Financial Activities - Banking

Nature of the Industry

Banks safeguard money and valuables and provide loans, credit, and payment services, such as checking accounts, money orders, and cashier’s checks. Banks also may offer investment and insurance products, which they were once prohibited from selling. As a variety of models for cooperation and integration among finance industries have emerged, some of the traditional distinctions between banks, insurance companies, and securities firms have diminished. In spite of these changes, banks continue to maintain and perform their primary role—accepting deposits and lending funds from these deposits.

Goods and services. Banking is comprised of two parts: Monetary Authorities—Central Bank, and Credit Intermediation and Related Activities. The former includes the bank establishments of the U.S. Federal Reserve System that manage the Nation’s money supply and international reserves, hold reserve deposits of other domestic banks and the central banks of other countries, and issue the currency we use. The establishments in the credit intermediation and related services industry provide banking services to the general public. They securely save the money of depositors, provide checking services, and lend the funds raised from depositors to consumers and businesses for mortgages, investment loans, and lines of credit.

Industry organization. There are several types of banks, which differ in the number of services they provide and the clientele they serve. Although some of the differences between these types of banks have lessened as they have begun to expand the range of products and services they offer, there are still key distinguishing traits. Commercial banks, which dominate this industry, offer a full range of services for individuals, businesses, and governments. These banks come in a wide range of sizes, from large global banks to regional and community banks. Global banks are involved in international lending and foreign currency trading, in addition to the more typical banking services. Regional banks have numerous branches and automated teller machine (ATM) locations throughout a multi-state area that provide banking services to individuals. Banks have become more oriented toward marketing and sales. As a result, employees need to know about all types of products and services offered by banks. Community banks are based locally and offer more personal attention, which many individuals and small businesses prefer. In recent years, online banks—which provide all services entirely over the Internet—have entered the market, with some success. However, many traditional banks have also expanded to offer online banking, and some formerly Internet-only banks are opting to open branches.

Savings banks and savings and loan associations, sometimes called thrift institutions, are the second largest group of depository institutions. They were first established as community-based institutions to finance mortgages for people to buy homes and still cater mostly to the savings and lending needs of individuals.

Credit unions are another kind of depository institution. Most credit unions are formed by people with a common bond, such as those who work for the same company or belong to the same labor union or church. Members pool their savings and, when they need money, they may borrow from the credit union, often at a lower interest rate than that demanded by other financial institutions.

Federal Reserve banks are Government agencies that perform many financial services for the Government. Their chief responsibilities are to regulate the banking industry and to help implement our Nation’s monetary policy so our economy can run more efficiently by controlling the Nation’s money supply—the total quantity of money in the country, including cash and bank deposits. For example, during slower periods of economic activity, the Federal Reserve may purchase government securities from commercial banks, giving them more money to lend, thus expanding the economy. Federal Reserve banks also perform a variety of services for other banks. For example, they may make emergency loans to banks that are short of cash, and clear checks that are drawn and paid out by different banks.

Interest on loans is the principal source of revenue for most banks, making their various lending departments critical to their success. The commercial lending department loans money to companies to start or expand their business or to purchase inventory and capital equipment. The consumer lending department handles student loans, credit cards, and loans for home improvements, debt consolidation, and automobile purchases. Finally, the mortgage lending department loans money to individuals and businesses to purchase real estate.

The money banks lend comes primarily from deposits in checking and savings accounts, certificates of deposit, money market accounts, and other deposit accounts that consumers and businesses set up with the bank. These deposits often earn interest for their owners, and accounts that offer checking provide owners with an easy method for making payments safely without using cash. Deposits in many banks are insured by the Federal Deposit Insurance Corporation, which guarantees that depositors will get their money back, up to a stated limit, if a bank should fail.

Recent developments. Technology is having a major impact on the banking industry. Direct deposit allows companies and governments to electronically transfer payments into various accounts. Debit cards, which may also be used as ATM cards, instantaneously deduct money from an account when the card is swiped across a machine at a store’s cash register. Electronic banking by phone or computer allows customers to access information such as account balances and statement history, pay bills, and transfer money from one account to another. Some banks also have begun offering online account aggregation, which makes available in one place detailed and up-to date information on a customer’s accounts held at various institutions.

Advancements in technology have also led to improvements in the ways in which banks process information. The use of check imaging allows banks to store photographed checks on the computer instead of paper files. Also, the availability and growing use of credit scoring software allows lending departments to approve loans in minutes, rather than days.

Other fundamental changes are occurring in the industry as banks diversify their services to become more competitive. Many banks now offer their customers financial planning and asset management services, as well as brokerage and insurance services, often through a subsidiary or third party. Others are beginning to provide investment banking services—usually through a subsidiary—that help companies and governments raise money through the issuance of stocks and bonds. As banks respond to deregulation and as competition in this sector grows, the nature of the banking industry will continue to undergo significant change.

Working Conditions

Hours. The average workweek for nonsupervisory workers in depository credit intermediation was 35.7 hours in 2006. Supervisory and managerial employees, however, usually work substantially longer hours. About 1 out of 10 employees in 2006, mostly tellers, worked part-time.

Employees in a typical branch work weekdays, some evenings if the bank is open late, and Saturday mornings. However, banks are increasingly expanding the hours that their branches are open and opening branches in nontraditional locations. For example, hours may be longer for workers in bank branches located in grocery stores and shopping malls, which are open most evenings and weekends. To improve customer service and provide greater access to bank personnel, banks are establishing centralized phone centers, staffed mainly by customer service representatives. Employees of phone centers spend most of their time answering phone calls from customers and must be available to work evening and weekend shifts.

Administrative support employees may work in large processing facilities, in the banks’ headquarters, or in other administrative offices. Most support staff work a standard 40-hour week; some may work overtime. Those support staff located in the processing facilities may work evening shifts.

Work environment. Branch office jobs, particularly teller positions, require continual communication with customers, repetitive tasks, and a high level of attention to security. Tellers also work for long periods in a confined space.

Commercial and mortgage loan officers often work out of the office, visiting clients, checking loan applications, and soliciting new business. Loan officers may travel to meet out-of-town clients, or work evenings if that is the only time at which a client can meet. Financial service-sales representatives also may visit clients in the evenings and on weekends to go over the client’s financial needs.

The remaining employees located primarily at the headquarters or other administrative offices usually work in comfortable surroundings and put in a standard workweek. In general, banks are relatively safe places to work. In 2006, the rate of work-related injury and illness per 100 full-time workers was 1.1 in central banks (monetary authorities) and 2.7 in other banking establishments (depository credit intermediation), both lower than the overall rate of 4.4 per 100 employees in the private sector.

