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Asset Valuation Using Discounted Cash Flows
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Course Instructions - How to use the Materials
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The EBook contains Review Questions section(s) that review the materials at the end of each significant chapter.
The Review Questions are an Interactive requirement of NASBA and are included to assist you in understanding the
material better. They are NOT graded however you will find them useful in reinforcing the materials that you have
read. The Explanations/Answers to the Review Questions sections are located at the end of the EBook before the
actual CPE exam.
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A copy of the CPE Exam is located at the end of the EBook.
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Course Description
The purpose of this course is to provide an overview of the Discounted Cash-Flow
(DCF) Method of valuation. The DCF method requires that an estimated cash-flow and
a risk-adjusted discount rate be determined. This course summarizes commonly-used
cash-flow proxies and discount rate estimation tools.
COURSE LEARNING OBJECTIVES
1. Define assets and equity
2. Compare cash-flow to U.S. GAAP-based net income
3. Compare commonly-used discount rate models
4. Recognize the relationship between risk and reward
5. Recognize commonly-used discount rate models
6. Compare commonly-used discount rate models
7. Understand how assets are valued using a discounted cash flow model
Level of Difficulty: Basic
Prerequisite: None
Field of Study: Accounting
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Contents
Introduction .................................................................................................................................................. 5
Review Questions...................................................................................................................................... 6
Chapter 1. Valuation – Background ............................................................................................................. 7
Review Questions.................................................................................................................................... 14
Chapter 2. Future cash flows ..................................................................................................................... 15
Cash Flow vs. U.S. GAAP-based Net Income ........................................................................................... 15
Cash Flow Estimation Models ................................................................................................................. 17
Cash Flow Model – EBITDA ................................................................................................................. 17
Cash Flow Model – Income from Continuing Operations ................................................................... 24
Cash Flow Model – S&P Core Earnings ............................................................................................... 27
Summary – Cash Flow Estimation ........................................................................................................... 30
Review Questions.................................................................................................................................... 30
Chapter 3. Discount rate. ........................................................................................................................... 32
Risk – Reward Relationship ..................................................................................................................... 32
Estimating the Discount Rate .................................................................................................................. 38
Capital Asset Pricing Model (CAPM) ................................................................................................... 38
Arbitrage Pricing Model (APM) ........................................................................................................... 46
Comparison of APT and CAPM ............................................................................................................ 50
Summary – Discount Rate ....................................................................................................................... 52
Review Questions.................................................................................................................................... 54
Chapter 4. The complete model - Value equals estimated cash-flow divided by the discount rate ......... 56
Review Questions.................................................................................................................................... 59
Conclusion ................................................................................................................................................... 60
Glossary ....................................................................................................................................................... 63
Index............................................................................................................................................................ 64
Review Question Answer Feedback ............................................................................................................ 65
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Introduction
By the end of this unit, learners should be able to -
Recognize commonly-used discount rate models
According to the Financial Accounting Standard Board’s Statement of Financial
Accounting Concepts number 6,
Assets are probable future economic benefits obtained or controlled by a
particular entity as a result of past transactions or events.
Assets can also be defined algebraically as -
Assets = Liabilities + Shareholders’ Equity (i.e. Net Assets).
Shareholders’ equity is often referred to as net assets since assets minus liabilities
equals shareholders’ equity (shareholders’ equity = assets – liabilities). With this in
mind, one can also state that, for a firm with no liabilities, total assets equals total net
assets (assets = 0 + net Assets).
For simplicity and consistency in this course, liabilities will be assumed to be zero and
firms’ equity structures will be assumed to include only common stock. No preferred
stock or hybrid securities will be considered in this course. So, the terms assets, net
assets and shareholders’ equity will be used interchangeably. Also, common stock
shareholders will be assumed to hold 100% of the value of the firm’s assets and 100%
of the firm’s net assets.
Across the globe, assets are bought and sold millions of times each day. These
transactions may involve the exchange of assets ranging from -
a single share of common stock (equity) representing ownership in a large
publicly-traded entity to
a 100% sale of a small privately-held company.
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Regardless of the size of the transaction or the nature of the asset(s) involved, one
important piece of information must be determined before any of these transactions can
take place. This vital statistic is the price or value of the asset(s) in question.
Review Questions
Review Questions are required by NASBA and are designed to enhance the learning process. They are
NOT graded. Answers can be found at the end of the EBook.
1. Assets can be defined as
liabilities + shareholders’ equity. liabilities less equity. probable future economic sacrifices. benefits resulting from future transactions or events.
2. Assets can equal shareholders’ equity if
A. no liabilities are present. B. shareholder equity is less than total liabilities C. liabilities are positive and equal to shareholders’ equity. D. only current assets are owned by the company.
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Chapter 1. Valuation – Background
By the end of this chapter, learners should be able to -
Recognize commonly-used discount rate models
Asset valuation methods have long been the focus of research and debate. Valuations
are necessary for a number of reasons. A few of the more common situations that call
for asset valuation techniques to be used involve -
Business Sales
Entire business
Partial interest
Common stock sales – primary and secondary
Estate Planning – Taxation
Real property
Collectibles
Family business
Litigation
Patent infringement
Breach of contract
Divorce
Many valuation methods are used. The choice often depends on the specific scenario
and the individual valuation analyst performing the service. Following is a review of
three commonly-used valuation methods.
Fair Market Value.
Market Comparison Method.
Earnings Method.
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Fair Market Value
Fair Market Value refers to an amount mutually and voluntarily agreed upon by both
buyer and seller. The Internal Revenue Service succinctly defines fair market value as,
“…the price that property would sell for on the open market. It is the price that would be
agreed on between a willing buyer and a willing seller, with both being required to act,
and both having reasonable knowledge of the relevant facts.” For instance, publicly-
held businesses may use the stock market to determine its fair market value. If a stock
is heavily traded and it is assumed that the market is efficient, then it can be assumed
that the total shares outstanding multiplied by the stock price per share would provide
one reasonable estimate of firm value. As of February 2012, General Mills, Inc. (GIS)
had nearly 650,000,000 shares of common stock outstanding. GIS closing price on
February 27, 2012 was $38.04 per share. Hence, the company’s market value – also
known as market capitalization - would be nearly $25 Billion.
[650,000,000 x $38.04]
Market Comparison Method
Valuation of privately-held firms may involve derivative methods. Using the market
comparison method, a private firm would be compared to a similar publicly-traded firm.
The value of the private firm would be based on the value of the public firm – adjusted
for the privately-held firm’s unique characteristics. For example, a privately-held
hardware store’s value could be derived by first calculating the value of Home Depot.
So, if Home Depot’s fair market value (market capitalization) is calculated at $72 billion,
then this figure could be compared to Home Depot’s revenue, net income, assets and
other key figures to determine a set of ratios. These ratios would then be used as a
basis to begin valuing the privately-held entity. Consider the following example which
compares Home Depot’s financial information with Handy-Helper Hardware -- a
fictional, privately-held retail hardware company.
Fair Annual Total Annual
Market Value Sales Assets Net Income
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Home
Depot:
$72,000,000,00
0
$70,000,000,00
0
$39,000,000,00
0
$3,500,000,00
0
Handy-
Helper
Hardware:
$???
$80,000,000
$28,000,000
$5,000,000
Ratio: 0.1143% 0.0718% 0.1429%
After gathering the necessary data and calculating ratios, the subjective part of the
process begins. Generally, each component ratio is given a relative weight. The final
weighted average ratio is used to determine Handy-Helper Hardware’s market value.
To find an estimated market value for Handy-Helper, assume these weights and ratios –
Annual Total Annual
Sales Assets Net Income
Ratio: 0.1143% 0.0718% 0.1429%
Weight 25% 20% 55%
The weighted average comparable ratio is then calculated as follows.
Comparable Ratio = (Annual Sales * Relative Weight) + (Total Assets * Relative
Weight) + (Annual Net Income * Weight)
Comparable Ratio = (.001143*.25) + (.000718*.20) + (.001429*.55)
Comparable Ratio = .1215%
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To solve for Handy-Helper’s market value, multiply Home Depot’s market value by the
Handy-Helper’s comparable ratio.
Handy-Helper Hardware Value = Home Depot Value * Comparable Ratio
Handy-Helper Hardware Value = $72,000,000,000 * .1215%
Handy-Helper Hardware Value = $87,480,000
This method has value but much subjectivity is involved. Mainly, the key components
(e.g., Revenue, Net Income) and the relative weights chosen can significantly affect the
calculated market value.
Earnings Method
The earnings method values assets, from individual shares of stock to entire business
enterprises, by discounting the expected future cash flows (i.e. economic earnings)
generated by the asset, then discounting those future cash flows by a risk-adjusted rate-
of-return.
