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A NOTEBOOK of FINANCE
NOTEBOOK12.1.doc
Edward Harding 26Feb2012
Table of Contents
Introduction
Chapter 1 NATURE of FINANCE
1) Academic Discipline
2) Profession
Investment Analyst
3) Other Disciplines
4) Ethics and Finance
Chapter 2 CORPORATE FINANCE
ANALYSIS:
1) RATIO ANALYSIS
FUNCTIONS:
1) CAPITAL STRUCTURE:
2) CAPITAL BUDGETING:
a) Net Present Value and Discounted Cash Flow
b) Modified Internal Rate of Return
3) DIVIDEND POLICY:
Chapter 3 PORTFOLIO MANAGEMENT
1) Standard Deviation of Returns
2) Hyperbolas
3) Efficient Frontier
4) Number of Firms
Chapter 4 STRATEGIES
1) Speculation
2) Hedging
3) Arbitrage
4) Active Management
5) Passive Management
6) Efficient Mkt Hypothesis
7) Capital Asset Pricing Model
8) Behavioral Finance
9) Fundamental Analysis
10) Technical Analysis
11) Black Swans & Fat Tails
12) Ponzi Schemes
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Chapter 5 VALUATION
1) Value of Stock
2) Valuation of Bonds
Risk on Bonds
3) Continuous Compounding
4) Loans
5) Expected Value
Chapter 6 DERIVATIVES
1) Mutual Funds
Index Funds
S&P500
DJIA
2) Options
Black-Scholes
Straddles &Strangles
3) Hedge Funds
4) Credit Default Swaps
5) Collateralized Debt Obligations
Chapter 7 FINANCIAL MODELS
1) Alpha
2) Beta
3) BlackSwan /Fat Tails
4) Brownian Motion
5) DJIA
6) LIBOR
7) MATLAB
8) Monte Carlo Simulation
9) Pi
10) Normalization
Chapter 8 INTERNATIONAL
1) Foreign currency Exchange
2) Arbitrage
3) Forward Contracts
4) Futures
5) Options
Appendix: STATISTICS
1) Correlation Coefficients
2) Regression Analysis
3) Z-Scores
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A NOTEBOOK of FINANCE
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Introduction: These notes were compiled from several course sketchbooks. They are not intended to
break new ground, to promote any particular investment philosophy, to be a compleat reference, or to
be the source of high entertainment. They are intended to replace the traditional textbook by touching
the peaks of topics considered essential in an introductory finance course but with an economy of
verbiage and economy of cost to the student. Note: There are several references in the guise of “see
also”, which refer to additional readings contained in a collection of “Readings in Finance”
Chapter 1 NATURE of FINANCE:
The nature of the broad discipline of finance may be regarded from slightly different perspectives: That
is, one might regard finance 1) as an academic study, or 2) as a professional activity with a particular set
of skills. The nature of finance is also revealed by noting 3) the relationship between finance and other
academic disciplines, and finally, by noting 4) the relationship between finance and ethics.
1) Academic Discipline
As a field of academic study, the primary focus of finance is often on the firm and the financial
management of the firm because such a large proportion of the total wealth in a capitalistic economy is
created within the context of the firm (i.e. the enterprise, the corporation, the business, the private
sector, etc.). For this reason, introductory finance courses are variously known as financial
management, corporate finance or managerial finance.
However, a secondary academic perspective is external to the firm and focuses on individuals and
institutions (acting as investors) that participate in the capitalization of the firm through the purchase of
equity securities (stocks) or debt instruments (bonds). As an academic study this second perspective is
referred to as investing or portfolio management.
There is yet a third perspective in the academic discipline of finance that regards the activities engaged
in by financial professionals who design new contracts (financial engineering), institutions that lend and
borrow capital (investment banking), and others who take speculative positions based on the
performance of other instruments (institutional finance). And in addition, more specialized courses, for
example in international finance, financial modeling, and financial accounting, cover various niches of
finance in greater detail.
2) Profession
As a professional activity, the practitioner needs to understand the nature of corporations, the laws
(regulations, standards, protocols) governing financial activities, the mathematics of the financial
instruments, the accounting standards adopted by the firm, and the economic theories that are inherent
in the actual management of the firm and the management of funds that may be held external to the
firm.
The particular skill set required to successfully participate in the financial profession include many of
those universally found in other professions, for example the ability to work with others, the ability to
communicate clearly in written and spoken discourse, and the ability to develop and practice productive
work habits. However, some skill sets are more essential to finance than to some other professions, for
example - the quantitative skills of mathematics and quantitative modeling and the skilled use of
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information technology. Below is a job description for an investment analyst at a large financial
institution. It reflects the skills sought in the real world:
Investment Analyst (IA)
[Entry-level investment/financial analyst]
Role: The Investment Analyst is responsible for participating in research and modeling work on
various traditional and alternative asset classes for institutional, private bank, domestic and
international retail, and high net worth businesses. The IA performs evaluation of products for
inclusion in asset allocation models and develops and maintains proprietary optimization and
simulation models and participates in asset allocation related projects with clients. The IA
provides general support of private bank and consumer bank with respect to risk management,
portfolio analytics, attribution analysis, and other general investment topics. The IA participates
in marketing efforts and client meetings.
Education and Experience: MBA in finance or similarly quantitative field, and/or CFA. Five to
ten years experience in portfolio management or quantitative analysis is required. Specific work
in asset allocation is a plus. Quantitative background and programming skills (e.g. VBA and
MATLAB) are required.
Additional qualifications:
Financial expertise: Practical understanding of financial principles and portfolio management
concepts; familiarity with, risk and valuation models.
Statistical understanding: Statistical methods must be understood well enough to judge their
efficacy and drawbacks in specific circumstances; must be comfortable dealing with numerical
and statistical concepts.
Computer understanding: Comfortable with statistical analysis and analytic programming;
familiarity with commercial analytic packages (e.g., VBA, MATLAB).
3) Other Disciplines
Finance has clear connections with other disciplines (e.g. law, marketing, information technology), but
often the distinction between finance and economics, and between finance and accounting gets fuzzy.
Finance has sometimes been considered a sub-type, a "grubby off-spring", of economics, owing to
finance's pre-occupation with actually making money for the practitioner. Economics, on the other
hand, tends to be more of an intellectual pursuit, seeking understanding of the cause and effect
relationships among variables.
The relationship between finance and accounting is similarly fuzzy, except that accounting is the more
mundane number-crunching activity that compiles the necessary statements for the higher-order
financial types. [The allusion to any actual superiority of one discipline over another is purely artistic
license used to illustrate a point.]
Psychology has more recently joined forces with finance as both disciplines strive to understand the
markets which are an aggregate of individuals making financial choices. Within the last decade, the
discipline of behavioral finance has emerged as a combination of these two disciplines.
Finally, mathematics, the pure science, provides the tools for financial calculations.
4) Ethics and Finance: In other cultures and in other times, the charging of interest was immoral and
illegal. Lending money to a neighbor was considered acceptable, but charging interest on the loan was
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considered harvesting money from a source into which no labor was provided. Ergo, interest must have
come from the Devil. Times and culture have changed to where “charging (reasonable) interest” is
universally accepted as fair play, yet there are still ample opportunities to take unethical advantage of
our imperfect economic system. Harry Markowitz has some thoughtful comments in "Markets and
Morality" related to ethical issues in capital markets. [See also: Markowitz ].
________________________________________________________________________________
Chapter 2 CORPORATE FINANCE
The practice of finance within the corporation is best initiated with a reading on the objective of the firm
[See also: Van Horne], in which the existential question of “why does the corporation exist?” is
addressed. In essence, widely accepted financial theory states that the firm exists for the benefit of the
owners of the firm. As a consequence, owners hire agents to manage the firm in a manner that will
attract new investors and create a demand for equity the firm. Owners benefit as the demand for the
stock goes up, as the price per share goes up, and the market value of their investment goes up.
ANALYSIS:
Financial analysis of the firm is practiced by outside investors [outside of the firm], owners of the firm,
and the management of the firm. The term "financial analysis" has different shades of meaning, but in
general, the analysis regards the financial health and value of the firm from various perspectives. For
example liquidity, leverage and operational efficiency are all measures of financial health, and
measuring those properties may reveal financial strengths and/or weaknesses.
Financial analysis may also refer to the process of stock valuation whereby the analyst comes up with
estimated price per share that the stock "should" be selling at. Benjamin Graham built a reputation on
techniques to analyze the inherent value of firms. Other techniques refer to "fundamental" analysis,
that regards fundamental factors such as market share, annual growth, technological advances, and
ratio analysis to establish a "fair market value". "Technical Analysis", in contrast to fundamental
analysis, regards quantitative and statistical patterns and trends in (primarily) the firm's stock price to
forecast future stock prices.
1) RATIO ANALYSIS:
Ratio analysis is the classic approach to conducting corporate financial analysis. The technique has been
used for over 100 years and many different ratios are commonly used. In the descriptions below, note
that when “one value (A) is compared to another value (B)”, the calculation of the ratio is A divided by B.
This calculation may also be framed as “A relative to B”. Some typical ratios include the following:
The Current Ratio shows the liquidity of the firm by comparing the current assets to the current
liabilities. Note: the word ”current” in this context refers to the accounting definition. That is, current
assets are assets that could reasonably be expected to be “convertible into cash within a year”. Current
assets are usually explicitly designated as such on the balance sheet. Current liabilities refer to
payments that are due within the year. A healthy firm would want a current ratio greater than 1.00, and
a ratio of 2 or 3 would certainly add a margin of comfort.
The Quick Ratio, or “Acid Test”, is a more stringent measure of liquidity, in that it subtracts inventory
from the current assets before doing the same comparison as with the Current Ratio (above). Inventory
is not considered to be as liquid as other current assets.
