LECTURE NOTES ON

INVESTMENT ANALYSIS AND PORTFOLIO MANAGEMENT

PART 3

SUDHANSHU PANI

DRAFT - JAN 2019

DISCLAIMER:

THIS IS A FIRST DRAFT AND HENCE ERROR PRONE. PLEASE DO NOT QUOTE. IT CONTAINS

MATERIAL THAT WILL BE FOLLOWED IN THE CLASSROOM. IT IS NOT THE RECOMMENDED TEXT

BOOK. RATHER A COMPANION TO CLASSROOM. DUE CREDITS HAVE BEEN GIVEN TO REFERENCES.

Chapter 3 The Markowitz Portfolio Optimisation Model

Creating a portfolio Optimisation model

Understanding the CAPM model

Concept of Efficient frontier

Markowitz argument was that a method that employs discounting the future returns will not

lead to a diversification and hence needs to be rejected. The maths from the paper is below:

In contrast he proposed the mean-variance method that he called as belief-choice rule. The

belief of expected returns and the choice of how much risk to take.

The Journal of Finance paper, Portfolio Selection (1952) introduced his ideas and later got

Harry Markowitz the Nobel Prize for this seminal contribution.

THEPROCESS OF SELECTING a portfolio may be divided into two stages. The first stage

starts with observation and experience and ends with beliefs about the future performances of

available securities. The second stage starts with the relevant beliefs about future

performances and ends with the choice of portfolio. This paper is concerned with the second

stage…

.. We next consider the rule that the investor does (or should) consider expected return a

desirable thing and variance of return an undesirable thing.

Law of large numbers does not apply to a portfolio of securities as the securities are

intercorrelated. Diversification cannot remove all variance.

RECALL:

Let be a random variable, i.e., a variable whose value is decided by chance. Suppose, Y can

take on a finite number of values . Let the probability that

. The expected value or mean of is defined to be

And the variance of Y is defined to be

Where, V is the average squared deviation of Y from its expected value.

If we have n number of random variables , a weighted sum of these random variables (R),

(linear combinations) is also a random variable. i.e.

How is the expected value and variance of R related to the distribution of .

The variance would also depend on the covariance between two random variables defined as

{[ ][ ]

The covariance may also be expressed as a product of the correlation coefficient and the

standard deviation of the variables. .

The variance of the weighted random variable is

∑

∑∑

We can now apply the above to our problem. Let be the return on ith security and the

weight /allocation assigned by the investor to the security. Let be the expected value of .

The return (yield) on the total portfolio is

∑

The weights are chosen by the investor and ∑ . The expected return of the portfolio

and the variance is as follows

∑

∑∑

For a set of fixed probability beliefs ( , the investor has a range of choices from (E,V)

depending on his choice of . If the E,V combinations are given as in the figure, then the

investor would like to go for a higher E for a given V or less or go for a lower V given an E.

An optimization for this is our next task.

Note that we have earlier plotted E vs V plots based on the text BKM. So in the latter plots

efficient combinations are in the north west of the plot and in the above plot the efficient

combinations are in the south east of the plot.

These efficient combinations are also known as efficient surfaces or efficient frontier. Two

conditions-at least-must be satisfied before it would be practical to use efficient surfaces in

the manner described above. First, the investor must desire to act according to the E-V

maxim. Second, we must be able to arrive at reasonable .

An illustration for a 3 security case.

In the plot of the weights of securities, the expected returns can be represented as parallel

lines known as isomean lines. Since the variance depends on square of the weights, the

variance is represented as elliptical curves. The center of such a series of curves is the point

of minimum variance. The triangle abc gives the set of attainable portfolios. The efficient

portfolios are a line that joins the point of minimum variance to the point of maximum

expected return.

The minimum variance point is outside the attainable set.

A 4 stock portfolio.

Moving from X plot to an E-V plot. The efficient portfolios are parabolas.

Not only does the E-V hypothesis imply diversification, it implies the "right kind" of

diversification for the "right reason.'' The adequacy of diversification is not thought by

investors to depend solely on the

number of different securities held. A portfolio with sixty different railway securities, for

example, would not be as well diversified as the same size portfolio with some railroad, some

public utility, mining, various sort of manufacturing, etc. The reason is that it is generally

more likely for firms within the same industry to do poorly at the same time than for firms in

dissimilar industries.