Most economic earnings valuation models employ a similar formula. These methods all
use some form of discount rate and a future cash flow projection. The goal of these
models is to discount all future cash flows to present value. Consistent with the
parameters set forth in the course introduction, equity valuation models refer specifically
to the value retained by the stockholder after debt obligations has been considered. A
generic discounted cash flow (DCF) equity valuation model follows:
n
tt
t
rExpected
CFExpectedP
1
01
][
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A simplified presentation of the model components –
Where -
0P = Price (Value) of the equity per share
1CF = net cash-flow in future period (1, 2, 3...) per share
r = discount rate (i.e., required rate-of-return)
This means that the current value of an asset (e.g., share of common stock) is directly
related to the future cash flows received by the shareholder and inversely related to the
uncertainty of receiving those future cash flows.
Note: This model assumes that the asset will be held indefinitely, so there is no
additional cash-flow component which represents the present value of the future
asset sale.
For example – to determine the value of one share of XYZ Co. common stock
Find -
The future periodic net cash flow
and an appropriate risk-adjusted discount rate.
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Assume that XYZ’s future periodic cash flow (1CF ) is $1.50/share. Next, assume that
12% is an appropriate discount rate. Based on these assumed variables, the generic
DCF equity valuation model calculates XYZ’s common stock value per share to be -
$12.50 [1.50/12%]
In theory, this value per share can be extended to encompass the value of the entire
business by multiplying the value per share times the total shares outstanding.
The generic DCF formula seems simple at first glance. After all, the equation only
contains a few basic parameters. However, accurately estimating each variable is
enormously complex.
Over the past several decades, many great financial minds have generated seminal
works related to equity valuation models.
A few notable authors are –
Eugene Fama – Significantly contributed to APT-related research
Fischer Black – One of two researchers – the other being Nobel Prize laureate,
Myron Scholes - credited with developing the Black/Sholes options pricing model.
Kenneth French - Significantly contributed to APT-related research
Stephen A. Ross – Credited with the development of the Arbitrage Pricing Theory
(APT).
Richard Roll – In accordance with fellow researcher, Stephen A. Ross, performed
research and published significant APT-related findings
William Sharpe – Credited with the initial development of the capital asset pricing
model (CAPM).
Jack Treynor – Significant CAPM follow-up research and consulting. President –
Treynor Capital Management.
Sample journal articles written by these researchers include:
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Black, F., 1986, Noise, Journal of Finance 41(3), pp. 529-543
Black, Fischer and M. Scholes, 1972, The valuation of option contracts and a test of
market efficiency. Journal of Finance, 27(2), pp.399-418.
Roll, R. and S. Ross (1980) An empirical investigation of the arbitrage pricing theory,
Journal of Finance 35(5), pp. 1073-1103.
Ross, S. (1976) The arbitrage theory of capital asset pricing. Journal of Economic
Theory, 13(3), pp341—360.
Sharpe, William (1964) Capital asset prices: A theory of market equilibrium under
conditions of risk, Journal of Finance,19 (3), pp. 425-442.
Summers, L., 1986, Does the stock market rationally reflect fundamental values?
Journal of Finance 41(3), pp. 591-601.
These researchers and other notable financial scientists continue to develop theoretical
valuation concepts. Yet, no one valuation concept has received full acceptance from the
academic and professional community.
A few of the more popular valuation theories and models incorporate one or more of the
following:
Black-Scholes Model – options valuation pricing model
Dividend Discount Model (DDM) – considers dividend payments and growth
rates to calculate equity valuation. This model considers common stock
dividends to be a reasonable proxy of cash flow available to common
stockholders.
Multi-stage Growth Model – similar to the DDM, this model considers changing
growth rates over time.
These and other valuation models and theorems are elaborate. This course will avoid
the more technical areas of this subject. Instead, this material will concentrate on the
major components shared by all discounted cash-flow models. The material will be
presented in three sections:
Future cash flows
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Discount rate
The complete model
Review Questions
Review Questions are required by NASBA and are designed to enhance the learning process. They are
NOT graded. Answers can be found at the end of the EBook.
1. Valuation methods are commonly used for all of the following except
A. estate planning. B. divorce litigation. C. patent litigation. D. compliance auditing.
2. Notable financial models used in asset valuation include all of the following
except the
A. CAPM. B. Dividend Discount Model. C. Arbitrage Pricing Model. D. Demaline Discount Model.
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Chapter 2. Future cash flows
By the end of this chapter, learners should be able to –
Compare cash-flow to U.S. GAAP-based net income
Compare commonly-used discount rate models
Cash Flow vs. U.S. GAAP-based Net Income
It is logical to conclude that one must first determine actual cash flow before estimating
future cash flow. Hence, determining actual periodic sustainable cash flows will be
the specific focus of this section.
The Financial Accounting Standards Board (FASB) and the International Accounting
Standards Board (IASB) have been working on a convergence project for several years.
One of their first completed projects was the release of the joint Conceptual Framework
for Financial Accounting (Phase A). FASB released this initial work on the framework
as Statement of Financial Accounting Concepts (SFAC) number 8.
SFAC 8 Conceptual Framework for Financial Reporting -
Chapter 1, The Objective of General Purpose Financial Reporting, and
Chapter 3, Qualitative Characteristics of Useful Financial Information
SFAC 8, suggests that financial information should be useful in predicting future cash-
flows. Under a subjection entitled, Financial Performance Reflected by Accrual
Accounting, SFAC 8 (OB.18) states,
Information about a reporting entity’s financial performance during a
period, reflected by changes in its economic resources and claims other
than by obtaining additional resources directly from investors and creditors
(see paragraph OB21), is useful in assessing the entity’s past and future
ability to generate net cash inflows…
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According to SFAC 5, paragraph 141, accrual accounting attempts to recognize
revenues when earned and expenses when incurred. Accrual accounting methods also
attempt to match expenses with the revenue that they help to produce.
Periodic accruals are a key part of providing timely financial information. However,
these accruals are subject to a significant amount of personal and professional
judgment. This often motivates company management to make accruals, not based on
their best estimate of economic reality, but based on what would provide the highest
compensation for management. Researchers in this field have coined the terms,
“discretionary accruals” and “earnings management” in reference to this opportunistic
behavior.
SFAC 8 suggests that one purpose of financial information is to provide an indication of
future cash-flows. However, there are several reasons that GAAP net income rarely
corresponds to actual cash-flows. Many of these differences result from accrual-based
adjustments as well as varying interpretation of GAAP from company-to-company.
Weaknesses
Manipulation is inherent in accrual-based financial information. In addition, many
components of GAAP-based earnings are the result of subjective, temporary and
volatile changes in asset and liability amounts. For example -
Depreciation and amortization
Estimation of warranty expenses
Estimation of bad debt allowance
Fair value adjustments
These are examples of items recognized in the financial statements. This type of
recognition is contingent upon the corporate accountant’s judgment and the
“independent” auditor’s approval, both of which could be skewed by the desires of
corporate management. This subjectivity often results in inaccurate balance sheet
reporting and related inaccuracies on the income statement.
Other changes in balance sheet account valuations may not be recognized in the
financial statements. For example,
An increase in land value does not appear on the balance sheet or as a
corresponding gain in net income.
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Only certain pension –related liabilities are recognized on the balance sheet.
Certain intangible assets, such as human capital (employee value), are not
directly reported on the balance sheet.
In response to confusion regarding U.S. GAAP-based valuation, FASB has changed
their focus in recent years. They have issued significant fair value guidance (see:
Accounting Standards Codification Topic 820). These changes offer guidance –
particularly as it relates to terminology and footnote disclosure.
While, theoretically, fair value accounting should result in a balance sheet with fair
market values for all assets, liabilities and equity items, in practice this is far from the
truth. The complexity of fair value calculations, along with the subjectivity, ulterior
motives and varied methods of calculations, often lead to balance sheets that are less
useful than the “historical cost” alternative. Some researchers have even suggested that
fair value accounting methods misled investor and contributed to the recent financial
crisis.
In addition, current fair value guidance does not adequately address many of the
specific areas that cause GAAP-based earnings to diverge from economic earnings.
Cash Flow Estimation Models
These limitations inherent in GAAP-based net earnings have created a need for
alternative earnings calculations. A discussion of a few common methods used to
convert GAAP-based income into a more useful cash-flow (economic earnings) figure
follow.
Cash Flow Model – EBITDA
Earnings Before Interest, Taxes, Depreciation and Amortization (EBITDA) has been
used as a cash-flow proxy for many years. EBITDA has become an increasingly
popular financial measure in recent years. It attempts to convert net income to a better
approximation of operating cash flow. EBITDA begins with the GAAP-based earnings
amount then adds back - interest, taxes, depreciation and amortization.
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Example:
Consider the following hypothetical income statement information.
Jones Sporting Goods Inc
Income Statement (Condensed)
For the Year ending December 31, 20XX
Net Sales $2,850,000
Cost of Goods Sold $1,400,000
Gross Margin $1,450,000
Sales, General & Administrative Expenses
Advertising $128,000
Depreciation and Amortization $210,000
Office Expense $85,000
Other… $300,000
Total Sales, General & Administrative Expenses $723,000
Income before Interest & Taxes $727,000
Interest
Expense $25,000
Income Taxes $230,000
Net Income $472,000
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Jones Sporting Goods EBITDA would be calculated by adding total depreciation and
amortization expenses to income before interest and taxes.
