current to the alternative debt/equity ratio would result in an identical expected return with identical risk. It would NOT seem logical for corporations to be able to increase the wealth of their shareholders simply by showing the corporation’s income in a particular way. This is made particularly unreasonable by the fact that single shareholders can alter the packaging to their own convenience, only by borrowing and/or lending homemade leverage. Profits of similar expected return and risk should be adequately priced in an  effective market with rational investors, irrespective of the packaging. The most simple balance sheet, with all entries expressed as current market value: Table 17 Assets Value of cash flow from the company’s real assets and operations Value of company

Liabilities Market value of debt Market value of equity Value of company

The right- and left-hand sides of the balance sheet are always equal. Consequently, if one add up the market values of all the corporation’s debt and equity security, one can estimate the value of the future cash flows from the real assets and operations.

60

7. CAPITAL STRUCTURE AND FINANCIAL LEVERAGE

For that reason Miller & Modigliani Proposition 1 argues that it is irrelevant how the debt and equity is structured in a corporation. Hence, the real assets (NOT the capital structure) determine the value of the corporation. If the value of the corporation is not affected by the financing techniques, the cost of capital is not decreased (or affected at all) by the introduction of leverage. The only matters on which the value of the corporation and its WACC depend are: (a) the cash flows which the corporation’s investments are expected to generate, and (b) their risk (i.e. their economic risk). This is another fact following the same model as Fisher separation model2. Miller & Modigliani were finally saying that management should concentrate all its attempts on finding and managing real investment opportunities, leaving the financing engineering for single shareholders to decide for themselves. 7.3.3 Cost of Equity and Financial Leverage: M&M Proposition II Although changing the capital structure of the company does not change the company’s total value, it does cause important changes in the company’s debt and equity. Let’s have a look at what happens to a company financed with debt and equity when the debt/equity ratio is changed. For the sake of simplicity of our examination, we will continue to ignore taxes. Miller & Modigliani Proposition II assumes that the value of the company depends on three things: 1) Required rate of return on the company’s assets (rA) 2) Cost of debt of the company (rD) 3) Debt/equity ratio of the company (D/E) As we know, the method for calculating weighted average cost of capital (WACC) can be expressed in the following formula (if we ignore taxes):

We recall that one way of interpreting the weighted average cost of capital is the required return on the company’s overall assets. We will use the symbol rA to stand for the WACC and write the WACC formula in another form:

The above formula can also be rewritten as:

Fisher’s separation theorem sates that in a  perfect capital market, it is possible to separate the company’s investment decisions from the owners’ consumption decisions.

2

rE  rA  (rA  rD )

D E

This formula shows the basic idea of Miller & Modigliani Proposition II. This is the famous Miller & Modigliani Proposition II, which shows that the cost of equity depends

61

on three things: the required rate of return on the company’s assets, r A, the company’s

This forofmula thecompany’s basic ideadebt-equity of Millerratio, & Modigliani Proposition II. This is the faD/E. cost debt, Rshows D, and the mous proposition, which argues that the cost of equity depends threerEthings: , againstthe therequired Figure 11 shows our arguments so far by plotting the cost of equityon capital, rate of return on the company’s assets, rA, the company’s cost of debt, rD, and the company’s debt-equity ratio. Miller & Modigliani Proposition II illustrates that the cost of equity, r E , debt-equity ratio, D/E. is represented by a straight line with a slope of (rA - rD). The y-intercept corresponds to a Figure 11 demonstrates our arguments so far by plotting the cost of equity capital, rE , against company with a debt-equity ratio of zero. Figure 11 shows that, as the company increases the debt-equity ratio. Miller & Modigliani Proposition II illustrates that the cost of equity, rE , is its debt-equity ratio, line the increase in leverage risk of the equity and therefore - rD).the The y-intercept corresponds tothe a company represented by a straight with a slope of (rAraises required returnratio or cost equity (rE). 11 shows that, as the company increases its debt-equity with a debt/equity of of zero. Figure ratio, the increase in leverage boosts the risk of the equity and therefore the required return or cost of equity (rE).11 M&M Preposition II Figure Figure 11 M&M Proposition II Cost of capital (%) Cost of Equity

