Corporate Finance. Bin Zou. University of Alberta

Transcription

1 Corporate Finance Bin Zou University of Alberta 2010 Fall Updated: February 22, 2015

2 Contents Preface I 1 The Corporation Organizational Types of Firms Goals of the Corporation Financial Statement Analysis Financial Statement Balance Sheet Income Sheet Time Value Mechanics Interest Rate Arbitrage and the Law of One Price Perpetuity and Annuity Investment Criteria NPV Rule Alternative Rules Comparison of Investment Opportunities Capital Budgeting Fundamentals of Capital Budgeting Case Study Bond and Stock Valuations Valuing Bonds Valuing Stocks Dividend-Discount Method i

3 ii CONTENTS Comparison Method Options Introduction Properties of Stock Options Factors Affecting Option Prices Bounds for Option Prices Put-Call Parity The Binomial Pricing Model Black-Scholes Model Risk and Return Statistics Background Measures of Risk and Return Portfolio Return and Risk Capital Asset Pricing Model and Portfolio Management Intuitive View of CAPM Portfolio Management Portfolio Selection without Risk-free Asset Portfolio Selection with Risk-free Asset Mathematical Proof of CAPM

4 Preface Those notes on Corporate Finance are mainly based on the lectures of FIN 501 taught by Professor András Marosi in Fall 2010 at the University of Alberta, and the reference book Corporate Finance (Canadian Edition) written by Jonathan Berk, Peter DeMarzo and David Stangeland [1]. If you find any typos or problems in the notes, please let me know by . I

5 Chapter 1 The Corporation In this chapter, we compare three major organizational types of firms: sole proprietorship, partnership and corporation. As the most common type of firms nowadays, corporation will draw our further attention. 1.1 Organizational Types of Firms There are three major types of firms: (i) Sole Proprietorship owned and run by one person easy to set up (pros) normally, small business with few employees, revenues and profits no separation between the firm and the owner no principal-agent problem, but the management largely depends on the owner s personal ability taxed at the personal level (cons) other investors cannot hold ownership difficult to raise money (cons) unlimited personal liabilities (huge cons) limited lifetime (cons) 1

6 2 1. The Corporation hard to transfer ownership (cons) (ii) Partnership like an extension of sole proprietorship with more than one owner taxed at the personal level (cons) all partners are liable for the firm s debt (huge cons) Special case limited partnership: at least one general partner with unlimited liabilities and several partners who have limited liabilities to their investment but has no authorities on management limited lifetime (cons) hard to transfer ownership (cons) (iii) Corporation legally defined and separate from the owners only responsible for its own obligations owners have limited liabilities (pros) articles of incorporations (or corporate charters) which stipulate the ownership, existence and other regulations of the corporation (possible cons) no limit on the number of owners, and ownership is easy to transfer (big pros) all the owners (stockholders, equity holders) have the right to share dividend according to their investment proportions (pros) all the owners are entitled to vote for firm s decisions with different weights (pros) no special expertise is required to be an owner easy to accumulate money (big pros) double taxation the corporation pays tax on its profits and the shareholders pay their personal income tax

7 1.2. Goals of the Corporation 3 Compensation for double taxation in Canada: income trusts which come in three forms business income trust, energy trust and real estate investment trust (REIT). In those firms, no corporate tax is collected. (However, the first two trusts will not exist anymore after 2011 in Canada.) 1.2 Goals of the Corporation A common goal of corporation is to maximize shareholders wealth, especially, in Canada, the United States and the UK. While in Japan and some European countries, corporations try to increase stakeholders satisfaction (The stakeholders of a corporation include its shareholders, debt holders, customers, suppliers, communities and the government). Other possible choices include maximizing sales/profit and maintaining steady growth. The separation of ownership and management in a corporation may lead to principal-agent problems in which the managers don t pursue the maximization of shareholders profits. Case. Liquor Barn Income Fund, a publicly traded company, used to own and operate liquor stores in Alberta. Liquor Barn had an agreement with Devco: Devco would purchase individual liquor stores and later resell them to Liquor Barn. In this case, the CEO of Liquor Barn was the owner of Devco and therefore as CEO, he should buy liquor stores as cheaply as possible; however, as the owner of Devco, he had an interest to ask a high price when Devco sold stores to Liquor Barn. Finally, Liquor Stores Income Fund, a competing firm, made an offer to purchase all Liquor Barn shares (at a premium) and Liquor Barn shareholders eventually accepted. Corporate control to address principal-agent problem minimize the number of interest-conflict decisions fire managers when they don t work well takeover (to replace the board of directions and the CEO)

8 Chapter 2 Financial Statement Analysis From the results in Chapter 1, we know that one advantage of corporation is that it can have many owners, especially shareholders. Therefore, financial managers and owners need to assess their companies performance while investors also want to estimate the firms performance to decide whether to buy or sell the firms stocks. A good way for all of them to evaluate the firms performance is through their financial statements. In this chapter, we discuss two major financial statements: the balance sheet and the income sheet, and study how to use these statements to evaluate a company s performance. 2.1 Financial Statement Financial statements are accounting reports with past performance information that a firm publishes periodically. Financial statements of Canadian companies can be found on the website of the System for Electronic Document Analysis and Retrieval (SEDAR Generally Accepted Accounting Principles (GAAP) provide a common set of rules and a standard format for public companies to prepare financial statements. In order to ensure the accuracy of financial statements, a neutral third party, known as an auditor, is usually involved. According to International Financial Reporting Standards (IFRS), every public company is required to produce five financial statements: the balance sheet, the income sheet, the statement of cash flows, the statement of shareholders equity and the statement of comprehensive income. 4