Similarly in trying to make variance small it is not enough to invest in many securities. It is

necessary to avoid investing in securities with high covariances among themselves. We

should diversify across industries because firms in different industries, especially industries

with different economic characteristics, have lower covariances than firms within an

industry.

The concepts "yield" and "risk" appear frequently in financial writings. Usually if the term

"yield" were replaced by "expected yield" or "expected return," and "risk" by "variance of

return," little change of apparent meaning would result. Variance is a well-known measure of

dispersion about the expected.

If instead of variance the investor was concerned with standard error, or with the coefficient

of dispersion, his choice would still lie in the set of efficient portfolios. Suppose an investor

diversifies between two portfolios (i.e., if he puts some of his money in one portfolio, the rest

of his money in the other. An example of diversifying among portfolios is the buying of the

shares of two different investment companies). If the two original portfolios have equal

variance then typically the variance of the resulting (compound) portfolio will be less than

the variance of either original portfolio.

The above figure from BKM. Plots E and Std deviation of 3 stocks.

How to derive the efficient set of risky assets.

Based on Markowitz – Efficient frontier. (from BKM)

Sketch of his approach:

First, determine the risk-return opportunity set. The aim is to construct the north-westernmost

portfolios in terms of expected return and standard deviation from the universe of securities.

The inputs are the expected returns and standard deviations of each asset in the universe,

along with the correlation coefficients between each pair of assets. These data come from

security analysis.

The graph that connects all the north-westernmost portfolios is called the efficient frontier of

risky assets. It represents the set of portfolios that offers the highest possible expected rate of

return for each level of portfolio standard deviation. These portfolios may be viewed as

efficiently diversified.

There are three ways to produce the efficient frontier.

For each method, first draw the horizontal axis for portfolio standard deviation and the

vertical axis for risk premium. We focus on the risk premium (expected excess returns), R,

rather than total returns, r, so that the risk-free asset will lie at the origin (with zero SD and

zero risk premium).

We begin with the minimum-variance portfolio—mark it as point G (for global minimum

variance).

The three ways to draw the efficient frontier are

(1) maximize the risk premium for any level of SD;

(2) minimize the SD for any level of risk premium; and

(3) maximize the Sharpe ratio for any level of SD (or risk premium).

A Dummy Example:

HDFC

Bank SBI RIL ITC HUL

AURO

BINDO TCS MARUTI DMART IBHSG

E 0.24 0.11 0.2 0.12 0.12 0.16 0.15 0.19 0.16 0.24

SD 0.16 0.15 0.13 0.1 0.14 0.17 0.15 0.12 0.15 0.3

CorrMatrix

HDFC Bank 1

SBI 0.6 1

RIL 0.4 0.4 1

ITC 0.1 0.1 0.2 1

HUL 0.15 0.15 0.2 0.8 1

AUROBINDO 0.2 0.2 0.4 -0.3 -0.3 1

TCS -0.2 -0.2 0.3 -0.3 -0.3 0.6 1

MARUTI -0.1 -0.1 -0.2 0.3 0.3 -0.1 -0.2 1

DMART 0.1 0.2 -0.1 0.5 0.5 -0.1 -0.1 0.3 1

IBHSG 0.6 0.4 -0.1 0.1 0.1 -0.1 -0.1 -0.05 0.05 1

Solution

E 0.153558 0.219342 0.228099 0.235247 0.24 0.277128 0.24

SD 0.002896 0.01 0.0532 0.08 0.1 0.12 0.2

HDFC Bank

0.0937 0.5648 0.2488 0.0047 0.0000 0.0000 0.0000

SBI

0.1201 0.0000 0.0915 0.0056 0.0000 0.0000 0.0000

RIL

0.0000 0.1102 0.0000 0.0072 0.0000 0.0000 0.0000

ITC

0.2974 0.0000 0.0000 0.0067 0.0000 0.0000 0.0000

HUL

0.0000 0.0000 0.0000 0.0066 0.0000 0.0000 0.0000

AUROBINDO

0.0000 0.0000 0.0000 0.0073 0.0000 0.0000 0.0000

TCS

0.2676 0.0000 0.0000 0.0073 0.0000 0.0000 0.0000

MARUTI

0.2212 0.3250 0.0000 0.0071 0.0000 0.0000 0.0000

DMART

0.0000 0.0000 0.0000 0.0068 0.0000 0.0000 0.0000

IBHSG

0.0000 0.0000 0.6597 0.9406 1.0000 1.1547 1.0000

0 0 1 1 1 1 1 1.1547 1 0

End of Dummy Example

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25

Efficient Frontier

Efficient Frontier

Refer the interesting 6 country investing example in BKM

In summary the Mean-Variance model simply implements Markowitz intuition that investors