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Days Sales Outstanding (DSO) is a measure of the firm's ability to manage one portion of its assets accounts receivable. Sometimes called "average collection period", it is a measure of how quickly
receivables can be converted into cash. Fewer days is better than more days, and an average of a
month and a half is normal for many industries. It is calculated by dividing sales into receivables to get
the proportion of sales still outstanding and then multiplying the results by the number of days in a year.
Inventory Turns is another measure of the firm's asset management prowess. In this case, the skill in
managing inventory is measured – having the right stuff on the shelf when needed, not having too much
in stock, and moving stuff quickly. Inventory Turns represents the number of times the firm has "filled
and emptied" its shelves in a year. More turns is better than fewer turns. There is a wide range of what
is a normal number of turns, depending on the nature of the business. Some firms use average
inventory in the calculation, others use year-end inventory. Some analyst divide inventory into sales,
and others divide inventory into cost of goods sold. [This course uses the latter e.g. COGS]
The Debt Ratio measures the proportion of debt (Total Liabilities) relative to the total capitalization
(Total Assets) of the firm. Somewhat counter-intuitively, more debt is not necessarily a bad condition
FOR A FIRM (debt is still something to be avoided for the individual). A firm that avoids borrowing, at
the expense of foregoing profitable projects, may be losing opportunities that its competitors end up
taking advantage of. Debt ratios will usually fall between 0 and 1.00. Over 1.00 is definitely a bad sign.
But between 0 and .80, it's difficult to say whether a change is better or worse without further
information.
Times Interest Earned (TIE) indicates a firm's ability to cover its interest payments due on its bonds with
its profits from operations. Said another way, TIE is the number of times the firm could pay its interest
obligations with its earnings before interest and taxes (EBIT). The calculation is EBIT relative to Interest.
Less than 1.00 would be a bad sign for sure, and multiples like 2, 3 and higher are healthy signs.
Profit Margin, Earnings relative to Revenues, is the percentage of the top line (sales=revenues) that
flows down to the bottom line (EAC=net profit=net income) after taking out all the expenses. A firm
that keeps two cents for every dollar of sales is typical, although there are certainly more profitable
firms, too.
Return on Assets (ROA) Pretty self-explanatory. ROA measures the firm's bottom line profits as
percentage of the total assets of the firm. Sometimes called "basic earning power", it suggests that a
firm should be able to earn a (relatively) fixed percentage of every dollar invested in the firm. A 12%
ROA is not unusual.
P/E Ratio is unlike the ratios above in that it is not generated solely from the income statement and
balance sheets – it also requires the price per share (pps) from current market data. This ratio regards
the price investors are willing to pay for a dollars worth of earnings. The earnings are usually historical,
trailing twelve months (ttm), although the P/E ratio often compares current pps to estimated (future)
earnings. The P/E ratio is often abbreviated to PE ratio and is sometimes called the firm's "multiple".
The reciprocal of the PE is called the "earnings yield", the percentage of net earnings generated each
year for a given price per share, and is perhaps a more intuitive way of understanding the relationship
between pps and earnings available to common stockholders (EAC). Remember, EAC belongs to the
owners of the firm – the shareholders. As of the closing bell on 17 July 2009, the average PE for the DJIA
was 13. As PE is a measure of "pricey-ness" of a stock, 13 is cheap, over 20 is getting pricey.
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Market to Book is similar to the PE ratio in that it relies on current pps data. In this ratio, "market"
refers to the value that investors have put on the firm, the "market capitalization" or MKT CAP (as seen
on Yahoo). Market cap is the total current market value of all outstanding stock, or pps times numberof-shares. Pretend you wanted to buy IBM, as in, buying the whole company. How much would it cost?
That's market cap. In mkt/book, "book" is how the accountants value the company and is found on the
balance sheet as "shareholders equity". Mkt/book is another measure of pricey-ness, and typical ratios
run 2.0 to 4.0.
FUNCTIONS:
As the primary objective of the firm is to maximize stockholders wealth as reflected in the stock price,
then the primary function of financial management is also to address that objective. Management
accomplishes this through convincing investors that the "quality of future earnings" is strong – that is,
that future earnings are both "likely" (have a high probability of coming to pass) and robust (exhibiting
healthy growth through time). If management is effectively convincing, then a demand for a share of
ownership in the firm will apply the upward pressure on the price, and management will have
succeeded in fulfilling their objective - for the time being. The function of a financial manager is to
support the primary objective through simultaneous actions in four major areas: 1) Capital Structure, 2)
Capital budgeting, 3) Dividend Policy, and 4) Cash Flow Management.
1) CAPITAL STRUCTURE:
Financial managers are responsible for the "capitalization of the firm", or identifying and executing
sources of funding. The two major sources are stocks (selling equity) and bonds (issuing debt), with a
third, and not inconsequential, source being the internally generated capitalization from earnings. This
area is also referred to as "the capital structure" decision, in which the financial manager tries to
anticipate the optimum mix of debt and equity (proportion of bonds to stock) to achieve the primary
objective of maximizing stock price. While there are many models addressing the capital structure
decision, the study by Miller [see also: Varian] raises doubts as to those models' efficacy.
In finance, capital structure refers to the way a corporation finances its assets through some
combination of equity, debt, or hybrid securities. A firm's capital structure is, then, the composition or
'structure' of its liabilities. For example, a firm that sells $20 billion in equity and issues (borrows) $80
billion in debt is said to be 20% equity-financed and 80% debt-financed. The firm's ratio of debt to total
financing, 80% in this example, is sometimes referred to as the firm's leverage. In reality, capital
structure may be highly complex and include multiple sources.
The Modigliani-Miller theorem, forms the basis for modern thinking on capital structure, though it is
generally viewed as a purely theoretical result since it assumes away many important factors in the
capital structure decision. The theorem states that, in a perfect market, how a firm is financed is
irrelevant to its value. This result provides the base with which to examine real world reasons why
capital structure is relevant, that is, a company's value is affected by the capital structure it employs.
These other real world reasons include bankruptcy costs, agency costs, taxes, information asymmetry,
to name some. This analysis can then be extended to look at whether there is in fact an optimal capital
structure: the one which maximizes the value of the firm. [Adapted from: Wikipedia "capital structure"]
2. CAPITAL BUDGETING:
The next major decision is in the use of the capitalization that was raised as a result of the capital
structure decision. The "uses" of capital in the corporate environment is called "capital budgeting", and
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the objective of this area is, again, the maximization of stock price. In cruder terms, the firm just raised
a lot of money. Now, what should they to do with it? They should invest it into projects that have the
greatest positive impact on the stock price.
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The terminology of capital budgeting refers to "capital projects", or those activities that require some
up-front investment in hopes of reaping longer-term rewards. Capital projects may take the form of
buying another company, building a new manufacturing plant, developing a new product, purchasing
equipment, entering a new market – there is a wide range of investments considered “capital projects”.
One could say that “attaining a college degree” is a capital project. Invest time and money now in hopes
of greater returns in the future.
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Also, recall the accounting distinction between capital spending and expense spending: capitalized
items become assets on the balance sheet and are depreciated over time and expense items are shown
as costs for the current time period on the income statement. The current US tax code favors capital
spending by allowing accelerated depreciation so as to realize tax savings sooner than later.
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The capital budgeting process involves 1) identifying potential projects, 2) estimating the incremental
cash flows related to the respective projects, 3) identifying and quantifying risk associated with each
project, 4) application of various financial models to quantify the projects value to the firm, prioritize the
project relative to other potential projects, and to make the final decision as to whether to pursue the
project.
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Net present value and discounted cash flow methods:
The net present value (NPV) method and the discounted cash flow (DCF) method are two names for the
same process. It is a technique used to take an estimated cash flow and to discount it to yield a net
present value.
1) Cash Flows (CF) are the marginal changes in cash to the firm that are anticipated as a result of the
adoption of a project. These cash flows would reflect estimated changes in revenue and direct expenses
and the magnitude of these changes are generally estimated by engineers and cost accountants.
Financial analysts assume the responsibility of calculating the net present value of the cash flows.
2) DISCOUNT RATES: To discount a cash flow (CF), one must first determine the discount rate. (Not to
be confused with the Fed's “Discount Rate” which is an entirely different concept.) For academic
purposes, the discount rate is often given as a constant, although sometimes it is a variable increasing
through time. In the following models, we use "k" as the variable for the discount rate. For corporate
purpose, there are various theoretical approaches to determine the rate's intrinsic value. For example,
one approach suggests using the weighted average cost of capital of the firm and then adding a risk
premium for the particular project to yield a discount rate.
3) ASCENDING DISCOUNT RATES: In capital budgeting spreadsheets, the discount rate used to discount
the cash flows is sometimes a variable rate rather than a fixed rate for the entire period of the project.
[In the Fox text you may notice a reference to how these discount rates are determined.] Typical of
ascending rates is a constant increase through time (ascending) to reflect the increasing uncertainty
about the cash flows actually materializing as forecasted. The assumption for the project may be
something like:
"The discount rate for this project, for this year (year 0), is 14%. However, with the increase in
uncertainty about years 1 through 5, we're adding ¼% each year to the discount rate."
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discount rate
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0.14
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0.1425
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0.145
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0.1475
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0.1525
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Adapted from Wikipedia: Another approach to choosing the discount rate is to decide the rate
which the capital needed for the project could return if invested in an alternative venture. If, for
example, the capital required for Project A can earn five percent elsewhere, use this discount
rate in the NPV calculation to allow a direct comparison to be made between Project A and the
alternative. Related to this concept is to use the firm's Reinvestment Rate. Reinvestment rate
can be defined as the rate of return for the firm's investments on average. When analyzing
projects in a capital constrained environment, it may be appropriate to use the reinvestment
rate rather than the firm's weighted average cost of capital as the discount factor. It reflects
opportunity cost of investment, rather than the possibly lower cost of capital.