9 2.2. Balance Sheet Balance Sheet The balance sheet lists a firm s assets and liabilities. A standard balance sheet should list the assets along with the liabilities and shareholder s equity. The firm s cash, inventory, property, plant, equipment and other investment appear in the column of assets while the liabilities include the firm s current and long-term obligations. Shareholder s equity, the firm s total assets less its liabilities, measures the net wealth of the firm. Therefore, we have the following identity: Assets = Liabilities + Shareholder s equity Figure 2.1: Balance Sheet Current assets are either cash or can be converted into cash within one year. Fixed assets are also called long-term assets, including real estate

10 6 2. Financial Statement Analysis and equipment. Depreciation is not an actual cash expense but is marked to recognize the fact that some assets like buildings become less valuable. The book value of an asset is the difference between its acquisition cost and accumulated depreciation. Liabilities that need to be paid off within one year are called current liabilities. Those liabilities with maturity longer than one year are long-term liabilities. One of the current liabilities is accounts payable, the amounts owned to suppliers for products or service purchased with credits. Shareholder s equity is also known as the book value of equity, however, it is not an accurate measurement of the firm s true value. For instance, the current value of the firm s land may be higher than its original acquisition value. Besides, there are other factors which play an important role in determining a firm s value, such as expertise of the firm s employees, and the firm s reputation in the industry. The total market value of a firm s equity equals the market price per share times the number of shares, referred to as the company s market capitalization, which indicates investors expectation in the future. The liquidation value of the firm is the value that would be left if its assets were sold and liabilities paid. From the balance sheet, we can obtain an estimate of the liquidation value. Market-to-Book Ratio (Price-to-Book [P/B] Ratio) Market-to-Book Ratio = Market Value of Equity Book Value of Equity Most successful companies have market-to-book ratio greater than 1, meaning that the value of firm s assets can produce more than their historical cost (or liquidation value). Normally, a higher market-to-book ratio is a good sign of the firm s future performance. Debt-Equity Ratio Debt-Equity Ratio = Total Debt Total Equity The debt-equity ratio is always used to assess a firm s leverage. In the above formula, we can use either book or market values for debt and equity to

11 2.3. Income Sheet 7 calculate this ratio. Since the book value of the firm s equity is not a precise assessment, so it s better to use market value of its equity for calculation. However, as the difference between the book value of debt and the market value of debt is basically slight, we always ignore such difference in practice. Enterprise Value Enterprise Value=Market Value of Equity + Debt - Cash The enterprise value of a firm assesses the value of the underlying business assets, also can be interpreted as the net cost to take over the business. Problem 2.1. On December 31, 2007, Maple Leaf Foods inc. (MFC) had a share price of $14.85, million shares outstanding, a market-to-book ratio of 1.66, a book debt-equity ratio of 0.76, and cash of $28 million. What was Maple Leaf s market capitalization? What was its enterprise value? Solution. Its market capitalization is $14.85 per share million shares= $1.92 billion. From its market-to-book ratio, we can obtain Maple Leaf s book value of equity, 1.92/1.66 = $1.16 billion. Then its total debt can be calculated as = $0.88 billion. Lastly, we calculate the Maple Leaf s enterprise value, = $2.77 billion. 2.3 Income Sheet The income sheet, also called statement of earnings, statement of operations, or profit and loss ( P & L ) statement, lists a firm s revenues and expenses over a period of time. The net income line of the income statement shows the firm s profitability during that period. Gross margin is the difference between the first two items, total sales and cost of sales. The following three items comprise operation expenses. The firm s gross margin less the operation expenses is called operating income. Other income lists all the other cash flows obtained from non-central part of the firm s business, like profits from financial investment. Earnings Before Interest and Taxes (EBIT) is the summation of operating income and other income. After deducting the interest paid on outstanding debt (Interest expense), we obtain Earnings Before Taxes (EBT), and then we subtract corporate taxes from EBT to obtain the firm s net income. We calculate

12 8 2. Financial Statement Analysis Earnings per share (EPS) through dividing net income by the outstanding shares. If there are stock options and/or convertible bonds in the market, then the number of outstanding shares may increase, which will lead to the decrease of EPS, referred as diluted EPS. Profitability Ratios EPS = Net Income Outstanding Shares Operating Margin = Operating Income Total Sales GLOBAL CONGLOMERATE CORPORATION Income Statement Year ended December 31 (in $ millions, except per share amount) Total sales Cost of sales (153.4) (147.3) Gross Margin Selling, general, and administrative expense (13.5) (13.0) Research and development (8.2) (7.6) Depreciation and amortization (1.2) (1.1) Operation Income Other income - - Earnings Before Interest and Taxes (EBIT) Interest expense (7.7) (4.6) Earnings Before Taxes (EBT) Taxes (0.7) (0.6) Net Income Earnings per share (EPS): $0.556 $0.528 Diluted earnings per share: $0.526 $0.500 The operating margin measures how much a company can earn before taxes and interest from each dollar of sales. Net Profit Margin = Net Income Total Sales

13 2.3. Income Sheet 9 The net income margin shows how much the shareholders can earn from each dollar of the firm s revenues. Accounts receivable days Accounts Receivable Days = Accounts Receivable Average Daily Sales which is used to measure a firm s efficiency of utilizing the net working capital. For instance, in 2011, the Global Conglomerate Corporation s yearly sales is $186.7 million. Then we can calculate its daily sales by 186.7/365 = $0.51 million. From its income statement, we know its accounts receivable is $18.5 million. Therefore, we obtain its accounts receivable days by 18.5/ days. This number tells us that the Global Conglomerate Corporation needs over a month to collect payment from its customers on average. Return on equity (ROE) Return on Equity = Net Income Book Value of Equity which is commonly used as an estimator of firm s return on new investment. We will go back to it when discussing capital budgeting. Price-earning ratio (P/E ratio) P/E Ratio = Market Capitalization Net Income = Share Price Earnings per Share P/E ratio can be used to assess whether a stock is over- or under-evaluated.

