Chapter A – Introduction

Introduction

Finance is about financial decisions; this book focuses on how people in corporations make financial decisions. The financial decisions, in your personal life and inside a business, are tied together in the Valuation Principle. This principle shows how to make the costs and benefits of a decision comparable so that we can weigh them properly.

The four types of firms

There are four major types of firms that financial managers run:

1. Sole proprietorships
A sole proprietorship is a business owned and run by one person. This type of firm is usually very small with few of none employees and is the most common type in the world.
Key features:

1. Easy to set up;

2. No separation between the firm and the owner (one owner who runs the business);

3. Owner has got unlimited personal liability for any of the firm’s debts;

4. Life of the firm is limited to the life of the owner.

2. Partnerships
A business owned and run by more than one owner is called a partnership.
Key features:

1. All partners are liable for the firm’s debt;

2. Death or withdrawal of any single partner means the end of the partnership;

3. If the partnership agreement provides for alternatives (e.g. buyout) partners can avoid liquidation.

When the partnership has got two kinds of owners, general partners and limited partners, the partnership is called a limited partnership. The general partners have the same rights and privileges as partners in any general partnership (personal liability). The liability of limited partners is limited to their investment, they have limited liability. Their private property is safe. The death or withdrawal of a limited partner does not mean the end of the partnership, and the interest of a limited partner is transferable. Limited partners have no management authority.

3. Limited liability companies
A limited liability company or corporation (LLC) is a limited partnership without a general partner. The owners’ liability is limited to their investment in shares. All the owners have limited liability, but they can run the business. There are two types of limited liability companies: private companies and public companies.

4. Corporations
A corporation is a legally defined, artificial being (a legal entity), separate from its owners. A corporation is solely responsible for its own obligations, this means: the owners of a corporation are not liable for any obligations the corporation enters into. The corporation is the most important type of firm.

Setting up a corporation is more costly, because a corporation must be legally formed. The number of owners is unlimited. The entire ownership stake or equity of a corporation is the stock, which is divided into shares. The equity is the collection of all outstanding shares of a corporation.

An owner of a share of stock is called a shareholder, stockholder or equity holder. Shareholders receive dividend payments: payments made at the discretion of the corporation to its equity holders. There is no limitation on who can own the stock. This is a unique feature of a corporation and this characteristic allows free trade in shares. Because corporations can sell ownership shares to anonymous outside investors, corporations can raise great amounts of capital.

An important difference among the types of corporate organizational forms is the way they are taxed. A corporation pays a tax on its profits. When the remaining profits are distributed to the shareholders, the shareholders pay their own personal tax over this income. In this way there is double taxation.

An alternative system to use is the imputation system:
S corporations: Corporations that elect subchapter S tax treatment and are allowed an exception from double taxation. Only the shareholders have to pay income taxes.
C corporations: Corporations that have no restrictions on who owns their shares or the number of shareholders; therefore, they cannot qualify for subchapter S treatment and are subject to direct taxation.

Table 1.1 (page 9) shows a brief overview of the characteristics of the different types of firms explained.

Financial Manager
The financial manager of a corporation makes the financial decisions of the business for the stockholders. He has got three tasks:

1. Making investment decisions: The financial manager must weigh the costs and benefits of all the investments/projects and decide which of them qualify as good use of the money stockholders have invested in the firm. It is the most important task of a financial manager.

2. Making finance decisions: The financial manager has to decide how to pay for an investment. Large investments may require raising additional money. Then the manager must decide whether to borrow the money or to raise money from new and existing owners by selling more shares.

3. Managing cash flow from operations activities: The financial manager must ensure that the firm has enough cash on hand to meet its obligations. This is also called managing working capital. It may seem straightforward but it can mean the difference between success and failure especially in young, growing firms.

The overall goal of the financial manager is to maximize the wealth of the owners; the shareholders (maximize the stock price). The financial manager is a caretaker of the stockholders’ money, making decisions in their interests.

The financial manager’s place in the corporation

The ownership and control of a corporation are separate. The shareholders exercise their control by electing the board of directors. The board of directors is defined as a group of people elected by shareholders who have the ultimate decision-making authority in the corporation. This board of directors makes the rules, sets the policy and monitors the performance. The board delegates tasks to its management.

The chief executive officer (CEO) is the person charged with running the corporation by instituting the rules and policies set by the board of directors.

Agency problems
Because of the separation of ownership and control in a corporation, the agency problem may occur. There is an agency problem when managers, despite being hired as the agents of shareholders, put their own self-interest ahead of the interests of those shareholders. There is a conflict of interests; the self-interest of managers against the interest of the shareholders. A given solution for this problem is providing bonuses to managers, thus the compensation is connected to the performance. But, by tying compensation too closely to performance, the shareholders might be asking managers to take on more risk.

Shareholders can also encourage managers to work in their interest, by disciplining them if they do not. The shareholders have the possibility to pressure the board to oust the CEO if they are unhappy with his performance.

However, more common is to sell your shares when you’re unhappy. If enough shareholders are dissatisfied, the price of shares will fall because this is the only way to entice investors to buy. When investors see a well-managed corporation, the stock price will rise because investors want to purchase them. Therefore, the stock price is a barometer for corporate leaders that continuously gives them feedback on the opinion of the shareholders of their performance.

In corporations in which the CEO is doing a poor job, the expectation of continued poor performance will cause the stock price to be low. This creates a profit opportunity. In a hostile takeover, an individual or organization, sometimes known as corporate rider, can purchase a large fraction of the equity and acquire enough votes to replace the board of directors and the CEO.

The stock market

Corporations can be private or public. A private corporation has got a limited numbers of owners and there isn’t an organized market for its shares. A public corporation has got many owners and its shares trade on an organized market: the stock market (or stock exchange or bourse). The stock market is defined as an organized market on which the shares of many corporations are traded. The market price of shares is determined on this market. These markets provide liquidity for a company’s shares. An investment that can easily be turned into cash because it can be sold immediately at a competitive market price is called liquid.

There is a difference between primary markets and secondary markets.

• The trade in shares starts at the primary market, when a corporation issues new shares of stock and sells them to investors.

• Secondary markets are markets where shares of a corporation are traded between investors without the involvement of the corporation (NYSE, NASDAQ).

A secondary market can be a physical market or an electronic market. The NYSE is an example of a physical market. In this kind of market, the market makers match the buyers with the sellers. They post two prices for every stock they make a market in: the bid price and the ask price. The bid price is the price at which a market maker or specialist is willing to buy a security. The ask price is the price at which a market maker is willing to sell a security.

Ask prices exceed the bid prices, this difference is called the bid-ask spread. The bid-ask spread is a transaction cost the investors have to pay in order to trade.

The NASDAQ is an example of an electronic stock market. These markets are called over-the-counter (OTC) markets. This are markets without a physical location and it is a collection of dealers connected by a network of computers and telephones. On NASDAQ, stocks can have multiple market makers. These market makers compete with each other.

Each exchange has its own listing standards; outlines of the requirements a company must meet to be traded on the exchange. These standards usually require a certain amount of shares outstanding a company must have.

Financial Institutions

Since the financial crisis of 2008 people pay more attention to financial institutions and their role in the economy. Financial institutions are entities that provide financial services, such as taking deposits, managing investments, brokering financial transactions or making loans. The role of financial institutions is to move funds from those who have extra funds (savers) to those who need funds (borrowers and firms). They also move funds through time.

The major categories of financial institutions are:

• Banks and credit unions

• Insurance companies

• Mutual funds

• Pension funds

• Hedge funds

• Venture capital funds

• Private equity funds

Chapter B – Financial Statement Analysis

Firms’ disclosure of financial information

Financial statements are accounting reports issued by a firm periodically that present past performance information and a snapshot of the firm’s financial position. Public companies must file their financial statements to a list of relevant authorities. The annual report is the yearly summary of business sent by publicly listed companies to their shareholders that accompanies or includes the financial statement. Private companies also prepare financial statements, but they don’t have to disclosure these reports to the public. The information in a financial statement is useful for investors, financial analysts, managers and other interested parties (e.g. creditors).

There are guidelines regarding the preparation of a financial statement. The general accepted accounting principles (GAAP) is a common set of rules and a standard format for public companies to use when they prepare their financial reports, e.g. US GAAP and IFRS. Because assurance about the accuracy is important for investors, a corporation has to hire a neutral third party: an auditor. An auditor checks the annual financial statements, ensures they are prepared according to GAAP and provides evidence to support the reliability of the information.

We will describe the three main types of financial statements: the balance sheet, the income statement and the statement of cash flows.

Balance sheet

The balance sheet is a list of the assets and liabilities of a firm. This list provides a snapshot of the financial position at a given point in time. A balance sheet is divided into two parts:

• (left) The assets: the cash, inventory, property, plant and equipment, and other investments a company has made. Shows how the firm uses its capital.

• (right) The liabilities: the obligations of a firm to its creditors.

The shareholders’ equity: an accounting measure of a firm’s net worth that represents the difference between the firm’s assets and its liabilities.

The right side summarizes the sources of capital.

The balance sheet identity: Assets = Liabilities + Shareholders’ equity

Assets
The assets of a firm are divided into current assets and long-term assets.
Current assets consist of cash or assets that could be converted into cash within one year:

1. Cash and other marketable securities. Marketable securities are short-term, low-risk investments that can be easily sold and converted to cash.