Employment

The banking industry employed about 1.8 million wage and salary workers in 2006. About 7 out of 10 jobs were in commercial banks; the remainder were concentrated in savings institutions and credit unions (table 1).

Table 1. Percent distribution of employment and establishments in banking by detailed industry sector, 2006

Industry segment	Employment	Establishments
Total	100.0	100.0
Monetary authorities - central bank	1.2	0.3
Depository credit intermediation	98.8	99.7
Commercial banking	72.5	69.7
Savings institutions	13.1	14.7
Credit unions	12.1	14.2
Other depository credit intermediation	1.2	1.1

In 2006, about 84 percent of establishments in banking employed fewer than 20 workers (chart 1). However, these small establishments, mostly bank branch offices, employed 36 percent of all employees. About 64 percent of the jobs were in establishments with 20 or more workers. Banks are found everywhere in the United States, but most bank employees work in heavily populated States such as New York, California, Illinois, Pennsylvania, and Texas.

Occupations in the Industry

Banks employ various types of financial and customer service occupations. Tellers make up the largest number of workers, and overall office and administrative support occupations make up the largest portion of jobs in the industry. Management, business, and financial occupations also employ a significant number of employees in the banking industry.

Office and administrative support occupations. These occupations account for 2 out of 3 jobs in the banking industry (table 2). Bank tellers, the largest number of workers in banking, provide routine financial services to the public. They handle customers’ deposits and withdrawals, change money, sell money orders and traveler’s checks, and accept payment for loans and utility bills. Increasingly, tellers also are selling bank services to customers. New accounts clerks and customer service representatives answer questions from customers, and help them open and close accounts and fill out forms to apply for banking services. They are knowledgeable about a broad array of bank services and must be able to sell those services to potential clients. Some customer service representatives work in a call or customer contact center environment, taking phone calls and answering emails from customers. In addition to responding to inquiries, these workers also help customers over the phone with routine banking transactions, and handle and resolve problems or complaints.

Loan and credit clerks assemble and prepare paperwork, process applications, and complete the documentation after a loan or line of credit has been approved. They also verify applications for completeness. Bill and account collectors attempt to collect payments on overdue loans. Many general office clerks and bookkeeping, accounting, and auditing clerks are employed to maintain financial records, enter data, and process the thousands of deposit slips, checks, and other documents that banks handle daily. Banks also employ many secretaries, data entry and information processing workers, receptionists, and other office and administrative support workers. Office and administrative support worker supervisors and managers oversee the activities and training of workers in the various administrative support occupations.

Management, business, and financial occupations. These occupations account for about 25 percent of employment in the banking industry. Financial managers direct bank branches and departments, resolve customers’ problems, ensure that standards of service are maintained, and administer the institutions’ operations and investments. Loan officers evaluate loan applications, determine an applicant’s ability to repay a loan, and recommend approval of loans. They usually specialize in commercial, consumer, or mortgage lending. When loans become delinquent, loan officers, or loan counselors, may advise borrowers on the management of their finances or take action to collect outstanding amounts. Loan officers also play a major role in bringing in new business and spend much of their time developing relationships with potential customers. Trust officers manage a variety of assets that were placed in trust with the bank for other people or organizations; these assets can include pension funds, school endowments, or a company’s profit-sharing plan. Sometimes, trust officers act as executors of estates upon a person’s death. They also may work as accountants, lawyers, and investment managers.

Securities, commodities, and financial services sales agents, who make up the majority of sales positions in banks, sell complex banking services. They contact potential customers to explain their services and to ascertain the customer’s banking and other financial needs. They also may discuss services, such as deposit accounts, lines of credit, sales or inventory financing, certificates of deposit, cash management, or investment services. These sales agents also solicit businesses to participate in consumer credit card programs. At most small and medium-size banks, however, branch managers and commercial loan officers are responsible for marketing the bank’s financial services. This has become a more important task in recent years.

Other occupations. Occupations used widely by banks to maintain financial records and ensure the bank’s compliance with Federal and State regulations are accountants and auditors, and lawyers. In addition, computer specialists maintain and upgrade the bank’s computer systems and implement the bank’s entry into the world of electronic banking and paperless transactions.

Table 2. Employment of wage and salary workers in banking by occupation, 2006 and projected change, 2006-2016.
(Employment in thousands)

Occupation	Employment, 2006		Percent change, 2006-16
	Number	Percent
All occupations	1,825	100.0	4.0
Management, business, and financial occupations	449	24.6	5.4
General and operations managers	34	1.8	-8.4
Marketing and sales managers	11	0.6	1.9
Financial managers	73	4.0	1.9
Human resources, training, and labor relations specialists	15	0.8	5.3
Management analysts	8	0.5	1.4
Accountants and auditors	27	1.5	1.7
Credit analysts	15	0.8	-8.3
Financial analysts	18	1.0	11.4
Personal financial advisors	24	1.3	22.3
Loan officers	133	7.3	12.1
Professional and related occupations	72	4.0	6.9
Computer specialists	56	3.0	8.8
Sales and related occupations	82	4.5	11.8
Securities, commodities, and financial services sales agents	50	2.7	17.2
Office and administrative support occupations	1,202	65.9	2.9
First-line supervisors/managers of office and administrative support workers	111	6.1	-5.2
Bookkeeping, accounting, and auditing clerks	63	3.5	1.7
Tellers	546	29.9	12.1
Brokerage clerks	9	0.5	-1.0
Customer service representatives	106	5.8	12.0
New accounts clerks	73	4.0	-18.4
Receptionists and information clerks	9	0.5	1.5
Couriers and messengers	6	0.3	-8.3
Executive secretaries and administrative assistants	36	2.0	1.9
Secretaries, except legal, medical, and executive	15	0.8	-9.6
Data entry keyers	8	0.5	-18.6
Office clerks, general	40	2.2	0.2
Office machine operators, except computer	12	0.6	-14.9
Note: Columns may not add to totals due to omission of occupations with small employment

Training and Advancement

A high school education is usually all that is needed for most office and administrative occupations, while management, business and financial occupations usually employ workers with at least a college degree. Good communication and customer service skills are necessary for all occupations in the banking industry.

Office and administrative support occupations. Bank tellers and other clerks usually need only a high school education. Most banks seek people who have good basic math and communication skills, enjoy public contact, and feel comfortable handling large amounts of money. Through a combination of formal classroom instruction and on-the-job training under the guidance of an experienced worker, tellers learn the procedures, rules, and regulations that govern their jobs. Banks are offering more products and spending more on reaching out to their customers. As a result, they will need more creative and talented people to compete in the market place. Banks encourage upward mobility by providing access to higher education and other sources of additional training.