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A NPV amount obtained using variable discount rates (if they are known for the duration of the
investment) better reflects the real situation than that calculated from a constant discount rate
for the entire investment duration.
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To some extent, the selection of the discount rate is dependent on the use to which it will be
put. If the intent is simply to determine whether a project will add value to the company, using
the firm's weighted average cost of capital may be appropriate. If trying to decide between
alternative investments in order to maximize the value of the firm, the corporate reinvestment
rate would probably be a better choice.
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For some professional investors, their investment funds are committed to target a specified rate
of return. In such cases, that rate of return should be selected as the discount rate for the NPV
calculation. In this way, a direct comparison can be made between the profitability of the
project and the desired rate of return.
2) DISCOUNT FACTORS: Once the discount rates have been determined, the discount factors can be
calculated using: discount factor = (1+k)-n , where "k" is the discount rate and "n" is the number of years
hence. You'll recognize this from PV=FV(1+k)-n, where "FV" is the future value, or the future cash flow.
The spreadsheet formula format for the discount factor is: =(1+k)^(-n), where actual cell locations are
substituted for k and n.
3) PRESENT VALUES: Using the PV formula above, each year's cash flow is multiplied by its respective
discount factor to get its present value. This step is applied to every year including "year 0".
4) NET PRESENT VALUES: The present values for all the years (including year 0) are added together to
get the "net present value(NPV)", or in other words, the "discounted cash flow (DCF)".
The NPV is the current value of all the project’s cash flows, current and future. A firm's total net value
increases by the NPV of a project when the firm commits to that project. This “increase” may seem
counter-intuitive: a firm writes a check for $100K, committing itself to a new project, and suddenly the
value of the firm goes up by $120K? Sure. The value of the project is $120K, and the firm is going to
engage in the project.
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This notion of increasing the value of the firm can be carried a little further. If the value of the firm goes
up, then so, too, goes the stock price. They are, after all, directly proportional. The value of firm divided
by number of shares equals stock price. Hence, the theoretical change in stock price for taking on a
project will be the NPV of the project divided by the number of shares.
The NPV relates to the rate of return of the project in the following way: If the discount rate (think "cost
of money") is less than the return of the project, then the project will "make money". More technically,
if the discount rate is less than the internal rate of return, then the NPV will be positive. And,
conversely, if the discount rate is greater than the internal rate of return, then the NPV will be negative.
Note also, that the discount rate is externally determined (e.g. by credit markets, and risk factors),
whereas the project's rate of return is internally determined by the project's cash flows and discount
rate. Finally, if the discount rate and the projects rate of return are equal, then the project has no
positive value and isn't worth doing. This will be reflected in an NPV=0 and is intuitively illustrated with
the scenario of borrowing money at 20% and putting the money in a 20% project.
Twisting the preceding paragraph around a bit, we could say that the internal rate of return (IRR) on a
project is that rate that when applied to the cash flow yields a net present value of zero. And that is the
definition of IRR.
Modified Internal Rate of Return
The IRR model referred to in the paragraph above has a short-coming. The IRR model assumes that cash
flows generated by one project can be re-invested into a second project that will yield the same rate as
the original cash-generating project – and this may not be a valid assumption. Thus, a modified internal
rate of return model has been concocted to address this shortcoming.
The IRR* model assumes that the positive cash flows generated by a project are reinvested at the firm's
discount rate to the end of the last year of the cash flow. This assumption differs from the reinvestment
assumption of the ordinary IRR model, which assumes implicitly that reinvestment of the cash flows are
made at the derived IRR.
In IRR*, the sum of the reinvested cash flows are an estimate of the future terminal value (TV) of the
project (not counting the original investment). The terminal value is subsequently reevaluated to reflect
its net present value, using a rate (the IRR*) that will discount the TV to an amount exactly equal to the
original investment. By definition, the hypothetical discount rate which yields a net present value (NPV)
of zero is the project's internal rate of return.
3. DIVIDEND POLICY:
The third major functional area in financial management is determining how much of the earnings
(generated by capital projects) should be distributed to shareholders in the form of dividends. This
determination may be based on the firm's "dividend policy" that is established by the board of directors.
A rough perspective of this decision is that 1) all the profits [earnings available to common stockholders
(EAC)] of the company belong to the stockholders, 2) but those earnings may be plowed back into the
firm or sent to the shareholders. The question is how much should go to each of the two destinations.
Dividend policy is discretionary, that is, the amount of the quarterly dividends is determined by the
board of directors. The criteria for determining a dividend policy should be "what policy will maximize
the stock price?" The accounting is straightforward: Net Profits (EAC) is split into dividends and
retained earnings. There is no third bucket where profits end up.
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Here are some abstracts of various theoretical dividend policy models:
1. Modigliani & Miller (MM) say that dividends don't matter, "dividends are irrelevant", that stock price
is determined by "basic earning power [ROA] of firm", not by dividend policy. MM acknowledge a
"signaling effect" and offer a "clientele theory".
2. Gordon & Lintner says dividends are everything, that investors have a preference for liquidity. MM
call this the "Bird in the hand theory". P0=D1/(ke-g) .
3. Harkavy did empirical test to find a slight positive correlation between payout ratios and stock price.
Both MM and Gordon/Lintner claim that Hakavey's study lends credibility to their respective theories.
4. Walter's theory of Residuals: Walters posited that firms should use EAC to fund projects whose IRR*
are expected to be greater than the stockholders required rate of return. Dollars not expended on these
projects should be paid out to stockholders in the form of dividends.
Here are some “real-world” models. That is, many actual companies follow the following models:
5.
6.
7.
8.
9.
Constant dollars: Pay out same amount of dollars per share every quarter.
Constant dollars with a kicker: Add a bonus every now and then.
Constant increases: Increase dollar amount every quarter.
Constant payout ratio: Hold the proportion of payout (div/EAC) constant each quarter.
Tie changes in dividends to changes in CPI: Increase div by cost of living.
4. CASH FLOW MANAGEMENT:
The first three functions (raising the funding, investing the capital, distributing the harvest) all require a
careful oversight of the flow of actual cash. It is this "cash flow management" function that is cited as
the fourth major function of the financial manager. Cash flow management requires planning and
oversight with diligent accounting skills.
The traditional functions of the financial manager of the firm include the four major areas described
above. In addition, the financial manager often has oversight responsibility for the non-financial areas
of accounting and information technology. Accountants are the first to gather the monetary
transactions of the firm and to put those transactions into some sort of order for the raw material for
the financial departments. Therefore the financial manager has vested interest in the integrity of the
work of the accountants. And information technology made its corporate debut in the accounting
department (for the processing of those same transactions) and likewise fell under the management
umbrella of the financial manager. But as for the clearly financial functions, the four functions above are
the most often mentioned.
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Chapter 3 PORTFOLIO MANAGEMENT
The practice of finance is not limited to the financial management of the firm. Considerable wealth is
held by unincorporated entities such as individuals, trusts, non-profit organizations, municipalities and a
wide range of other institutions. And financial institutions, while often technically incorporated, manage
their financial assets with portfolio management models more than with "corporate" revenue-to-profit
models. The wealth held in portfolios, or collections of the financial instruments, and the management
of these portfolios requires financial professionals with portfolio management skills.
Modern Portfolio Theory (MPT): The discipline of portfolio management had a major incarnation with
the introduction of Modern Portfolio Theory (MPT). See also the first section of the Varian article on
Markowitz. Prior to MPT, portfolio management focused on active stock picking – that is, speculating
and selecting those stocks with the highest expected returns. This is an intuitive and still popular focus
of many investors. However, with MPT, Markowitz quantified the notion of minimizing risk through
diversification, and lowering the probability of losing principal. This is not to say that "maximizing
returns" is wrong-headed or undesirable, but it suggests that "minimizing risk" is a worthwhile and
simultaneous endeavor. His model identifies a set of portfolios that have the lowest risk for a given
return - or said another way, a set of portfolios with the highest return for a given level of risk. This set
is the “efficient frontier”.
1) The development of the MPT model can be illustrated with a series of
charts. [Note: Ctrl + click on a chart to see larger version]. This first chart
shows daily stock prices, or price per share (pps), through time. In this
example the time span is one year and shows pps at the close of each trading
day. The blue dash line is a projection of the original value at the start of the
measured year. The short red vertical line represents the change (Δ) in pps
over the course of the year.
2) The second chart shows returns (K) for each of those same trading days
using the classic “K = Δ/orig” model, or “returns equal the change in price
divided by the original price”. “K” is a percentage, and is expressed either as a
percent (i.e. 25%) or as a decimal (i.e. .25). Although dividends are ignored in
this return calculation example, total stock returns are more correctly
calculated with dividends, taxes and transaction costs factored in.
3) The third chart illustrates the calculation of the variability, (the
"bounciness" of the data, or volatility) of the returns. The statistic that
reflects this volatility is the standard deviation (σ) of the returns, and is, by
definition, the risk of the stock. There is a natural attraction to think of risk
as the volatility of the price, and indeed, a bouncy price will cause bouncy
returns, however, the volatility of the returns is the accepted measure. Note
that data in this third chart is the same as the second chart, but a line has been added representing the
mean and a normal distribution of returns based on the mean and standard deviation.