2. Accounts receivable: amounts owed to a firm by customers who have purchased goods or services on credit.

3. Inventories: a firm’s raw materials as well as its work-in-progress and finished goods.

4. Other current assets, e.g. prepaid expenses.

Long-term assets are assets that produce tangible benefits for more than one year. Depreciation is the yearly deduction a firm makes from the value of its fixed assets over time, according to a depreciation schedule that depends on an asset’s life span. Depreciation isn’t an actual cash expense. The book value is the valuation of an asset on the balance sheet: the acquisition cost of an asset less its accumulated depreciation.

Liabilities
Like assets, liabilities are also divided into current liabilities and long-term liabilities.
The current liabilities are the liabilities that will be satisfied within one year.

1. Accounts payable: the amounts owed to creditors for products or services purchased with credit.

2. Notes payable or short-term debt: loans that must be repaid in the next year.

3. Accrual items that are owed but have not yet been paid.

The net working capital is the difference between a firm’s current assets and current liabilities that represent the capital available in the short term to run the business. So,

Net working capital = Current assets – Current liabilities.

The long-term liabilities are defined as any loan or debt obligation with a maturity of more than a year.

Stockholders’ equity
As mentioned, the difference between the assets and the liabilities is the stockholders’ equity. The stockholders’ equity is also called the book value of equity, it represent the net worth of a firm from an accounting perspective. However, the book value of equity isn’t an accurate assessment of the actual value of the firm’s equity. The book value of equity differs from the market value of the firm’s equity, because of the way assets and liabilities are recorded for accounting purposes. The total market value of a firm’s equity equals the market price per share times the number of shares (market capitalization).

Balance sheet analysis

The book value of equity is sometimes used as an estimate of the liquidation value; the value of a firm after its assets are sold and the liabilities are paid.

For comparing the market and book values of equity, we use the market-to-book (or price-to-book) ratio: the ratio of a firm’s market capitalization to the book value of its stockholders’ equity.

Market-to-book ratio = Market value of equity / Book value of equity.

The market-to-book ratio of a successful firm typically exceeds 1.
This ratio provides feedback to its managers on the market’s assessment of their decisions. Value stocks are firms with low market-to-book ratios. Growth stocks are firms with high market-to-book ratios.

From the balance sheet, we can also get information about the leverage of a firm. Leverage is a measure of the extent to which a firm relies on debt as source of financing. The ratio of a firm’s total amount of short- and long-term debt to the value of its equity is called the debt-equity ratio.
Debt-equity ratio = total debt / total equity.

We can calculate this ratio using either book or market values for equity and debt.

The enterprise value is the total market value of a firm’s equity and debt, less the value of its cash and marketable securities. The enterprise value measures the value of the firm’s underlying business assets, unencumbered by debt and separate from any cash and marketable securities.

Enterprise value = Market value of equity + Debt – Cash.

To measure whether the firm has sufficient working capital to meet its short-term needs, creditors compare the current assets and the current liabilities.
The current ratio is the ratio of current assets to current liabilities.
Current ratio = current assets / current liabilities.
 

The quick ratio is the ratio of current assets other than inventory to current liabilities.
Quick ratio = (current assets – inventory) / current liabilities.
A low ratio means high risk.

Income statement

The income statement reports a list of the revenues and expenses of the firm over a period of time. This statement computes the firm’s bottom line of net income/earnings, this is a measure of the profitability during a period.

An income statement consists of different categories:

• Gross profit: the difference between a firm’s sales revenues and its costs.

• Operating income: the gross profit of a firm less its operating expenses. Operating expenses are expenses form the ordinary course of running the business. These expenses aren’t directly related to producing the goods or services being sold.

• Earnings before interest and taxes (EBIT): the firm’s earnings before interest and taxes are deducted.

• Pretax and Net income. Net income represents the total earnings of the firm’s equity holders. It is often reported on a per-share basis as the firm’s earnings per share. You compute the earnings per share (EPS) dividing the net income by the total number of shares outstanding.
  EPS = net income / shares outstanding.

   

There are two ways to issue more shares:

1. To give stock options to employees or executives. Share (stock) options give the holder the right to buy a certain number of shares of stock by a specific date at a specific price.

2. To issue convertible bonds. Convertible bonds are corporate bonds with a provision that gives the bondholder an option to convert each bond owned into a fixed number of shares.

In the case of share options and convertible bonds the increase in the total number of shares, because there will be more shares to divide into the same earnings, is called dilution. The diluted EPS shows the EPS the company would have if the stock options were exercised.

Income statement analysis

The income statement provides useful information about the profitability of the firm and how it relates to the value of the firm’s shares. Some profitability ratios that are often used to evaluate a firm’s performance and value:

• Gross margin, this is the ratio of gross profit to revenues (sales). It reflects the ability of the company to sell a product for more than the sum of the direct costs of making it.
  Gross margin = Gross profit / sales.

• Operating margin, this is the ratio of operating profit to revenues. This ratio indicates how much a company earns from each dollar of sales, before interest and taxes are deducted.
  Operating margin = Operating Profit / Sales

• Net profit margin, this is the ratio of net income to revenues. The net profit margin shows which part of each dollar in revenues is available to equity holders, after the firm pays its expenses plus interest and taxes.
  Net profit margin = Net income / Sales

Asset efficiency
By combining the information from the balance sheet and the income statement, you can measure how efficiently a firm is utilizing its assets.

The ratio of sales to total assets is called the asset turnover, a broad measure of efficiency.

Asset turnover = Sales / Total assets.

When the asset turnover is low, you can conclude that the firm in not generating much revenues per euro of assets.

Another efficiency ratio is the fixed asset turnover. The fixed asset turnover is the ratio of sales to fixed (non current) assets.

Fixed asset turnover = Sales / Fixed assets.

Working capital ratios
Besides the information about the assets, information about the net working capital is very important for financial managers.

Accounts receivable days: an expression of a firm’s accounts receivable in terms of the number of day’s worth of sales that the accounts receivable represents. Also called: average collection period or days sales outstanding.

Accounts receivable days = Accounts receivable / Average daily sales.

Accounts payable = accounts payable / average daily cost of goods sold
Inventory days = inventory / average daily cost of goods sold.

There is also a ratio to measure how efficiently a firm turns his inventory into sales: the inventory turnover ratio. In general, a higher level of inventory turnover is better.

Inventory turnover = Sales / inventory.

EBITDA is a computation of a firm’s earnings before interest, taxes, depreciation and amortization are deducted. The EBITDA reflects the cash a firm has earned from its operations.

Lenders often asses a firm’s leverage by computing an interest coverage ratio or times interest earned (TIE) ratio.

TIE = operating income (earnings) / interest expense

A high outcome shows that the firm is earning more than necessary to pay the interest. Financial managers watch these ratios carefully because they asses how easily the firm will be able to cover its interest payments.

Analysts and financial managers often evaluate the firm’s return on investment by comparing its income to it investment using ratios such as ROE and ROA;

Return on equity (ROE) = Net income / Book value of equity
Return on assets (ROA) = Net income / Total book value of assets.

The DuPont identity expresses return on equity as the product of profit margin, asset turnover and a measure of leverage. This identity is a way to delve deeper into the sources of return on equity. By doing this, a financial manager can gain a clear sense of the financial picture of the firm.

The final expression of the DuPont identity states that the ROE is equal to the net firm’s profit times asset turnover times the equity multiplier. The equity multiplier is a measure of leverage equal to the total assets divided by total equity.

Valuation ratios
The price-earnings ratio (P/E) is the ratio of the market value of equity to the firm’s earnings, or its price to its earnings per share. The valuation ratios give information about the market value of the firm.
P/E ratio = Market capitalization / Net income
or
P/E ratio = Share Price / Earnings per share

When the firm’s earnings per share are negative it is not meaningful to look at the P/E ratio. In this case it is common to look at the firm’s enterprise value relative to sales.

The PEG ratio is the ratio of a firm’s P/E to its expected earnings growth rate. Investors consider PEG ratios of 1 or below as indicating that the stock is fairly priced. When the PEG ratio is higher than 1, investors question whether the company is overvalued.

Statement of cash flows

The income statement provides information about profits but this statement doesn’t give any information about the amount of cash the firm has earned. The statement of cash flows does. The statement of cash flows is an accounting statement that shows how a firm had used the cash it earned during a set period. A cash flow statement is divided into three sections: operating activities, investment activities and financing activities.

• The operating activities start with net income, to this number they add back all non-cash entries related to the firm’s operating activities (this means: deducting the depreciation, deducting an increase in accounts receivable, adding an increase in accounts payable and deducting an increase in inventory).

• The investing activities list the cash used for investment. Purchases of new property, plant and equipment are capital expenditures.

• The financing activities reflect the flow of cash between investors and the firm. The retained earnings of a firm is the difference between net income and the amount a firm spends on dividends.

Retained earnings = Net income – Dividends

The payout ratio is the ratio of a firm’s dividends to its net income.

Payout ratio = dividends / net income.

Other financial statement information

We have discussed the most important elements of a firm’s financial statements (balance sheet, income statement and statement of cash flows). Now we will discuss several other pieces of information contained in the financial statements: the management discussion and analysis, the statement of changes in shareholders’ equity and notes to the financial statement.