14 Chapter 3 Time Value Mechanics In this chapter, we will introduce the time value of money and show how to discount and compound the value among different time points. By the argument of no-arbitrage theory, we will deduce an important principle: law of one price. Then we shall use this principle to price annuities and perpetuities. 3.1 Interest Rate For most financial decisions, incomes and expenditures occur at different time. Generally speaking, a certain amount of money at present time should worth more than the same amount in the future. Such difference in value between different time is referred as time value of money. Hence we should discount or compound values to the same time in order to compare them. The rate we use for discounting or compounding through different time is interest rate. For simplicity, let s consider a discrete time model, in which time is numbered by natural numbers 1, 2,. Let r be the risk-free interest rate, for instance, we can use three-month treasury rate as a replacement in practice. Therefore, future value is obtained by P t = P t 1 (1 + r) = P 0 (1 + r) t, 1 t N +. (3.1) By the same token, we can also discount value from future time t to current/past time s (s < t) by dividing interest rate correspondingly. Notice that the above formula assumes that all the gained interest will be re-invested in the same asset P. So rate r is actually the compound interest 10

15 3.1. Interest Rate 11 rate. Alternatively, we have simple rate, and using simple interest rate r, future value is calculated by P t = P 0 (1 + rt). (3.2) In the above discussion, we have the same time interval (time unit) and r is actually the interest rate of one period. However, if we have different scales, for instance, annual rate r a, semi-annual rate r s, quarterly rate r q, monthly rate r m, weekly rate r w and daily rate r d, then we need to find the relationship among them. If we assume there are 365 days per year, then we can convert r d into r a by r a = (1 + r d ) (3.3) In general case, assume that r i and r j are two interest rates of time length i and j, and i = N j. For instance, in the previous example, period i is one year while period j is one day, so N = 365. Then we shall have: r i = (1 + r j ) N 1. (3.4) Notice that all the above mentioned compound interest rates are effective rates. However, most quoted interest rates (the ones you can find through banks) are not effective rates. For instance, a commonly used one is annual percentage rate (APR), which is the amount of simple interest earned in one year. In order to convert the APR to effective annual rate (EAR), we need to know the compounding times per year. Example. Bank of Montreal issues a security with interest rate of 5% per year with monthly compounding, then the EAR of this security is given by: ( EAR = 1 + APR ) 12 1 = 5.12%. 12 Given APR with K compounding periods per year, the effective interest rate per compounding period r p is: r p = APR K (3.5) while the EAR is simply: ( EAR = 1 + APR ) K 1 = (1 + r p ) K 1. (3.6) K

16 12 3. Time Value Mechanics Many loans, such as consumer loan and car loans, have monthly payments and are quoted in APR with monthly compounding. This type of loans is known as amortizing loans because in each payment, you pay both principal and interest. However, in Canada, home mortgage is quoted in APR with semiannual compounding but have monthly payments. Therefore, the effective monthly rate r m is calculated as r m = ( 1 + APR ) 1/6 1. (3.7) 2 In (3.6), when K, 1+EAR will converge to e APR, which is corresponding to continuous models. So if compounding happens at every point in time axis (verbal explanation of continuous model), then if we deposit 1$ dollar today, we will get e APR dollars one year later. So far, all the interest rates we have discussed before so-called nominal interest rates, which indicates how much money you will earn by investing in a saving account (other security with certain return) for a certain period. But in real life, there is inflation/deflation which will alter the purchasing value of money. The interest rate after adjusting for inflation reflects the real change of purchasing power of your money, and this rate is called real interest rate. Let i be the inflation rate and r be the nominal interest rate, then real interest rate r r is determined by (all rates are quoted in same time period.) 1 + r r = 1 + r 1 + i. (3.8) An approximation is given by r r r i. (3.9) 3.2 Arbitrage and the Law of One Price Arbitrage is an investment portfolio which can guarantee positive profits for sure, or mathematically defined as V (0) = 0, V (1) 0 and P (V (1) > 0) > 0, (3.10) where V (t) denotes the value of the portfolio at time t and P is the actual probability measure.