WACC

Cost of Debt

Debt (D)

Summary of Miller & Modigliani Proposition II: • required rate of return on the company (rE) is a straight line with a slope of (rA – rD), ofcurved Miller &and Modigliani Proposition II: • rSummary is linear upwards sloping because as a company borrows more debt (and E increases its debt/equity ratio), the risk of bankruptcy is even higher. As adding more debt is risky, the shareholders demand a higher rate of return (rE) from real economic operations in company (rE is upwards sloping), • as debt/equity ratio increases, rE will increase (upwards sloping), - 81 - is a straight line with no slope. Thus WACC • weighted average cost of capital (WACC) does not have any relationship with the debt/equity ratio. This is the basic formula of Miller & Modigliani Proposition I and II, that the capital structure of the company does not affect its total value, • for that reason WACC remains unchanged even if the company borrows more debt (and increases its debt/equity ratio, table 18).

62

7. CAPITAL STRUCTURE AND FINANCIAL LEVERAGE

Table 18 Capital structure A 1000 10 10000 0

Shares Price per share Equity value Debt value Asset value Operating results Net income before interest Interest expense Net income Earnings per share Return on equity rA rD rE WACC

Capital structure B 500 10 5000 5000

10000

500 0 500 0,5 5%

scenarios 1500 0 1500 1,5 15%

rD= 10%

10000

2000 0 2000 2 20%

500 500 0 0,0 0%

15% 10% 15% 15%

scenarios 1500 500 1000 2,0 20%

2000 500 1500 3,0 30%

15% 10% 20% 15%

Miller & Modigliani’s disagreement with the traditionalists was that Miller & Modigliani noticed the capital market’s return expectations increase when leverage is introduced, and that they increase in proportion to the amount of leverage. The traditionalists believed that this would not take place at lower levels of leverage.

7.4 Taxation, Financial Distress and Bankruptcy Cost Debt has two distinguishing features that we have not taken proper account of. First, as we have emphasized many times, interest paid on debt is tax deductible. This is good for the company, and it may be an added benefit of debt financing. Second, the failure to meet debt obligations can result in bankruptcy. This is not good for the company, and it may be an added cost of debt financing, called the cost of financial distress. Since we have not explicitly considered either of these two features of debt, we realize that we may arrive at different conclusions about capital structure once we do. We can start by considering what happens to Miller & Modigliani’s Propositions I  and II when we consider the effect of corporate taxes. In order to do this, we will examine two companies, Company U (Unlevered) and Company L (Levered). These two companies are identical on the left-hand side of the balance sheet, so their assets and operations are the same.

63 7.4.1 The Interest Tax Shield To make things simpler we assume that (1) capital expenditure is zero, (2) there are no changes in Net Working Capital and (3) depreciation is zero. Cash flows from Capital structure A (Unlevered) and Capital structure B (Levered) are not the same even if the two companies have identical assets. Total cash flow to Levered company is 175 more. This occurs because tax bill levered (which is a cash outflow) is 175 less. Due to the fact that interest is deductible for tax purposes it generates a tax saving equal to the interest payment 500 multiplied by the corporate tax rate 35%: $500*35%=$175. We call this tax saving the interest tax shield. Table 19 Capital structure A Shares Price per share Equity value

Capital structure B

rD= 10%

1000

500

10

10

10000

5000

Debt value

0

5000

Asset value

10000

10000

1500

1500

Operating results Net income before interest Interest expense

0

Taxable income

1500

500 flow 1000

Income tax

525

350 -175

Net income

975 flow

650 flow

Value of Tax Shield rD

10,0%

t

35%

D

5000

TS

175

PV(TS)