Income Before Interest & Taxes $ 727,000
Add: Depreciation & Amortization $ 210,000
$ 937,000
GAAP-Based Net Income $ 472,000
Difference Between Net Income & EBITDA $ 465,000
Note the significant difference between GAAP-based Net income and EBITDA.
Benefits.
EB I TDA
Interest
Interest is removed from the calculation for comparability purposes. When interest
expense is removed, one can compare firms with different leverage. Removing interest
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also allows analysts to review potential future cash flows on the assets if the debt were
paid in full (e.g., asset acquisition/takeover).
EBI T DA
Taxes
The reason for removing taxes is that it is an uncontrollable expense. In theory,
management decisions will not affect tax rates. Hence, the tax expense should be
removed from all net income calculations to create comparable operating figures.
Removing taxes also permits acquiring firms to review potential future cash flows based
on their own tax rates.
EBIT D A
Depreciation
Depreciation and Amortization are “non-cash” expenses. Adding those expenses back
to net income provides a better proxy for cash flow in the current period. In fact, this is
why the same reconciliation is made when preparing the Statement of Cash Flows
(Indirect Method).
According to SFAC 5, depreciation is a systematic and rational allocation of a long-lived
asset over more that one accounting period. Depreciation is a system of allocation not
valuation. Since depreciation is arbitrary it is not considered in the cash flow
calculation using EBITDA.
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Depreciation Example – Differing GAAP methods result in differing amounts of periodic
depreciation.
Capital Asset: Delivery Truck
Cost: $45,000
Estimated useful life: 5 years or 25,000 miles
Salvage Value: $5000
Annual Mileage: 1: 22,000 2: 28,000 3: 26,000 4: 27,000 5: 21,000
Comparison of three acceptable GAAP depreciation methods:
1 2 3 4 5
straight line $ 8,000 $ 8,000 $ 8,000
$
8,000 $ 8,000
double-declining
balance $ 18,000 $ 10,800 $ 6,480
$
3,888 $ 832
units of production
($.32/mi ) $ 7,040 $ 8,960 $ 8,320
$
8,640 $ 7,040
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As depicted in the table and graph, differing GAAP-approved methods result in differing
amounts of periodic depreciation. This arbitrary allocation of capital asset costs results
in a net income figure that is not consistent with economic earnings.
EBITD A
Amortization
As technology continues to play a more prominent role in the economy, amortization
and the accounting methods used to recognize amortization are beginning to have a
material effect on overall GAAP-based earnings.
Accounting for intangibles and amortization is covered in FASB ASC topic 350.
For instance, FASB ASC 350-20-25-3, Intangibles – Goodwill and Other – Goodwill –
Recognition requires that costs associated with developing intangible assets that are
not specifically identifiable, have indeterminate lives, or are inherent in continuing
operations be expensed. This treatment results in skewed financial reports that show a
lower net income in the current period and a lower asset valuation going forward. It also
suggests that periodic amortization may be lower than economic reality, since many
intangible assets were immediately expensed.
Depreciation Comparison
$-
$5,000
$10,000
$15,000
$20,000
1 2 3 4 5
Year
$
straight line
double-declining
balance
units of production ($.32/mile )
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Due to the difficulty in accurately presenting intangible asset information, FASB and the
IASB regularly consider adjustments to intangible asset guidance. However, the
inherent complexity in the calculations and relatively high cost associated with
performing these appraisals makes the issue of intangible asset valuation an ongoing
challenge.
For the reasons presented, the EBITDA method of cash flow estimation eliminates
some of the subjectivity present in GAAP-based net earnings.
Limitations.
EB I TDA
Interest
The amount of interest expense is dependent upon the firm’s capital structure. This
structure is likely a significant and ongoing characteristic of the company. So, removing
the interest expense from the calculation is not logical for most investors. However, if a
firm is a takeover target, the potential buyer may remove the interest cost, and then
replace it with the finance costs of the acquiring firm. So, interest expense is not being
removed but recalculated based on assumed changes in the company’s controlling
shareholders.
EBI T DA
Taxes
Even though – for the most part – income tax expense is not influenced by management
behavior, it is still (assuming a firm will earn future net income) an ongoing cash
expense. Much as with interest, a potential investor would only remove taxes from the
calculation if they were planning on acquiring the entire firm. Thus, the projected tax
expense would replace the reported tax expense.
EBIT D A
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Depreciation
While it is accurate to say that depreciation and amortization are non-cash expenses.
They both are still expenses. Depreciation systematically expenses a capital asset over
the course of multiple years. At some point these assets need to be retired and
replaced. Financial analysts often account for this by considering capital expenditures
(CAPEX) as part of their cash flow model. In other words, EBITDA removes the firm’s
depreciation estimate and financial analysts replace that expense figure (depreciation)
with their own estimate (CAPEX). Finally, EBITDA does not account for some of the
same limitation associated with the operating income method. For instance, EBITDA
does not consider certain components of GAAP-based earnings resulting from
temporary, subjective and volatile changes in asset and liability amounts.
EBITD A
Amortization
ASC Topic 350, Intangibles - Goodwill and Other is not a cure-all for the inherent
problems associated with accounting for intangibles. However, it is a marked
improvement over previous GAAP. According to Topic 350, amortization is no longer
considered a completely arbitrary expense. Firms no longer regularly amortize the cost
of intangibles that have indefinite life. Instead, firms are now required to regularly
evaluate their reported intangible assets (e.g., goodwill) to determine if the reported
amount is accurate. Thus, if a company has recognized amortization expense, financial
statement users can reasonably infer that the firm’s intangible assets have lost real
value. It can then be assumed that future cash flows associated with those intangible
have also been hampered.
Cash Flow Model – Income from Continuing Operations
GAAP-based income statements contain intermediate measures of income. Income
from continuing operations is separately presented from net income if the two figures
differ. The two figures may differ when an entity experiences one or more economic
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events that are not related to the firm’s going concern. Examples include extraordinary
items and discontinued operations. Before May 2005, cumulative effect of accounting
changes would have also been included in this section. However, pre-codification
Statement of Financial Accounting Standard (SFAS) 154 virtually eliminated this income
statement category, instead requiring retrospective treatment of voluntary accounting
changes. The purpose of this division between income from continuing operations and
net income is to provide a net income figure that better characterizes the firm’s future
cash-flow potential.
Hence, a straightforward way to adjust net income to net cash-flow is to use the income
from continuing operations figure provided on the income statement. The following
scenario exemplifies this technique.
Jordon & Nile Company
Income Statement (condensed)
For the Year Ended December 31, 20XX
($000)
Net Sales Revenues: 15,000
Cost of goods sold: 6,200
Gross profit on sales 8,800
Operating expenses: 3,500
Income from continuing operations before income tax 5,300
Income tax expense 1,800
Income from continuing operations 3,500
Discontinued operations (net of tax): (820)
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Extraordinary items: 255
Net income 2,935
Jordon & Nile’s income from continuing operations is already calculated on the income
statement.
Income from continuing operations 3,500,000
Net income 2,935,000
Difference Between Net Income & Income from operations 565,000
This, rather simplistic adjustment, suggests that actual cash flows are nearly 20%
higher than GAAP-based net income. More importantly, the income from operations
figure should not fluctuate due to the firm’s peripheral economic activity.
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Benefits
The primary benefit of the income from continuing operations method is its ease of
calculation. Actually, since a firm’s income from continuing operations is already
presented on their GAAP-based income statement, the figure is readily available without
any calculation being necessary.
Income from continuing operations – as its name suggests – also eliminates many of
the effects of discontinued or otherwise nonrecurring activities. This adjustment leaves a
consistent basis from which to estimate future economic earnings
Limitations
While the phrase, “income from continuing operations” may lead one to believe that this
figure should closely mirror continuing net cash flows, there are a few things that may
cause net cash flows and income from continuing operations to diverge. First, income
from continuing operations is a GAAP-based accrual figure. It does not make
adjustments for non-cash allocations such as depreciation and amortization. Second,
not all unusual or infrequent economic activity is excluded from the continuing
operations figure. Per ASC 225-20, Income Statement – Extraordinary and Unusual
Items, an item must be both unusual and infrequent to be excluded from continuing
operations figures. Accordingly, certain peripheral non-systematic revenue and
expenses may be encompassed in the continuing operations figure.
Cash Flow Model – S&P Core Earnings
In November 2001, Standard and Poor’s released a white paper delineating their new
profit measure – core earnings. Similar to the Income from operations measure
presented previously, the core earnings measure starts with GAAP-based net income
adjusted for three items -
extraordinary items,
cumulative effect of accounting changes, and
discontinued operations
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This figure, called “as reported earnings”, is then adjusted to eliminate the effects of
other peripheral economic items. The desired result is a periodic profit figure that will
more accurately reflect current economic profit.