Some banks have their own training programs which result in teller certification. Experienced tellers qualify for certification by taking required courses and passing examinations. Experienced tellers and clerks may advance to head teller, new accounts clerk, or customer service representative. Outstanding tellers who have had some college or specialized training are sometimes promoted to managerial positions.

Management, business, and financial occupations. Workers in management, business, and financial occupations usually have at least a college degree. A bachelor’s degree in business administration or a liberal arts degree with business administration courses is suitable preparation, as is a bachelor’s degree in any field followed by a master’s degree in business administration (MBA). Many management positions are filled by promoting experienced, technically skilled professional personnel—for example, accountants, auditors, budget analysts, credit analysts, or financial analysts—or accounting or related department supervisors in large banks.

There are currently no specific licensing requirements for loan counselors and officers working in banks or credit unions. Training and licensing requirements for loan counselors and officers who work in mortgage banks or brokerages vary by State, depending on whether they are employed by a mortgage bank or mortgage brokerage.

Various banking-related associations and private schools offer courses and programs for students interested in lending, as well as for experienced loan officers who want to keep their skills current. Completion of these courses and programs generally enhances the individual’s employment and advancement opportunities. The Banking Administration Institute offers the Loan Review Certificate program for persons who review and approve loans. The Mortgage Bankers Association (MBA) offers the Certified Mortgage Banker (CMB) program. A candidate who earns the CMB exhibits a deep understanding of the mortgage business. To obtain the CMB, one must have at least 3 years of experience, earn educational credits, and pass an exam.

Financial services sales agents usually need a college degree; a major or courses in finance, accounting, economics, marketing, or related fields serve as excellent preparation. Experience in sales also is very helpful. These workers learn on the job under the supervision of bank officers. Sales agents selling securities need to be licensed by the National Association of Securities Dealers, and agents selling insurance also must obtain an appropriate license.

Advancement to higher level executive, administrative, managerial, and professional positions may be accelerated by taking additional training. Banks often provide opportunities and encourage employees to take classes offered by banking and financial management affiliated organizations or other educational institutions. Classes often deal with one of the different phases of financial management and banking, such as accounting management, budget management, corporate cash management, financial analysis, international banking, and data processing systems procedures and management. Employers also sponsor seminars and conferences, and provide textbooks and other educational materials. Many employers pay all or part of the costs for those who successfully complete courses.

In recent years, the banking field has been revolutionized by technological advancements in computer and data processing equipment. Learning how to apply this technology can greatly improve one’s skills and advancement opportunities in the banking industry.

Outlook

The number of local branches and offices in the United States has been steadily increasing, and this trend is expected to continue to result in moderate growth in employment in banking.

Employment change. Wage and salary employment in banking is projected to increase by about 4 percent between 2006 and 2016, compared with the 11 percent growth projected for wage and salary employment across all industries. Growth will result from banks refocusing on the local branch as a critical means of servicing customers, because branch location is often the most important factor for customers in selecting a bank. New branches also will be appearing more frequently in nontraditional locations, such as inside local grocery stores or shopping malls. Growth will likely be greatest in areas where the population is growing.

The combined effects of deregulation, technology, demographic changes, and mergers will continue to affect total employment growth and the mix of occupations in the banking industry. Deregulation of the banking industry allows banks to offer a variety of financial and insurance products that they were once prohibited from selling. The need to develop, analyze, and sell these new services will spur demand for securities and financial services sales representatives, financial analysts, and personal financial advisors. Demand for “personal bankers” to advise and manage the assets of wealthy clients, as well as the aging baby-boom generation, also will grow. However, banks will continue to face considerable competition—particularly in lending—from nonbank establishments, such as consumer credit companies and mortgage brokers. Companies and individuals now are able to obtain loans and credit and raise money through a variety of means other than bank loans. Therefore, some loan officers may be replaced by financial services sales representatives, who sell loans along with other bank services.

Advances in technology should continue to have a significant effect on employment in the banking industry. Demand for computer specialists will grow, as more banks make their services available electronically and eliminate much of the paperwork involved in many banking transactions. On the other hand, these changes in technology will reduce the need for some office and administrative support occupations. Employment growth among tellers will be limited as customers increasingly use ATMs, direct deposit, debit cards, and telephone and Internet banking to perform routine transactions. The number of electronic payments has increased and checks now account for less than half of consumers’ monthly bill payments. In addition, technological improvements, such as digital imaging and computer networking, are likely to lead to a decrease or change in the nature of employment of the “back-office” clerical workers who process checks and other bank statements. Employment of customer service representatives, however, is expected to increase as banks hire more of these workers to staff phone centers and respond to e-mails.

The increasing number of retired baby boomers should have a beneficial effect on total employment in the banking industry. They are more likely than younger age groups to hold bank deposits and visit branches to do their banking. Many also may need help in retirement planning and investing wealth inherited from their parents and so may seek the services of the various financial professionals in banking, such as financial managers, and securities, commodities, and financial services sales agents.

In the past, consolidation within the banking industry contributed significantly to employment declines, but the effect of mergers on employment within the industry is expected to be minimal in the years ahead. Merger activity has slowed recently, and a balance is beginning to develop between the numbers of new banks established and existing banks lost due to mergers and acquisitions.

Job prospects. Job opportunities should be favorable for tellers and other administrative support workers because they make up a large proportion of bank employees and many individuals leave these positions for other jobs that offer higher pay or greater responsibilities. The need for skilled workers will create good job opportunities for individuals with up-to-date computer skills and financial services backgrounds.

Earnings

Industry earnings. Earnings of nonsupervisory bank employees involved in depository credit intermediation averaged $535 a week in 2006, compared with $738 for all workers in finance and insurance industries, and $568 for workers throughout the private sector. Relatively low pay in the banking industry reflects the high proportion of low-paying administrative support jobs.

Greater responsibilities generally result in a higher salary. Experience, length of service, and, especially, the location and size of the bank also are important. Earnings in the banking industry also vary significantly by occupation. Earnings in the largest occupations in banking appear in table 3.