1. Management discussion and analysis
The management discussion and analysis (MD&A) is a preface to the financial statements in which a company’s management discusses the recent year (or quarter), providing a background on the company and any significant events that may have occurred. Management may also discuss the coming year and outline goals and new projects.

Management is also required to disclose any off-balance sheet transactions: transactions or arrangements that can have a material impact on a firm’s future performance yet do not appear on the balance sheet.

2. Statement of changes in shareholders’ equity
The statement of stockholders’ equity is an accounting statement that provides details of changes in share capital and reserves. It breaks down the stockholders’ equity computed on the balance sheet into the amount that came from issuing new shares versus retained earnings. It is not always useful to financial managers because of the use of book values rather than market value.

3. Notes to the financial statement
This notes generally contain important details regarding the numbers used in the main statements. They also often provide information specific to a firm’s subsidiaries or its separate product lines.

Financial reporting in practice

Financial statements have proved to be important, because of recent accounting scandals. The penalties for fraud have increased and new legislation had tightened the procedures firms must use to assure that statements are accurate.

In 2002 the U.S. congress passed the Sarbanes-Oxley act (SOX), which was intended to improve the accuracy of financial information given to both boards and shareholders.
 

Chapter C – Time value of money: An introduction

Cost-benefit analysis

When evaluating a decision, the valuation of the incremental costs and benefits, associated with that decision, is important. Identifying the costs and benefits of a decision is the first step in decision making. If the value of the benefits exceeds the value of the costs, the value of the firm will increase and a decision can be regarded as good. If you would like to compare costs and benefits that occur at different points in time, you have to put all costs and benefits in common terms. For this reason, we convert costs and benefits into cash today. If it’s possible to express costs and benefits in terms of “cash today”, it’s a straightforward process to compare them and to determine whether the financial decision will increase the value of the firm.

A competitive market is a market in which the good can be bought and sold at the same price. The price is given and determines the cash value of the good. Because the good can be bought and sold at the same price, the personal preference and the opinion of the buyer/seller do not matter. It’s easy to determine the best decision for the firm, if we use market prices to evaluate the costs and benefits of a decision in terms of cash today.

Valuation principle

The valuation principle provides the basis for decision making. The valuation principle is: the competitive market price determines the value of a commodity or an asset to the firm or its investors. The benefits and costs of a decision should be evaluated using those market prices. A decision will increase the market value of the firm, when the value of benefits exceeds the value of costs.

The valuation principle relies on using a competitive market price to value a cost or benefit. We cannot have two different competitive market prices for the same good. This established the Law of one price; in competitive markets, securities with the same cash flows must have the same price. According to this law, arbitrage opportunities cannot exist. The law of one price implies the price of a security should equal the present value of the future cash flows obtained from owning that security.

Arbitrage is the practice of buying and selling equivalent goods to take advantage of a price difference. We call the situation in which it is possible to make a profit without taking any risk or making any investment an arbitrage opportunity.

Time value of money and interest rates

There is a difference in value between money today and money in the future. Today a euro is worth more than a euro in one year, in general. This difference is called the time value of money; this is the observation that two cash flows at two different points in time have different values.

It’s possible to convert money today into money in the future by depositing money into a savings account. It’s also possible to convert money in the future for money today: borrowing money from the bank. The current interest rate determines the rate at which we can exchange money today for money in the future (like an exchange rate across time).

The interest rate is defined as the rate at which money can be borrowed or lent over a given period. The interest rate factor is the rate of exchange between dollars today and dollars in the future: ( 1 + r ). This factor has units of “€ in one year/€ today”. For example, when the interest rate is 4%, the interest rate factor is 1.04.

{Example}
Suppose a firm has got an investment opportunity. The cost will be €100.000, today. The benefit will be €102.000, in one year. Because of the time value of money, the cost and benefits aren’t directly comparable (the net value isn’t €2.000). To calculate the net value, we need to know the cost of the investment in one year. We can calculate the cost in one year by multiplying the cost today by the interest rate factor. The cost in one year will be €104.000 (if the interest rate is 4%). This means: the firm gives up the €104.000 it would have had in one year if it had left the money in the bank. Or, if the firm borrows the €100.000 from the same bank, the firm has got a debt of €104.000 in one year. When we have converted the costs and benefits in “dollars in one year”, we can use the Valuation Principle to compare them and the compute the net value. The cost was €104.000 in one year, the benefit was €102.000, so the net value of this investment will be -€2.000 in one year. Because the net value is negative, the firm would reject the investment.

In the example we calculated the net value by computing the “cost in one year”. In the same example, it’s also possible to calculate the net value by computing the “benefit today”. You can compute the “benefit today” by dividing the “benefit in one year” by the interest rate factor.

The present value (PV) is the value of a cost or benefit computed in terms of cash today. The future value is the value of a cash flow that is moved forward in time.

The discount factor is the value today of a dollar received in the future: .
This factor provides the discount at which we can purchase money in the future.
The appropriate rate to discount a stream of cash flows to determine their value at an earlier time is called the discount rate = r.

To visual the benefits and costs in time we often use a time line; a linear representation of the timing of (potential) cash flows. To differentiate between inflows and outflows on a timeline, inflows are positive cash flows (cash flows received) and outflows are negative cash flows (cash flows paid out).

Valuing cash flows at different points in time

There are three important rules central to financial decision making that allows us to compare or combine values across time.

1. Rule 1: Comparing and combining values

The first rule is that it is only possible to compare or combine values at the same point in time. Rule 2 and 3 show how to move cash flows on the timeline.

2. Rule 2: Compounding

The second rule states that to calculate a cash flow’s value, you must compound it. Compounding is computing the return on an investment over a long horizon by multiplying the return factors associated with each intervening period. The compounding interest is the effect of earning ‘interest on interest’.

Future value of a cash flow: FV_n= C x ( 1+r )ⁿ

3. Rule 3: Discounting

The third rule states that to calculate the value of a future cash flow at an earlier point in time, we must discount it. Discounting is finding the equivalent value today of a future cash flow by multiplying by a discount factor, or equivalently, dividing by 1 plus the discount rate. Present value of a cash flow: PV = C / ( 1+r )ⁿ

Chapter D – Valuing cash flow streams

Valuing a stream of cash flows

Almost all investment opportunities have multiple cash flows that occur at different points in time. Using the rules of cash flow valuation, we compute the present value of a stream of cash flows in two steps.

1. Compute the present value of each individual cash flow.

2. When the cash flows are in common units of dollars today, we can combine them.

This leads to the following formula for the present value of a stream of cash flows:

We will consider two types of cash flow streams; perpetuities and annuities.

Perpetuities

A perpetuity is a stream of equal cash flows that occurs at regular intervals and lasts forever. One example is the consol, a British government bond. The consol promises the owner a fixed cash flow every year, forever.
The first cash flows of a perpetuity arrives at the end of the first period:
 

The present value of a perpetuity:

Annuities

An annuity is a stream of equal cash flows arriving at a regular interval and ending after a specified time period. A perpetuity is infinite, while an annuity ends after some fixed number of payments.

The present value of an annuity:

The future value of an annuity:

Growing cash flows

A growing perpetuity is a stream of cash flows that occurs at regular intervals and grows at a constant rate forever.

We follow the same logic to calculate the present value of a growing perpetuity as for a regular perpetuity. But now we also take the growth rate, g, in consideration.

The present value of a constant growing perpetuity:

A growing annuity is an annuity with the cash flow growing at a fixed growth factor.

The present value of a constant growing annuity:

Other variables

Sometimes we know the PV or FV, but we don’t know one of the other variables (e.g. C, n or r). In such situations, we can use the PV or FV as inputs to solve for the variable we are interested in. Some examples:

1. We don’t know the cash flow, but we know the PV.
   To solve this problem, you enter all the known variables (also the PV) into the formula and you solve the equation for C.

In case of a loan payment (annuity) : The periodic payment on an N-period loan with principal P and interest rate r is:

2. We don’t know the interest rate, but we know the PV and the cash flows.
   In a situation with an unknown interest rate, the interest rate is called the internal rate of return (IRR). The IRR is the interest rate at which the PV of the benefits exactly offsets the PV of the costs. Solving this problem is the same as the previous one: you enter all the known variables into the same formula as with the annuity formulas and solve the equation for r.

3. We don’t know the number of periods “n”, but we know the interest rate, present value and future value. It is most easy to calculate the years, n, with a formula in excel.

Chapter E – Interest Rates

Interest rate quotes

Interest rates are set by market forces, especially the supply and demand of funds. When the savings are high (high supply) and the borrowing is low (low demand), the interest rates will be low. Besides market forces, interest rates are also influenced by expected inflation and risk. Interest rates are often quoted for different time intervals, such as monthly or annual, and therefore it is necessary to adjust the interest rate to a time period that matches the cash flows.

The effective annual rate (EAR) or the annual percentage yield (APY) indicates the total amount of interest that will be earned at the end of one year. The EAR can be used as a discount rate for annual cash flows.