The adjustments to GAAP-based earnings are summarized below.
Included in Core Earnings –
Employee stock option grant expense
Restructuring charges from ongoing operations
Write-downs of depreciable or amortizable operating assets
Pension costs
Purchased research & development expenses
Excluded from Core Earnings –
Goodwill impairment charges
Gains/losses from asset sales
Pension gains
Unrealized gains/losses from hedging
Merger/acquisition related expenses
Litigation or insurance settlements and
Items included in core earnings are those items which are put into the calculation of
core earnings if they were not already included in the firm’s as reported figure. Items
excluded from core earnings are removed from the as reported earnings figure
calculation.
Generally core earnings are less than GAAP-based earnings. Consider the comparison
for General Mills from fiscal 2007 – fiscal 2011.
Per Share Data ($)
2011 2010 2009 2008 2007
GAAP Earnings: 2.70 2.24 1.90 1.85 1.59
S&P Core Earnings: 2.62 2.09 1.44 1.65 1.52
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S&P Core Earnings are lower each year. In 2009, core earnings were nearly 25% lower
than GAAP earnings.
Benefits
Proponents of core earnings figures suggest that the core figure provides a more
accurate measure of continuing earnings potential. One could infer from the Standard &
Poor’s core earnings white paper that the core earnings calculation is more reliable and
more useful than as stated earnings. By removing the item not central to operations, the
core earnings figure provides a better starting point for which to estimate future
economic earnings.
Limitations
Compared to the other profit measures presented thus far, the core earnings calculation
is the measure that best mirrors actual cash flow. However, to a lesser degree it shares
limiting characteristics with the income from continuing operations figure.
First, core earnings do not make adjustments for non-cash allocations such as
depreciation. Second, much but not all unusual or infrequent economic activity is
excluded from the continuing operations figure.
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Summary – Cash Flow Estimation
The discounted-cash flow method (DCF) is one common equity/business valuation
approach. This course summarizes a few of the more popular cash-flow proxy models
used to complete that portion of the DCF.
EBITDA
Income from Continuing Operations
Standard & Poor’s Core Earnings
Each model presented has its own set of benefits and limitations. Most importantly, no
one method is perfect. Accordingly, any discounted-cash flow valuation model that
implements one of these cash-flow proxies is subject to some level of error. Finally,
note that these cash-flow proxies are only the starting point for developing cash-flow
forecasts to use in a DCF valuation model. Once a single-period cash-flow proxy is
determined, that cash-flow estimate must then be combined with other information (e.g.
a series of historical periodic cash-flow estimates and forecasted economic changes) to
better forecast future cash-flows. It is that estimate of future cash-flows which must be
discounted by an appropriate risk-adjusted rate, to determine an asset’s value. A
discussion of these calculations is the topic of section two.
Review Questions
Review Questions are required by NASBA and are designed to enhance the learning process. They are
NOT graded. Answers can be found at the end of the EBook.
1. Fair value accounting is directly addressed in
A. ASC Topic 450. B. ASC Topic 220. C. ASC Topic 820. D. ASC Topic 330.
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2. Examples of GAAP-based expenses that are commonly subject to estimation
include
A. cash. B. warranty expense. C. gain on publicly-traded securities available for sale. D. cash dividends payable.
3. Income from continuing operations
E. excludes amortization expense. F. is not included on GAAP-based income statements. G. can be used as a proxy for economic earnings (cash-flow). H. excludes all adjustments for voluntary accounting method changes.
4. Before calculating an asset value using the DCF method, the single period cash-
flow estimate must be
A. multiplied by two. B. combined with other data to determine future cash-flows adjusted for probable
economic changes. C. multiplied by the risk-adjusted discount rate-of-return. D. adjusted for the tax benefit associated with future depreciation charges.
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Chapter 3. Discount rate.
By the end of this chapter, learners should be able to –
Recognize the relationship between risk and reward
Recognize commonly-used discount rate models
Compare commonly-used discount rate models
Risk – Reward Relationship
Previously, the reader was introduced to a simple generic discounted cash-flow (DCF)
equity valuation model:
Where:
0P = Price (Value) of the equity per share
1CF = net cash-flow in future period (1, 2, 3...) per share
r = required rate-of-return (i.e. risk-adjusted discount rate)
Two methods are commonly used to determine the discount rate (the denominator)
used in the discounted cash-flow equation.
Capital Asset Pricing Model (CAPM) – [a risk calculation model which calculates
equity risk by adjusting market-level risk by individual company risk as measured
by the individual equity’s relative volatility (beta).] and
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Arbitrage Pricing Theory (APT) – [Like CAPM, APT is a risk calculation model.
However, individual equity risk is calculated by combining multiple risk factors as
opposed to simply considering the equity’s beta.]
These risk estimation methods are rather complex. This section will focus on the
specifics that comprise the two theories and summarize their
components,
benefits and
limitations
in a straightforward, comprehendible manner.
Before addressing the specifics of CAPM and APT, the general risk-reward
relationship will be explored.
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Note: In financial terms, “risk” is the interplay linking peril and progress.
When referring to the DCF model, risk entails the uncertainty related to the amount and
timing of projected cash-flows used in the numerator of the discounted cash-flow
projection. The goals of CAPM and APT are to accurately reflect equity risk so that the
required rate-of-return (discount rate) can be determined. Note that as the cash-flows
become more uncertain, the perceived risk increases.
By decomposing these concepts, one arrives at the following:
Risk increases as uncertainty increases. Conversely, as uncertainty decreases, risk
decreases. Required rate-of-return increases as risk increases. So, assuming cash-
flow projections are held constant, one can surmise that the value of an asset
decreases as the level of projected cash-flow become more uncertain.
This relationship is graphically illustrated by the Security Market Line (SML). The SML
displays the relationship between the relative sensitivity of a particular equity’s return
compared to market returns (beta) and that equity’s expected return. As the stock’s beta
increases the expected (required) rate-of-return also increases.
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Security Market Line – Graphic Representation
Where:
E(rtn) – Expected return (i.e., required rate-of-return)
MRP – Market Risk Premium
Rf – Risk-free rate
Beta ( ) – Sensitivity to Market returns
These terms will be further discussed in subsequent sections of this course.
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Uncertainty.
Merriam-Webster (www.m-w.com), suggests that the term “uncertain” is synonymous
with the term, “doubtful”. For use in valuation analysis, “uncertainty” refers to the
likelihood of receiving future cash-inflows from an investment of current cash out-flows.
One of Ben Franklin’s famous quotes, “Nothing in life is certain except death and taxes”,
could be slightly amended. Most modern American financial researchers agree that
repayment of United States treasury debt is also certain – or at least – close enough.
The interest rate on United States Treasury debt is commonly considered to be a “risk-
free” rate of return. Some disagreement lies in the term that should be used. Many
researchers use the 3-month U.S. T-bill as a risk-free (certain) benchmark. Still others
use the 10-year T-bond, 30-year T-bond or some other U.S. treasury debt instrument.
Whichever rate is chosen to represent the benchmark of “certainty”, this rate is then
theoretically considered to be the least amount that an investor will accept in exchange
for the use of their funds. An investor will expect to receive a premium over and above
the “risk-free” rate for investing in any project that is less than guaranteed to return a
certain, specific future cash flow.
Integrating this information into the DCF model, one can see that the purchase price of
U.S. treasury debt will be relatively high, resulting in a relatively low return (i.e., yield or
interest rate).
Where -
billtP = Amount paid for treasury debt instrument
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Now consider a similar debt security investment with the same anticipated cash flows –
interest plus return of face value – and timeline. However, this 3-month security, known
affectionately as the, “Z-bill”, is issued by the mythical national government of Zualau –
a nearly bankrupt small island country located 200 miles southwest of Nowhere. The
following DCF valuation model displays the “Z-bill’s” higher expected risk and its related
lower valuation.
Where:
billzP = Amount paid for “risky” debt instrument
Based on the information provided thus far, one could infer that,
billzbillt PP [All else remaining the same]
In an efficient (or even semi-efficient) financial market, it can be presumed that
individuals will expect to be compensated commensurate with the risk that they are
willing to bear.
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Once an effectively risk-free security is determined, then its total return can be used as
a risk-free rate-of-return proxy. The total return to be used is another area of
controversy. One must decide to use either the current yield, a historic average yield, a
moving average yield or some other return calculation. For simplicity, this course will
assume that the 3-month U.S. T-bill is an acceptable proxy for risk-free rate. The 3-
month U.S. Treasury bill has had an approximately 3.5% average yield over the past 50
years. So, this 3.5% figure will be assumed to be a reasonable estimate of the risk-free
rate for any calculations performed henceforth.