1750

tax bill less

64 shield is generated by paying 7. CAPITAL ANDasFINANCIAL LEVERAGE Tax interest, itSTRUCTURE has the same risk the debt, so10% (the cost of debt) is thus the appropriate discount rate. The value of the tax shield is: 7.4.2 Taxes and M&M Proposition I * 0,35) For the reason that  the(5000 debt*is0,1perpetual, same 175 shield =1750 will be generated every = the (5000*0,1*0,35)/0,1 PV(TS)= n ( 1  0 , 1 )  i 1 year onwards. The after tax cash flow to Levered company will thus be the same 975 that UnLevered Company earns plus the 175 tax shield. Because Levered company’s cash flow is always $175 greater, worth more than UnLevered Company, the differThe another famous Levered result ofCompany Miller &is Modigliani’s Proposition I (after-tax version) ence being the value of this 175 perpetuity. states that market forces must Tax shield is generated by cause: paying interest and it has the same risk as the debt, so 10% (the cost of debt) is thus the appropriate discount rate. The value of the tax shield is: n

Vlevered = Vunlevered + t D where D is the value of the company’s debt and t is the relevant Corporation Tax rate, at Another famous result of Miller & Modigliani’s Proposition I (after-tax version) states that which interest be relieved. marketdebt forces mustwill cause: The effect of borrowing in this caseVis illustrated = V in Figure + TD 12. We have plotted the value levered

unlevered

of the levered company, VL, against the amount of debt, D. Miller & Modigliani where D is the value of the company’s debt and T is the relevant Corporation Tax rate, at Proposition I with corporate taxes implies that the relationship is given by a straight line which debt interest will be relieved. with a slope of TC and a y-intercept of VU. We have also drawn a horizontal line The effectVof borrowing in this case is illustrated in Figure 12. We have plotted the value representing U. As pointed out, the distance between the two lines is TC * D, the present of the levered company, VL, against the amount of debt, D. Miller & Modigliani Proposition value of the tax shield. I with corporate taxes implies that the relationship is given by a straight line with a slope of TC and a y-intercept of VU. We have also drawn a horizontal line representing VU. As pointed out, the distance between the two lines is TC * D,I with the present value of the tax shield. Figure 12 Miller & Modigliani Proposition corporate taxes Figure 12 Miller & Modigliani Proposition I with corporate taxes Company value (VL) Value of levered company VL = VL + TC D

Tax shield

VL

TC D Value of unlevered company (VU)

VL Debt (D)

- 85 -

65 7.4.3 Taxes and the WACC The Miller & Modigliani after-tax proposition implies that the value of the levered company is greater than the value of the unlevered one. Consequently the higher the level of leverage, the higher the value of the company. The inevitable conclusion is that the value of the company is the highest and WACC the lowest, when leverage is at 100% (all-debt company), i.e. all finance is provided by debtholders. If we consider the effect of taxes, the WACC is:

To calculate this WACC, we need to know the cost of equity. Miller & Modigliani Proposition II with corporate taxes states that the cost of equity is:

Table 20 shows some examples of WACC with taxes. Let us assume that we have $60 debt, so that levered company is worth $124 total. Because the debt is worth $60, the equity must be worth $124–$60=$64. For this company, the cost of equity is thus:

rE = 0,16 + (0,16 - 0,10) *(60/64) * (1 - 0,40)= 19,4%

The weighted average cost of capital is:

WACC = (60/124) * 19,4% + (64/124) * 10% * (1 - 0,40)= 12,9% The WACC without debt is over 16%, and, with debt, it is 12,9%. Consequently, the company is better off with debt. Table 20 Assumptions

Beta Risk-free rate Equity risk premium Cost of unlevered Long Term Borrowing Rate Tax rate