Table 3. Median hourly earnings of the largest occupations in depository credit intermediation, May 2006

Occupation	Depository credit intermediation	All industries
General and operations managers	$40.89	$40.97
Financial managers	34.89	43.74
Loan officers	23.51	24.89
First-line supervisors/managers of office and administrative support workers	19.66	20.92
Executive secretaries and administrative assistants	18.05	17.90
Loan interviewers and clerks	14.35	14.89
Customer service representatives	13.68	13.62
New accounts clerks	13.60	13.65
Office clerks, general	11.82	11.40
Tellers	10.63	10.64

Benefits and union membership. In addition to common benefits offered by many industries, equity sharing and performance-based pay increasingly are part of compensation packages for some bank employees. As banks encourage employees to become more sales-oriented, incentives are increasingly tied to meeting sales goals, and some workers may even receive commissions for sales or referrals. As in other industries, part-time workers do not enjoy the same benefits that full-time workers do.

Very few workers in the banking industry are unionized—only 2 percent are union members or are covered by union contracts, compared with 13 percent of workers throughout private industry.

Free Cash Flow

When valuing the operations of a firm using a discounted cash flow model, the operating cash flow is needed. This operating cash flow also is called the unlevered free cash flow (UFCF). The term "free cash flow" is used because this cash is free to be paid back to the suppliers of capital.

Calculating Free Cash Flow

For a particular year, the unlevered free cash flow is calculated as follows:

1. Start with the annual sales and subtract cash costs and depreciation to calculate the earnings before interest and taxes (EBIT). The EBIT also is referred to as the operating income and represents the pre-tax earnings without regard to how the business is financed.

2. Calculate the earnings before interest and after tax (EBIAT) by multiplying the EBIT by one minus the tax rate. Note that the EBIAT represents the after-tax earnings of the firm as if it were financed entirely with equity capital.

3. To arrive at the UFCF, add the depreciation expense back to the EBIAT, and subtract capital expenditures (CAPEX) that were not charged against earnings and subtract any investments in net working capital (NWC).

The free cash flow calculation in equation form:

Operating Income (EBIT) = Revenues – Cash Costs – Depreciation Expense

EBIAT = EBIT – Taxes, where Taxes = (tax rate)(EBIT)

UFCF = EBIAT + Depreciation Expense – CAPEX – Increase in NWC

Capital expenditures are calculated by solving for CAPEX in the following equation:

BV of Assets at Year End	=	BV of assets at Beginning of Year
		+ CAPEX
		– Depreciation

An additional cash adjustment may be necessary for an increase in deferred taxes that would have a positive impact on cash flow.

Financial Ratios

A firm's performance can be evaluated using financial ratios. Referencing these ratios to those of other firms allows a comparison to be made. The following is a listing of some useful ratios.

Leverage : Assets / Shareholder's Equity

Gross Margin = Gross Profit / Sales.
Gross margin measures the profitability considering only variable costs and is a measure of the percentage of revenue that goes to fixed costs and profit.

Net Profit Margin = Net Income / Sales

Total Asset Turnover = defined as Sales / Total Assets

Return on Assets (ROA) = Net Income / Assets
ROA is a measure of the return on money provided by both owners and creditors, and is a measure of how efficiently all resources are managed.

Return on Equity (ROE) = defined as Net Income / Equity
where the equity value is the shareholder's equity at the end of the period in which the income was earned. ROE is a measure of the return on money provided by the firm's owners.

ROE can be calculated indirectly as:

ROE = ( Net Income / Total Assets ) ( Total Assets / Equity )
ROE also can be calculated using DuPont analysis : ROE = (Net Income / Sales)(Sales / Total Assets)(Total Assets / Equity)

This states that ROE is determined by multiplication of three levers:

ROE = (net profit margin) (total asset turnover) (leverage)

These levers are readily viewed on the company's financial statements. While ROE's may be similar among firms, the levers may differ significantly.

Liquidity

The term working capital is used to describe the current items of the balance sheet. Working capital includes current assets such as cash, accounts receivable, and inventory, and current liabilities such as accounts payable and other short term liabilities. Net working capital is defined as non-cash current operating assets minus non-debt current operating liabilities. Cash, short-term debt, and current portion of long-term debt are excluded from the net working capital calculation because they are related to financing and not to operations.

Two commonly used liquidity ratios are the current ratio and the quick ratio.

Current Ratio : defined as Current Assets / Current Liabilities.
The current ratio is a measure of the firm's ability to pay off current liabilities as they become due.

Quick Ratio : defined as Quick Assets / Current Liabilities.

The quick ratio also is known as the acid test. Quick assets are defined as cash, accounts receivable, and notes receivable - essentially current assets minus inventory.

Security Analysis

Security analysis is about valuing the assets, debt, warrants, and equity of companies from the perspective of outside investors using publicly available information. The security analyst must have a thorough understanding of financial statements, which are an important source of this information. As such, the ability to value equity securities requires cross-disciplinary knowledge in both finance and financial accounting.

While there is much overlap between the analytical tools used in security analysis and those used in corporate finance, security analysis tends to take the perspective of potential investors, whereas corporate finance tends to take an inside perspective such as that of a corporate financial manager.

Equity Value and Enterprise Value

The equity value of a firm is simply its market capitalization; that is, the market price per share multiplied by the number of outstanding shares. The enterprise value, also referred to as the firm value, is the equity value plus the net liabilities. The enterprise value is the value of the productive assets of the firm, not just its equity value, based on the accounting identity:

Assets = Net Liabilities + Equity

Note that net values of the assets and liabilities are used. Any cash and cash-equivalents would be used to offset the liabilities and therefore are not included in the enterprise value.

As an analogy, imagine purchasing a house with a market value of $100,000, for which the owner has $50,000 in equity and a $50,000 assumable mortgage. To purchase the house, the new owner would pay $50,000 in cash and assume the $50,000 mortgage, for a total capital structure of $100,000. If $20,000 of that market value were due to $20,000 in cash locked in a safe in the basement, and the owner pledged to leave the money in the house, the cash could be used to pay down the $50,000 mortgage and the net assets would become $80,000 and the net liabilities would become $30,000. The "enterprise value" of the house therefore would be $80,000.

Valuation Methods

Two types of approaches to valuation are discounted cash flow methods and financial ratio methods.

Two discounted cash flow approaches to valuation are:

1. value the cash flow to equity, and
2. value the cash flow to the enterprise.

The "cash flow to equity" approach to valuation directly discounts the firm's cash flow to the equity owners. This cash flow takes the form of dividends or share buybacks. While intuitively straightforward, this technique suffers from numerous drawbacks. First, it is not very useful in identifying areas of value creation. Second, changes in the dividend payout ratio result in a change in the calculated value of the company even though the operating performance might not change. This effect must be compensated by adjusting the discount rate to be consistent with the new payout ratio. Despite its drawbacks, the equity approach often is more appropriate when valuing financial institutions because it treats the firm's liabilities as a part of operations. Since banks have significant liabilities that are owed to the retail depositors, they indeed have significant liabilities that are part of operations.