Estimating the Discount Rate
Capital Asset Pricing Model (CAPM)
Background
The CAPM method was originally developed by renowned financial researcher, William
F. Sharpe. Sharpe’s theory was initially published in the mid 1960’s; it has received
great notoriety ever since.
Sharpe postulated that an asset’s risk of future cash-flows can be calculated by
comparing the historic variability of its price to a benchmark.
Components
Specifically, Sharpe developed the CAPM to determine the risk associated with an
individual equity by comparing its price variability with the variability of the stock market
as a whole. This relative sensitivity to market returns is referred to as stock market-
related beta coefficient, stock beta or simply, beta.
Beta ( )
Beta is a measure of relative systematic risk. It is commonly used to compare the
volatility of an individual common stock’s price to the volatility of the stock market as a
whole. Normally, a widely-known stock index is used as a stock market proxy, Example
indexes include:
Dow Jones - Wilshire 5000 Total Stock Market Index
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o billed as the most comprehensive United States stock market
measurement index - represents the performance of all U.S. equity
securities with readily available price data
The New York Stock Exchange Composite
o represents about 80% of the total market capitalization of all U.S publicly-
traded companies
Value Line Index
o stock market index containing 1,700 companies representing various U.S.
stock exchanges
Standard and Poor’s 500 Index – most widely-used by finance researchers
o Consists of 500 of the largest U.S. companys’ stocks – chosen
considering characteristics such as: market value, liquidity and sector
group.
These indexes are frequently used as a surrogate for the equity market as a whole. An
individual stock’s beta is determined by comparing the periodic changes in the stock
index to the periodic changes in the individual stock’s price. By definition, the stock
market as a whole has a beta equal to 1. An individual stock whose periodic price
changes more than the index has a beta of greater than 1. A stock price that is less
volatile than the index has a beta of less than 1. The chart below depicts hypothetical
periodic price changes of two individual stocks and a market index. This chart
graphically displays the volatility difference between stocks with a high beta (> 1) and
stocks with a low beta (< 1).
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The CAPM first suggests that common stock equities – such as those that compose the
Standard & Poor’s 500 Index – are relatively more risky than U.S. treasury securities.
Hence, a stock market risk premium (MRP) is automatically incorporated into the value
of common stocks that are traded in an efficient market. This risk premium is generally
calculated by subtracting the risk-free rate from the stock market expected (mean)
return. Certain institutions, for example, Ibbotson Associates, specialize in providing
historical data and analysis of financial information such as current and historical:
market risk premium (MRP)
Relative Stock Volatility
-40.0%
-30.0%
-20.0%
-10.0%
0.0%
10.0%
20.0%
30.0%
40.0%
1 4 7 10 13 16 19
Time
% c
han
ge
(Vo
lati
lity
)
Market Index
1.5 beta stock
.5 beta stock
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T-bill yields,
T-bond yields,
dividend yields and stock market returns, including various
- Dow Jones equity averages,
- NASDAQ information, and
- Standard & Poor’s index data.
According to Ibottson’s web site (www.ibbottson.com),
…Ibottson products and personnel will supply the data, explain and
illustrate the concepts, and provide the analytical tools you need to help
you gather grow and retain assets...
Depending on which source is used to assemble the MRP, and which figures are used
to represent the risk-free rate and the stock market return, this risk premium usually
ranges from 5 – 8% per annum. For simplicity, a 7% MRP will be assumed to be
sufficiently accurate for any future calculations presented in this course.
CAPM is mathematically described by the following equation:
market stock torelativebeta
a
fafa RMRPRR
premium risk marketMRP
rate freeriskf
R
rate) (discount asset individual anofriskaR
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So, the risk of a specific asset, such as a share of a firm’s common stock, can be
determined by adding the risk-free rate to an adjusted MRP based on the individual
asset’s relative price stability.
Illustration – CAPM
Consider the following hypothetical information relating to XYZ Corporation’s common
stock equity shares.
stock beta = 1.25
Now, one only needs to link this information to the Capital Asset Pricing Model. The
solution is the risk-adjusted discount rate.
This discount rate (7.875%) would then be used as a basis for the denominator in the
discounted cash-flow model.
Pxyz = CF1/Rxyz
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Once the expected future cash-flow (CF1) is properly calculated the empirical value of
XYZ’s common stock (Pxyz) can be calculated.
The following graph and table compare an asset’s price based on the uncertainty of one
particular cash-flow.
Value of $1 - Expected Cash-Flow
$-
$20.00
$40.00
$60.00
$80.00
$100.00
$120.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Required Rate-of-Return (risk)
Pri
ce
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The graph and table show that as the required rate-or-return (risk) increases, the price
of the related asset decreases.
Benefits
CAPM’s primary benefit is its apparent simplicity. One only needs to determine a few
simple variables. These factors include a risk-free rate and a stock beta. Both of these
figures are widely available. Many publishers provide both current and historical U.S.
Treasury security yields including Moody’s, Standard and Poor’s and Ibottson. The
official U.S. treasury website, www.treasury.gov, also provides a wide variety U.S.
treasury securities information. Stock betas are available from a variety of sources
including financial research publishers, Standard and Poor’s and Value Line.
One may assume that use of any historical financial data in an attempt to forecast future
financial information is flawed. However, empirical evidence suggests that stock betas
tend to remain relatively constant over time. In addition, even though treasury rates
vary over time, they tend to invariably revert toward their long-term mean.
Limitations
Even though empirical evidence suggests that betas tend to remain relatively constant
over time, certain events may permanently and significantly change a firm’s future stock
beta. A few examples include:
Price $ CF risk
$ 100.00 1.00 1%
$ 50.00 1.00 2%
$ 33.33 1.00 3%
$ 25.00 1.00 4%
$ 20.00 1.00 5%
$ 16.67 1.00 6%
$ 14.29 1.00 7%
$ 12.50 1.00 8%
$ 11.11 1.00 9%
$ 10.00 1.00 10%
$ 9.09 1.00 11%
$ 8.33 1.00 12%
$ 7.69 1.00 13%
$ 7.14 1.00 14%
$ 6.67 1.00 15%
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Top Management change,
Financial characteristics changes (leverage adjustments)
Industry and competitive changes
New Legislation
Another significant CAPM limitation is that a stock beta cannot be determined unless a
legitimate market for the equity exists. In other words, privately-held securities or thinly-
traded securities may not have an accurately developed stock market-related beta
coefficient.
Finally, the determination of the discount rate may be overly simplistic. The CAPM
assumes that all individual, non-diversifiable components of an individual asset’s risk
are linked to the stock market. Hence, the only adjustment to the risk-free rate is an
addition of a general market risk premium manipulated by the asset’s volatility as it
relates to the volatility of a stock market index.
Summary of CAPM
Stephen Sharpe’s Capital Asset Pricing Model was a significant advancement in the
field of finance. The CAPM is a succinct mathematical model that offers a reasonable
proxy of an asset’s risk and subsequent required rate-of-return. CAPM is a useful risk
measurement model but a model that has its limitations. Many contemporary financial
researchers feel that the CAPM is too simplistic to fully reflect all of the variables that
determine an individual equity’s risk characteristics. One, potentially more useful, risk
calculation method was developed by Stephen Ross, Richard Roll and other finance
researchers. This risk evaluation method is known as the Arbitrage Pricing Theory.
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Arbitrage Pricing Model (APM)
Background
The Arbitrage Pricing Theory (APT) was originally developed by finance professor,
Stephen A. Ross. Ross’ theory was initially published in the mid-1970’s, It has received
much attention and has been the topic of great debate ever since.
The Arbitrage Pricing Theory can be expressed mathematically as a multi-factor risk
model. For precision this model will be referred to as the Arbitrage Pricing Model
(APM). This course will discuss the APT and express the APM in elementary terms by
minimizing the discussion of many of its more complex components. The specifics of
the APT can become rather convoluted. However, its basic assumptions are
straightforward. APT gets its name from one of its preliminary assumptions. That
assumption is that, in an efficient equity market, arbitrage (riskless profits) will be quickly
and effectively eliminated by knowledgeable investors. In simple terms, all available
information including:
Consensus forecasts of future developments such as anticipated
changes in interest rates,
inflation and
other economic factors
are already priced into the equity market.
Components
The APM, unlike the Capital Asset Pricing Model, does not depend upon equity-relative
stock market volatility. Arbitrage Pricing Theory implies that a particular equity risk can
be determined by summing multiple sources of macroeconomic risk. The main
components of APT can be mathematically represented with the APM:
Where:
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Ri = Risk of the individual equity
rf = risk-free rate
bi = risk factors
i = weight of related risk factor (factor loading)
e = random volatility = 0 --- (presupposed by experience)
APT assumes that all random variability can be removed from a stock portfolio if the
portfolio of stock consists of enough complementary stock holding. Hence, APT
assumes that the random volatility (e) can be completely removed (e=0).
Thus, an individual stock’s total risk is the risk-free rate plus the summed and properly
weighted total of each non-diversifiable risk factor. That concept is simple enough.