We can convert a discount rate for one period to an equivalent discount rate for n periods. The discount rate for an n-year time interval, given an EAR r, can be calculated with this formula:

Equivalent n-period discount rate = (1 + r)ⁿ - 1

When you compute a rate over more than one period, n is larger than 1. n is smaller than 1 if you compute a rate over a fraction of a period.

The annual percentage rate (APR) indicates the amount of simple interest; which is the interest earned in one year without the effect of compounding. APR this is the most common way to note interest rates. The APR quote is less than the actual amount of interest you’ll earn, because APR doesn’t include the effect of compounding. For this reason, the APR itself cannot be used as a discount rate.

Interest rate per compounding period = APR / number of compounding periods per year (m).

We can convert an APR to an EAR with this formula:

This formula gives the effective annual rate corresponding to an APR. The EAR increases with the compounding frequency, for a given APR. For example, given an 6% APR and a annual compounding interval, the effective annual rate is 6% . Given the same APR and a monthly compounding interval, the effective annual rate is 6.1678% .

Before you’re able to evaluate the PV or FV of set of cash flows, you have to convert the APR to a discount rate per compounding interval (use: (1 + r)ⁿ - 1) or convert the APR to an EAR (use: )

Loans

Many loans, such as mortgages and car loans, are amortizing loans. An amortizing loan is a loan on which the borrower makes monthly payments that include interest on the loan plus some part of the loan balance. Amortizing loans are quoted in terms of an APR with monthly compounding.

The amount you pay for your loan every month includes interest and repayment of part of the principal, reducing the amount you still owe (the loan balance).

The loan balance decreases each month. For that reason, the accruing interest on that loan balance will decrease too. Because the loan payment is equal every month, the repayment component will increase. So, the outstanding balance on an amortizing loan differs every month. You can calculate the outstanding loan balance by determining the PV of the remaining loan payments, using the loan rate as the discount rate.

Determinants of interest rates

We will discuss some determinants of interest rates, such as inflation, current economic activity, and expectation of future growth.

Inflation affects the evaluating of interest rates being quoted by banks and other financial institutions. Those quoted interest rates are nominal interest rates: interest rates that indicate the rate at which money will grow if invested in a certain period of time. The real interest rate indicated the rate of growth of purchasing power after adjusting for inflation. You can compute the rate of growth of purchasing power with this formula:
Growth in purchasing power = 1 + real rate
= (1 + nominal rate ) / (1+inflation rate )
= growth of money / growth of prices

The real interest rate:
Real rate = (nominal rate – inflation rate) / (1+ inflation rate)
≈ nominal rate – inflation rate

Nominal interest rate tends to move with inflation. The nominal interest rate is high when inflation is high and vice versa. Savings depends on the growth in purchasing power individuals can expect, this growth in purchasing power is given by the real interest rate. So, when the inflation rate is high, the country needs a higher nominal interest rate to induce individuals to save.

Interest rates also affect the incentive of firm’s to raise capital and invest. When the interest rate rises, the PV of the benefits will fall and companies will not invest because the investment isn’t profitable. When the interest rates are high, you’re discounting the positive cash flows at a higher rate, which reduces the PV.

The Federal Reserve in the USA and central banks in other countries try to guide the economy by influencing interest rates. If the economy is slowing, they will lower the interest rates in attempt to stimulate investment in the economy. They attempt to reduce investment in the economy by raising the interest rates.

Interest rates depend on the horizon of the investment or the loan according to the term structure of interest rates. The term structure of interest is the relationship between the investment term and the interest rate. The yield curve is a plot of bond yields as a function of the bonds’ maturity date, with on the x-axis the term and on the y-axis the interest rate. A risk-free interest rate is the interest rate at which money can be borrowed or lent without risk over a given period.

To compute the PV and the FV of a risk-free cash flow over different investment horizons, we can use the term structure. You should discount cash flows using the discount rate that is appropriate for their horizon. So, you should discount a cash flow received in two years at the two-year interest rate and a cash flow received in eight years at the eight-year interest rate. You need to match the term of the cash flow with the term of the interest rate.

The present value of a cash flow stream using a term structure of discount rates:

Note that you use a different discount rate for each cash flow. This discount rate is based on the rate from the yield curve with the same term. So, you cannot use the formulas discussed before (e.g. annuity and perpetuity) when discount rates vary with the horizon.

The federal funds rate is the overnight loan rate charged by banks with excess reserves at a Federal Reserve bank (called federal funds) to banks that need additional funds to meet reserve requirements.

A yield curve changes over time. The shape of the yield curve depends to a larger extent on the expectations of investors of future economic growth and interest rates, because these expectations have a major effect on the willingness to lend or borrow for longer terms. To attract investors, long-term interest will tend to be higher than short-term interest rates, if interest rates are expected to rise. Then the yield curve will be steep. Otherwise, if interest rates are expected to fall, short-term interest rates will tend to be higher than long-term interest rates to attract borrowers. In this case, the yield curve will be descending, this is a negative forecast for economic growth.

Opportunity cost of capital

To evaluate cash flows we will base the discount rate that we use on the opportunity cost of capital of the investor. The opportunity cost of capital is the best available expected return offered in the market on an investment of comparable risk and term to the cash flow being discounted. It is the return the investor forgoes on an alternative investment of equivalent risk and term when the investor takes on a new investment.

Chapter F – Bonds

Bond terminology

A bond is defined as a security sold by governments and corporations to raise money from investors today in exchange for a promised future payment. The bond certificate states the terms of a bond as well as the amounts and dates of all payments to be made. The maturity date of a bond is the final repayment date, until this date payments will continue. The term of a bond is the time remaining until the final repayment date of a bond is known. The notional amount of bond used to compute its interest payment is called the face value, or principal value, or par value. This value is usually repaid on the maturity date. Coupons are promised interest payments, in addition to the face value. Coupons are paid periodically until the maturity date. The coupon rate determines the amount of each coupon payment of a bond.

Coupon payment:

Zero-coupon bonds

Bonds without coupon payments are known as zero-coupon bonds, the investor only receives the face value at the maturity date. Treasury bills are an example of zero-coupon bonds. These bonds are issued by the U.S. government with a maturity of up to one year.

A zero-coupon bond has got two cash flows: the payment of the current market price when purchasing and the receiving of the face value at the maturity date. The market price of a zero-coupon bond is always lower than the face value; these bonds always trade at a discount. Zero-coupon bonds are also known as pure discount bonds.

The IIR of an investment in a bond is the yield to maturity (YTM). The YTM, or yield, is the discount rate that sets the present value of the promised bond payment equal to the current market prices of the bond. In other words, it’s the return you will earn if you buy the bond at its current market price, hold the bond to maturity and receive the promised face value payment.

Yield to maturity of an n-year zero-coupon bond:

Often people will refer to the yield to maturity of the zero-coupon risk-free bond as the risk-free interest rate. Some financial professionals also use the term spot interest rate.

The risk-free interest rates for investments until date n correspond to the yields of risk-free zero-coupon bonds that mature on date n. The zero-coupon yield curve is the plot of the yield of risk-free zero-coupon bonds as a function of the bond’s maturity date.

Coupon bonds

Bonds that pay regular coupon interest payments up to maturity, when the face value is also paid are called coupon bonds. There’re two types of U.S. Treasury coupon securities:

• Treasury notes are currently traded in financial markets, with original maturities from one to ten years.

• Treasury bonds are currently traded in financial markets, with original maturities of more than ten years.

The zero-coupon bonds have only two cash flows, but coupon bonds have got many cash flows. This makes it more complicated to calculate the yield to maturity. The return on a coupon bond comes from two sources:

1. Differences between the purchase price and the face value

2. The periodic coupon payments.

The following timeline represents the cash flows of a coupon bond:

Yield to maturity of a coupon bond:

The yield to maturity of the bond is the single discount rate, y, that equates the PV of the bond’s remaining cash flows to its current price.

The first part [] is the present value of all the periodic coupon payments.

Where is the annuity factor using the YTM (y).

The last part [] is the present value of the face value repayment using the YTM (y).

When we calculate the yield to maturity with this formula, the yield we compute is a rate per coupon interval. Yields are typically quoted as APRs so we multiply by the number of coupons per year, thereby converting the answer into an APR quote with the same compounding interval as the coupon rate.

The change of bond prices

Zero-coupon bonds always trade for a discount, but coupon bonds may trade at a discount or at a premium. A premium is a price at which coupon bonds trade that is greater than their face value.

When the bond price is equal to the face value the bond trades at par. This occurs when the coupon rate is equal to the yield to maturity. When the coupon rate is higher than the yield to maturity the bond trades above par, or at a premium. In this case, the bond price is greater than the face value of the bond. When the bond price is less than the face value the bond trades below par, or at a discount. When a bond trades at a discount, the coupon rate is lower than the yield to maturity.

As interest rates and bond yields rise, bond prices will fall and vice versa, so that interest rates and bond prices always move in opposite direction.

The market price of a bond can change over time for two reasons.

1. The bonds get closer to the maturity date, as time passes.

2. The YTM and the price will are affected by changes in market interest rates.

As a coupon payment nears, the price will slowly rise. After the payment is made, the price will drop immediately.