The "cash flow to the enterprise" approach values the equity of the firm as the value of the operations less the value of the debt. The value of the operations is the present value of the future free cash flows expected to be generated. The free cash flow is calculated by taking the operating earnings (earnings excluding interest expenses), subtracting items that required cash but that did not reduce reported earnings, and adding non-cash items that did reduce reported earnings but that did not result in cash expenditures. Interest and dividend payments are not subtracted since we are calculating the free cash flow available to all capital providers, both equity and debt, before financing. The result is the cash generated by operations. The free cash flow basically is the cash that would be available to shareholders if the firm had no debt - the cash produced by the business regardless of the way it is financed. The expected future cash flow then is discounted by the weighted average cost of capital to determine the enterprise value. The value of the equity then is the enterprise value less the value of the debt.

When valuing cash flows, pro forma projections are made a certain number of years into the future, then a terminal value is calculated for years thereafter and discounted back to the present.

Free Cash Flow Calculation

The free cash flow (FCF) is calculated by starting with the profits after taxes, then adding back depreciation that reduced earnings even though it was not a cash outflow, then adding back after-tax interest (since we are interested in the cash flow from operations), and adding back any non-cash decrease in net working capital (NWC). For example, if accounts receivable decreased, this decrease had a positive effect on cash flow.

If the accounting earnings are negative and the free cash flow is positive, the carry-forward tax benefit is in effect realized in the current year and must be added to the FCF calculation.

Leverage

In 1958, economists and now Nobel laureates Franco Modigliani and Merton H. Miller proposed that the capital structure of a firm did not affect its value, assuming no taxes, no bankruptcy costs, no transaction costs, that the firm's investment decisions are independent of capital structure, and that managers, shareholders, and bondholders have the same information. The mix of debt and equity simply reallocates the cash flow between stockholders and bondholders, but the total amount of the cash flow is independent of the capital structure. According to Modigliani and Miller's first proposition, the value of the firm if levered equals the value if unlevered:

V_L = V_U

However, the assumptions behind Proposition I do not all hold. One of the more unrealistic assumptions is that of no taxes. Since the firm benefits from the tax deduction associated with interest paid on the debt, the value of the levered firm becomes:

V_L = V_U + t_cD

where t_c = marginal corporate tax rate.

When considering the effect of taxes on firm value, it is worthwhile to consider taxes from a potential investors point of view. For equity investors, the firm first must pay taxes at the corporate tax rate, t_c, then the investor must pay taxes at the individual equity holder tax rate, t_e. Then for debt holders,

After-tax income = ( debt income )( 1 – t_d )

For equity holders,

After-tax income = ( equity income )( 1 – t_c )( 1 – t_e )

The relative advantage (if any) of equity to debt can be expressed as:

Relative Advantage (RA) = ( 1 – t_c )( 1 – t_e ) / ( 1 – t_d )

RA > 1 signifies a relative advantage for equity financing.
RA <>

One can define T as the net advantage of debt :

T = 1 – RA

For T positive, there is a net advantage from using debt; for T negative there is a net disadvantage.

Empirical evidence suggests that T is small; in equilibrium T = 0. This is known as Miller's equilibrium and implies that the capital structure does not affect enterprise value (though it can affect equity value, even if T=0).


Calculating the Cost of Capital

Note that the return on assets, r_a, sometimes is referred to as r_u, the unlevered return.

Gordon Dividend Model:

P₀ = Div₁ / ( r_e – g )

where

P₀ = current stock price,
Div₁ = dividend paid out one year from now,
r_e = return of equity
g = dividend growth rate

Then:

r_e = ( Div₁ / P₀ ) + g

Capital Asset Pricing Model:

The security market line is used to calculate the expected return on equity:

r_e = r_f + β_e ( r_m – r_f )

where

r_f = risk-free rate,
r_m = market return
β_e = equity beta

However, this model ignores the effect of corporate income taxes.

Considering corporate income taxes:

r_e = r_f ( 1 – t_c ) + β_e [ r_m – r_f ( 1 – t_c ) ]

where t_c = corporate tax rate.

Once the expected return on equity and on debt are known, the weighted average cost of capital can be calculated using Modigliani and Miller's second proposition:

WACC = r_e E / ( E + D ) + r_d D / ( E + D )

Taking into account the tax shield:

WACC = r_e E / ( E + D ) + r_d ( 1 – t_c ) D / ( E + D )

For T = 0 (no tax advantage for debt), the WACC is equivalent to the return on assets, r_a.

r_d is calculated using the CAPM:

r_d = r_f + β_d [ r_m – r_f ( 1 – t_c ) ]

For a levered firm in an environment in which there are both corporate and personal income taxes and in which there is no tax advantage to debt (T=0), WACC is equal to r_a, and the above WACC equation can be rearranged to solve for r_e:

r_e = r_a + (D/E)[ r_a – r_d(1 – t_c) ]

From this equation it is evident that if a firm with a constant future free cash flow increases its debt-to-equity ratio, for example by issuing debt and repurchasing some of its shares, its cost of equity will increase.

r_a also can be calculated directly by first obtaining a value for the asset beta, β_a, and then applying the CAPM. The asset beta is:

β_a = β_e ( E / V ) + β_d ( D / V )( 1 – t_c )

Then return on assets is calculated as:

r_a = r_f ( 1 – t_c ) + β_a [ r_m – r_f ( 1 – t_c ) ]

In summary, for the case in which there is personal taxation and in which Miller's Equilibrium holds ( T = 0 ), the following equations describe the expected returns on equity, debt, and assets:

r_e = r_f ( 1 – t_c ) + β_e [ r_m – r_f ( 1 – t_c ) ]

r_a = r_f ( 1 – t_c ) + β_a [ r_m – r_f ( 1 – t_c ) ]

r_d = r_f + β_d [ r_m – r_f ( 1 – t_c ) ]

The cost of capital also can be calculated using historical averages. The arithmetic mean generally is used for this calculation, though some argue that the geometric mean should be used.

Finally, the cost of equity can be determined from financial ratios. For example, the cost of unleveraged equity is:

r_e,U = [ r_{e, L} + r_f,debt ( 1 – t_c ) D/E ] / ( 1 + D/E )

r_e,L = b(1+g) / (P/E) + g

where b = dividend payout ratio

g = ( 1 – b ) (ROE)

where (1 – b) = plowback ratio.