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4) MPT suggests that the standard deviation of the returns, the risk, can be
mitigated by the introduction of a second security whose return pattern is
significantly different than the original security. In the adjacent chart the
original security (from the previous charts) is shown as "A" in blue, and the
second security is shown as "B" in red. Their respective distributions, based
on their respective means and standard deviations, are shown on the right
side of the chart. The two distributions might be similar, but probably won't
be identical.
5) As the two securities are combined into a single portfolio, the resulting
composite return (the return of the portfolio) is shown in blue in this chart. A
correlation coefficient is used to compare return patterns, with a low
correlation indicating greater differences in patterns, and results in a portfolio
with less volatility (less risk) than either of the original securities. [This is a
good chart to click on to better see the low risk portfolio].
6) Chart 6 illustrates this same (as 5) phenomenon from a different perspective. Again, consider the
same securities, A and B. A has some historical risk/return profile, and let's assume
that B has a somewhat higher risk and higher return (consistent with the CAPM, as
shown here). A portfolio with 100% of the dollar invested in A and 0% in B, would
have a risk/return profile ( σ vs. K%) at point A on the chart. As B is incrementally
added to the portfolio, the return of the portfolio will rise and the risk will be
reduced, the risk/return profile starts moving toward the upper left in the chart,
beginning to trace a hyperbola. Around the apex of the hyperbola, there is a 50/50
dollar mix of A and B in the portfolio, and at this mix, the portfolio of A &B has lower risk than either A
or B. Magic. The returns of the portfolio will always be a weighted average of the returns of the
components.
7) A set of portfolios composed of six securities will trace five hyperbolas (A to B, B
to C, etc.), but will also define an area representing every possible weightcombination of A,B,C,D,E,F. The upper left boundary of this set of portfolios is a line
that represents those portfolios with the lowest risk for a given return (or the highest
return for a given level of risk.) This boundary is the "efficient frontier".
8) Tobin posited that a line emanating from the risk-free rate (Krf)and laying on
the efficient frontier will have a point of contact at the risk/return point of
market portfolio (e.g. an S&P 500 index fund). The line between the risk-free
rate and the market portfolio represents a set of portfolios comprised of varying
weights of risk-free assets and market funds. The line extending beyond the
market represents a set of portfolios, all fully invested in the market funds, but
with ever increasing amounts of borrowing at the risk-free rate.
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9) The last chart shows that as more firms are added to a portfolio, the risk of the
portfolio tends to decline to the level of the market's risk (systematic risk). The
first point on the left on the red line represents a portfolio with one security. The
risk of the portfolio is thus the same as the risk of that single security. The next
point shows how the addition of a second security to the portfolio reduces the
risk of that portfolio. With the addition of each additional security, the portfolio
tends to diminish until it reaches the market risk. Note that the market risk is the same as the risk of a
portfolio with every publically traded security included.
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Chapter 4 STRATEGIES
This topic explores different financial strategies that are used by firms, individuals, and institutions.
Items in this list are not necessarily mutually exclusive (an investor might use a combination of several
strategies), but they are representative of currently popular approaches. Excluded from the list are
some well-known and commonly practiced approaches that are rational methods for saving and
investing (e.g. 401Ks, dollar-cost-averaging, buy-and-hold, buy low-sell high) but whose underlying
assumptions are too obvious to warrant further academic discussion in this context.
1) Speculation: This strategy assumes that the investor has some special insight into what might
happen in the future. As the investor believes that the price of a security will go up, he/she will "buy
low" and wait for the security to go up and "sell high". This special insight may be due to the recognition
of an historical pattern that other investors either do not perceive or perhaps other investors believe
the pattern does not apply to this particular environment. The notion of "alpha", excess returns above
market returns, are tied to this strategy. Speculators believe they have a good chance of generating
alpha.
2) Hedging: This strategy requires Investing in multiple instruments that are likely to perform
differently from each other, such that if one investment under-performs, the other is likely to overperform. A passive investor whose entire portfolio is comprised of market index funds has essentially
hedged all the firm-specific risk and is guaranteed, by definition, not to generate any alpha, but to
achieve the market return. Hedge funds derive their name from this strategy, as they originally invested
in instruments that were negatively correlated with existing portfolios, however, through the years,
many hedge funds have tended to follow more speculative strategies
3) Arbitrage: This strategy requires highly sophisticated computer systems to identify relatively small
dis-equilibriums in a market and trading with very large sums of cash. For these two reasons, arbitrage
is engaged in by institutions rather than individuals. A simple example of an arbitrage would entail
noticing a small price difference in the price of gold in two markets, buying at the low price in one
market and immediately selling in the other market at the higher price. Large differentials in prices tend
not to occur, and small differentials tend to disappear quickly. But imbalances do happen in the normal
course of changing prices and it is in the process of arbitrage that prices tend to seek equilibrium.
4) Active Management: [a term used in the context "active versus passive"] An active investment
management strategy implies an underlying belief that through rigorous analysis, quality information
and rational decision making, an investor can, on average, beat the market. A disparaging phrase for
this approach is "chasing alpha". Stock brokers and professional portfolio managers tend to fall into the
category of "active" managers, as they sell their clients on the concept that their active management
adds value to the portfolio and justifies their management fees. Traditional portfolio management fees
are about 1% of the portfolio's value per year. Carried to extremes, an overly active manager who
continually re-balances a client's portfolio, presumably for the additional commissions related to the
trades, might be accused of "churning", not for higher returns, but for excessive fees.
5) Passive Management: [in contrast to "active" above] A passive investment strategy implies an
underlying belief in the efficiency of markets and the futility of chasing alpha. This strategy manifests
itself in the investment in market index funds, such that the returns of the investor will invariably
approximate the market, except for the minimal management expense fees (typically .25%)charged by
index funds. Professional portfolio managers may use this strategy with clients who have little interest
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in the details of the allocation of assets. The value added in passive management is in the knowing
which index funds to invest in, how to mechanically implement the investments, and in being able to
justify the reduced activity in the portfolio.
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7) Capital Asset Pricing Model (CAPM): Relate to EMH (above) is the Capital Asset Pricing Model
(CAPM) that was developed by William Sharpe. He argued that in an efficient market the return on an
asset (a security, a stock) should be a function of the risk of that security, where risk is measured by
beta. Here are the underlying mechanics of CAPM:
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When historic return data for a particular security (Ke) are regressed against historic return data for the
market (Kmkt), the resulting regression line illustrates graphically how much of the changes in the
returns of the security are "caused by" the changes in the returns of the market. Remember, in its
general form, regression analysis compares a dependent variable (y) to an independent variable, (x) and
yields a "regression line" of the form y = mx + b. In this particular application related to financial
returns, the dependent variables are the returns on the security (Ke) and the independent variables are
the returns of the market (Kmkt), where the market is usually defined as the S&P 500 . Thus, when the
returns of an individual security are regressed against the returns of the market, the resulting slope of
the regression line (the line of best fit, or the line of least squares) is defined as "beta" (or the volatility
of a security's return relative to the volatility of the returns of the market). Obviously, when the returns
of the market are regressed against the returns of the market, the slope of the line will be 1.000.
Therefore, by definition, the beta of the market is 1.00.
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The CAPM model estimates the return on a security as being the return on a "risk free security", plus a
"risk premium". Traditionally, a US Treasury instrument, perhaps a 3-month bill or a 10-year note, was
used to represent a risk free security. But contemporary derivative models are tending to use the
London Inter-Bank Offer Rate (LIBOR) as a more global risk-free instrument. In CAPM the "risk
premium" is estimated to be β(Kmkt-Krf), or beta times the difference between the expected return of
the market and the risk free rate.
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Thus the CAPM model: Ke=Krf+β(Kmkt-Krf). When the data from the model is graphed to a quadrant in
which the horizontal axis is "risk", or "beta", and the vertical axis is "returns", the result is a straight line,
the so called "Security Market Line" or (SML). [Take care not to confuse this graph with the graph
described further above that generates the values for beta]. This line, the SML, goes through two
points, or two "risk/return profiles" – one point being that of the risk free instrument (beta=0, Krf), the
second being that of the market (beta=1.00, Kmkt). The SML represents the straight line relationship
between risk and return, and suggests that all securities should fall on this line, thus showing a higher
expected return for securities with higher risk.
6) Efficient Market Hypothesis (EMH): Eugene Fama of the University of Chicago is credited for having
brought EMH into focus. His premise is that financial markets (and equity markets in particular) are
efficient in the sense that prices in the markets respond instantaneously to new information, and as a
consequence, prices are always reflective of the fair market value. These assumptions imply that an
investor cannot consistently beat the market - with the following caveat: Given the great number of
investors in the world, there exists the very real probability that a few investors will consistently beat
the market simply through blind dumb luck. And the fact that these lucky investors may also be
intelligent, honest, hard working, and convinced (as well as being convincing) of their special prowess
does not negate the randomness of their golden touch. The Mann interview [unavailable at the time of
this writing] with Eugene Fama elaborates on this theory.
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Sample problem: Use CAPM to find the expected returns of a security, given that a 3-month Treasury
bill yields .93 %, that the expected returns on the S&P500 are 17 % and the beta of the security is 1.50.
Draw the related Security Market Line (SML).
Solution: The CAPM set up would be Ke=.0093 + 1.5(.17-.0093) for an expected yield of 25.035%. The
graph should have points (X,Y)=(βeta, K) at (0, .93%) for the Risk Free instrument (the US T-Bill);
(1.0,17%) for the market; and (1.5,25%) for the security in question.