The effect of time on bond prices is predictable. On the other hand, the changes in interest rates are unpredictable. Because bonds have got different characteristics, bonds will respond differently to changes in interest rates. Long-term bonds are more sensitive to changes than short-term bonds. Long-term bonds are evaluated more riskier. The sensitivity also depends on the coupon rate of a bond. Bonds with low coupon rates are more sensitive to changes in the interest rates.

Bond traders make a difference between the dirty price and the clean price of a bond. The dirty price or invoice price is the actual cash price of a bond. Bond traders quote bonds in terms of a clean price, this is the bond’s cash price less an adjustment for accrued interest, the amount of the next coupon payment that has already accrued.

Clean price = dirty price – accrued interest.

Accrued interest = coupon amount x (days since last coupon payment / days in current coupon period).

Corporate bonds

Bonds issued by corporations are called corporate bonds. The bonds we discussed before, U.S. Treasury bonds, have no risk of default. Treasure bonds are risk-free because there is practically no change the government will fail to pay the interest and default. For that reason, the interest rates will be low. However, corporate bonds have risks or defaults. There is a credit risk: the risk of default by the issue of any bond that is not default free; it is an indication that the bond’s cash flows are not known with certainty. As compensation, investors in corporate bonds demand a higher interest rate.

General truths:

1. An investor pays more for a default-free bond than for a bond with credit risk.

2. The yield of bonds with credit risk will be higher than the yield of default-free bonds, because the yield is calculated using the promised cash flows instead of the excepted cash flows.

Now we can conclude that the yield to maturity of a bond with credit risk will always be higher than the expected return of investing in the bond, because the promised cash flows will always be higher than the expected cash flows.

The creditworthiness of bonds for investors is summarized in bond ratings. Investment-grade bonds are bonds in the top four categories of creditworthiness with a low risk of default. The bonds in one of the bottom five categories of creditworthiness are speculative bonds, bonds with a high risk of default.

The default spread or credit spread is the difference between the risk-free interest rate on U.S. Treasury notes and the interest rates on all other loans. The magnitude of the credit spread will depend on investors’ assessment of the likelihood that a particular firm will default.

Chapter G – Investment decision Rules

The NPV decision rule

Most companies prefer to measure values in terms of cash today: the present value. The net present value (NPV) of a project or investment is the difference between the present value of the benefits and the present value of the costs.

Net present value (NPV) = PV (benefits) – PV (costs).

The present value is the amount you need to put in the bank today (at the current interest rate) to recreate the cash flow. If the NPV is positive, the decision is a good decision regardless of your current cash needs or preferences regarding when to spend the money. This decision will increase the value of the firm.

The NPV decision rule states: when choosing among investment alternatives, take the alternative with the highest NPV. Choosing this alternative is equivalent to receiving its NPV in cash today.

The NPV decision rule can help by accepting or rejecting a project. According to the NPV decision rule we should accept positive-NPV projects and reject negative-NPV projects. Furthermore, you can use the NPV decision rule to choose among alternatives. You will generally choose the alternative with the highest NPV.

Maximizing NPV should always be the first priority, regardless of the preferences for cash today versus cash in the future. When you maximize the NPV, you can borrow or lend to shift cash flows through time. By doing this, you can find the most preferred pattern of cash flows.

You can compute the future value of a stream of cash flows directly (compute the bank balance each year) or you can compute the present value and move this value to the future.

NPV of a stream

To evaluate an investment decision, the central goal is calculating the NPV of future cash flows. Earlier we defined the NPV as: NPV = PV (benefits) – PV (costs). The benefits are the cash inflows (positive cash flows) and the costs are the cash outflows (negative cash flows).
NPV = PV (benefits) – PV (costs) = PV (benefits – costs).
The NPV of an investment opportunity, is the present value of the stream of cash flows of the opportunity.

Using the NPV rule

The NPV of the project depends on its appropriate cost of capital. When there is some uncertainty regarding the project’s cost of capital it may be helpful to compute the NPV profile, a graph of a project’s NPV over a range of discount rates. You can also determine the IIR of an investment by constructing the NPV profile. The appropriate cost of capital is very important in the NPV decision. The difference between the cost of capital and the IRR of an investment makes clear the amount of estimation error in the cost of capital estimate that can exist without altering the investment decision.

Alternative decision rules

The NPV is the most accurate and reliable investment decision rule but in practice also other rules are applied. Alternative decision rules may conflict or agree with the NPV decision. When the rules conflict, always base the decision on the NPV. We will focus on two alternative decision rules for single, stand-alone projects within the firm; the payback rule and the IRR rule.

1. The payback investment rule says that only projects that pay back their initial investment within the payback period are undertaken. This is the simplest investment rule. To apply the rule the following steps are undertaken:

• Calculate the payback period: the amount of time until the cash flows from a project offset the initial investment. The time it takes to pay back the initial investment.

• If this period is less than a prespecified length of time, accept the project.

• If this period is greater than the prespecified length of time, reject the project.

The payback investment rule doesn’t care about the time value of money, it ignores the cash-flows after the payback period and the rule has no decision criterion which is grounded in economics. Some economics addressed the first failing of the rule by computing the payback period using discounted cash flows. It is called the discounted payback rule; only accept projects where the sum of the discounted cash flows within the payback period is greater than or equal to the initial investment.

2. The internal rate of return (IRR) investment rule is a decision rule that accepts any investment opportunity where the IRR exceeds the opportunity cost of capital and otherwise rejects the opportunity. The IRR shows the sensitivity of the investment decision to uncertainty in the estimation of the cost of capital.

This rule will give the correct answer in many but not in all situations. The IRR is hard to compute. Multiple IRRs can occur, this lead to ambiguity. The IRR cannot be used to choose among projects. If project has got future liabilities, the IRR can be incorrect.

There is a ‘solution’ for the great disadvantage of multiple IRRs: the modified internal rate of return (MIRR): the discount rate sets the NPV of modified cash flows of a project equal to zero. Cash flows are modified so there is only one negative cash flow (and one sign-change) to ensure that only one IRR exists.

Two other approaches to solve the multiple IRR problem are:

1. Discount all negative cash flows to time 0 and leave the positive cash flows alone.

2. Leave the initial cash flow alone and compound all the remaining cash flows to the final period of the project. In this approach, you are implicitly reinvesting all the cash flows from the project at your compound rate until the project is complete.

Choosing between projects

Managers often have to choose between different investment options. They are facing mutually exclusive projects: projects that compete with one another; by accepting one, you exclude the others. The manager wants to choose only the best one by ranking the projects. In this case, only computing the NPVs isn’t enough. Besides, only computing the IRR can also lead to mistakes: problems will arise when the projects have got differences in scale and differences in the timing of cash flows. In this situation you should always rely on the NPV.
 

Projects with different lives

When you have to choose among projects with different lives, you need a standard basis of comparison. One method to evaluate project with different lives is the equivalent annual annuity: the level annual cash flows that has the same present value as the cash flow of a project.

• Step one: compute an annuity with an equivalent PV to the NPV of each project ().

• Step two: compare the projects on their cost or value created per year.

When using the equivalent annual annuity you have to think about the: required life and the replacement costs over time.

Choosing among projects when resources are limited

Before, we compared projects with identical resource needs. Now, we will discuss projects with different resource needs (e.g. budgets or max. hour labor). We usually assume that you will be able to finance all positive NPV projects that you have. In reality, managers deal with the constraint of a budget that restricts the amount of capital they may invest in a given period.

In this situation, managers often use the profitability index to help identify the optimal combination of projects to undertake. The profitability index measures the NPV per unit of resource consumed.
Managers choose the set of projects with the highest profitability indices that can still be undertaken given the limited resource. A critic is that the index breaks down when there’s more than one resource constraint.
 

Chapter H – Capital Budgeting

The capital budgeting process

When analyzing various investment opportunities, the first step you have to make is compiling a list of potential projects. A capital budget lists all of the projects that a company plans to undertake during the next period. The process of analyzing investment opportunities and deciding which ones to accept is called capital budgeting. The goal of capital budgeting is to determine the effect of the decision to accept or reject a project on the cash flows of the firm, and to estimate the consequences of the decision for the value by evaluating the NPV.

The process of capital budgeting begins with forecasting, which is frequently challenging. By forecasting, you can rely on different experts or you can look at past projects of the firm or projects of other firms in the same industry. Another important part of forecasting is determining the incremental earnings: the amount by which a firm’s earnings are expected to change as a result of an investment decision. Note that earnings are not actual cash flows.

Forecasting incremental earnings

Projects require some form of upfront investment (e.g. launch an ad campaign, design a prototype, marketing survey). You count these costs as operating costs in the year they are incurred. The investment in plant, property and equipment are capital costs. These capital costs are cash expenses, but you spread these costs over several years (depreciation). The straight-line depreciation is a method of depreciation in which an asset’s costs is divided equally over its life. Note that a depreciation expense does not correspond to an actual cash outflow.

The goal of capital budgeting is to evaluate how the project will change the cash flows of the firm. For that reason, we’re only looking at the incremental revenues and costs. This means that we only account for the additional sales and costs, generated by the project.

Incremental earnings before interest (EBIT) =
incremental revenue – incremental costs – depreciation.

We also have to account for corporate taxes. The marginal corporate tax rate is the tax rate a firm will pay on an incremental dollar of pre-tax income.