1,50 10,0% 4,0% 16,0% 10,0% 40%

100 100 0

108 88 20

116 76 40

124 64 60

132 52 80

140 40 100

Debt-to-Equity ratio Debt-to-Assets ratio rE

0% 0% 16,0%

23% 19% 16,8%

53% 34% 17,9%

94% 48% 19,4%

154% 61% 21,5%

250% 71% 25,0%

WACC no tax WACC

16,0% 16,0%

16,0% 14,8%

16,0% 13,8%

16,0% 12,9%

16,0% 12,1%

16,0% 11,4%

Assets=Equity+TaxShield Equity Debt

66

7. CAPITAL STRUCTURE AND FINANCIAL LEVERAGE

Figure 13 shows the Miller & Modigliani after-tax view of capital leverage. There is a difference with the pre-tax proposition. In the after-tax example the cost of debt is low enough (due to higher tax relief on loan interest) for increasing amounts of loan finance to reduce WACC at a greater rate than the increasing demand of stockholders are raising it. Thus the WACC line slopes downward. Nevertheless, it is unclear if this 100% debt finance conclusion is rational in practical terms. At extremely high levels of leverage the debtholders would recognize that their safety had been substantially eroded and that, while they theoretically might be debtholders, as risk takers they are stockholders in reality. They would therefore demand a level of return which would compensate them for this risk, a  level of return similar to the one the stockholders demand. This shows that at very high levels of leverage, both in the pre-tax and in the more upwards at somepropositions, high level of leverage, thedebt WACC linerise can significantly. only continue its downward important post-tax the cost of would If path Miller & Modigliani this as requires thetherate of increase in the cost if the cost of equityconclusions line becomeshold, less steep WACCthat line is average of the other of equity starts to fall. In Figure 13, if the cost of debt line is going to start turning upwards at two. some high level of leverage, the WACC line can only continue its downward path if the cost of equity line becomes less steep as WACC line is the average of the other two. Figure 13. Miller & Modigliani after-tax view of capital leverage ( gearing)

Figure 13. Miller & Modigliani after-tax view of capital leverage (gearing) Cost of capital (%) Cost of Equity

WACC

Cost of Debt

Debt (D)

7.5 Financial Distress and Bankruptcy Cost The concept of the rate of increase in the cost of equity, as leverage is increased, rapidly 7.5 Financial distress and bankruptcy cost Why should stockholders start decelerating at high leverage levels, seems not to be sensible. to behave differently to all the concepts and evidence of investor feedback to growing risk? This would undermine in the M&M theorem, which possibly means that their deduction in The concept of the rate of increase in the cost of equity, as leverage is increased, rapidly the after-tax situation is not reasonable. starting to reduce at high leverage levels seems not to be sensible. Why should Cost of debt increases fixed costs and when a company has higher fixed costs then the stockholders start todistress behaveincreases. different to all theinconcepts and evidence of investor cannot probability of financial It arises the situation where a company feedback to growing risk?toThis proposes the weakness in the M&M theorem which meet its financial obligations its debtholders. possibly means that their deduction in the after-tax situation is not reasonable. Cost of debt increases fixed costs and when a company has higher fixed costs then the probability of financial distress increases. It rises the situation where a company cannot