The payout ratio can be calculated using dividend and earnings ratios:

b = ( Dividend / Price ) ( Price / Earnings)

Share Buy-Back

Take a firm that is 100% equity financed in an environment in which T is not equal to zero; i.e., there is a net tax advantage to debt. If the firm decides to issue debt and buyback shares, the levered value of the firm then is:

V_L = V_U + T (debt)

The number of shares that could be repurchased then is:

n = (debt) / ( price per share after relevering)

where the price per share after relevering is:

V_L / (original number of outstanding shares)

The buyback will lower the firm's WACC.


Project Valuation

The NPV of a capital investment made by a firm, assuming that the investment results in an annual free cash flow P received at the end of each year beginning with the first year, and assuming that the asset is financed using current debt/equity ratios, is equal to:

NPV = – P₀ + P / WACC

Warrant Valuation

Warrants are call options issued by the firm and that would require new shares to be issued if exercised. Any outstanding warrants must be considered when valuing the equity of the firm. The Black-Scholes option pricing formula can be used to value the firm's warrants.

Valuation Calculation

Once the free cash flow and WACC are known, the valuation calculation can be made. If the free cash flow is equally distributed across the year, an adjustment is necessary to shift the year-end cash flows to mid-year. This adjustment is performed by shifting the cash flow by one-half of a year by multiplying the valuation by ( 1 + WACC )^1/2.

The enterprise value includes the value of any outstanding warrants. The value of the warrants must be subtracted from the enterprise value to calculate the equity value. This result is divided by the current number of outstanding shares to yield the per share equity value.

PEG Ratio

As a rule of thumb, the P/E ratio of a stock should be equal to the earnings growth rate. Mathematically, this can be shown as follows:

P = D / r_e + PVGO

where

P = price

D = annual dividend

r_e = return on equity

PVGO = present value of growth opportunities.

For high growth firms, PVGO usually dominates D / r_e. PVGO is equal to the earnings divided by the earnings growth rate.

Treatment of Goodwill

Prior to 2002, amortization of goodwill was an expense on the income statement, but unlike depreciation of fixed assets, amortization of goodwill is not tax deductible.

In 2002, FASB Statement No. 142 discontinued the depreciation of goodwill and specified that it be kept on the books as a non-depreciating asset and written off only when its value is determined to have declined.

Glossary

APV: Adjusted Present Value

CAPM: Capital Asset Pricing Model

EBIT: Earnings Before Interest and Taxes

EBITDA: Earnings Before Interest, Taxes, Depreciation, and Amortization

Enterprise Value: Market value of a firm's equity plus the net market value of its debt.

• Enterprise value = market cap + LTD - net cash & investments

FCF: Free Cash Flow

LTD: Long-Term Debt

MRP: Market risk premium, defined as r_m – r_f , unless it specifically is referred to as tax-adjusted market risk premium, in which case there would be a factor to adjust r_f for taxes.

NOPLAT: Net Operating Profits Less Adjusted Taxes

OLS: Ordinary Least Squares (method of regression)

PEG: The ratio of P/E to growth rate in earnings.

RADR: Risk Adjusted Discount Rate

RAYTM: Rating-Adjusted Yield-To-Maturity

ROE: Return On Equity; equivalent to the expected return on retained earnings

YTM: Yield To Maturity

Recommended Reading

McKinsey & Company, Inc., Thomas E. Copeland, et al., Valuation: Measuring and Managing the Value of Companies

Valuation is a practitioner's guide to valuation. It begins by making the case for valuation as a performance metric and then introduces valuation frameworks focusing on discounted cash flow, concluding with applications to real-world environments. Required reading for many MBA security analysis courses.

Corporate Finance

Arguably, the role of a corporation's management is to increase the value of the firm to its shareholders while observing applicable laws and responsibilities. Corporate finance deals with the strategic financial issues associated with achieving this goal, such as how the corporation should raise and manage its capital, what investments the firm should make, what portion of profits should be returned to shareholders in the form of dividends, and whether it makes sense to merge with or acquire another firm.

Balance Sheet Approach to Valuation

If the role of management is to increase the shareholder value, then managers can make better decisions if they can predict the impact of those decisions on the firm's value. By observing the difference in the firm's equity value at different points in time, one can better evaluate the effectiveness of financial decisions. A rudimentary way of valuing the equity of a company is simply to take its balance sheet and subtract liabilities from assets to arrive at the equity value. However, this book value has little resemblance to the real value of the company. First, the assets are recorded at historical costs, which may be much greater than or much less their present market values. Second, assets such as patents, trademarks, loyal customers, and talented managers do not appear on the balance sheet but may have a significant impact on the firm's ability to generate future profits. So while the balance sheet method is simple, it is not accurate; there are better ways of accomplishing the task of valuation.

Cash vs. Profits

Another way to value the firm is to consider the future flow of cash. Since cash today is worth more than the same amount of cash tomorrow, a valuation model based on cash flow can discount the value of cash received in future years, thus providing a more accurate picture of the true impact of financial decisions.

Decisions about finances affect operations and vice versa; a company's finances and operations are interrelated. The firm's working capital flows in a cycle, beginning with cash that may be converted into equipment and raw materials. Additional cash is used to convert the raw materials into inventory, which then is converted into accounts receivable and eventually back to cash, completing the cycle. The goal is to have more cash at the end of the cycle than at the beginning.

The change in cash is different from accounting profits. A company can report consistent profits but still become insolvent. For example, if the firm extends customers increasingly longer periods of time to settle their accounts, even though the reported earnings do not change, the cash flow will decrease. As another example, take the case of a firm that produces more product than it sells, a situation that results in the accumulation of inventory. In such a situation, the inventory will appear as an asset on the balance sheet, but does not result in profit or loss. Even though the inventory was not sold, cash nonetheless was consumed in producing it.

Note also the distinction between cash and equity. Shareholders' equity is the sum of common stock at par value, additional paid-in capital, and retained earnings. Some people have been known to picture retained earnings as money sitting in a shoe box or bank account. But shareholders' equity is on the opposite side of the balance sheet from cash. In fact, retained earnings represent shareholders' claims on the assets of the firm, and do not represent cash that can be used if the cash balance gets too low. In this regard, one can say that retained earnings represent cash that already has been spent.

Shareholder equity changes due to three things:

• net income or losses
• payment of dividends
• share issuance or repurchase.

Changes in cash are reported by the cash flow statement, which organizes the sources and uses of cash into three categories: operating activities, investing activities, and financing activities.