However, the difficulty arises when one attempts to determine what the individual risk
factors are and - subsequently - the relative weights of those risk factors. Many
research studies have been performed in an attempt to quantify these parameters and
variables.
Over the years, Richard Roll and Stephen Ross have found that a few particular
economic factors seem to correlate to general equity market risk. These factors
included shocks in:
inflation,
yield curve shifts,
aggregate net production (GNP) and
investor confidence
Other researchers have tested similar factors in different ways.
For example, another multi-factor model was discussed in the following article,
Berry, M., Edwin Burmeister and Marjorie B. McElroy (1988) Sorting out risks using
known APT factors. Financial Analysts Journal. March-April, pp 29-41.
The authors consider the validity of using unanticipated changes in certain
macroeconomic variables including:
default risk,
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the term structure of interest rates,
inflation,
the long-run expected growth rate of profits and
While it is commonly assumed that the risk factors range from such things as inflation,
unemployment and credit risk to individual company financial performance volatility, no
one universal set of factors and factor loads has been developed.
So, while the multi-factor risk theory is very impressive, it is very difficult to implement in
practice. Even so, consider the following very basic hypothetical scenario.
Illustration -
ABC Company – risk analysis using the Arbitrage Pricing Model (multi-factor risk model)
as developed using APT.
Risk Factors* and
Relative weights** (factor loading)
Risk factor 1 9.5%
Risk factor 2 8%
Risk factor 3 11.5%
Relative weight of risk factor 1 0.5
Relative weight of risk factor 2 0.3
Relative weight of risk factor 3 0.2
Risk-free rate 3.50%
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2.035.115.3.035.08.5.035.095.035. abcR
%5.9abcR
*Required additional rate- of-return due to this generic factor
**Total weight must equal 1 (100%)
Benefits – APT/APM
The Arbitrage Pricing Theory is considered by most finance researchers to be a
theoretically sound risk determination method. Some empirical evidence suggests that
Arbitrage Pricing Models provide more accurate risk information than the Capital Asset
Pricing Model. This is because APT considers a broad range of relevant factors in
determining a particular equity’s unique risk profile. This theory also makes the
standard adjustment of adding a “benchmark” or risk-free rate to the other factors in
order to determine an individual stock’s required rate-of-return.
Limitations– APT
While the Arbitrage Pricing Theory is considered by most finance researchers to be a
theoretically sound risk determination method, APT has several practical limitations.
Most notably, the particular risk factors and the relative weights of those risk factors are
difficult if not impossible to determine. Hence, APT’s practical implementation is
difficult. In “real-life” situations, CAPM offers a much more feasible risk measurement
process.
While a list of multiple risk factor proxies seems to relate to much of the equity market
risk characteristics, there is still no definite way of knowing if these are the actual risk
factors. In other words, these factors may or may not directly represent the stock market
risk.
Also, APT does not consider risk associated with one specific firm. A few examples of
firm-specific risk factors not included in APT are changes in the particular companies -
product mix,
pricing structure,
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marketing strategy and
personnel
Summary of APT/APM
APM is a multi-factor mathematical model that estimates an individual asset’s particular
risk factors and subsequently provides a proxy of that asset’s risk and required rate-of-
return.
While not perfect, Multi-factor Models, as derived from Stephen Ross and Richard Roll’s
Arbitrage Pricing Theory, are considered by many to be a significant improvement over
CAPM in the asset risk measurement field.
Comparison of APT and CAPM
Multi-factor Arbitrage Pricing Models - as derived from Arbitrage Pricing Theory - are
generally considered to be a more scientific approach to risk measurement than the
single-factor Capital Asset Pricing Model
Distinguished researchers, Eugene Fama and Kenneth French co-authored the
following article -
Fama E. and K. French, (1996) Multifactor Explanations of Asset Pricing Anomalies.
Journal of Finance 51(1) pp55-84.
This article displayed results of empirical research showing that a multi-factor (3 factors)
model provided more predictive discount rate information than a single-factor model.
This, in turn resulted in a more useful discounted cash-flow valuation model than that
provided by a single-factor beta model (CAPM).
However, other researchers have offered empirical defenses of the CAPM as compared
to APT. Consider the following article authored by notable researcher and consultant,
Jack Treynor.
Treynor, Jack L. (1993). In defense of the CAPM, Financial Analysts Journal, 49 (3),
p11-14.
In this report, Treynor suggests that CAPM and APM are very similar in their specific
characteristics. Through a series of rather complex mathematical computations, he
attempts to show that both models have similar limitation. Treynor stops short of
suggesting that the CAPM is the better model, he simply suggests that both models are
equally imperfect.
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It is clear that both the APT method and the CAPM suffer from certain, related practical
and theoretical limitations. Both models assume that,
An efficient stock market exists, which means that
knowledgeable investors make rational decisions that
continuously create “fair” asset prices and
the risk-return trade-off is systematic.
Regardless of finance theorists’ varying opinions, in practice, CAPM is more commonly
used than APM because –
CAPM is considered to be more straightforward model than APT/APM and
CAPM’s necessary components are more widely understood and more
commonly available.
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Summary – Discount Rate
In this section, two common risk (required rate-of-return) proxy models were discussed -
1. Capital Asset Pricing Model.
Where -
Ra = risk of individual asset
Rf = risk-free rate
MRP = market risk premium
aβ = beta relative to stock market
2. Arbitrage Pricing Theory-based multi-factor model.
Where -
Ri = Risk of the individual equity
rf = risk-free rate
bi = risk factors
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I = weight of related risk factor (factor loading)
e = random volatility = 0
The background and specific components of each supposition were described. The
background of each section was then followed by a hypothetical illustration using the
related model. Finally, the benefits and limitations of each model were described.
After careful analysis, one can see that neither the Arbitrage Pricing Theory nor the
Capital Asset Pricing Model is irreproachable. Yet, both models were originally - and
still are considered to be - significant advancements in finance and the specialized craft
of asset valuation.
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Review Questions
Review Questions are required by NASBA and are designed to enhance the learning process. They are
NOT graded. Answers can be found at the end of the EBook.
1. When referring to the DCF model, risk refers to the uncertainty of the _____ of
projected cash-flows used in the numerator of the discounted cash-flow
projection.
A. Accuracy and Liquidity B. Amount and Solvency C. Pros and Cons D. Amount and Timing
2. In financial terms, “uncertainty” refers to the likelihood of
A. receiving future cash-inflows from an investment of current cash out-flows. B. current cash inflows being received from future cash out-flows. C. incurring risk associated with present cash in-flows. D. obtaining current cash-inflows as a result of past cash-outflows.
3. The CAPM method was originally developed by which financial researcher?
A. William Sharpe B. Myron Scholes C. Stephen Ross D. Ken Blanchard
4. The following is used as a proxy for the U.S. stock market as a whole.
A. Dow Jones Utility Index [DJUI] B. S & P 500 Index C. Value Line Investment Survey D. TOPIX
5. A significant limitation associated with CAPM is that a stock beta cannot be
determined unless
A. a legitimate and efficient market for the asset exists. B. the risk-free rate is known. C. the MRP is known.
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D. future price changes are known.
6. The CAPM differs from the APM in that the CAPM includes a specific parameter
that represents
A. inflation. B. interest rates. C. stock beta. D. factor loading.
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Chapter 4. The complete model - Value equals estimated cash-flow divided by the
discount rate
By the end of this chapter, learners should be able to –
Understand how assets are valued using a discounted cash flow model
The relationship presented in the graphic above is, by now, quite familiar to the reader.
It is a mathematical expression that can be interpreted to mean -
An asset’s value today (P0) equals estimated cash-flows (CF1) divided by
the discount rate (r).
As was discussed earlier in the course, value can be determined once the cash flow
projection and the discount rate are determined.
Consider the following simple hypothetical information:
Alpha-Omega, Inc
Financial Information
Cash-Flow Estimates (per share):
Using EBITDA: $1.25
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Income from Continuing Operations: $1.31
Discount Rate Estimates:
CAPM: 15%
APT: 13%
Using the above information, an analyst can proceed to calculate value using the DCF
model. Several computation methods could be employed. For instance, an analyst
could choose to use -
EBITDA and CAPM figures,
EBITDA and APT figures,
income from Cont. Operations and APT figures or
many other potential combinations of factors
Oftentimes, analysts will choose to combine multiple factor calculations to form an
arithmetic mean.
To illustrate, the parameters - “CF1” and “r” - could be calculated as follows:
CF1 = [EBITDA + Income from Continuing Operations]/2
r = [CAPM + APT]/2
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Substituting the values provided in the example, the stock price (value) per share would
be determined as –
Thus, the estimated stock price per share would be $9.14.
While this example is simplistic, it adequately displays how DCF can be used to
estimate asset valuation.
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Review Questions
Review Questions are required by NASBA and are designed to enhance the learning process. They are
NOT graded. Answers can be found at the end of the EBook.