67 7.5.1 Cost of Capital in Practice When financial distress cannot be handled, it can lead to bankruptcy. Financial distress is typically associated with certain costs to the company (financial distress costs). A typical case of a cost of financial distress are bankruptcy costs. Such financial direct costs include management fees, legal fees, auditors’ fees and other expenditures. Cost of financial distress can occur even if bankruptcy is avoided (financial indirect costs). If a situation of financial distress arises indirect costs of financial distress makes higher costs of capital (banks generally increase the interest rates). High levels of debt expose the company to the risk. Financial distress in companies involves management attention and might lead to less attention to the processes of the company. If the company is not able to meet its payment obligations to creditors, or at least not using its operating cash flows, the company should experience a particularly adverse period of trading. While it is equally true that the 100% equity company might have difficulty paying dividends in analogous economic conditions, there is a key difference. Creditors have a contractual right to collect interest payments and capital repayment on the payable dates. If debtholders do not collect these payments, they have the right and power to legally enforce payment. The implementation of such right and power can in practice lead to the process of selling the company’s assets, paying off creditors, distributing any remaining assets to the principals and then dissolving the company. This will typically be a disadvantage to the ordinary shareholder to a significant extent, but neither in a levered, nor in an unlevered company do ordinary shareholders have any rights to enforce the declaration and dividend payment. The risk of bankruptcy is perhaps unimportant at low levels of leverage, because any lack of cash for interest expenses could be covered from some other source – an option probably not so readily available to highly levered companies in distress. At very small levels of financial leverage the position of debtholders is one of great security, with the value of their bonds probably covered many times by the value of the company’s assets. Since the level of leverage rises, this situation erodes until at very high levels creditors, because they deliver most of the finance, bear most of the risk. Assume that the undeveloped company were financed 90% by equity and 10% by debt. The value of company’s assets would have to decrease by 90% before the security of the debtholders would be vulnerable. Though the equity/debt ratio changed to 80:20, the debtholders’ security, while in theory somewhat weakened, is not less in practical terms than had they only supplied 10% of the finance. If still the ratio changed to 10:90, only a small fall (10%) in the value would leave the debtholders bearing all the risk. Unsurprisingly, debtholders would request high returns to encourage them to buy bonds in such a highly levered company, apparently something like the returns expected by equityholders. Understandably, the debtholders would not see risk (and required return) increasing significantly with increases in leverage at the lower end. Finally, unless the asset on which security rests is extremely volatile in its value, an assets/debt ratio of 5:1 is probably as good as 10:1; the debtholders only need to be paid once. If this ratio increases to nearer 1:1, debtholders

68

7. CAPITAL STRUCTURE AND FINANCIAL LEVERAGE

would no doubt start to see things in different way. It is remarkable that neither of these two ‘weak’ assumptions of Miller & Modigliani theorem drastically affect the situation at lower levels of leverage, but they start to appear very large at higher levels. Let us make a  provisional conclusion: in acceptable, sensible levels of leverage the tax compensation of debt finance will make the WACC decrease as more leverage is introduced. Away from reasonable levels, bankruptcy risk (to equity shareholders) and the introduction of real risk to debtholders will move up the returns required by each group, making WACC a very high figure at high leverage levels. It is more likely that, in real economy, reasonable is not a permanent point for any particular company; it is rather a range below which Miller & Modigliani proposition is valid, but above which it obviously is not. The problem is that it is difficult to define a ‘reasonable level’ and, per se, it is a matter of opinion of financial management. It has to be the point at which the tax gain on the one hand unbalance bankruptcy cost and increase cost of debt on the other. It is different in different business and will to some degree depend upon the industry Figure 14 shows this conclusion with D*, WACC, and Cost of Debt all following the same risk of the investments in which the particular company is involved. pattern as Miller & Modiguani post-tax up to a *reasonable level of leverage and then all Figure 14 shows this conclusion with D , WACC, and cost of debt all following the same startingastoMiller increase&toModigliani very high levels as additional leverage pattern post-tax proposition upistomade. a reasonable level of leverage and then all starting to increase to very high levels as additional leverage is made. Figure 14 Cost of capital in practice

Figure 14 Cost of capital in practice Cost of capital (%) Cost of Equity

WACC

Cost of Debt

Sensible level of debt Debt (D) *

Optimal debt (D )

The connection between the value of the levered and unlevered companies can be expressed as: Formally the connection between the value of the levered and unlevered companies can

V

levered be expressed as:

= Vunlevered + TD – Present value of the expected cost of bankruptcy

Since leverage increases, the value of the tax shield (TD) increases but also does the probable cost of bankruptcy. Vlevered = Vunlevered + t D – Present value of the expected cost of bankruptcy

Since leverage increases, the value of the tax shield (TL) increases but also does the probable cost of bankruptcy.