Incremental income tax expense = EBIT x firm’s marginal corporate tax rate

Putting all the pieces together we can forecast the incremental earnings:

Incremental earnings
= (incremental revenues – incremental cost – depreciation) x (1 – tax rate)
= EBIT x ( 1 – tax rate)

The table of calculating incremental earnings is called a pro forma statement: a statement that is not based on actual data but rather depicts a firm’s financials under a given set of hypothetical assumptions.

Interest expenses are not relevant in capital budgeting, because we evaluate the project as if the firm will not use any debt to finance the project.

Any incremental interest expenses will be related to the firm’s decision regarding how to finance the project, which is a separate decision. We can calculate the unlevered net income: the net income that does not include interest expenses associated with debt.

Incremental free cash flow

Once we have determined the earnings, we have not determined the actual cash flow. If we want to evaluate a capital budgeting decision, we must determine the consequences of the decision for the available cash. The incremental free cash flow of a project is the incremental effect of a project on a firm’s available cash.

There are important differences between earnings and actual cash flows. We have to adjust for these differences if we want to determine the free cash flow of a project from its incremental earnings.

Depreciation: isn’t a cash flow, we have to add it back to the incremental earnings. But there is a cash flow associated with depreciation, because depreciation affects taxes.

Calculation of incremental free cash flow:

• Incremental gross profit € … … (positive cash flow)

• Depreciation - € … …

• EBIT = € … …

• Tax - € … … (negative cash flow)

• Incremental earnings = € … …

• Add back depreciation + € … …

• Incremental free cash flow = € … … (positive cash flow)

Depreciation is not the only difference between incremental earnings and free cash flow: if there are changes in net working capital the earnings and cash will also differ.

Net working capital = current assets – current liabilities
= cash + inventory + receivables – payables

Trade credit is the difference between receivables and payables that is the net amount of a firm’s capital consumed as a result of those credit transactions; the credit that a firm extends to its customers.

The change in net working capital (NWC) in year t = NWC_t – NWC_t-1

General rule: Increases in net working capital are deducted. Decreases in net working capital are added. Actual capital expenditures (CapEx) are deducted.

The capital budgeting process usually starts by forecasting earnings. We can calculate a project’s free cash flow directly by using the following formula:

Free cash flow = (revenues – costs – depreciation) x (1 – tax rate) + depreciation – CapEx – change in NWC.

= (revenues – costs) x (1 – tax rate) – CapEx – change in NWC + tax rate x depreciation

The last part ‘Tax rate x depreciation’ is the depreciation tax shield: the tax savings that result from the ability to deduct depreciation. Depreciation has got a positive impact on the free cash flow.

The goal of forecasting the incremental free cash flow was to calculate the NPV of the project. We can calculate the present value of each free cash flow in the future as follows:

The t-year discount factor is:

So this results in the formula

Other effects

There are more differences between the cash flows of a firm with the project and cash flows of a firm without the project:

1. Firms often use a resource that the company already owns for a project. Because the firm does not need to pay cash to acquire this resource for a new project, it is tempting to assume that the resource is available for free. The Opportunity costs: the value a resource could have provided in its best alternative use. When you use the resource by another project, the value will be lost.

2. Project externalities: are the indirect effects of a project that may increase or decrease the profits of other business activities of a firm. Cannibalization is the situation when sales of a firm’s new product displace sales of one of its existing products. The lost sales are an incremental cost to the firm.

3. A Sunk cost is any unrecoverable cost for which a firm is already liable. Sunk costs have been or will be paid regardless of the decision whether or not to proceed with the project. A rule to remember: “if your decision does not affect a cash flow, then the cash flow should not affect your decision”.
Two examples of sunk costs:

• Overhead expenses: those expenses associated with activities that are not directly attributable to a single business activity but instead effect many different areas of a corporation.

• Past research and development expenditures.

4. Estimating a project’s free cash flow, the complications:

• Timing of cash flow: cash flows will be spread throughout the year. We can forecast at an annual level, a monthly level or even a weekly level. How riskier the project, how shorter the intervals.

• Accelerated depreciation: a firm wants to use the most accelerated depreciation method, because depreciation contributes positively to the cash flows. An example: MACRS depreciation: the most accelerated cost recovery system allowed by the IRS. The system is based on the recovery period, MACRS depreciation tables assign a fraction of the purchase price that the firm can depreciate each year.

• Liquidation or salvage value: when you sell an asset, you have to adjust the free cash flow to account for the after-tax cash flow that would result.
  Capital gain = sale price – book value.
  Book value = purchase price – accumulated depreciation.

After-tax cash flow from an asset sale = sale price – (tax rate x capital gain).

• Tax carryforwards: tax loss carryforwards and carrybacks are two features of the U.S. tax code that allow corporations to take losses during a current year and offset them against gains in nearby years. Since 1997, companies can carry back losses for two years and carry forward losses for twenty years.

Analyzing the project

Estimating the cash flows and cost of capital is the most difficult and uncertain part of capital budgeting. Now we’re going to look at methods that assess the importance of this uncertainty and identify the drivers of value in the project.

A sensitivity analysis is an important capital budgeting tool that determines how the NPV varies as a single underlying assumption is changed. A sensitivity analysis breaks the NPV calculation into its component assumptions. The NPV profile is a type of sensitivity analysis.

We can extend the sensitivity analysis by asking at what level of each parameter the project would just break-even. Break-even is the level for which an investment has an NPV of zero. The break-even analysis is a calculation of the value of each parameter for which the NPV of the project is zero. We can graph the NPV as a function of each of the critical assumptions (parameters).

Another break-even is the EBIT break-even: the level of a particular parameter for which a project’s EBIT is zero. EBIT = 0 

Units sold x (sale price – cost per unit) – SG&A – depreciation = 0.

SG&A are the selling, general and administrative expenses.

Scenario analysis is an important capital budgeting tool that determines how the NPV varies as a number of the underlying assumptions are changing simultaneously.

Real options

Most projects contain real options. Real options are options to make a business decision, often after gathering more information. It’s the right to make this decision. The presence of real options in a project increases the project’s NPV, because you can build greater flexibility into your project.

Common real options:

• Option to delay commitment: the option to time a particular investment. It is almost always present.

• Option to expand: the option to start with limited production and expand only if the project is successful.

• Abandonment option: an option for an investor to cease making investments in a project. Abandonment options can add value to a project because a firm can drop a project if it turns out to be unsuccessful.

Chapter I – Stock valuation

Stock basis

A common stock is a share of ownership in the corporation, which confers rights to any common dividends as well as rights to vote on election of directors, mergers or other major events. Shares trade at the public market under a ticker symbol; a unique abbreviation assigned to each publicly traded company. It is used when its trades are reported on the ticker (a real time electronic display of trading activity).

We will now take a look at the rights of common stock holders.

• A company can have straight voting; voting for directors where shareholders must vote for each director separately, with each shareholder having as many votes as shares held.

• Another voting form a company can have is cumulative voting. That is voting for directors where each shareholder is allocated votes equal to the number of open spots multiplied by his or her number of shares. With this voting even shareholders with minority blocks have a change at representation on the board.

• Some companies have different types of common stock, called classes. The different classes of common stock carry different voting rights.

Directors and other proposals as well as ask managers questions. All shareholders have the right to attend this meeting. But in practice most allow the board to vote for them or direct that their shares be voted for them via a proxy. A proxy is a written authorization for someone else to vote your shares. A proxy contest is when two or more groups are competing to collect proxies to prevail in a matter up for shareholder vote (such as election of directors).

Some companies have an additional issue of stock called preferred stock. Preferred stock is stock with preference over common shares in payment of dividends and liquidation. Dividends are periodic payments that are made to shareholders as a partial return on their investment in the corporation. The firm must pay this dividend to preferred shareholders before common shareholders can receive a dividend.

There are two types of preferred stock:

• Cumulative preferred stock: preferred stock where all missed preferred dividends must be paid before any common dividends may be paid.

• Non-cumulative preferred stock: preferred stock where missed dividends do not accumulate. Only the current dividend is owed before common dividends can be paid.

The mechanics of stock trades

If you make a large share trade at a physical stock market such as the NSYE, your trade would be transmitted electronically to the exchange but it would be sent to the wireless handheld of a terminal of a floor broker. A floor broker is a person at the NSYE with a trading license who represents orders on the floor, balancing speed and price to get the best execution.

Dividend-discount model

A shareholder has got two cash flow possibilities from owning a stock:

1. Payout cash in form of dividends

2. Generating cash by selling the shares

The investment horizon of the investor determines the total amount of the cash flows. Consider an investment horizon of one year. The timeline for this investment will be:

When an investor buys a stock, the investor has to pay the current market price: P0. Div1 are the total dividends paid per share during the year. If the investor sells the share at the end of the year, the investor receives the new market price: P1.

The dividend payments and new market price at the end of the year are not known with certainty. Hence, we cannot discount the cash flows using a risk-free interest rate. You must use the cost of capital for the firm equity. The equity cost of capital (rE) is the expected rate of return available in the market on other investments that have equivalent risk to the risk associated with the firm’s shares.