Cash Cycle

The duration of the cash cycle is the time between the date the inventory (or raw materials) is paid for and the date the cash is collected from the sale of the inventory. A company's cash cycle is important because it affects the need for financing. The cash cycle is calculated as:

days in inventory + days in receivables - days in payables

Financing requirements will increase if either of the following occurs:

• Sales increase while the cash cycle remains fixed in duration. Increased sales increase the value of assets in the cycle.

• Sales remain flat but the cash cycle increases in duration.

While financially it makes sense to reduce the length of the cash cycle, such a reduction should not be done without considering the impact on operations. For example, one must consider the impact on customer and supplier relations as well as the impact on order fill rates.

Revenue, Expenses, and Inventory

A firm's income is calculated by subtracting its expenses from its revenue. However, not all costs are considered expenses; accounting standards and tax laws prohibit the expensing of costs incurred in the production of inventory. Rather, these costs must be allocated to inventory accounts and appear as assets on the balance sheet. Once the finished goods are drawn from inventory and sold, these costs are reported on the income statement as the cost of goods sold (COGS). If one wishes to know how much product the firm actually produced, the cost of goods produced in an accounting period is determined by adding the change in inventory to the COGS.

Assets

Assets can be classified as current assets and long-term assets. It is useful to know the number of days of certain assets and liabilities that a firm has on hand. These numbers are easily calculated from the financial statements as follows:

Accounts Receivable (A/R)

Number of days of A/R = ( accounts receivable / annual credit sales ) ( 365 ).
This also is known as the collection period.

Inventory

Number of days of inventory = ( inventory / annual COGS ) ( 365 ).
This also is known as the inventory period.

On the liabilities side:

Payables

Number of days of accounts payable = ( accounts payable / COGS ) ( 365 ), assuming that all accounts payable are for the production of goods. This also is known as the payables period.

Financial Ratios

A firm's performance can be evaluated using various financial ratios. Ratios are used to measure leverage, margins, turnover rates, return on assets, return on equity, and liquidity. Additional insight can be gained by comparing ratios among firms in the industry.

Bank Loans

Bank loans can be classified according to their durations. There are short-term loans (one year or less), long-term loans (also known as term loans), and revolving loans that allow one to borrow up to a specified credit level at any time over the duration of the loan. Some revolving loans automatically renew at maturity; these loans are said to be "evergreen."


Sources and Uses of Cash

It can be worthwhile to know where a firm's cash is originating and how it is being used. There are two sources of cash: reducing assets or increasing liabilities or equity. Similarly, a company uses cash either by increasing assets or decreasing liabilities or equity.


Sustainable Growth

A company's sustainable growth rate is calculated by multiplying the ROE by the earnings retention rate.


Firm Value, Equity Value, and Debt Value

The value of the firm is the value of its assets, or rather, the present value of the unlevered free cash flow resulting from the use of those assets. In the case of an all-equity financed firm, the equity value is equal to the firm value. When the firm has issued debt, the debt holders have a priority claim on their interest and principal, and the equity holders have a residual claim on what remains after the debt obligations are met. The sum of the value of the debt and the value of the equity then is equal to the value of the firm, ignoring the tax benefits from the interest paid on the debt. Considering taxes, the effective value of the firm will be higher since a levered firm has a tax benefit from the interest paid on the debt. If there is outstanding preferred stock, the firm value is the sum of the equity value, debt value, and preferred stock value, plus the value of the interest tax shield.

The debt holders and stock holders each have a claim on the cash flows of the firm. In a given time period, the debt holders have a claim equal to the interest payments during that period plus any principal payments that are due. The stock holders then have a claim equal to the unlevered free cash flow in that period plus the cash generated by the interest tax shield, minus the claims of the debt holders.


Capital Structure

The proportion of a firm's capital structure supplied by debt and by equity is reported as either the debt to equity ratio (D/E) or as the debt to value ratio (D/V), the latter of which is equal to the debt divided by the sum of the debt and the equity.

One can quickly convert between the D/E ratio and the D/V ratio by using the following relationships:

D / V = ( D / E ) / ( 1 + D / E )

D / E = ( D / V ) / ( 1 - D / V )


Risk Premiums

• Business risk is the risk associated with a firm's operations. It is the undiversifiable volatility in the operating earnings (EBIT). Business risk is affected by the firm's investment decisions. A measure for the business risk is the asset beta, also known the unlevered beta. In terms of the discount rate, the return on assets of a firm can be expressed as a function of the risk-free rate and the business risk premium (BRP):

r_A = r_F + BRP

• Financial risk is associated with the firm's capital structure. Financial risk magnifies the business risk of a firm. Financial risk is affected by the firm's financing decision.

• Total corporate risk is the sum of the business and financial risks and is measured by the equity beta, also known as the levered beta. The business risk premium (BRP) and financial risk premium (FRP) are reflected in the levered (equity) beta, and the return on levered equity can be written as:

r_E = r_F + BRP + FRP

Debt beta is a measure of the risk of a firm's defaulting on its debt. The return on debt can be written as:

r_D = r_F + default risk premium


Cost of Capital

The cost of capital is the rate of return that must be realized in order to satisfy investors. The cost of debt capital is the return demanded by investors in the firm's debt; this return largely is related to the interest the firm pays on its debt. In the past some managers believed that equity capital had no cost if no dividends were paid; however, equity investors incur an opportunity cost in owning the equity of the firm and they therefore demand a rate of return comparable to what they could earn by investing in securities of comparable risk.

The return required by debt holders is found by applying the CAPM:

r_D = r_F + beta_debt ( r_M - r_F )

The required rate of return on assets (that is, on unlevered equity) can be found using the CAPM:

r_A = r_F + beta_unlevered ( r_M - r_F )

Using the CAPM, a firm's required return on equity is calculated as:

r_E = r_F + beta_levered ( r_M - r_F )

Under the Modigliani-Miller assumptions of constant cash flows and constant debt level, the required return on equity is:

r_E = r_A + (1-τ)(r_A - r_D)(D / E)

where τ is the corporate tax rate.

The overall cost of capital is a weighted-average of the cost of its equity capital and the after-tax cost of its debt capital. The weighted average cost of capital (WACC) then is given by:

WACC = r_E (E / V_L) + r_D (1-τ)(D / V_L)

Assuming perpetuities for the cash flows, the weighted average cost of capital can be calculated as:

WACC = r_A [ 1 - τ(D / V_L)]

Neglecting taxes, the WACC would be equal to the expected return on assets because the WACC is the return on a portfolio of all the firm's equity and all of its debt, and such a portfolio essentially has claim to all of the firm's assets.