1. The DCF model contains a factor represented by “r” which stands for
A. Risk-adjusted discount rate. B. Cash flow. C. Net Income. D. Current Value.
2. A financial analyst could estimate an asset’s value with a DCF model by
A. using EBITDA to calculate future cash-flow proxy and CAPM to calculate the risk-adjusted discount rate.
B. using EBITDA as a risk-adjusted discount rate proxy and APT to calculate the estimated future cash-flow.
C. using Income from Operations as a cash-flow proxy and EBITDA to calculate the risk-adjusted discount rate.
D. using APT as a cash-flow proxy and CAPM to calculate the risk-adjusted discount rate.
3. One could calculate expected future cash-flows by
A. using the CAPM. B. using a figure calculated as an average of APT and CAPM. C. observing the Security Market Line. D. using a figure calculated as an average of income from continuing operations
and EBITDA.
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Conclusion
This course presented a discussion of the art and science of asset valuation, including a
brief discussion of valuation methods applications. Next, the discounted-cash-flow
model was introduced as a common asset/equity valuation method.
Where:
0P = Price (Value) of the equity per share
1CF = net cash-flow in future period (1, 2, 3...) per share
r = required rate-of-return (i.e. risk-adjusted discount rate)
A few of the more popular cash-flow proxy models were used to determine the
numerator.
Income from Continuing Operations,
EBITDA and
S&P core earnings
Each model presented has its own set of benefits and restrictions. No one method is
perfect. Thus, any discounted-cash flow valuation model that implements one of these
cash-flow proxies is subject to some level of fault.
Once a single-period cash-flow proxy is determined, that cash-flow estimate must then
be combined with other information (e.g. a series of historical periodic cash-flow
estimates and forecasted economic changes) to better forecast future cash-flows. It is
that estimate of future cash-flows which is discounted by an appropriate risk-adjusted
discount rate to determine an asset’s value.
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Next, the discounted cash-flow model’s discount rate (i.e., denominator) was
considered. The discount rate was shown to be inversely related to the uncertainty of
the future estimated cash-flows. Two common risk models were discussed:
1. Capital Asset Pricing Model.
Where -
Ra = risk of individual asset
Rf = risk-free rate
MRP = market risk premium
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aβ = beta relative to stock market
2. Arbitrage Pricing Theory-based multi-factor model.
Where -
Ri = Risk of the individual equity
rf = risk-free rate
bi = risk factors
I = weight of related risk factor (factor loading)
e = random volatility = 0
The background and specific mechanism of each supposition was explained. The
background of each section was followed by a hypothetical illustration using the related
model. Finally, the advantages and disadvantages of each model were described.
It was determined that neither the Arbitrage Pricing Theory nor the Capital Asset Pricing
Model was perfect. However, both models are generally considered to be significant
asset valuation tools.
A presentation of the interchange between the cash-flow estimation methods and the
discount rate proxy models showed how the figures can be incorporated into the DCF
formula to determine a reasonable estimate of an asset’s value.
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Glossary
Arbitrage Pricing Theory (APT) – Risk calculation model whereby individual equity risk
is calculated by combining multiple risk factors
Beta – Relative systematic risk measure
CAPEX – Capital expenditures
Capital Asset Pricing Model (CAPM) - A risk calculation model which calculates equity
risk by adjusting market-level risk by individual company risk as measured by the
individual equity’s relative volatility
Discount rate – Rate at which future cash flows are adjusted to present value
EBITDA – Earning before interest, taxes, depreciation and amortization
Economic profit – Represent sustainable cash flows
Fair Market Value (FMV) - Amount mutually and voluntarily agreed upon by both buyer
and seller.
Market comparison method – Valuation method that compares relative firm values
Market Risk Premium (MRP) – Expected additional return from stocks that relate to the
stocks’ additional risk.
Required rate-of-return – Return necessary to compensate for the related risk.
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Index
Arbitrage Pricing Theory, 12, 33, 46, 47, 50,
51, 53, 54, 62, 64
beta, 33, 34, 35, 38, 39, 42, 45, 46, 51, 53, 62,
64
CAPEX, 24, 64
Capital Asset Pricing Model, 33, 38, 42, 46,
47, 50, 51, 53, 54, 61, 62, 64
Discount rate, 1, 10, 11, 12, 14, 32, 34, 42, 43,
46, 51, 56, 60, 61, 63, 64
EBITDA, 17, 19, 20, 23, 24, 30, 57, 58, 60, 64
Economic profit, 28, 64
Fair Market Value, 7, 8, 64
Market comparison method, 7, 8, 64
Market Risk Premium, 35, 65
Required rate-of-return, 11, 32, 34, 35, 46, 50,
51, 53, 60, 65
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Review Question Answer Feedback
Introduction
1. Answer Feedback:
A. The double entry accounting equation is defined as “Assets = Liabilities + Shareholders’ Equity.
B. A = L + SE C. This is a component of the liabilities definition. D. Assets result from PAST transactions or events. 2. Answer Feedback:
A. Assets = Liabilities + Shareholders’ Equity. Hence, “Assets = $0 (Liabilities) + Shareholders’ Equity” is equivalent to “Assets = Shareholders’ Equity”.
B. If total liabilities are greater than shareholder equity then liabilities are not equal to $0. If liabilities are not equal to $0, then assets cannot equal shareholder equity per the accounting equation: Assets = Liabilities + Shareholders’ Equity.
C. If liabilities are not equal to $0, then assets cannot equal shareholder equity per the accounting equation: Assets = Liabilities + Shareholders’ Equity. Assets = Liabilities + Shareholders’ Equity so if Liabilities = 0, the Assets = Shareholders’ Equity.
D. Assets = $0 (Liabilities) + Shareholders’ Equity” is equivalent to “Assets = Shareholders’ Equity”. The firm could own current assets and/or noncurrent assets
Chapter 1. Valuation – Background
1. Answer Feedback:
A. Valuation methods are often used for estate planning. B. Valuation methods are often used for divorce litigation. C. Valuation methods are often used for patent litigation. D. While it is conceivable that valuation methods may be used in certain types of
compliance auditing it is common for valuation methods to be used in estate planning, divorce litigation and patent litigation.
2. Answer Feedback:
A. Capital Asset Pricing Model - often used to determined discount rate B. A version of the DCF valuation model C. Arbitrage Pricing Model - often used to determined discount rate
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D. Neither I (the course author) nor anyone else with my surname has yet to invent a revolutionary valuation model.
Chapter 2. Future cash-flows
1. Answer Feedback:
A. ASC Topic 450 addresses contingencies. B. ASC Topic 220 addresses comprehensive income. C. ASC Topic 820 addresses fair value. D. ASC Topic 330 addresses inventory.
2. Answer Feedback:
A. Cash is valued at its face amount by definition. B. Expenses related to warranty offers should be recognized in the same period that
the related product or service is sold. Hence, an estimate of future warranty cost is necessary.
C. This is a gain not an expense. D. This is a liability previously determined by the board of directors – it is not subject
to estimation.
3. Answer Feedback:
A. This method includes amortization expense. B. This line item is included on GAAP-based income statements. C. Income from continuing operations can be used as a proxy for economic
earnings (cash-flow) because it excludes certain “non-continuing” items. D. Voluntary accounting method changes are accounted for retroactively.
4. Answer Feedback:
A. No specific cash-flow multiple is an assumed part of the DCF method. B. The initial cash-flow estimate may need to be adjusted for probable future
economic changes. C. Once the future cash-flows are determined then they are divided by the discount
rate. D. Tax-related cash-flow are already considered in the initial calculation of cash-
flows.
Chapter 3. Discount rate determination
1. Answer Feedback:
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A. Uncertainty and level of accuracy are synonymous – liquidity is a risk factor but it is not related to the amount and timing of future cash flows.
B. Amount is correct but solvency is a term referring to a firm’s ability to pay debt. C. Risk relates to uncertainty of the amount and timing of project cash flows not the
pros and cons. D. Risk relates to uncertainty of the amount and timing of project cash flows, not
the pros and cons.
2. Answer Feedback:
A. “Uncertainty” refers to the likelihood of receiving future cash-inflows from an investment of current cash out-flows.
B. “Uncertainty” refers to the likelihood of receiving future cash-inflows from an investment of current cash out-flows.
C. Risk is equivalent to uncertainty”. Also, “uncertainty” is related to future cash-flows, not current cash-flows.
D. “Uncertainty “related to future cash-flows not current cash-flows.
3. Answer Feedback:
A. William Sharpe is credited for developing the CAPM and publishing his findings in the mid-1960’s.
B. Myron Scholes is credited for developing the Black-Scholes options valuation model, not the CAPM.
C. Stephen Ross was not involved is developing the CAPM. D. Ken Blanchard is a notable business management theorist. He is not a finance
researcher.
4. Answer Feedback:
A. DJUI is an index of utility stocks only. It is not considered to be a proxy for the stock market as a whole.
B. S & P 500 Index is a measure including a broad range of equities. The S & P 500 Index is commonly used as a proxy for the U.S. stock market as a whole.