8) Behavioral Finance: This branch of finance is not so much a strategy itself, but is the study of
financial strategies from a psychological perspective. Given that most financial transactions are the
result of a human being making a decision, then understanding the process of human decision making
becomes an essential component of understanding financial markets. Amos Tversky and Daniel
Kahneman are credited with extensive studies of decision making, and Richard Thaler has juxtaposed
their work against the efficient market hypothesis to expose some significant conflicts between these
two widely accepted theories. Perhaps the significance of behavioral finance to students is that this
field reveals how little we really know about the hows and whys of human financial decision making, and
this may help mitigate the intimidation of the vast amount of technical knowledge that appears to have
accumulated over the years. The Kolbert article lends substance to this discussion and the Hilsenrath
article gives a summary of the Fama/Thaler dispute.
9) Fundamental Analysis: In the taxonomy of financial analysis, there is often a distinction made
between fundament analysis and technical analysis. These two approaches are perhaps not so much
strategies, per se, as they are techniques or methods used in the pursuit of a speculation strategy.
There exist certain empirical dimensions of a firm that are called the "fundamentals" of the firm, similar
in concept to a broader view of the "fundamentals" of the economy. The fundamentals of the firm
include, for example, market share, revenue growth, earnings, and leverage. And fundamental analysis
is the process of regarding the trends (historical long term and short term) of these variables, plus their
relative and absolute levels, and ultimately makes a subjective judgment about the health of the firm
based on the fundamentals.
10) Technical Analysis: A technical analysis takes a different approach by regarding stock price data and
the patterns imbedded therein and further using these patterns to forecast price trends. An implicit
assumption is that past patterns will tend to repeat in the future, so the critical step is to discover and
recognize the patterns. And if an analyst can find a new, previously undiscovered pattern, then acting
on that new pattern will give the analyst (now, the investor) a competitive edge over the rest of the
market. Related to technical analysis are "event studies" in which the analyst tries to identify a
previously undiscovered relationship between some variable, or variables, and stock prices. An example
that has become a historical cliché is the relationship between sun spots and stock prices, in which there
was for many years a tight correlation between the two data sets and the resulting predictive model
enjoyed surprising success. That there was no rational causal relationship between the two variables
did not negate the strong statistical significance of this finding.
11) Black Swans and Fat Tails: For hundreds of years "black swans" have been a symbol of completely
unanticipated events. In recent years, Nissam Taleb has popularized the symbol in the context of
finance, using as examples the collapse of Long Term Capital Management (LTCM) and 9/11 and the
effects of these events on financial markets. The impact of black swans on financial strategies is that
non-believers will tend to bet on probable outcomes, where probabilities are estimated from past
occurrences. Believers, on the other side of the bet, will assume that the tails of the probability
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distributions, while historically "thin", contain unpredictable and calamitous outcomes. Hence, the tails
are fat.
12) Ponzi Schemes: Named for Charles Ponzi of Boston, this illegal strategy involves soliciting assets
from investors on the grounds that the "investment manager" has created a unique combination of
trades that can yield extraordinary returns for the investor. The hoax attains credibility as the manager
pays extraordinary returns to the original investors using assets from new investors and may be further
reinforced by issuing fictitious statements that reflect those extraordinary returns. The fraud may
continue for years, continually harvesting new money, avoiding regulatory scrutiny, until enough
investors try to withdraw their non-existent funds. When the pyramid scheme collapses, it falls quickly.
But given the frailties of human nature, new Ponzi schemes will probably continue to emerge.
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Chapter 5 VALUATION
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The process of valuation in finance is consistent with the more general meaning of valuation – that is,
assigning a value to, assessing, or appraising some asset. And, not surprisingly, the assets commonly
valued in finance are stocks and bonds.
Consider that a share of stock (or a bond, too, in this case) is simply a contract (a written legal
agreement) between two parties that entitles one party to certain rights in consideration of an agreed
upon purchase price. The valuation issue is: how much is that contract (the stock/the bond) worth?
What is its value? How much should the buyer pay for the contract?
The answer is: The buyer should pay the current value of the future cash flows related to that contract.
Let's embellish our terms and instead of referring to the "contract", let's refer to "financial instruments",
or a "share of stock", or a "bond" and switch from "purchase price" to "price per share" (pps) or the
price of a bond. And while we're at it, let's change "current value" to "present value", as that is more
consistent with the common name of the models used in this valuation process.
1) Valuation of Stock:
Myron Gordon offered a model for the valuation of stock based on the premise as that shown above.
That is, the value of a share of stock is the present value of the future cash flows associated with that
particular security. Gordon suggested that the only real cash flow associated with a share of stock is the
dividends. Further, he assumed that dividends can be expected to grow at a constant growth rate (often
tied to expected growth in earnings) and that these cash flows should be discounted at the required rate
of return of the stockholder. For more on Gordon's model, see gordon.html
Several inherent problems with stock valuation models in general can be illustrated using Gordon's
model as a straw man. First, some input data is historical, or empirical, in nature. The data itself is true
enough, but there are no guarantees that the data will hold true in the future. Second, some input data
is speculative, and looking into the future is a foggy view at best. And third, even if the historical data
holds true for the future AND the speculative data is luckily "dead-on", the resulting perfect answer of
what the intrinsic value of the stock should be, as often as not, is not likely to be the same as the actual
current market price. This leads to an investor's valuation dilemma.
The dilemma is that regardless of the integrity of a valuation, there is little assurance that the market
will tend towards that valuation. For example, if an analyst determines that a particular stock is worth
$60, and the spot price (current market price) is $50, the rational investor would buy the stock (at $50)
and wait for the rest of the market to wise up and drive the stock to $60. But the nature of the market
is that stocks do not consistently trend toward their valuations.
2) Valuation of Bonds:
The following illustrates how the value of a bond is determined. The sample bond, issued by Sample,
Inc. promises to pay the bondholder a fixed 5 1/4 % (annual) coupon rate, that is, 5.25% each year of the
face value (denomination). By convention, the actual payments are paid every six months in amounts
equal to half of the amount due annually. The denomination of corporate bonds is typically $1000. And
for this example, let's assume that the bond matures (that is, the firm returns the principal to the
bondholder) in 8 years. [Note: This valuation was done in 2012 when there was 8 years to maturity.
Through time, the date of maturity does not change, but the years to maturity changes every year.] This
bond would be listed as:
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SMPL 5.25% 2020 (or some variation of company_name , coupon_rate, date_of_maturity) These three
properties of the bond are fixed – they do not change over the life of the bond. If this sounds like an
I.O.U. that's because it is an I.O.U.. Further, assume that current market rate for comparable (same risk
category) bonds is only 4%. [See also: “Risk on Bonds” (below)] How much should this investor pay for
the Simple, Inc. bond? What is its value?
The "future cash flow" will be 1) the interest payments of $26.25 every 6-months for 8years, plus 2) the
face value of the bond, $1000, at maturity. The firm, Sample, Inc. will eventually pay $26.25x16=$420 in
coupon payments, plus $1000 back to the investor, for a total of $1420. But the $1420 isn't all paid
today. The present value of that future cash flow can be calculated using "time value of money"
concepts. Use the "Present Value of an Annuity" model [see also:……], where the 6-month discount rate
is .04/2=.02, the number of 6-month compounding periods is 16, and the annuity payment amount is
$26.25. See the spreadsheet bonds.xls for the calculations. The answer is $1084.86.
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Moody's
Aaa
Aa
A
Baa
Ba
B
Caa
Ca
C
--
Risk on Bonds:
The risk on a bond is the probability of the firm not being able to pay the full amount of the interest
payments due to the bondholder, or not being able to return the face value of the bond when due at the
date of maturity. This risk is measured by Moody's, Standard & Poor's and Fitch Ratings who use the
"probability of default" as the primary criteria for grading the bond. Their rating scale is similar to an
academic scale of A,B,C, & D, where A is good, and D is not. Specifically,
S&P
AAA
AA
A
BBB
BB
B
CCC
CC
C
D
Best quality, low risk, low return
High quality, but some long term risk
Still "investment grade"
Long term uncertainty, medium risk, medium returns
Speculative attributes
Not considered investment grade
In poor standing, high risk of default, high returns
Highly speculative, high probability of default
Lowest rated class
In default
3) Continuous Compounding: The classic FV=PV(1+k)n model illustrates how more frequent
compounding yields greater future values than less frequent compounding over the same period with
the same rate. For example: Given PV=$1000, annual rate = 5%, for 2 years yields $1102.50 when
compounded annually (twice over the period), but yields $1105.16 when compounded daily.
In this example, the frequency of compounding went from 2 to 730 over the two years. Imagine the
frequency going to a million, or to a gazillion, or to infinity. That's continuous compounding. In the
model FV=PV(1+k)n , k would be equal to the annual rate divided by infinity, and n would equal two
times infinity. One needs a little Calculus to handle infinity in an equation, but it can be done, and the
resulting model looks like FV=PV erT, where r=annual rate, T = time, and e is the natural log. To execute
this in a spreadsheet, try FV=PV*(EXP(r*T)). You should get an extra penny for all your work, $1105.17.
This model is commonly used in valuation of derivatives, for example BSOPM.
4) Loans: Here are some of the mechanics behind the calculation of payments on loans, and the
construction of amortization tables.
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Typical “vanilla” loans (so called, recently, to differentiate traditional loans from exotic adjustable rate,
up-front points, etc.) have payments equal to the sum of the interest due on the outstanding balance
plus a payment that contributes to paying off the principal. The actual formula is:
PMTS= (PVa x k) / [1-(1+k)^-n] where PVa=present value of the annuity, or the amount that the bank is
willing to give to you if you sign a contract promising to pay them a fixed amount (the annuity) every
month (the timing of the payments doesn’t HAVE to be monthly, but that’s the traditional timing). K=
the rate for the period of compounding , usually the annual (quoted) rate divided by 12 months of the
year. And n= the number of periods of compounding.