If you use the equity cost of capital, you can calculate the stock price with the following formula:

We can rewrite this equation into:
The first part is this equation is the dividend yield: the percentage return an investor expects to earn from the dividend paid by the stock: the expected annual dividend divided by its current market price. The second part is the capital gain rate: an expression of capital gain as a percentage of the initial price of the asset. The capital gain is the amount by which the selling price of an asset exceeds its initial purchase price. The total return is the sum of a stock’s dividend yield and its capital gain rate.

The NPV of this investment is positive when the P0 exceeds the current stock price. If the NPV is positive, investors will buy stocks and the stock price will rise. Otherwise, if the P0 is less than the stock price, the NPV of selling the stock will be positive and the stock price will fall.

Suppose the time horizon is two years. The investor pays the same current market price P0, but will receive dividends two times. The investor will sell the share at the new market price in year 2: P2. In this case:

Continuing this for any number of years results in the dividend-discount model: a model that values shares of a firm according to the present value of the future dividends the firm will pay.

This equation is applicable to investors who collect dividends for N year and sell the share after N years, and to a series of investor reselling the share after shorter periods.

If the time horizon of the investment isn’t limited (in this case it is possible to hold the shares forever) the price of a stock will be equal to the present value of all the expected future dividends.

Estimating dividends with the dividend-discount model

In the long run, we assume dividends will grow at a constant rate: growth rate g. If the time horizon isn’t limited, the dividends are a constant growth perpetuity. Then the following model is used; Constant dividend growth model:

According to this model, the value of the firm depends on the dividend level next year, divided by the equity cost of capital adjusted by the growth rate. The constant dividend growth model provides insight into an important tradeoff: a firm tries to maximize the current dividend level and the expected growth rate, because maximizing these quantities is equal to maximizing the share price. But, increasing growth may require investment, and if a firm spends money to investment there is less money to pay dividends.

The total dividend each year is the earnings per share multiplied by the dividend payout rate:

The dividend payout rate is the fraction of a firm’s earnings that the firm pays out as dividends each year. This formula shows that a firm can increase the dividends by:

1. increasing earnings (net income)

2. increasing the dividend payout rate and

3. decreasing the number of shares outstanding

Change in earnings = New investment x Return on new investment

New investment = Earnings x Retention rate

Assume the number of shares is fixed and the firm decides to keep the dividend payout rate constant, the growth rate will be:

The retention rate is the fraction of a firm’s current earnings that the firm retains.

Only if the new investment of the firm has got a positive NPV, cutting the dividends to increase investment will raise the stock price.

Note: the constant dividend growth model is only applicable when growth rates are constant. It is not possible for a discount cash flow (dividends) with a changing growth rate.

Limitations of the model

The dividend-discount model has two fundamental limitations:

1. Reliance on dividend forecasts. The model values a stock based on a forecast of the future dividends paid to shareholders. But a firm’s future dividends carry a high amount of uncertainty.

2. Lack of applicability to non-dividend-paying stocks. Often companies do not pay dividends. When this is the case it is hard to value the stocks.
    

Total payout model

In the constant dividend growth model we assumed the only cash an investor receives was dividend. However, firms replace these dividends with share repurchases: a situation in which a firm uses cash to buy back its own stock. There are two consequences for our previous model:

• There is less cash available to pay dividends because the firm spend more cash to repurchase and

• The earnings and dividend per share will increase because the share count decreases.

In the case of share repurchases, the total payout model is more reliable: this is a method that values shares of a firm by discounting the firm’s total payouts to equity holders (dividends and share repurchases) and then dividing by the current number of shares outstanding:

Discounted free cash flow model

The discounted free cash flow model is a method for estimating a firm’s enterprise value by discounting its future free cash flow. We calculated the enterprise value as:

Enterprise value = Market value of equity + Debt - Cash

The free cash flow (FCF) is the cash generated by the firm before any payments to debt or equity holders are considered:

FCF = EBIT x (1 – Tax rate) + Depreciation – CapEx – Increases in NWC

The present value of this free cash flow determines the current enterprise value (V0):

V0 = PV (future FCF)

So the share price will be:

Instead of the equity costs of capital rE, we use the weighted average cost of capital (WACC) as the discount rate in the discounted free cash flow model. The WACC is the cost of capital that reflects the risk of the overall business, which is the combined risk of the firm’s equity and debt: rWACC.

A constant long-run growth rate, gFCF, implies:

Valuation based on comparable firms

The method of comparables is an estimate of the value of a firm bases on the value of other, comparable firms or other investments that are expected to generate very similar cash flows in the future. If you want to use comparables to value identical firms, you need to adjust for scale differences. This is possible by expressing the value of the firms in term of a valuation multiple: a ratio of a firm’s value to some measure of the firm’s scale or cash flow.

Valuation multiples:

1. The price-earnings ratio. It’s possible to estimate the value of a share by multiplying the current earnings per share by the average P/E ratio of comparable firms. We make a distinction between the trailing P/E and the forward P/E. Trailing P/E uses trailing earnings: earnings over the prior 12 months. The forward P/E uses forward earnings: earnings over the coming 12 months.

2. Enterprise value multiples. These multiples value the entire firm and so they are closely related to the discount cash flow model.

3. Other multiples.

These valuation multiples also have limitations:

• Because firms are not identical in practice, the usefulness of valuation multiples depends on the nature of the differences between firms and the sensitivity of the multiples to these differences.

• Comparables only provide information regarding the value of a firm relative to other firms.

No valuation model provides a definitive value for the stock. To identify a reasonable range for the value, it is best to use several methods.

Information, competition and stock prices

Stock prices aggregate the information of many investors. Therefor if our valuation disagrees with the stock’s market price, it is most likely an indication that our assumptions about the firm’s cash flows are wrong.

Competition between investors tends to eliminate positive-NPV trading opportunities. Competition will be strongest when information is public and easy to interpret. Privately formed traders may be able to profit from their information, which is reflected in prices only gradually.

The efficient markets hypothesis states that competition among investors eliminates all positive-NPV trading opportunities, which is equivalent to stating that securities with equivalent risk have the same expected returns. It implies that securities will be fairly priced, based on their future cash flows, given all information that is available for investors.

In an efficient market, investors will not find positive-NPV trading opportunities without some source of competitive advantage. By contrast, the average investor will earn a fair return on his or her investment. And in an efficient market, to raise the stock price, corporate managers should focus on maximizing the present value of the free cash flow from the firm’s investments, rather than accounting consequences or financial policy.

Individual biases and trading

Individual investors display many biases, including overconfidence, disposition effect, limited attention and mood affects.

The overconfidence hypothesis is the tendency of individual investors to trade too much based on the mistaken belief that they can pick winners and losers better than investment professionals.

The disposition effect is the tendency to hold on to stocks that have lost value and sell stocks that have risen in value since the time of purchase.

In an efficient market, these biases can lead to trading losses through excessive trading or biases in valuations.

Chapter J – Risk and Return

A first look

While in hindsight some investments have had very high returns, they have also had the most volatility over time.

Historical risks and returns

To estimate the possible future returns for investors, it can be useful to use historical stock market data about the distribution of past returns.

The realized return is the total return that occurs over a particular time period. The realized return from your investment in the stock from period t to period t+1 can be calculated with this formula:

So, if you invest one euro in period t, you will receive 1 + R (t+1) in period t+1.

The average annual return (R) is the arithmetic average of an investment’s realized returns for each year:

R = )

The average annual return is an estimate of the return an investor can expect (the expected return), if the distribution is the same over time. You can calculate the standard deviation of the distribution of realized returns to determine the variability. The standard deviation is a common method used to measure the risk of a probability distribution, it is the square root of the variance. Variance is a method to measure the variability of returns, it is the expected squared variation of returns from the mean. The standard deviation is an indication of the tendency of the historical returns to be different from their average.
Variance estimate using realized returns:

(In this formula: R is the average annual return R).

The standard deviation is:

The standard deviation can be used to describe a normal distribution: a symmetric probability distribution that is completely characterized by its average and standard deviation.

Using a normal distribution, we can conclude with 95% confidence that next year’s return will be within two standard deviations of the average: the 95% prediction interval =
average ± (2 x standard deviation) = R ± (2 x SD(R)).

Historical tradeoff

There is a historical tradeoff between risk and reward. Risk is measured by price volatility and rewards are measured by returns. There is no clear relationship between returns and the volatility of individual stocks. However, there is a relationship between size and risk: larger stocks have got lower volatility, but large stocks are always more risky than a portfolio of large stocks.

Common versus independent risk

Different types of risk:

• A common risk is a risk that is linked across outcomes.

• Independent risks are risks that bear no relation to each other: knowing the outcome of one provides no information about the other. Diversification is the averaging of independent risks in a large portfolio.

Diversification in stock portfolios

The type of individual risks (common or independent) determines the risk of a portfolio. Common risks are not diversified in a large portfolio, independent risks are.

The realized return of a stock is risky, because the dividends plus the final stock price can vary. Dividends and stock prices can fluctuate due to company or industry-specific news and market-wide news. Fluctuations due to company or industry-specific news are independent risks, also called: unsystematic risk. These risks are independent risks unrelated across stocks. Fluctuations due to market-wide news are common risks: systematic risk. Unsystematic risk will be eliminated by diversification. Systematic risk will not be eliminated and will affect the entire portfolio.

Some rules to remember about risk premiums:

• A risk premium of a stock isn’t affected by its unsystematic, diversifiable risk.