C. Value Line Investment Survey is an equity research publication not an index. D. TOPIX is an index of Japanese equities not U.S. equities.
5. Answer Feedback:
A. A legitimate/efficient market for the asset needs to exist in order to calculate the stock’s beta coefficient.
B. The risk-free rate does not determine beta. C. The MRP does not determine beta. D. Future price changes are never known – they can only be predicted or estimated.
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6. Answer Feedback:
A. The CAPM does not include a separate parameter that represents inflation. B. The CAPM does not include a separate parameter that represents interest. C. The CAPM includes a separate parameter that represents beta. The APM does
not include this parameter. D. The CAPM does not include a separate parameter that represents factor loading
since only one factor (beta) is considered.
Chapter 4. The complete model - Value equals estimated cash-flow divided by the
discount rate
1. Answer Feedback:
A. The “r” stands for risk-adjusted discount rate. B. Cash flow is represent by CF1 C. Net Income is not used in the DCF formula D. Current Value (Price) is represented by “P0” in the model
2. Answer Feedback:
A. An asset’s valuation can be calculated using EBITDA to calculate future cash-flow proxy and CAPM to calculate the risk-adjusted discount rate.
B. An asset’s valuation cannot be calculated using EBITDA as a risk-adjusted discount rate proxy and APT to calculate the estimated future cash-flow because EBITDA is a cash-flow estimate and APT is a discount rate estimate.
C. An asset’s valuation cannot be calculated using Income from Operations as a cash-flow proxy and EBITDA to calculate the risk-adjusted discount rate because EBITDA is a cash-flow estimate not a discount rate estimate
D. An asset’s valuation cannot be calculated using APT as a cash-flow proxy and CAPM to calculate the risk-adjusted discount rate because APT is used to calculate the risk-adjusted discount rate, not the cash-flow estimate.
3. Answer Feedback:
A. One could NOT calculate expected future cash-flows by Using the CAPM because the CAPM is used to calculate the discount rate not the expected cash-flow amount.
B. One could NOT calculate expected future cash-flows by sing a figure calculated using an average of APT and CAPM because the CAPM and APT are used to calculate the discount rate not the expected cash-flow amount.
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C. One could NOT calculate expected future cash-flows by observing the Security Market Line (SML) because the SML displays the relationship between risk and reward. The SML does is not used to calculate expected future cash-flows.
D. One could calculate expected future cash-flows by using a figure calculated using an average of income from continuing operations and EBITDA.
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CPE Exam
Instructions: Please select the single best answer and transfer it to the preceding
sheet.
1. Assets are best defined as
A. probable past obligations of a firm. B. probable future economic benefits C. liabilities minus equity. D. equity – expenses.
2. Fair market value can be defined as
A. an amount mutually and voluntarily agreed upon by both buyer and seller. B. sales price of thinly-traded stock. C. an amount determined by one person. D. net book value.
3. The market method of valuation is best described by which of the following?
A. Trading prices of heavily-traded stock B. Wholesale store price index C. Discounted valuation of future cash-flows based on a market-adjusted discount
rate D. Deducing the value of one firm by comparing it to a similar firm in the same
industry
4. The DCF valuation method considers ______ and their associated risks.
A. estimated future cash-flows B. historical cash-flows C. current net assets D. discontinued operations
5. The DCF valuation method considers estimated future cash-flows and the
______ associated with the related equity shares.
A. risk B. dividends C. expenses D. EBITDA
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6. Income from continuing operations
A. includes all expenses related to discontinued operations. B. does not include depreciation. C. does not include effects of extraordinary items. D. includes an adjustment for permanent working capital increases.
7. Which cash-flow proxy method adds back depreciation to GAAP-based net
income?
A. Continuing Operation B. EBITDA C. Direct Equity D. Both b and c
8. One difference between the “Income from Continuing Operations” method and
the EBITDA method is that the “Income from Continuing Operations” method
does not consider
A. depletion. B. risk-flow assumptions. C. CAPEX. D. the randomness of depreciation allocations.
9. The income from continuing operations method adjusts GAAP-based net income
for all of the following except
A. depreciation. B. discontinued operations. C. extraordinary items. D. depreciation and extraordinary items.
10. Valuation methods are often used during
A. party planning. B. compliance auditing. C. patent litigation. D. marriage counseling services.
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11. The general Discounted Cash-Flow valuation model can be expressed by which
of the following mathematical models?
A. i jrx
divdiv 12
B. r
CFP o0
C. r
CFP 1
0
D. r
divCF 1
0
12. The market comparison method of valuation is best described by which of the
following statements.
A. Trading prices of heavily-traded stock. B. Wholesale store price index compared to retails prices. C. Discounted valuation of future cash-flows based on a market-adjusted discount
rate. D. Deducing the value of one firm by comparing it to a similar firm in the same
industry.
13. When referring to the DCF model, risk refers to the level of uncertainty of _____
cash-flows used in the numerator of the discounted cash-flow projection.
A. GAAP-based B. historical C. projected future D. known
14. The discount rate [i.e., required rate-of-return] increases as
A. economic profits increase.
B. risk increases.
C. uncertainty decreases.
D. certainty of cash-flows increases.
15. The return of a short-term U.S. treasury security, such as the 3-month T-bill, is
often used as a proxy for
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A. investor confidence. B. real estate values. C. risk-free rate of return. D. market risk premium.
16. The Security Market Line employs ___as a proxy of the sensitivity of equity’s
returns compared to market returns.
A. return B. theta C. sigma D. beta
17. Which researcher is credited with developing the Arbitrage Pricing Theory?
A. William Black. B. Myron Scholes. C. Stephen Ross. D. W. F. Sharpe.
18. The ___ uses beta as a proxy for a security’s relative risk.
A. APT B. Security Market Line C. SLM D. DCF
19. The CAPM can be modeled as
A.
faifaRMRPxRR .
B. r
CFP 1
0
.
C.
fafaRMRPRR
.
D.
fafaRMRPRPRICE
.
20. The Arbitrage Pricing Model (APM) includes a parameter “Rf”. This parameter
characterizes
A. risk factors. B. the numeric value zero. C. a risk-free rate.
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D. a stock market related beta coefficient.
21. The CAPM includes a parameter “ ”. This parameters represents
A. dividend payments. B. a numeric value between zero and one. C. discounted cash-flows. D. a stock market related beta coefficient.
22. One limitation of the Arbitrage Pricing Theory is that it
A. does not use stock market-related beta. B. includes too many risk factors. C. does not include a risk-free rate component. D. does not identify particular risk factors or relative factor loads.
23. The CAPM resembles the APM in that both include
A. adjustments to account for preferred stock returns. B. an adjustment to account for a risk-free rate. C. an adjustment to account for the stock market-related beta. D. multiple risk factors.
24. In practice, CAPM is more commonly used than APT/APM because
A. CAPM’s component values are more commonly available. B. APM’s necessary components are better understood and more widely available. C. APM is less accurate risk determinant than CAPM. D. CAPM considers an individual risk characteristic and factor loadings.
25. APT proposed a ____ risk proxy model.
A. single-factor B. constant C. multi-factor D. redundant
26. The Arbitrage Pricing Model is illustrated by which of the following equations?
A. erbrbrbrbc i nfi ni nfiifiifii ...
32211 .
B. erbrbrbrbrR i nfi ni nfiifiififi ...
32211 .
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C.
fafaRMRPRR
.
D. r
CFP 1
0 .
27. The model: fafa
RMRPRR represents
A. the SML B. the CAPM C. the APT D. the Dividend Growth Model
28. Which discount rate estimation model is considered to be virtually faultless?
A. No discount rate model is without fault. B. APM/APT. C. Black-Scholes. D. Arbitrage Pricing Theory.
29. The DCF model contains a factor represented by “CF1” which stands for
A. Risk-adjusted discount rate. B. Cash flow. C. Net Income. D. Current Value.
30. One could estimate an equity price using DCF by
A. using EBITDA as a risk-adjusted discount rate proxy and APT to calculate the estimated future cash-flow.
B. using EBITDA to calculate future cash-flow proxy and CAPM to calculate the risk-adjusted discount rate.
C. using Income from Operations as a cash-flow proxy and EBITDA to calculate the risk-adjusted discount rate.
D. using APT as a cash-flow proxy and CAPM to calculate the risk-adjusted discount rate.
31. An expected risk-adjusted discount rate could be calculated by taking the _____
of the discount rate as calculated by the APT and the EBITDA.
A. average B. sum
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C. difference D. covariance
32. An expected risk-adjusted discount rate could be calculated by taking the
average of the ________ as calculated by the Arbitrage Pricing Model and the
Capital Asset Pricing Model.
A. discount rate B. inflation rate C. risk-free rate D. prime rate
33. Earnings manipulation is inherent in
A. management reports B. the CAPM C. accrual-based financial statements D. dividend payment policy