The amortization of the loan can be expressed in a table in which every period (or month) the interest is
calculated on the outstanding loan balance using the following logic:
Beginning Balance = for the 1st period, is the original amount of the loan. In subsequent periods, the
beginning balance is the ending balance from the previous period, i.e. the outstanding balance.
Interest $ = (annual rate/12 months of the year) X outstanding balance of the loan.
Payments (PMTS)= (see above)
Reduction in Principal: The interest$ are subtracted from the fixed monthly payments to yield the
"reduction in the balance of the loan" (Red'n Bal).
Ending Balance: When the Red'n Bal is subtracted from the "Beginning Balance" the result is the ending
balance for that period. The ending balance of one period is the beginning balance of the next period.
5) Expected Value:
Beyond present value/future value concepts, the notion of expected value is often seen in the valuation
process. Expected value is the product of the anticipated cash flow multiplied by the probability the
cash flow actually materializing. For example, if I roll a die (that's singular for dice) and will pay you $100
if I roll "6", then the value of that game to you equals $100 x .166666…. = $16.67
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5. DERIVATIVES
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Derivatives are a general class of financial contracts that have grown to huge economic significance as
measured in dollars of positions held and transactions conducted. They are currently highlighted in the
financial news as being at least partially responsible for the larger economic meltdown. Yet, for all their
detractors, there are many who defend the existence of derivatives as being essential for the flow of
cash in a free market system. Regardless of whether derivatives are good or evil, some derivatives will
likely survive the character assassination and thus they are entitled to a close regard.
1) Mutual Funds:
Mutual funds are incorporated entities created by a parent company. Two dominant examples of
parent companies are Vanguard and Fidelity. As the parent company creates a new mutual fund, the
contributors (investors) in the fund become the stockholders. With capitalization from the investors,
the fund manger re-invests in financial instruments with characteristics that were decided upon a priori
into a portfolio that is owned mutually [hence the name] by the stockholders/investors. As a
hypothetical example, if Fidelity decided to start a "Green Energy Fund", they would incorporate the
fund, sell shares in the fund, and buy shares of companies serving the environmentally-friendly energy
market. As the share prices of the companies fluctuated, so too would the value of the portfolio
fluctuate. The net asset value (NAV) of the portfolio and the value of the mutual fund company (not the
parent, but the individual fund) are the same, so the share price of the mutual fund changes
proportionally also. Hence the term "derivative" – the value of a share of the mutual fund is derived
from the underlying assets, the green companies.
1.1) Index Funds:
Index funds are a sub-group of mutual funds. Their unique characteristic is that their portfolios are
constructed to mirror the performance of a particular market index, for example the Dow Jones
Industrial Average (DJIA) or the Standard & Poor's 500 index. Index funds are popular vehicles for
passive investors because capital put into index funds are likely to have the same yield as the entire
market – with the following exception: fund managers need to be compensated. Typical management
expense fees for index funds average about .25% of invested assets per year, so net yields to the fund
investor will tend to be lower than the market yield by that fractional percent.
1.1.1) S&P 500 Index Funds:
The S&P 500 Index is constructed by S&P (the firm) selecting the top 500 firms according to level of
market capitalization. And because each firms influence on the index is weighted by that firm's market
cap, then a portfolio (an S&P 500 index fund) that endeavors to match the performance of the index
must hold positions in each firm according to the firm's weighting in the index. As a firm's stock price
fluctuates continually, the firm's weighting in the index will also fluctuate. But, conveniently, the
weighting in the index fund will also fluctuate by the same proportions, as they are based on the same
dynamic market value.
1.1.2) DJIA Index Funds:
The DJIA is constructed differently than the S&P500. In the DJIA, the firms' influences on the index are
weighted according to price per share. Those firms with the higher stock prices have the higher
influence. Thus, a DJIA Index fund would be constructed with the same number of shares for each firm
in order to match the performance of the index.
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2. Options:
Options are derivative financial instruments whose values are a function of the price of the underlying
asset. The more common underlying assets are stocks, but options may be written on other financial
asset classes such as commodities and currencies.
Options give the purchaser of the contract the right, but not the obligation, to buy (or sell – depending
on the type of option) shares from (or “to”) the seller (the writer) of the contract at a predetermined
price, the strike price.
The two variations of options are calls and puts. Calls are options to buy shares at the strike price.
Puts are options to sell shares at the strike price.
An investor may take either side of an option trade – that is, they may buy or sell (write) a call, and they
may buy or sell a put. A single option contract is for 100 shares.
An option has an expiration date, as a specification of the contract, that is either 1) the one future date
at which the option must be exercised or allowed to expire (a so called "European" option), versus 2) the
last date of a period during which the option may be exercised (the "American" option).
Buying and writing options is a “zero sum game”. That is, the buyer’s profit (or loss) will be the same as
the writer’s loss (or profit). The buyer pays the writer for the options up front and may or may not make
that cost back through the exercise of the options. The writer receives the price of the option up front
and may or may not be forced to buy or sell shares at a loss. These transactions are all done through the
options broker at an exchange and buyers and sellers will not generally be aware of the identity of the
counterparty.
Because of the dynamic nature of the market price of the stock, the spot price, will usually be different
than the strike price, the option will either have value (said to be “in the money”) or will be worthless.
1) a) That is, for a call, if the spot is greater than the strike, the investor can exercise her option, buy at
the strike and turn around immediately and sell at the higher spot, and take home the difference.
b) If the spot is less than the strike, the call is worthless and will not be exercised.
2)a) For a put, if the spot is less than the strike, the investor can buy at the lower spot, and then exercise
her option, i.e. sell at the higher strike.
b) If the spot is greater than the strike, the put is worthless and will not be exercised.
Having options “in the money” does not necessarily ensure that the holder of the option will actually
realize a net gain from the entire transaction. In order to break even, the spot needs to be far enough
“in the money” to cover the original cost of the option. Then, if and when the spot passes the
breakeven price, the investor can realize a gain.
The valuation of options is generally determined by the Black-Scholes Option Pricing Model (BSOPM) as
seen in the next section.
BLACK and SCHOLES OPTION PRICING MODEL (BSOPM):
BSOPM has been referred to as a milestone in the development of financial theory on the grounds that
it significantly raised the level of sophistication in the quantitative valuation of complex financial
instruments. While options have been around for hundreds of years, their market value was mostly
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determined by the forces of supply and demand prior to BSOPM. Now, options are valued based on
empirical data, which tends to give them a truer value (perhaps closer to an inherent value). The result
of BSOPM is the price (or value) of a European call, or an American call on a non-dividend-paying stock.
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The intermediate variables are:
d1 = the z-score, or number of standard deviations away from the mean of the distribution of stock
prices. This is a measure of the spot price relative to the mean stock price.
d2 = the z-score of the strike price relative to the mean stock price.
N(d1) = the cumulative standard normal distribution for d1, or the area under the curve given d1, or the
probability related the spot price. To read more about converting z-scores (d1) to the cumulative
standard normal distributions (N(d1)), see http://oz.plymouth.edu/~harding/BU5120/stats.doc.
N(d2) = the cumulative standard normal distribution for d2.
The input variables are:
S = Current price ("spot" price) of the underlying stock. As this pps changes through time, so too will the
price of the option.
K = The "strike" price of the call. The strike is fixed (does not change) throughout the life of the option.
It is the price that the buyer of the option will pay for the stock if the call is exercised.
r = The "risk free rate". As with many derivative models, the current LIBOR (for a time period equivalent
to the remaining life of the option) is used as the risk free rate.
T = The amount of time remaining to the expiration date. T is in years, so an option with 3-months
remaining would have T = .25.
σ = The standard deviation of the underlying asset's annualized returns based on historical data. This
variable is necessarily based on historical data, and as such, is an imperfect estimator of future volatility
– a weak link in the model.
The output variable is:
C = The price of the call.
A more detailed model with the calculations are listed on a separate pages at:
http://oz.plymouth.edu/~harding/BU5120/bsopm1.jpg
http://oz.plymouth.edu/~harding/BU5120/bsopm2.jpg
Straddles and Strangles: Investors will often buy calls and puts on the same stock with the same
expiration data and the same strike – this combination is a straddle. With a straddle, either the call or
the put will be “in the money” (unless the spot is exactly the same as the strike, which would be quite a
coincidence). And in order to realize a net gain, the spot must exceed a breakeven stock price that
covers the original cost of both the calls and the puts, even though only one of the two will be exercised.
The strangle, a variation of the straddle, is a position with calls and puts on the same stock, with the
same expiration date, but with a spread between the strike of the call and the strike of the put. (The
strike of the call will be higher than the strike of the put.) Strangles are cheaper to buy than straddles
because they have a lower probability of being in the money and the calls and puts will be cheaper.
A Strangle example:
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An investor creates a strangle on stock with a strike price of $32.00 on the calls, with 10 call contracts
(@100 shares per contract) and 10 put contracts with strike of $28.00, selling at $2.00 and $1.60 per
share for the calls and puts respectively. On the final day of the option, the spot price is $33.33.
a) Which of the options does she exercise (if either)? Answer: CALLS
[When the spot exceeds the strike, the calls will be exercised. The investor buys at the lower strike and
sells at the higher spot. Actually, the broker will do all that for her.]
b) How much did the original strangle cost her? $3600
[$2.00 per share for the calls, times 1000 shares (10 contracts at 100 shares per contract)
Plus $1.60 per share for the puts, times 1000 shares (10 contracts at 100 shares per contract)]
c) How much did she make (or lose) on the entire transaction? ($2270)
[She buys at the strike, $32.00 /share, sells at the spot, $33.33 /share, makes $1.33 /share, on 1000
shares (10 contracts at 100 shares per contract), for a gross profit of $1330. But she paid $3600 for the
strangle, yielding a net loss of $2270 [=-3600+1330].
d) How much did the writer of the strangle make or lose? $2270
[Options are a zero sum game. Assume the writer doesn’t own the stock. He needs to buy it at the spot,
sell it at the strike to the investor, losing $1330. But he still gets to keep the original price of the calls
and puts that he sold for $3600. So his net gain is $2270 [=3600-1330]
e) What were the breakeven stock prices for the entire transaction? High $35.60 Low $24.40
[The spot needs to exceed the call’s strike by $3.60 [$2.00 =$1.60] in order to cover the cost of the calls
and the puts and to breakeven. That is, $32.00 + $3.60 = $35.60. Or, the spot needs to be below the
put’s strike by $3.60 in order to cover the calls and puts. That is, $28.00 - $3.60 = $24.40.]