17 3.3. Perpetuity and Annuity 13 By the definition of arbitrage, it can be considered as free-lunch, and all rational investors will buy arbitrage portfolios as much as they could. Those trading activities will for sure drive up the prices of the arbitrage portfolios, which will ultimately cause the arbitrage opportunities to vanish. Based on such argument, we can assume that an efficient market is arbitrage-free. Remark. It can be shown the sufficient and necessary condition for a market to be arbitrage-free is that there exists at least one risk-neutral probability measure P. Under this probability measure P, all assets have the same return as the risk-free asset and so bear the same risk, which is the reason why this probability measure is called risk-neutral. For further reference, please read [10]. Law of one price. If an investment opportunity is traded simultaneously in different competitive markets, then the price is the same in all markets. According to the law of one price, for any security with certain future price S(1), its present price S(0) is determined by S(0) = S(1)/(1 + r), where r is the risk-free rate. Or equivalently, we can say that financial investment doesn t increase or decrease any extra value. (Such conclusion is made under the assumption that there is no risk/uncertainty) This conclusion is known as Separation principle, which gives the reason for raising capital through issuing stocks or bonds. 3.3 Perpetuity and Annuity A perpetuity is a stream of cash flows C n that occur at regular intervals and last forever. The real-life example is British government bond (called consol bond). Normally, when taking about perpetuity, we assume that all C n are equal, denoted by C. Suppose that the cash flows start from time 1 and the risk-free interest rate in the same time interval is denoted by r (such assumption will hold unless otherwise specified, for instance, the staring point will be 0 only when we consider perpetuity/annuity due). Therefore, we can

18 14 3. Time Value Mechanics calculate the present value of such defined perpetuity: P V = C 1 + r + C (1 + r) + C 2 (1 + r) + = C 3 (1 + r) t = C 1 + r r t=1 = C r. (3.11) For a perpetuity with payment beginning at time 0, we call it perpetuity due. If we still use (3.11), then we will obtain the value of perpetuity due at time 1, so to calculate its present value (at time 0), we need to accumulate to time 0 by multiplying 1 + r, thus the present value of a perpetuity due is: P V = C(1 + r). (3.12) r A growing perpetuity is a perpetuity with growing cash flows at growth rate g, so C n = C 1 (1 + g) t 1, t 1. r > g Under the same assumption, the present value of growing perpetuity can be calculated as follows: P V = C r + C 2 (1 + r) 2 + C 3 (1 + r) 3 + = C g 1 + g = C g 1 + r g 1 + r t=1 (1 + g) t (1 + r) t = C 1 r g. (3.13) r g The convergence radius of infinite power series is 1 and it is evident that the power series doesn t converge at point 1. According to this proposition, it s easy to see that in this case the growing perpetuity has a infinite large present value.

19 3.3. Perpetuity and Annuity 15 An annuity is a stream of N equal payments that occur at regular time intervals. Using the same method, its present value is given by P V = C 1 + r + C (1 + r) + + C 2 (1 + r) = N C N (1 + r) t 1 = C r (1 + r) N r = C r t=1 ( ) 1 1. (3.14) (1 + r) N Correspondingly, we also have growing annuity and its present value can be obtained according to two different cases: r = g r g P V = C r N 1 t=0 ( ) t 1 + g = N C r 1 + r. (3.15) P V = C r + C 2 (1 + r) + + C N 2 (1 + r) = C 1 N 1 + r = C r ( 1 + g 1 + r g 1 + r ) N = C 1 r g [ 1 N 1 t=0 ( ) t 1 + g 1 + r ( ) ] N 1 + g. (3.16) 1 + r Example. Your six years old wants to go to Princeton. This will cost you $30, 000 per year for 4 years. On her next birthday you will start to put money into an account paying 14% annual rate and continue to deposit the same amount every year until her 17 th birthday. The first tuition payment will occur on her 18 th birthday, the second on her 19 th birthday, etc. How much do you need to deposit to this account every year? Solution. In this example, we have two annuities, the first one will last from time 1 to 11 with payment C (the value we need to calculate) while the second will begin at time 12 and end at time 15 with payment $30, 000, where we label 6 th birthday as time 0 and time interval is one year. Since

20 16 3. Time Value Mechanics we use the first annuity to finance the second one, so obviously, their value should coincide at any time. Firstly, by (3.14), we can get the value of second annuity at time 11: [ ( 1 P11 2 = )] = $87, (1.14) 4 Therefore, by discounting, we can calculate its present value: P0 2 = P 11 2 = $20, (1.14) 11 According to the above argument, we know that the first annuity should also have the same present value of $20, , and so [ ( )] 1 P0 1 1 = C 1 = $20, (1.14) 11 C = $3,

21 Chapter 4 Investment Criteria In corporate finance, one of the most important topics is to study the investment decision rules. Among all these rules, net present value (NPV) is the most widely used one while there are also some other alternative rules commonly used in real industries like the payback investment rule. In this chapter, we will discuss major decision rules in details. Before we introduce decision rules, we should answer a question first: what should a good criterion do? Basically, a good evaluation should Consider all cash flows Account for time differences Provide unambiguous decision Measure wealth created for shareholders 4.1 NPV Rule When the payoff of an investment is discounted back to present time, then such discounted value is called present value. Then we can define net present value of an investment as: NPV = PV(Benefits) - PV(Costs). If an investment opportunity has a strictly positive NPV, then such investment can generate profits and thus should be taken. In addition, if there 17

22 18 4. Investment Criteria are several investment opportunities which generate positive NPV, then we should take the one with the highest NPV. These arguments then lead to the famous NPV rule: When making investment decision, take the alternative one with the highest positive NPV. Notice that if there are many options, we cannot just take the one with the highest NPV. Because it is possible that even the highest one is negative, then we should reject all these options. For instance, consider a project with C 0 outlay instantly and cash flows C t, 1 t T. Assume the cost of capital is r, then the NPV of this project is: T C t NPV = C 0 + (1 + r). t t=1 Define the profitability index of a project by PI = PV I 0, (4.1) where PV is the present value of all the benefits of the project and I 0 is the present value of all the cost of the project. Since P I > 1 NP V > 0, we should take the project when its P I > 1. Internal rate of return (IRR) is the interest rate which makes the N- PV of a project equal to zero. When making decisions, we need to know the cost of capital, but we can only estimate this rate, thus IRR can be used to measure the sensitivity of our estimation. In order to make the correct decision, the biggest error of estimating the cost of capital is the difference between the cost of capital and IRR. 4.2 Alternative Rules The Payback Rule To use the payback rule, we need to calculate the payback period (T p ) of a project and then compare the payback period with the pre-specified time period (T s ). If T p < T s, then we accept the project. Example. Assume that we will accept the project when its payback period is less than 5 years. Consider a project with investment C 0 = $200 million