should like returns and dislike the variability of the returns. An alternative to this approach

for investors who watch the market closely is to decompose the returns from the stock of a

particular firm into factors attributable to the firm and factors attributable to the market. Since

we already have a price dataset for an index, it would be easier to obtain the returns and

variance of this index. So the investor needs to work only on the specific stocks that he wants

to evaluate for his portfolio.

Let be the excess return from ith security [ ]. This is returns in excess from

the risk free rate. This return can be decomposed into factors that are specific to the firms

performance and factors that are common to the market or economy. [Let us assume for this

course that such a decomposition is technically feasible. Such an assumption is based on

strong assumption about the equilibrium of security markets and statistical theory. If the

returns in a set of securities are joint normally distributed, the returns in each asset is linear. ]

i.e Excess Returns of Firm = Returns from factors that explain Systematic /Market risk +

Returns from factors that explain unsystematic or firm risk + Excess return from Security (a

surprise that investors try to hunt)

Excess Returns of Firm = Excess return explained using sensitivity to Market excess return +

Excess return from events at firm + firm specific outperformance

This approach gives certain additional levers to the investor:

Beta – sensitivity of the stock performance to the market performance. This helps investors

evaluate the risk that they want to take relative to the market and choose securities

accordingly.

Alpha – Having chosen a beta, a security with a positive and higher alpha is chosen

Variance ( ) = Variance ( ) = Systematic Risk + Unsystematic Risk

=

Plotting the excess returns of a firm as per above equation. The slope is the beta and y-

intercept is the alpha. We would get alpha if the market gives zero return. This kind of

regression analysis is possible using easily available historical data. Such a line is called the

security characteristic line. The regression line gives the expectation of the returns

|

What do you understand of Beta and how can you use this concept?

The correlation coefficient between the firm and the market can be explained using the

variance as

Diversification in the Index Model

Suppose you invest as per securities and the weights in an index. What is the variance of

returns in this portfolio. For a single security we have seen the following:

Variance-Ri =

.

Similarly, the variance of the portfolio comprises the variance from market and variance from

firm specific factors. The beta of the portfolio is a simple average of the betas of the

underlying stocks. Hence, market risk is

. Thus, there is no benefit from

diversification in case of market risk.

What happens to the firm risk. Assume there are n stocks and the allocation in the portfolio is

. The returns of portfolio explained by firm specific risk parameters of the portfolio is

∑ . The variance of this component, Since the firm specific risk is not

correlated among the firms, is the weighted sum of the firm specific risk. Since, the weights

are a square of the allocation weights, it leads to substantial diversification benefits in terms

of reduction of portfolio variance.

∑

Hence, by building a diversified portfolio, the firm specific can be reduced substantially.

Investors with a diversified portfolio are then only bothered by the systematic part of the risk

in the portfolio. Of this part they can only influence the portfolio beta in terms of stock

selection. Beta remains the key source of the risk in such portfolios.

Summarising: When we control the systematic risk of the portfolio by manipulating the

average beta of the component securities, the number of securities is of no consequence. But

for nonsystematic risk the number of securities is more important than the firm-specific

variance of the securities. Sufficient diversification can virtually eliminate firm-specific risk.

Understanding this distinction is essential to understanding the role of diversification. As we

had seen earlier, diversification between 20-100 securities completely eliminates the firm-

specific risk.

Comment: Trenor-Black model

Use security analysis to add a few stocks to the index portfolio to improve sharp ratio.

Calculate weight of active portfolio:

1) (alpha/variance) / (market return/Market variance)

2) Adjust above for beta. W= w0/(1+w0(1-beta))

Weight of a security in active portfolio

(Firm alpha/firm specific variance) / sum(alpha/firmspecific variance)

CAPITAL ASSET PRICING MODEL In finance CAPITAL ASSET PRICING MODEL (CAPM) is associated with the model

proposed by Sharpe (1964), Lintner (1965) and extended by Black (1972). It marked the

beginning of Asset pricing theory. Sharpe won a Nobel prize for his work on CAPM.