3. Hedge Funds: Hedge funds are "derivatives" in the sense that the value of a share of a hedge fund
is derived from the value of the underlying assets held by the fund , similar to the value of a share of a
mutual fund. Hedge funds differ from mutual funds on several dimensions, for instance 1) minimum
initial investment in a hedge fund tends to be higher than in a mutual fund by a factor of perhaps a 100.
2) hedge funds are sold privately versus public offering of mutual funds 3) consequently, hedge funds
are subject to less scrutiny and less regulation 4) management expense fees for hedge funds tend to be
much higher than mutual funds. 5) Hedge funds provide an environment for financial engineering, or
the construction and design of new derivatives. (See also: the article "Hedge Clipping" by Cassidy)
4. Credit Default Swaps (CDS): A swap is contractual arrangement between two parties whereby a
trade is conducted, no cash is exchanged, and the items traded are similar entities, for example, and in
this case, credit default contracts. No cash is exchanged initially because both parties are swapping cash
flows of equal net present value. A credit default contract is similar to an insurance policy against the
defaulting on a loan obligation, such that if the loan goes into default, the insurer would then be
obligated to cover the loan payments. In a credit default swap, the two counterparties exchange their
risk (and coverage obligation) of their respective third party defaults.
5. Collateralized Debt Obligations (CDO): Large collections of loans, most of which are typically backed
by (collateralized) real estate are called CDOs. After the loans are aggregated, shares of the whole are
sold much like shares of stock. The ability to measure the probability of default on an individual loan is
challenging enough, let alone measuring the probability of default on a bundle of hundreds, perhaps
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thousands of loans. Many CDOs received excellent (but totally baseless) default risk ratings from
reputable rating agencies, and paid higher yields than instruments of comparable ratings and were
popular with large investors. When the individual loans start defaulting, the values of these CDOs
become impossible to measure. See Patterson "Math Wizards".
6. FINANCIAL MODELS
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Here is a sampling, in alphabetical order, of some other financial models that appear frequently in the
academic literature, in the real world and in this course.
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5. Dow Jones Industrial Average (DJIA): The DJIA is the standard measure of US stock prices, not so
much for its accuracy in reflecting changes in market value, but more for its longevity and exposure in
the popular media. It is somewhat narrow in scope, using price information on a stable set of 30 large
cap firms. A major flaw in its logical integrity is in its price weighted property – the stocks with the
higher nominal price-per-share have greater impact on the index than those of more modest pps.
1. Alpha: Alpha is the excess return over a benchmark return. For US stocks in general, the benchmark
return is the S&P500, or similarly, an S&P500 index fund. The concept is used to measure performance,
often of a fund manager, such that a manager who consistently "generates" alpha is "beating the
market".
2. Beta: Beta is a measure of risk of a stock relative to the risk of the market. Technically, it is the slope
of the regression line y=mx + b derived from the least-squares method where the dependent variable (y)
is the return of the stock being measured, versus the independent variable (x), the return of the market.
When market returns are regressed against market returns, the slope will equal 1.000 by definition, in
other words, the beta of the market is 1.000. Stocks with greater volatility than the market have betas
greater than one [ > 1.00], and stocks with volatility less than the market have betas less than one.
3. Black Swans & fat tails: A black swan is an unusual, unanticipated, statistically unlikely, and
particularly ugly market event. A fat tail refers to the tails of the probability distribution of returns (or
sometimes prices). A fat tail refers to the likelihood of the unlikely happening. See also blackswan.
4. Brownian Motion:
Brownian motion was originally observed and defined as a physical phenomenon, specifically, the
motion of dust particles in suspension in water. The motion was later associated with bond price
movements (See also: Bachelier), mathematically defined by Weiner and Einstein, and is now regarded
as the classic model of random movement in stock prices.
6. LIBOR:
The "London Inter Bank Offer Rate" is a composite index of rates charged by large banks to other banks.
7. MATLAB:
The name "MATLAB" is short for "matrix laboratory", where a matrix is a table of rows and columns and
the basis of a branch of mathematics – matrix algebra. MATLAB is a proprietary software package
owned by Mathworks, Inc. and licensed to PSU for educational use. Within MATLAB there are financial
modules that can plot efficient frontiers of portfolios and perform many of the calculations related to
the valuation of derivatives.
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8. Monte Carlo Simulation:
"Monte Carlo" refers to the city in Monaco with the grand casino, and Monte Carlo simulation models
use random variables, not unlike the randomness on the roulette wheels at the casino. While there are
several variations of the model, they all have the property of being able to integrate probabilities into
the outcomes, thus distinguishing them from deterministic models. They are used in financial
forecasting because the outcomes expressed as a distribution of results (with related probabilities) can
be more useful than outcomes expressed without their related probabilities.
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For example, assume that a portfolio manager wanted to equally weight two stocks in her portfolio, i.e.
"to have a 50/50 dollar mix of StockA and StockB. Let's further assume that the portfolio's total value is
$20K, which implies $10K of StockA and $10K of StockB. And finally, assume that the most recent price
per share for StockA is $32.75 and the pps for StockB is $56.25. How many shares of each does she
need to make $10K for each?
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Chapter 7 INTERNATIONAL FINANCE
9. Pi ( ):
One of the most recognized mathematical relationships, pi's connection to finance in its use with the
Normal distribution model. Specifically, it is used in computing the area under the curve, or the
cumulative distribution function (cdf). While practitioners of finance rarely have the need to actually
calculate pi's value, they will see that it is an essential piece in some financial models, for example
BSOPM. Students should be comfortable pi's definition and memorize pi to at least a few places past
the decimal point. See also Pi.
10. Normalized number of shares: In portfolio management modeling, the "quantity" of the diverse
assets (the different stocks) is often measured in terms of dollars of market value (or other currency) or
in the proportion of dollars relative to the whole (sometimes the proportions are referred to as the
"weights"). And frequently it becomes necessary to translate those dollars, or sometimes the weights,
into "Number of Shares". The process of calculating the number of shares required to equate to a given
dollar amount is the process of normalization. While it is perhaps more common and more intuitive to
go from number of shares to dollar amount (by multiplying no of shares times pps), nonetheless, it is not
uncommon to have to "go backwards" from dollars to number of shares.
The Mechanics: Divide $10,000 by $32.75. Answer: 305.343511450382 [At this point the financial
analyst might want to make some decision about "appropriate level of precision". Is this an academic
exercise? Is this amusing theoretical research? Is this somebody's real money? Let's go 3 places past
the decimal – it looks precise and it really doesn't matter.] = 305.344 shares. And not to forget StockB:
$10,000/$56.25=177.778 shares.
The Results: The normalized number of shares for a 50/50 mix of StockA and StockB, when trading at
$32.75 and $56.25 respectively, equals 305.344 and 177.778. To calculate the no. of shares for a 75/25
mix of A and B, cut the 305.344 in half and raise the 177.778 by half [152.672 shares of A and 266.667
shares of B?]. That's "normalization of shares".
A variety of new financial issues arise when one broadens their perspective from concentrating on the
internal domestic scene to regarding the financial interaction among the various other countries of the
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world. While many of the traditional financial models still hold true (e.g. portfolio management, time
value of money, and corporate finance), these models are made increasing complex with the
introduction of differences in currency exchange rates, national interest rates and foreign accounting
conventions. With the increasing complexity there is also an expansion of opportunity for new
variations of financial activity with foreign investment instruments. Here are some of the more
significant financial instruments related to international finance.
1) Foreign Currency Exchange: Every country has an official currency for transactions. Each country's
currency's value is stated in terms of every other country's currency. The relationship between the
currencies is established by government policy and/or by world markets. For example, an exchange rate
between one country's currency and another may be "fixed" by policy, whereby the value relationship
doesn't change between it and some other currency (e.g. the Chinese yuan pegged to the US dollar). Or
currencies may "float", whereby the value relationship is constantly changing as determined by supply
and demand in the currency markets. At any point in time all exchange rates are more or less in
equilibrium, and this can be expressed in a "cross rate table".
A student grappling with exchange rates might do well to start with the simple $/€ format. This fractionlike looking expression means "the number of US dollars required to buy one euro". It does NOT mean
dollars divided by euros. Further, the exchange rate of $/€ = 1.35 means $1.35 = €1.00. It is true that,
similar to a fraction, that the reciprocal of the exchange rate format equals the reciprocal of the value,
that is, €/$ = .74074.
Another format for expressing exchange rates is the cross rate table. These tables used to be printed
daily in The Wall Street Journal although the numbers are in a constant state of flux. Understanding
how to read and how to construct a cross rate table is a useful exercise in understanding foreign
exchange.