For arbitrary cash flows, and under the assumption that the debt to value ratio is held constant, the following relationship derived by James A. Miles and John R. Ezzell is applicable:

WACC = r_A - τ r_D (D / V_L)(1+r_A) / (1+r_D)

Under the same assumptions, the cost of equity capital can be calculated from r_A and r_D using the following relationship from Miles and Ezzell:

r_E = r_A + [ 1 - τ r_D / (1+r_D)] [ r_A - r_D ] D/E

For low values of r_D, [ 1 - τ r_D / (1+r_D)] is approximately equal to one, and the expression can be simplified if high precision is not required.

If one cannot assume a constant debt to value ratio, then the APV method should be used.

Estimating Beta

In order to use the CAPM to calculate the return on assets or the return on equity, one needs to estimate the asset (unlevered) beta or the equity (levered) beta of the firm. The beta that often is reported for a stock is the levered beta for the firm. When estimating a beta for a particular line of business, it is better to use the beta of an existing firm in that exact line of business (a pure play) rather than an average beta of several firms in similar lines of business that are not exactly the same.

Expressing the levered beta, unlevered beta, and debt beta in terms of the covariance of their corresponding returns with that of the market, one can derive an expression relating the three betas. This relationship between the betas is:

beta_levered = beta_unlevered­ [ 1 + (1 - τ) D/E ] - beta_debt(1- τ) D/E

beta_unlevered = [ beta_levered + beta_debt(1- τ) D/E ] / [ 1 + (1 - τ) D/E ]

The debt beta can be estimated using CAPM given the risk-free rate, bond yield, and market risk premium.


Unlevered Free Cash Flows

To value the operations of the firm using a discounted cash flow model, the unlevered free cash flow is used. The unlevered free cash flow represents the cash generated by the firm's operations and is the cash that is free to be paid to stock and bond holders after all other operating cash outlays have been performed.

Terminal Value

The value of the firm at the end of the last year for which unique cash flows are projected is known as the terminal value. The terminal value is important because it can represent 50% or more of the total value of the firm.

Three Discounted Cash Flow Methods for Valuing Levered Assets

APV (Adjusted Present Value) Method
The APV approach first performs the valuation under an unlevered all-equity assumption, then adjusts this value for the effect of the interest tax shield. Using this approach,

V_L = V_U + PVITS

where V_L = value if levered
V_U = value if financed 100% with equity
PVITS = present value of interest tax shield

The unlevered value is found by discounting the unlevered free cash flow at the required return on assets. The present value of the interest tax shield is found by discounting the interest tax shield savings at the required return on debt, r_D.

The APV method is useful for valuing firms with a changing capital structure since the return on assets is independent of capital structure. For example, in a leveraged buyout, the debt to equity ratio gradually declines, so the required return on equity and the weighted average cost of capital change as the lenders are repaid. However, when calculating the terminal value it may be appropriate to assume a stable capital structure, so in calculating the terminal value in a leveraged buyout situation the WACC method may be a better approach.


Flows to Equity Method
The flows to equity method sums the NPV of the cash flows to equity and to debt.

Then, V_L = E + D


WACC Method
The WACC method discounts the unlevered free cash flow at the weighted average cost of capital to arrive at the levered value of the firm.


Cash Flows to Debt and Equity

When calculating the amount of cash flowing to debt and equity holders, it is not appropriate to use the unlevered free cash flows because these cash flows do not reflect the tax savings from the interest paid. Starting with the UFCF, add back the taxes saved to obtain the total amount of cash available to suppliers of capital.


Hurdle Price

At times a firm may wish to know at what price it would have to sell its product for a particular investment to have a positive net present value. A procedure for determining this price is as follows:

• Express the operating cash flow in terms of price. There may be multiple phases such as a short start-up period, a long operating period, and a final year in which the terminal value is calculated.
  

• Write out the expression for the NPV using the appropriate discount rate. For the longer operating period, one can calculate an annuity factor to multiply by the operating cash flow expression. Solve the expression for the cash flow that would result in an NPV of zero.
  

• Since the operating cash flow was written in terms of price, the price now can be found.
  

Debt Valuation

While debt may be issued at a particular face value and coupon rate, the debt value changes as market interest rates change. The debt can be valued by determining the present value of the cash flows, discounting the coupon payments at the market rate of interest for debt of the same duration and rating. The final period's cash flow will include the final coupon payment and the face value of the bond.


Investment Decision

If the unlevered NPV of a project is negative, aside from potential strategic benefits, the project is destroying value, even if the levered NPV is positive. The firm always could benefit from the tax shield of debt by borrowing money and putting it to other uses such as stock buybacks.


Optimal Capital Structure

The total value of a firm is the sum of the value of its equity and the value of its debt. The optimal capital structure is the amount of debt and equity that maximizes the value of the firm.


Share Buyback

If a firm has extra cash on hand it may choose to buy back some of its outstanding shares. One interesting aspect of such transactions is that they can be based on information that the firm has that the market does not have. Therefore, a share buyback could serve as a signal that the share price has potential to rise at above average rates.


Mergers and Acquisitions

Companies may combine for direct financial reasons or for non-financial ones such as expanding a product line. The target firm usually is acquired at a premium to its market value, with the hope that synergies from the merger will exceed the price premium. Mergers and acquisitions do not always achieve their goals, as promised syngeries may fail to materialize.

Appendix

Compounding and Discounting

Compound annual growth rate (CAGR): ( FV/C )^1/T - 1

Continuous compounding: FV_t = C e^{r t}

Perpetuity: PV = C / r

Growing perpetuity: PV = C / ( r - g )

T-year annuity (T equally spaced payments): PV = ( C / r ) [ 1 - 1/(1+r)^T ]

T-year growing annuity: PV = [C / (r - g)] { 1 - [(1+g) / (1+r)]^T }


HP 19BII Calculator Tip

IRR Calculation:

Press the yellow button then "EXIT" to reset the calculator.

Press button under "FIN"

Pess button under "CFLO"

Press yellow button then INPUT to clear list

Press button under "YES"

Enter the initial cash inflow (negative number for outflow). Press "INPUT".

For "FLOW(1)", enter the cash flow value for the end of year 1, then press "INPUT".

Enter the number of periods for that value, then press "INPUT".

For "FLOW(2)", enter the next cash flow value, then press "INPUT".

The number of times will default to the previous number. Press "INPUT" to keep, or enter a new value.

When the cash flow entries are complete, press the button under "CALC".

Press the button under "IRR%" to calculate the IRR of the cash flow.