69 7.5.2 The Static Theory of Capital Structure from an additional debt is precisely equal to the cost that comes from the increased

The idea ofofcapital structure havethat presented is known as intheterms static of capital possibility financial distress. that We we assume the company is fixed of theory its structure. According theory companies borrowratio upchanges. to the point where the tax gains operations and assetsto andthis we only considers the debt-equity fromFigure an additional debt is precisely equal to the cost that comes the increased possibility 15 illustrates the static theory of capital structure and plotsfrom the value of the of financial distress. We assume that the company is fixed in terms of its operations and assets company, VL, against the total debt, D. In Figure 15, we have drawn lines matching to and we only consider the debt-equity ratio changes. three different scenarios. The first scenario represents Miller & Modiguani Proposition I Figure 15 illustrates the static theory of capital structure and plots the value of the compawith no taxes. This is the horizontal line extending from VU, and it shows that the value ny, VL, against the total debt, D. In Figure 15, we have drawn lines matching to three different of the company is not affected by its capital structure. The second scenario, Miller & scenarios. The first scenario represents Miller & Modigliani Proposition I with no taxes. This Proposition I with corporate taxes, is represented by the upward-sloping is theModiguani horizontal line extending from VU, and it shows that the value of the company is not straight line. affected by its capital structure. The second scenario, Miller & Modigliani Proposition II with corporate taxes, is represented by the upward-sloping straight line. Figure 15 Optimal Capital Structure

Figure 15 Optimal Capital Structure Company value (VL)

Value of levered company (VL)

Maximum company value (VL*)

Financial distress cost

Present value of tax shield on debt

Current company value Value of unlevered company (VU)

Debt (D) Optimal debt (D*)

In third scenario in Figure 15 the value of the company rises to a maximum and then deIn third scenario in Figure 15 the value of the company rises to a maximum and then clines outside that point. This is the situation that we get from a static theory of capital strucdeclines outside that point. This is the situation that we get from static theory of capital ture. The maximum value of the company, VL*, is reached at D* and this point represents the structure. The maximum value of the company, VL*, is reached at D* and this point optimal amount of debt in company. Put differently, that company’s optimal capital structure represents optimal amount of debt in company. We can say differently, that the is composed oftheD*/V * in debt and (1 – D*)/V * in equity. L L and (1 the – D*/V in the company company’s optimal capital of difference D*/VL* in debt L*) of The final obserwation instructure Figure is 15composed is that the between value equity. in static theory and the Miller & Modigliani (value of the company) with taxes is the loss in value as financial distress is rising. Moreover, the difference between the static theory value of the company and the Miller & Modigliani value with no taxes is the gain from leverage, net - 92 of distress costs.

70

List of Figures Figure 1 Organization chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Figure 2 Components of the return on investment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Figure 3 Systematic and Unsystematic Risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Figure 4 Market risk and size of portfolio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Figure 5 Capital structure.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 6 Security Market Line (SML) and Weighted Average Cost of Capital WACC. . . . . . 49 Figure 7 The impact of leverage. EPS versus EBIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Figure 8 ROE for levered company with fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Figure 9 Cost of capital for different levels of leverage (gearing) - traditional view. . . . . . . . 58 Figure 10 Capital structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Figure 11 M&M Preposition II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Figure 12 Miller & Modigliani Proposition I with corporate taxes. . . . . . . . . . . . . . . . . . . . . . 64 Figure 13. Miller & Modigliani after-tax view of capital leverage (gearing) . . . . . . . . . . . . . . 66 Figure 14 Cost of capital in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Figure 15 Optimal Capital Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

71

List of Tables Table 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Table 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Table 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Table 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Table 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Table 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Table 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Table 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Table 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Table 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Table 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Table 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Table 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Table 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Table 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Table 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Table 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Table 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Table 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Table 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

72

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