• The risk premium for diversifiable risk is zero. This means: investors don’t get a compensation for holding unsystematic risk (because investors can eliminate unsystematic risk).

• The risk premium of a security is determined by its systematic risk and does not depend on its diversifiable risk. For this reason, the risk premium depends upon the amount of its systematic risk (not the total risk).

• Volatility and average returns for individual securities are not related.

Chapter K – Equity Risk Premium

Expected return of a Portfolio

A portfolio can be described by its portfolio weights: the fraction of the total investment in the portfolio held in each individual investment in de portfolio.

The return on a portfolio (Rp) is the weighted average of the returns on the investments in the portfolio, where the weights correspond to the portfolio weights.

Rp = w1 R1 + w2 R2 + … + wn Rn

The expected return of a portfolio is the return you can expect to earn on your portfolio, it is the weighted average of expected returns of the investments in a portfolio, where the weights correspond to the portfolio weights.

E [Rp] = w1 E [R1] + w2 E [R2] + … + wn E [Rn].

Volatility of portfolio

The volatility of a portfolio is the total risk, measured as a standard deviation, of a portfolio. Recall that risk is reduced through diversification by combining stocks into a portfolio. The degree to which the stocks face common risks and move together determines the amount of risk that is eliminated.

Correlation is a measure of the degree to which returns share common risk. It’s a degree to which stocks’ returns move together. Correlation is always between -1 and 1. A correlation of -1 means: returns move in opposite directions. A correlation of 0: uncorrelated. A correlation of 1: returns tend to move together.

Now we can compute the portfolio variance:

The first part accounts for the risk of stock 1, the second part for the risk of stock 2. The third part is the adjustment for how much the two stocks move together.

The benefits of diversification will rise when the amount of stocks in a portfolio increases. In a large portfolio, only risk that is common to all of the stocks (systematic risk) will matter.

An equally weighted portfolio is a portfolio in which the same dollar amount is invested in each stock.

Systematic risk

Only systematic risk is related to return, unsystematic isn’t. However, standard deviation measures total risk (systematic and unsystematic risk). We need a way to measure just the systematic risk.

If all investors diversify their portfolios, only systematic risk will remain. If all investors do this optimally, the aggregate portfolio held by all investors is a fully-diversifies, optimal portfolio: market portfolio: the portfolio of all risky investments, held in proportion to their value. This portfolio contains all shares outstanding of every risky security.

Market capitalization is the total market value of equity.

Market capitalization = (number of shares outstanding) x (price per share).

A value-weighted portfolio is a portfolio in which each security is held in proportion to its market capitalization. Because a market portfolio only contains systematic risk, we can calculate the amount of systematic risk a stock has by looking at the sensitivity of a stock’s return to the overall market.

Because it isn’t possible to collect and update returns on all risky assets in the world, we use a market proxy: a portfolio whose return is believed to closely track the true market portfolio. Market indexes are the most common proxy portfolios. A market index is the market value of a broad-based portfolio of securities.

To calculate the amount of systematic risk, we use the relationship between individual stock’s returns and the market portfolio’s returns. A stock is highly sensitive to systematic risk, if the return of the stock is highly sensitive to the return of the market portfolio. A stock has got little systematic risk, if the returns of the stock do not depend on the market’s returns.

To estimate the sensitivity of a stock to the market portfolio (the systematic risk), we use the stock’s beta (β): the expected percent change in the excess return of a security for a 1% change in the excess return of the market portfolio. Beta represents the amount by which risks that affect the overall market are amplified or dampened in a given stock. A security has got a high beta if it tends to move more than the market, a security has got a low beta if it tends to move less than the market.

The beta is the Cov (Ri,Rmkt) divided by the Var (Rmkt). The beta of a risk-free investment is always zero, because this investment has no correlation and volatility with the market (the return is known in advance). The beta of a market portfolio is always 1.

The capital asset pricing model
The expected return on any investment is the sum of a risk-free rate of return, as a compensation for inflation and the time value of money, and a risk premium for systematic risk.

Expected return = risk-free rate + risk premium for systematic risk.

Expected return for investment i = risk-free rate + βi x risk premium per unit of systematic risk.

The risk premium per unit of systematic risk can be estimated with the market risk premium (equity risk premium): the historical average excess returns on the market portfolio.

The capital asset pricing model (CAPM) is an equilibrium model of the relationship between risk and return that characterizes a security’s expected return based on its beta with the market portfolio.

In this formula represents the risk premium for security i. This premium is multiplied by the beta. Investors will not invest unless they can expect the return E [Ri], for that reason this return is also called the required return: the expected return of an investment that is necessary to compensate for the risk of undertaking the investment.

Putting the outcomes in a graph: there is no relationship between the standard deviation (total risk) and the expected return of a stock. But there is a linear relationship between the beta and the expected return: the security market line (SML). This is the pricing implication of the CAPM, it specifies a linear relation between the risk premium of a security and its beta with the market portfolio. The SML applies to all securities, also to portfolios.

The beta of a portfolio is the weighted average beta of the securities in the portfolio:
βp = w1 β1 + w2 β2 + … + wn βn
 

Chapter L – The Cost of Capital

A first look

Capital is the firm’s sources of financing: the debt, equity and other securities that it has outstanding. The capital structure is the relative proportions of debt, equity and other securities that a firm has outstanding. Financial managers use the capital structure to determine the overall cost of capital. Investors of each type of capital have a required return. Providing this return is the cost a company bears to obtain capital from investors: cost of capital.

We calculate the firm’s overall cost of capitals as the weighted average cost of capital (WACC): the average of a firm’s equity and debt costs of capital, weighted by the fractions of the firm’s value that correspond to equity and debt, respectively. The weights used in the WACC are the proportions of debt and equity, used in the capital structure. Important: use the market values of the proportions, not the book values mentioned at the balance sheet. The market-value balance sheet is similar to an accounting balance sheet, but all values are current market values rather than historical costs.

A firm that does not have debt outstanding is called unlevered. A firm that has debt outstanding is called levered. The leverage is the relative amount of debt on a firm’s balance sheet.

Unlevered firm: In this case, the cash flows to equity holders are the same as the cash flows from the assets. According to the Valuation Principle, the market value, risk and the cost of capital for the firm’s equity must equal the corresponding amounts for its assets. The equity cost of capital in this firm can be estimated using the Capital Asset Pricing Model (CAPM).

Levered firm: If the firm has got debt, the weighted average cost of capital (pre-tax) are:

r wacc = (fraction of firm value financed by equity) x (equity cost of capital) + (fraction of firm value financed by debt) x (debt cost of capital)

r wacc = asset cost of capital.

Cost of debt and equity capital

The interest a firm would have to pay to refinance the existing debt is the firm’s cost of debt. This interest rate isn’t the same as the coupon rate; the coupon rate is the interest rate the firm had to offer at the time the debt was issued. The yield to maturity can be used to estimate the firm’s current cost of debt: it is the yield that investors demand to hold the firm’s debt. Because the interest paid on debt is a tax deductible expense, the return paid to debt holders isn’t the same as the cost of debt to the firm. A firm’s net cost of interest on its debt after accounting for the interest tax deduction is called the effective cost of the debt. Tax deductibility of interest lowers the effective cost of the debt: the effective after-tax borrowing rate is: rD (1-Tc).

Cost of preferred stock capital:
Holders of preferred stock get a fixed dividend, this dividend must be paid ‘in preference to’ any dividends paid to common stockholders.
Cost of common stock capital:
Method 1: Capital Asset Pricing Model:
Method 2: Constant Dividend Growth Model:

The CAPM model is more popular than the CDGM model, because the latter has got many difficulties.

Weighted average cost of capital

The final formula of the WACC:

r wacc = rE x E% + r pfd x P%+ rD (1-Tc) x D%

A weighted average costs of capital can vary widely across industries and companies, because it’s driven by the risk and the leverage.

The net debt is de total debt outstanding minus any cash balances.

Valuing a project

The levered value is the value of an investment, including the benefit of the interest tax deduction, given the firm’s leverage policy. We can calculate the levered value with the WACC method: discounting future incremental free cash flows using the firm’s WACC. The intuition for the WACC method is that the firm’s WACC represents the average return a firm must pay to its investors (debt and equity holders) on an after-tax basis.

The levered value of an investment can be calculated with this formula:

It’s important to keep in mind the underlying assumptions, when using the WACC as the discount rate in capital budgeting.

1. The market risk of the project equals the average market risk of the firm’s investments.

2. The debt-equity ratio is constant, so the firm adjusts its leverage continuously.

3. The effect of leverage on valuation is limited: the main effect follows from the interest tax deduction.

Key steps in the WACC valuation method:

1. Determine the incremental free cash flow.

2. Compute the WACC

3. Discount the incremental free cash flow using the WACC to compute the value of the investment, including the tax benefit of leverage.

We assumed that the market risk of the project equals the average market risk of the firm’s investments. This assumption isn’t always correct.

If the risk of the project differs from the average market risk, the WACC isn’t the appropriate discount rate for the project. In this case, you have to estimate the WACC from the WACC of other firms operating in the same line of business as the new project.

The calculation of the WACC doesn’t take into account the direct costs of raising external financing. You have to subtract the present value of these costs from the NPV of the project.