CAPM offers a powerful prediction about how to measure risk and the relation between

expected return and risk. The empirical record of the model, however, is not good and

hence not recommended to be used in applications. The CAPM’s empirical problems may

reflect theoretical failings, the result of many simplifying assumptions or may also be

caused by difficulties in implementing tests of the model. For example, what should be

considered as a comprehensive market portfolio.

CAPM – The Intuition (From Fama-French 2004)

The CAPM builds on the model of portfolio choice developed Markowitz (1959). In

Markowitz’s model, an investor selects a portfolio at time (t – 1) that produces a

stochastic return at t. The model assumes investors are risk averse and, when choosing

among portfolios, they care only about the mean and variance of their one-period

investment return. As a result, investors choose ―meanvariance-efficient‖ portfolios. We

have earlier seen such portfolios. Such portfolios minimize the variance of portfolio

return, given expected return, and given variance, they maximize expected return.

The portfolio model provides an algebraic condition on asset weights in meanvariance-

efficient portfolios. The CAPM turns this algebraic statement into a testable prediction

about the relation between risk and expected return by identifying a portfolio that must be

efficient if asset prices are to clear the market of all assets.

Sharpe (1964) and Lintner (1965) add two key assumptions to the Markowitz model to

identify a portfolio that must be mean-variance-efficient. The first assumption is complete

agreement: given market clearing asset prices at t - 1, investors agree on the joint

distribution of asset returns from t - 1 to t. And this distribution is the true one—that is, it

is the distribution from which the returns we use to test the model are drawn. The second

assumption is that there is borrowing and lending at a risk-free rate, which is the same for

all investors and does not depend on the amount borrowed or lent.

A Std-dev vs Returns plot as given above. The curve abc, called the minimum variance

frontier, traces combinations of expected return and risk for portfolios of risky assets that

minimize return variance at different levels of expected return. (These portfolios do not

include risk-free borrowing and lending.)

There is a trade-off between risk and expected return for minimum variance portfolios.

An investor who wants a high expected return, say at point a, must accept high volatility.

At point T, the investor can have an intermediate expected return with lower volatility. If

there is no risk-free borrowing or lending, only portfolios above b along abc are mean-

variance-efficient, since these portfolios also maximize expected return, given their return

variances.

Adding risk-free borrowing and lending turns the efficient set into a straight line.

Consider a portfolio that invests the proportion x of portfolio funds in a risk-free security

and 1 - x in some portfolio g. If all funds are invested in the risk-free security—that is,

they are loaned at the risk-free rate of interest—the result is the point Rf in above figure, a

portfolio with zero variance and a risk-free rate of return. Combinations of risk-free

lending and positive investment in g plot on the straight line between Rf and g. Points to

the right of g on the line represent borrowing at the risk-free rate, with the proceeds from

the borrowing used to increase investment in portfolio g. In short, portfolios that combine

risk-free lending or borrowing with some risky portfolio g plot along a straight line from

Rf through g.

To obtain the mean-variance-efficient portfolios available with risk-free borrowing and

lending, one swings a line from Rf in the Figure up to the tangency portfolio T. We can

then see that all efficient portfolios are combinations of the risk-free asset (either risk-free

borrowing or lending) and a single risky tangency portfolio, T. This key result is Tobin’s

(1958) ―separation theorem‖ which we have seen earlier.

With complete agreement about distributions of returns, all investors see the same

opportunity set, and they combine the same risky tangency portfolio T with risk-free

lending or borrowing. Since all investors hold the same portfolio T of risky assets, it must

be the value-weight market portfolio of risky assets. Specifically, each risky asset’s

weight in the tangency portfolio, which we now call M (for the ―market‖), must be the

total market value of all outstanding units of the asset divided by the total market value of

all risky assets. In addition, the risk-free rate must be set (along with the prices of risky

assets) to clear the market for risk-free borrowing and lending.

In short, the CAPM assumptions imply that the market portfolio M must be on the

minimum variance frontier if the asset market is to clear. This means that the algebraic

relation that holds for any minimum variance portfolio must hold for the market portfolio.

Specifically, if there are N risky assets, the minimum variance condition for M

[ ]

Thus, is the covariance risk of asset i in M measured relative to the

average covariance risk of assets, which is just the variance of the market return. is

proportional to the risk each dollar invested in asset i contributes to the market portfolio.