23 4.2. Alternative Rules 19 and positive cash flows C i = $50 million in year i, 1 i 6. Should we accept this project? Solution. To fully pay back the initial investment, we need C 0 /C i = 4 years. Since the payback period of this project is less than 5, so we should launch this project. From the above example, we can easily see that the payback rules doesn t consider the time value of money and thus it is not a reliable rule. To make the payback rule more reasonable, we can use discounted cash flows to calculate the payback rule. Example continued. In the previous example, suppose cost of capital is r = 10%. Then we discount all the cash flows to the present time and will obtain: C 1 = $45.45, C 2 = $41.32, C 3 = $37.57, C 4 = $34.15, C 5 = $31.05, C 6 = $ Since 5 t=1 C t = $ < C 0 and 6 t=1 C t = $ > C 0, so in this case, the payback period is T p = 6 > 5. According to the payback rule, we should turn down this project. But obviously, the NPV of this project is = $17.76 million, and based on NPV rule, we should accept this project. This example tells us that even the modified payback rule is not trustful, and the reason is because the pre-set time period is not entirely objective and accurate. However, the most favorable advantage of the payback rule is simplicity, we even don t need to know the cost of capital to make the decision. The Internal Rate of Return Rule IRR rule: we should only take the project when its IRR is greater than the project s cost of capital. This rule directly comes from the definition of IRR, since IRR is the rate such that NPV=0, so if IRR > r, then we should have positive NPV if the cash flows are normal. Regarding the term normal, we mean the cost occurs right now and benefits come in the future time. Therefore, IRR rule coincides with NPV rule when the cash flows of the project is normal. However, the above IRR rule will give the reverse result when cash flows are unconventional. Example. Consider the following two projects: project A requires investment $12, 000 today and will give $15, 600 next year while project B offers

24 20 4. Investment Criteria $12, 000 cash right now but needs outlay $15, 600 next year. Then which project should we take according to different cost of capital r? Solution. Firstly, it s easy to calculate the IRR for two projects and both of them have the same IRR 30%. We can plot the NPV-r figure for two projects and find that their NPV have totally different correlation to the cost of capital. From the picture, we know that IRR rule works for project A since the NPV of project A is positive when IRR exceeds the cost of capital. Another problem of IRR rule is that the project may have no IRR or have more than one IRR. Economic Value Added Rule Consider a project with initial cost I dollars today and cash flows C n in each period n. The economic value added (EVA) in period n is given by EV A n = C n ri. (4.2) If the investment is a changing variable, use I n to denote the investment needed in period n and D n represents the depreciation in period n. Then the EVA is defined as EV A n = C n ri n 1 D n. (4.3) The EVA rule states that we should accept the project if the sum of present value of all EVAs. Example. Consider a project with C 0 = $300, 000 and C 1 = = C 5 = $75, 000. The cost of capital is r = 7% per year and the initial cost on equipment will be equally worn out over five years. Should we accept this project?

25 4.3. Comparison of Investment Opportunities 21 Solution. Firstly, it s easy to identify that D n = $60, 000 and so I n = C 0 nd n, 0 n 5. Then using EV A n = C n ri n 1 D n, we can obtain the streams of EVAs. For instance, EV A 1 = 75, 000 7% 300, , 000 = 6, 000 and EV A 4 = 75, 000 7% 120, , 000 = 6, 600. Therefore, the sum of present value of all EVAs is given by 6, 000 1, 800 2, 400 6, , 800 P V = = $7, So according to EVA rule, we should take this project. 4.3 Comparison of Investment Opportunities In the above two sections, we only deal with single project problem, whether we should take the project or not?. But in this section, we face more options and need to decide which one to take or turn down all of them. For mutually exclusive investment opportunities, we should take the one with the highest positive NPV and reject all of them if none of them have positive NPV. Projects with Different Lifetime In this case, we cannot simply compare the NPV but have to calculate the effective annual cost, EAC and take the one with the least EAC. Example. Firm A needs to purchase one machine and has two options right now. The information of these two options are listed below: Machine Initial Outlay Operating Cost per year Lifetime 1 10,000 3, ,000 4,000 5 Which machine should Firm A buy? (Cost of capital is given as r = 10%.) Solution. Firstly, let s calculate the present value of the cost of two machines: ( 3, 000 P V 1 = 10, ) = 17, , P V 2 = 9, 000 = 24, , ( 1 1 ) 1.1 5

26 22 4. Investment Criteria If we just consider the PV, then we should choose Machine 1 because of less cost in total. But we need to notice that P V 1 is the present value of cost over 3 years while P V 2 covers 5 years. So the fair game is to compare the EAC not the total cost. To calculate the EAC for Machine 1, we can use the following formula: P V 1 = C ( 1 1 ) C = 7, Similarly, we can also get the EAC of Machine 2 as C 2 = 6, Since Machine 2 has the lower EAC, we should purchase Machine 2 instead of Machine 1. Projects with Resource Constraints If there are resource constraints, then NPV rule has to be applied with constraints. Basically, we rank the projects according to their PI and launch integer optimization programming to find the optimal combination of projects. Example. Assume a firm has $1, 000 cash available and is unable to raise any more capital. Currently, it has several projects available, which are listed in the following table: Project Cost PI A B C Solution. If we take Project B, the NPV will be (1.8 1) 800 = 640. If we take Projects A and C, then what we will get is (2 1) (1.5 1) 400 = 700. Therefore, under the budget constraint, we should choose Projects A and C.