Cross rates are calculated using the following logic: If $/€ =1.35 and $/C$ = .80, then
€/C$ = ($/C$)/($/€) or €/C$ = .80/1.35 = .592593
It should be noted that some countries also have "unofficial" currencies that are used in addition to their
official currency. A common example can be found in developing countries in which transactions are
conducted with both the official national currency and also with the US dollar. Transactions with
unofficial currencies are often illegal (but not always) in the countries where the practice is common.
2. Arbitrage: Currency arbitrage is conducted by professional traders who take advantage of the small
dis-equilibrium that occurs among currencies as a result of the floating valuations between the many
different currencies. The profits generated by arbitrage trading are small as a percent of the amounts
traded, but the amounts traded are huge, transaction costs are small and thus actual dollar profits can
be significant. The process of arbitraging currencies has the effect of restoring equilibrium to the
currency markets. Arbitrage opportunities are recognized by computers and trades are executed within
fractions of a second. There is fierce ongoing competition for faster and faster trading systems among
trading houses, because being second best means losing the trade.
Two way currency arbitrage is conducted by professional traders who use super-fast computer systems
to scan bank quotes looking for even the slightest imbalance in exchange rates. When banks quote their
exchange rates, they list both the bid and ask prices. See also the example below. That is,
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€/C$ @ .71076-78 means that Deutsche Bank is offering to buy Canadian dollars at € .71076 per C$ and
is willing to sell Canadian dollars at €.71078 per C$. The two-way arbitrage opportunity is created when
a second bank, in this case, Royal Bank of Canada, quotes bid/ask prices outside the range, either above
or below, the first bank, as in the example below. Over-lapping ranges do not create an arbitrage
opportunity.
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In the example, a trader could buy Canadian dollars from Royal Bank of Canada at €.71074 and sell to
Deutsche Bank at €.71076. The profits are small (.00002/.71074), working out to the equivalent of
$28.14 for every $1 million traded, or .002814%. However, if a bank trades in quantities of several
million dollars per trade, and trades several times per day, the profits are enough to buy lunch. Note
also that the first trade will signal both banks that their pricing structures are out of balance, and they
will react quickly to closing the gap and ending the arbitrage opportunity. On the other hand, trades are
not guaranteed by law or by circumstances to be immediately transparent to all parties, and thus
arbitrage opportunities continue to emerge.
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Example: At 13:45 hrs GMT on 9 November 2010 Deutsche Bank quotes €/C$ @ .71076-78 and Royal
Bank of Canada quotes €/C$ @.71072-74
a) Is there an arbitrage opportunity? Y/N? _yes___
b) If so, one would buy from _RBoC___ and sell to _Deutsche Bank_?
c) Calculate the return (i.e. the profit as a %) _.002814__%
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Three way currency arbitrage is similar to the two way arbitrage, in that it is conducted by professional
traders who use computer systems to watch multiple exchange rates simultaneously. When the
opportunity arises, the system conducts the trade quickly. In the following example it suggests that a
human is doing the trading, whereas the reality is that systems are actually executing the trades.
The mechanics: The trader starts with some amount of currency A. Trader uses A to buy currency B,
then, uses B to buy currency C. And finally, uses C to buy currency A. Or, A to B, B to C, C to A. If the
currencies were in perfect equilibrium, then the trader would end up with the same amount of currency
A as when he/she started. However, when the currencies are in disequilibrium, there is a potential to
make a profit through the three-way trade.
There is also the opportunity to lose money after the three trades. If A to B , B to C, and C to A
generates a profit, then it is certain that trading in the other direction (A to C, C to B, B to A) will
generate a loss. One should not assume that even an experienced trader can identify disequilibrium in
currency rates by merely glancing at the various data. So, as an academic exercise, if A to B, B to C, and
C to A generates a loss, then a student should recalculate in the reverse order (A to C, C to B, B to A). In
the real world, the system does these calculations.
Example: Assume that $/Ps=.092 in Mexico City, and Ps/€ = 16.06 in Zürich and €/$=.675 in NYC. Start
with $1M and show the trades (a,b,c) that would yield an arbitrage profit. Note: in sell& buy answers,
show both currency and amount. The "trial" is for you to see if your path will make a profit. If it does,
then that is your answer. If it doesn't, then reverse direction and put the results in "final".
Trial:
a) city__NYC_____ sell_$1,000,000_ buy___________
b) city___________ sell__________ buy____________
c) city___________ sell__________ buy____________
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Final:
a) city___________
b) city___________
c) city___________
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Calculate the arbitrage profit as a percent of original
investment.__________%
sell__________
sell__________
sell__________
buy____________
buy____________
buy____________
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3. Forward Contracts: Forward currency contracts are a popular investment vehicle used for hedging
against future changes in exchange rates. An investor with a known obligation to provide funds in a
foreign currency at a future date (for example, to pay for imported goods scheduled to be delivered in
several weeks) may be motivated to buy forward contracts to lock in the cost of obligation. Forwards
are sold for only a few currencies and are sold at a premium or discount relative the current spot price.
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4. Futures: Futures are similar to forward contracts in that they represent advance sales of currencies
at an agreed upon price. However, only a few specific currencies are traded, they are only traded in a
few physical markets (e.g. IMM and CME), and they are only sold in the following fixed denominations:
GBP 62,500 Great Britian pound
CAD 100,000 Canadian dollar
JPY 12,5000,000 Japanese yen
CHF 125,000 Swiss franc
MXP 500,000 Mexican peso
EUR 125,000 euro
The currencies that have forward contracts available are the Canadian dollar (C$), Japanese yen (¥),
Swiss franc (SF), and the UK pound (£). Within each of these currencies, there are 1 month, 3 month,
and 6 month forwards available. That is, the currency delivery date is one month (or 3 or 6 months)
from the date of purchase.
For a given currency, for example the Swiss franc, the forwards may be selling at premium or discount to
the recent market or spot price. The forward prices are set by supply and demand and change based on
currency traders' aggregate assumptions regarding future exchange rates between, in this case, the US
dollar and the Swiss franc. A clipping from the 8 July 2009 issue of The Wall Street Journal from their
table called "Currencies July 7, 2009 U.S. –dollar foreign-exchange rates in late New York trading" shows
in part the following:
Switzerland franc
1-mos forward
3-mos forward
6-mos forward
.9177
.9180
.9188
.9203
The first quote is the spot price of the SF (from the day before) and is in US$ , or number of US$ to buy
one Swiss franc, ($/SF). The 6-mos forward is selling at a premium over the spot, and this is generally
expressed as a percentage. (.9203 - .9177) / .9177 = 0.00283317 or "the 6-mos forward is selling at a
.28% premium". Forwards selling at a premium reflect the widely held assumption that the value of the
currency (the SF in this case) is more likely to rise than to fall in relation to the US dollar over the next
few months.
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5. Options : Options on currency are mechanically similar to options on stocks, indices, and
commodities. There are calls and puts, strike prices and expiration dates as with a stock option.
Straddle and strangle strategies can be created as with stock options and individual investors as well as
institutions engage in this popular market for speculating and hedging.
Appendix: STATISTICS
(and other scary math concepts)
1. Correlation Coefficients:
A "coefficient" is a fancy word for an "index", or a "measurement", and "correlation" might be thought
of as "co-relation" – or the relation between two streams (or series) of data. In finance, correlation
coefficients are used to measure how closely (or how differently) two data sets follow each other. A
common example is the measurement of the returns, through time, of two equities; for instance, how
closely do the daily returns of Ford match the daily returns of GM?
The variable "r" is used for the correlation coefficient and the range of possible values goes from r = 1.00
to r = -1.00, where 1.00 indicates "perfect positive correlation", -1.00 shows "perfect negative
correlation", and r = 0.00 indicates that there is no positive or negative correlation at all. Perfect
positive correlation occurs when the percent change in one series is matched by the same exact percent
change in the second series. Perfect negative correlation is when the changes in the two series are the
same in magnitude, but opposite in sign.
Using the Ford/GM example above, assume that the two data series consisted of daily returns for 250
consecutive trading days (about one calendar year). Further, assume that on most days Ford and GM's
stock prices move in the same direction as the market (sometimes up, sometimes down), but that their
respective "percent change from the previous day" are usually slightly different from one another. And
further assume that on some days the percentage changes are in the opposite direction from each
other. The correlation coefficient of Ford and GM's daily returns is likely to be "mildly positive", for
example r = .342 Note: Correlating Ford with GM
In MS Excel, the correlation coefficient function is "CORREL". Assume Ford's daily returns are in column
A, and GM's returns are in B. The correlation coefficient expression would be:
=CORREL(A1:A250,B1:B250)
See also: Wikipedia "Correlation Coefficient"
2. Regression Analysis:
Regression analysis also regards two series of data as does the calculation of correlation coefficients,
however, regression analysis treats the two series differently by designating one data series a dependent
variable, and the other an independent variable. The regression analysis seeks to measure how much of
the change in the dependent variable can be explained by changes in the independent variable.
In finance, the daily returns of the market are often considered the independent variable and the
individual security (e.g. Ford) considered the dependent variable. Hence, the question becomes "how
much of the daily change in Ford's stock price can be explained by the daily change in the market"
3. Z-Scores: In the Normal distribution, the number of standard deviations away from the mean is
called the z-score. The z-score is used to find the percent of the total area under the curve, which in
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turn, represents the probability of occurrence of some condition (or event) – in this case, the probability
of a stock being over (or under or between) some finite price. The area is more technically called the
cumulative standard normal distribution. To convert z-score to area/probability, one may use a z-score
table.xls, a z-score table.doc, the Excel function "=NORMSDIST(…)", or on a TI-83 there is a fairly easy
routine to generate the area/probability (2nd function /DIST. Select #2 on menu. NORMCDF(lower limit,
upperlimit) [where lower limit = -1E99 and upper limit=0] and E="EE" on the TI-83 keypad.