The last step is to use the assumption of risk-free borrowing and lending to compute the

, the expected return on zero-beta assets. A risky asset’s return is uncorrelated

with the market return—its beta is zero—when the average of the asset’s covariances

with the returns on other assets just offsets the variance of the asset’s return. Such a risky

asset is riskless in the market portfolio in the sense that it contributes nothing to the

variance of the market return. When there is risk-free borrowing and lending, the

expected return on assets that are uncorrelated with the market return, , must

equal the risk-free rate, Rf. The relation between expected return and beta then becomes

the familiar Sharpe-Lintner CAPM equation,

[ ]

In words, the expected return on any asset i is the risk-free interest rate, Rf , plus a risk

premium, which is the asset’s market beta, , times the premium per unit of beta risk,

[ ].

Unrestricted risk-free borrowing and lending is an unrealistic assumption. Fischer Black

(1972) develops a version of the CAPM without risk-free borrowing or lending. He shows

that the CAPM’s key result—that the market portfolio is meanvariance-efficient—can be

obtained by instead allowing unrestricted short sales of risky assets. In brief, back in

Figure .., if there is no risk-free asset, investors select portfolios from along the mean-

variance-efficient frontier from a to b. Market clearing prices imply that when one

weights the efficient portfolios chosen by investors by their (positive) shares of aggregate

invested wealth, the resulting portfolio is the market portfolio. The market portfolio is

thus a portfolio of the efficient portfolios chosen by investors. With unrestricted short

selling of risky assets, portfolios made up of efficient portfolios are themselves efficient.

Thus, the market portfolio is efficient, which means that the minimum variance condition

for M given above holds, and it is the expected return-risk relation of the Black CAPM.

The relations between expected return and market beta of the Black and Sharpe-Lintner

versions of the CAPM differ only in terms of what each says about E(RZM), the expected

return on assets uncorrelated with the market. The Black version says only that E(RZM)

must be less than the expected market return, so the premium for beta is positive. In

contrast, in the Sharpe-Lintner version of the model, E( ) must be the risk-free interest

rate, Rf , and the premium per unit of beta risk is E( )- Rf.

The assumption that short selling is unrestricted is as unrealistic as unrestricted risk-free

borrowing and lending. If there is no risk-free asset and short sales of risky assets are not

allowed, mean-variance investors still choose efficient portfolios—points above b on the

abc curve in Figure ... But when there is no short selling of risky assets and no risk-free

asset, the algebra of portfolio efficiency says that portfolios made up of efficient

portfolios are not typically efficient. This means that the market portfolio, which is a

portfolio of the efficient portfolios chosen by investors, is not typically efficient. And the

CAPM relation between expected return and market beta is lost. This does not rule out

predictions about expected return and betas with respect to other efficient portfolios—if

theory can specify portfolios that must be efficient if the market is to clear. But so far this

has proven impossible.

In short, the familiar CAPM equation relating expected asset returns to their market betas

is just an application to the market portfolio of the relation between expected return and

portfolio beta that holds in any mean-variance-efficient portfolio. The efficiency of the

market portfolio is based on many unrealistic assumptions, including complete agreement

and either unrestricted risk-free borrowing and lending or unrestricted short selling of

risky assets. But all interesting models involve unrealistic simplifications, which is why

they must be tested against data.

CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER

CONDITIONS OF RISK – JF(1964) – William F Sharpe

It attempts to build a micro-economic theory dealing with conditions of risk which did not

exist at that point of time.

Optimal investment policy for the individual

The investors preference function

Assume the investor sees the outcome of his investments in terms of some probabilistic

distribution. And this distribution has parameters mean and standard deviation and he is

willing to act on these. In terms of utility this is represented as . Here E is the

expected future wealth and sigma the predicted standard deviation of the actual future

wealth from the expected. Investors are assumed to expect higher future wealth,

.

Given the level of E they exhibit risk aversion,

. The indifference curves relating E

and sigma would be upward sloping. The marginal rate of substitution between the two

would be diminishing.

Assume the investor allocates an initial wealth of Wi for investing. Wt is his final wealth.

So his return is

. Investors utility can then be represented in terms of R,

.

The investment opportunity curve

From a set of investment opportunities the investor chooses that which maximizes his

utility. Every investment plan available to him can be represented by a point in the plot.

This will transform into a compact area. His best opportunities lie along AFBDCX.