27 Chapter 5 Capital Budgeting A capital budget lists all the projects that a company plants to take during the coming year while the process of constructing capital budget is called capital budgeting. 5.1 Fundamentals of Capital Budgeting Firstly, notice that earnings are accounting concept and different from actual cash flow. The incremental earnings are the amount of money expected to gain as a result of investment. To estimate the incremental earnings, we need to forecast the revenue and cost. The difference of sales and cost of goods sold is gross profit, Gross Profit=Sales - Cost of Goods Sold. Generally, operating cost includes advertising, marketing and support costs but excludes equipment cost. Besides, there might be other costs in the project like research cost. But among them, there are two special costs: sunk cost and opportunity cost. A sunk cost is any unrecoverable cost which firm already paid, hence the firm s decision won t make any change to this cost. For this reason, this cost should be excluded from capital budgeting. The opportunity cost of using a resource is the value it could have provided in its best alternative use. For instance, a company has some spare warehouse which can be rented out for 1 million per year but it will use the warehouse if it decides to take a certain project. In capital budgeting of this project, 23

28 24 5. Capital Budgeting we should include the opportunity cost 1 million because taking this project will cost us a loss of 1 million income. Net Working Capital (NWC) is the difference between current assets and current liabilities. To better interpret NWC, we introduce the following terms: Receivables collected (RC) VS Sales booked (S) Expenses paid (EP) VS Expenses incurred (E) Apparently, RC S and EP E since the firms cannot get all the money of sales immediately and won t pay all the cost at the same when purchasing the goods/materials. So we have: Then NWC is given by: S RC = AR (Account Receivable) E EP = AP (Account Payable) NWC = Current Assets Current Liabilities = Cash + Inventory + Receivables - Payables (5.1) Therefore, the change of NWC in year t is NW C t = NW C t NW C t 1 = AR t + I t AP t, (5.2) where we assume there is no change in cash and I stands for inventory. When we sum all NW C t together, we should get zero. Capital expenditures (CapEx) are the investments in plant, property and equipment. They are cash expenses but will not be listed as expenses when calculating earnings. Instead, such expenditures will be deducted gradually for tax purpose as capital cost allowance (CCA). Assume an asset is in class d and the undepreciated capital cost (UCC) of the asset pool is denoted by UCC t in year t, then CCA in year t is given by: CCA t = UCC t d. (5.3) The formula for calculating UCC is listed as follows: t = 1 UCC 1 = 0.5 CapEx, ( t 2 UCC t = CapEx 1 d ) (1 d) t 2, (5.4) 2

29 5.1. Fundamentals of Capital Budgeting 25 where the first formula comes from the so-called half-year rule. Use S t denote sales and E t be all the costs except depreciation (CapEx), then the earnings before interest and taxes (EBIT) is: EBIT t = S t - E t - CCA t, and unlevered net income (UNI) (excluding interest expense, so no borrowing to finance project) is then given by: Unlevered Net Income = EBIT (1 τ c ). To compute free cash flow, we should add CCA back to unlevered net income (since CCA is not an actual cash flow), subtract actual capital expenditure (CapEx) and less the change of NWC. Therefore, free cash flow is calculated by CF t = UNI t + CCA t CapEx t NW C t = (S t E t CCA t ) (1 τ c ) + CCA t CapEx t NW C t = (S t E t ) (1 τ c ) + CCA t τ c CapEx t NW C t (5.5) In order to decide whether to take a project, we need to discount all CF t to present time and add them together to see whether the NPV is greater than zero. Thus we need to calculate the present value of all CCA tax shield. To do this, we shall decompose the cash flows of CCA tax shields into two growing perpetuities: the first one starts in year 1 with first cash flow C 1 = 0.5 CapEx d τ c and growth rate d while the second one is same to the first one except starting in year 2. Therefore, we have the present value of CCA tax shield: P V = C 1 r + d + C 1 r + d r = CapEx d τ c r + d 1 + r r (5.6) Notice that (5.6) only holds when the asset is not sold. If assets are no longer needed and company decides to sell it in future time year t, then it will gain salvage value (liquidation value). If the assets are sold at a higher

30 26 5. Capital Budgeting price than the acquisition price, then the company needs to pay capital gain tax, which is defined by: Capital Gain Tax = 1 2 (Sale Price - CapEx) τ c. (5.7) Capital gain tax should be paid in the same year when the assets are sold and will appear negative in free cash flow. Another tax effect of selling assets is that the firm will lose the remaining CCA tax shields. For simplicity, we assume that the asset pool is not liquidated, which means the firm still holds some other same class assets. (such assumption is called continuing pool). We further assume that future purchase of assets in the same class won t exceed the asset s sale price, which is referred as negative net addition. Under these two assumptions, we can calculate the present value of the CCA tax shields we lose because of sale: P V lost CCA tax shields = min{sale Price, CapEx} d τ c r + d 1 (1 + r) t. (5.8) Recall the original formula (5.6), we then obtain the modified present value of CCA tax shields as: P V CCA tax shields = CapEx d τ c r + d 1 + r r min{sale Price, CapEx} d τ c r + d 1 (1 + r) t. (5.9) With the help of CCA tax shields formula (5.6) and (5.9), we can compute the free cash flow excluding CCA tax shields and directly add the present value of CCA tax shields back when calculating the NPV of a project. 5.2 Case Study Case 1 Company A is facing a project of buying a machine. This machine (class 10, d = 30%) will cost Company A $800, 000 outlay and has an expected lifetime of 3 years. When it s sold 3 years later, Company A will acquire salvage value $125, 000. The production of this machine per is 20, 000, 25, 000, 20, 000