Anything within the shaded area such as Z is inferior to B,D,C. The investor will choose

from among all possible plans the one placing him on the indifference curve representing

the highest level of utility (point F). The decision can be made in two stages: first, find

the set of efficient investment plans and, second choose one from among this set.

The only plans which would be chosen must lie along the lower right-hand boundary

(AFBDCX)- the investment opportunity curve.

To understand the nature of this curve, consider two investment plans -A and B, each

including one or more assets. Their predicted expected values and standard deviations of

rate of return are shown in Figure 3.

Place proportion , in A and ( ) in B.

E=alpha*ERa+(1-alpha)ERb

\

Depending on the correlation coefficient the curve would vary from straight line

(perfectly correlated) to zero correlation (the curve) to inversely correlated (more U

shaped).

Bring the risk free rate and the risky portfolio together. Lend in risk free rate and allocate

remaining to risky portfolio. The tangential combination dominates

Similarly borrowing.

EQUILIBRIUM IN THE CAPITAL MARKET

To establish the equilibrium we invoke two assumptions:

First, we assume a common risk free rate of interest, with all investors able to borrow or

lend funds on equal terms.

Second, we assume homogeneity of investor expectations: investors are assumed to agree

on the prospects of various investments- the expected values, standard deviations and

correlation coefficient.

Under these assumptions, given some set of capital asset prices, each investor will view

his alternatives in the same manner.

A(LEND+INVEST), B(INVEST), C(BORROW+INVEST)

In any event, all would attempt to purchase only those risky assets which enter

combination phi.

Since all investors seek to invest in , the prices of assets in will increase and prices

not in will fall. This has consequences on the expected returns in the future for these

assets. This will lead to assets in phi moving left and other assets moving right in the

figure. In this way there will be a continuous churn of the asset prices and opportunities

set.

Capital asset prices must continue to change until a set of prices is attained for which

every asset enters at least one combination lying on the capital market line.

We are familiar with the following figures and we have already dealt with the concepts

these represent.

EFFICIENT MARKET HYPOTHESIS (from KBM) Attempt to find recurrent patterns in stock price movements fail. A forecast about

favorable future performance leads instead to favorable current performance, as market

participants all try to get in on the action before the price increase. Any information that

could be used to predict stock performance should already be reflected in stock prices. As

soon as there is any information indicating that a stock is underpriced and therefore offers

a profit opportunity, investors flock to buy the stock and immediately bid up its price to a

fair level, where only ordinary rates of return can be expected. These ―ordinary rates‖ are

simply rates of return commensurate with the risk of the stock.

However, if prices are bid immediately to fair levels, given all available information, it

must be that they increase or decrease only in response to new information. New

information, by definition, must be unpredictable; if it could be predicted, then the

prediction would be part of today’s information. Thus stock prices that change in

response to new (unpredictable) information also must move unpredictably.

This is the essence of the argument that stock prices should follow a random walk, that is,

that price changes should be random and unpredictable. Far from a proof of market

irrationality, randomly evolving stock prices would be the necessary consequence of

intelligent investors competing to discover relevant information on which to buy or sell

stocks before the rest of the market becomes aware of that information.

Therefore, a random walk is the natural result of prices that always reflect all current

knowledge. Indeed, if stock price movements were predictable, that would be damning

evidence of stock market inefficiency, because the ability to predict prices would indicate

that all available information was not already reflected in stock prices. Therefore, the

notion that stocks already reflect all available information is referred to as the efficient

market hypothesis (EMH).

The response of stock prices to new information in an efficient market. The graph plots

the price response of a sample of 194 firms that were targets of takeover attempts. In most

takeovers, the acquiring firm pays a substantial premium over current market prices.

Therefore, announcement of a takeover attempt should cause the stock price to jump. The

figure shows that stock prices jump dramatically on the day the news becomes public.

However, there is no further drift in prices after the announcement date, suggesting that

prices reflect the new information, including the likely magnitude of the takeover

premium, by the end of the trading day.

minute-by-minute stock prices of firms that are featured on CNBC’s ―Morning‖ or

―Midday Call‖ segments

weak-form EMH

The assertion that stock prices already reflect all information contained in the history of

past trading.

semistrong-form EMH

The assertion that stock prices already reflect all publicly available information.

strong-form EMH

The assertion that stock prices reflect all relevant information, including inside

information.