31 5.2. Case Study 27 (units) in the coming 3 years and each unit is worth $100 and costs $80. Besides the costs of production, there is another fixed cost $50, 000 per year to maintain the machine. The initial working capital is $30, 000 and needs 10% of sales for maintenance. The corporate tax rate for Company A is 40% and the cost of capital is 10%. Firstly, based on the production information and unit price, we can compute the sales in 3 years as: S 1 = S 3 = 20, = $2, 000, 000, S 3 = 25, = $2, 500, 000. Similarly, the costs each year can be obtained by: E 1 = E 3 = 20, , 000 = $1, 650, 000, Then we can get the gross profits by: E 2 = 25, , 000 = $2, 050, 000. (S 1 E 1 ) = (S 3 E 3 ) = $350, 000, (S 2 E 2 ) = $450, 000. Secondly, the NWC of the project is given as: NW C 0 = $30, 000, NW C 1 = S 1 10% = $200, 000, NW C 2 = S 2 10% = $250, 000, NW C 3 = S 3 10% = $200, 000. and so we can easily get the change of NWC NW C 0 = $30, 000, NW C 1 = $170, 000, NW C 2 = $50, 000, NW C 3 = $250, 000. Notice that we use same year rule when calculating NWC and NW C. However, in textbook [1], they use the next year rule in order to keep consistent with CCA tax shields calculation. Besides, we know CapEx 0 = $800, 000 and CapEx 3 = $125, 000. Then we can calculate the free cash flows excluding CCA tax shields as: CF 0 = CapEx 0 NW C 0 = 800, , 000 = $830, 000, CF 1 = (S 1 E 1 ) (1 τ c ) NW C 1 = 350, % 170, 000 = $40, 000, CF 2 = (S 2 E 2 ) (1 τ c ) NW C 2 = 450, % 50, 000 = $222, 000, CF 3 = (S 3 E 3 ) (1 τ c ) CapEx 3 NW C 3 = 350, % + 125, , 000 = $585, 000.

32 28 5. Capital Budgeting Furthermore, using (5.9), we can get the present value of CCA tax shields: 800, % 40% P V = , % 40% 1 = $200, Therefore, the NPV of this project is: NP V = 3 t=0 CF t + P V = $28, (1 + r) t Since the NPV is positive, so Company A should take the project and purchase the machine. t=0 t=1 t=2 t=3 S t $2, 000, 000 $2, 500, 000 $2, 000, 000 E t $1, 650, 000 $2, 050, 000 $1, 650, 000 (S t E t ) (1 τ c ) $210, 000 $270, 000 $210, 000 NW C t $30, 000 $170, 000 $50, 000 $250, 000 CapEx t $800, 000 $125, 000 CF t (ex. CCA) $830, 000 $40, 000 $220, 000 $585, 000 CCA tax shields $200, NPV $28, Case 2 Olympia, Inc. is considering the replacement of an existing machine with a new, higher capacity model The existing machine was purchased 3 years ago, its current book value is $83, 000, and its expected remaining useful life is 4 years. Olympia could sell the old machine for $100, 000 today, but if it is used for another four years the salvage value will be $25, 000. The new machine costs $440, 000, has an estimated useful life of 4 years, and an estimated salvage value of $50, 000. Both the old and the new machine belong to asset class 10 with a CCA rate of 30%, and manager believes that Olympia will have other class 10 assets in four years when the new machine would be sold. The new machine is expected to reduce annual expenses by $150, 000. The new machine will require an immediate (i.e. at t = 0) $10, 000 increase in our investment in working capital Olympia uses a 11% discount rate to

33 5.2. Case Study 29 evaluate similar projects and its marginal tax rate is 40%. Should Olympia replace the existing machine? Applying the same method used in Case 1, we can get the capital budget of the replacement project as the following table: Year (S t E t ) 0 $150, 000 $150, 000 $150, 000 $150, 000 (S t E t ) (1 τ c ) 0 $90, 000 $90, 000 $90, 000 $90, 000 NW C t $10, $10, 000 CapEx $340, $25, 000 CF t (ex. CCA) $350, 000 $90, 000 $90, 000 $90, 000 $125, 000 PV (ex. CCA) $350, 000 $81, 081 $73, 046 $65, 807 $82, 341 PV CCA $89, 761 NPV $42, 037 To compute the CapEx, we notice that if we purchase the new machine to replace the old machine, then we pay $440, 000 for the new machine but get $100, 000 from the sale of the old machine, so CapEx 0 = $440, 000 $100, 000 = $340, 000. For the same reason, CapEx 4 = $50, 000 $25, 000 = $25, 000. The present value of CCA tax shields is given by P V CCA = 340, % 40% , % 40% = $89,

34 Chapter 6 Bond and Stock Valuations A Bond is a financial debt security, which stipulates the issuer of the bond (debtor) is obligated to pay the owners of the bond (creditor) certain interest (coupon) and/or repay the principal at maturity date. Generally, instruments with maturities less than one year are called Money Market Instruments instead of bonds. A stock (also known as an equity or a share) is an instrument that signifies an ownership position (called equity) in a corporation, and represents a claim on its proportional share in the corporation s assets and profits. Bonds and stocks are both securities, but a major difference is that (capital) stockholders have an equity stake in the company (i.e., they are owners), whereas bondholders have a creditor stake in the company (i.e., they are lenders). Another difference is that bonds usually have a defined term, or maturity, whereas stocks may be outstanding indefinitely. An exception is a consol bond, which is a perpetuity (i.e., bond with no maturity). [11] 6.1 Valuing Bonds The principal of a bond (face value) is the notational amount we use to calculate the interest payments. For instance, if a bond contract has a coupon rate r c (annual rate), face value C and the number of payments n per year, then its each coupon payment (CPN) is given by: CP N = r c C. (6.1) n The simplest bond is zero-coupon bond. From its name, we know that 30

35 6.1. Valuing Bonds 31 such bond won t pay interest but only repay the principal at maturity time. Examples are treasury bills. Since there is no coupon and money has time value, zero-coupon bond will always be sold at a price less than its face value, thus we can call it pure discount bond. A very important function of zerocoupon bond is to provide a good estimator of spot interest rate, but here it has a special name, yield to maturity or simply yield. For example, a newly issued one-year government zero-coupon bond is sold at P 0 = $ (face value is C = $1000.), then we can calculate the implied interest rate (investors expected interest rate): r = C P 0 1 = 5%. According to such argument, if we have infinitely many zero-coupon bonds with nearly continuous maturities (we can find zero-coupon bond with any maturity), then we can obtain the approximately continuous interest rate, which is very useful in asset pricing. But in real life, we don t have enough zero-coupon bonds to plot a continuous line of interest rate versus time, so we use numerical methods and finite (time, yield) data to construct yield curve. Probably, the most effective method is B splint interpolation while a commonly used one is simple polynomial interpolation. Besides, stochastic model (called term structure model) can also be used to capture the dynamics of yield. In contrary, the pricing of zero-coupon bond is simple, as long as we know the discount rate with the same maturity of bond, then we can simply discount back to get the price. Let C be the face value, remaining lifetime of bond is T (years) and interest rate of T years is r T or annual interest r, then the price is: P = C 1 + r T or P = C (1 + r) T. (6.2) Basically, all the other types of bond can be called coupon bond. Here, for simplicity, we only discuss fixed rate bonds. But in financial markets, there are indeed floating rate bonds (linked to the interest rate of some other assets, like LIBOR, even inflation rate). Consider a coupon bond with face value C, coupon rate r c, number of payments n per period and maturity N periods. Assume yield rate per period is r, the cash flows of this bond can be decomposed into an annuity plus a fixed inflow at maturity time. So the

36 32 6. Bond and Stock Valuations present value (issue price) is given by: ( 1 P 0 = CP N r 1 (1 + r) N ) + where CP N can be calculated through (6.1). Example. We have the following there bonds: Bond C r c T n r Bond 1 $1, % 4 2 4% Bond 2 $1, % 4 2 5% Bond 3 $1, % 4 2 6% C (1 + r) N, (6.3) Solution. Using (6.1) to calculate CPN, for bond 1, we have CP N 1 = % = $50. Notice r is the yield per period, (half year in this example), 2 so we can directly use (6.3) to obtain the price of bond 1 as: (N = T n = 8) P 1 0 = ( ) = $ > C By the same method, we can also calculate the prices of bond 2 and 3 by: P0 2 = 50 ( 1 1 ) = $1000 = C, P0 3 = 50 ( 1 1 ) = $ < C From the above example, we can summarize that r < r c n P 0 > C (premium) r = r c n P 0 = C (par) r > r c n P 0 < C (discount) The rigorous proof is easy. Denote rc by a, then we can rewrite (6.3) as n r ( ) 1 C P 0 = ac 1 + (1 + r) N (1 + r) ( ) N 1 = C + (a 1)C 1 (1 + r) ( N ) 1 P 0 C = (a 1)C 1 (1 + r) N Sign(P 0 C) = Sign(a 1),

37 6.1. Valuing Bonds 33 which Sign(x) is the sign of x. Since yield is not constant through the whole maturity period of a bond, bond value will change as yield changes. An obvious relation is that when yield rate goes up, bond value will decrease and vise versa. However, we still want to find a measurement for bond s sensitivity to the change of interest rate. An intuitive idea is use maturity, but the following example will show that maturity is an imperfect measure of sensitivity against interest rate. Example. Consider two bonds and list the information as follows: Bond C r c T n r 1 r 2 Bond A $1, % % 4% Bond B $1, 000 4% % 4% Case 1. r = r 1 = 10% Since r = rc A, P0 A = $1, 000. For bond B, its price is given by: P0 B = 40 ( Case 2. r = r 2 = 4% In this case, r = rc B, so P0 B by: P0 A = 100 ( ) ( ) 9 ( ) = $ = $1, 000. For bond A, we can calculate its price ) 1 + ( ) = $1, ( ) 10 Therefore, we can obtain the percent of change of prices for two bonds: % A = = 48.7% % B = = 52.8% This example shows that the bond with shorter maturity time can be more sensitive to the change of interest rate. So we have the accurate measure duration, which is defined by: D = 1 P 0 T t=1 CF t t, (6.4) (1 + r) t where CF t = CP N, 1 t T 1 and CF T = CP N + C.