CHAPTER 2: PORTFOLIO THEORY

Portfolio theory is concerned with determining the optimal combination of securities or assets
that investors should purchase in order to minimize risk per given return and maximize return
per given risk.
Assumptions of portfolio theory
 Investors are rational – they are mean variance optimizers
 Investors are solely influenced by expected returns and standard deviation of assets
 Investors have homogeneous expectations of returns and standard deviation of
assets- as markets are informationally efficient.
 Investors have a single and same period investment horizon
 A risk free asset exist that can be invested in or short sold by all investors

Return
Financial assets are expected to produce cash flows and the return of an asset is judged in terms
of the cash flows it produces. The concept of return provides investors with a convenient way of
expressing the financial performance/reward of investments eg you buy 10 shares at $1000
each and the stock pays no dividend. After one year you sell your shares at $1100 each. What is
your return in dollar terms?

Measures of return.
 Required rate of return-is the minimum return or profit to induce interest in you to
invest generally it is measured and referred to as the cost of common equity.
 Holding period return- refers to the total return from an investment over a given
investment horizon
 It is calculated as follows:

 Holding period yield- refers to the holding period return as a percentage

HPY = [HPR – 1]%

 Annualized holding period return

 Arithmetic mean- good in making future estimates

An average of the subperiod returns, calculated by summing the subperiod returns and
dividing by the number of subperiods.

 Geometric mean – good for historical returns

The geometric mean is the average of a set of products, the calculation of which is
commonly used to determine the performance results of an investment or
portfolio. It is primarily used for investments that are compounded. It is used to
calculate average rate per period on investments that are compounded over
multiple periods.
NB: The challenge of investment analysis is that future returns can never be predicted exactly.
The investor may have beliefs about what the return will be, but the market never fails to
deliver surprises. Looking at future returns, it is necessary to accommodate their
unpredictability by determining the range of possible values for the return and the likelihood of
each. This provides a value for the expected return from the investment. What remains is to
determine just how uncertain the return is. The measure that is used to do this, the variance of
return, is the analyst’s measure of risk. Together the expected return and variance of alternative
portfolios provide the information needed to compare investment strategies.

 Expected return
It is the expected value of the variable. In case of an investment decision, it is the reward for
making the commitment of your funds [undertaking project]. No investment should be
undertaken unless the expected return is high enough to cover the investor for the perceived
risk.
Expected return of a single asset/project/npv
i. With different probabilities- if all possible events/outcomes are listed and each event
assigned a probability, the listing is called the probability distribution. It is calculated as follows;
Calculate the expected NPV of the above projects

ii. Same probabilities / no probabilities- take note of the fact that in calculation of expected
return, it is assumed that the states/events are mutually exclusive.

Portfolio holding period return

The calculation of the holding period return on a portfolio can be accomplished in two ways.
Firstly, the initial and final values of the portfolio can be determined, dividends added to the
final value, and the return computed. Alternatively, the prices and payments of the individual
assets, and the holding of those assets, can be used directly. Focusing on the total value of the
portfolio, if the initial value is V0, the final value V1; and dividends received are d, then the
return is given by:
Expected return of a portfolio
A portfolio is an asset holding which is made up of at least two different securities or assets
[held/owned by an investor]. Expected return on a portfolio is the weighted average of expected
return from each asset in the portfolio. Hence, there is need to know/compute the return of
each individual asset. It is calculated as follows;

Where; n is the number of assets in a portfolio, Ri is [expected] the return on the ith asset in
the portfolio, Wi is the weight of the ith asset in the portfolio

Total return = expected return + unexpected return

The unexpected comes about because of unanticipated events and can be greater or less than
the expected, it can also be any number [positive or negative]. The difference between risky
returns and risk free rate is called the excess return [positive/negative] on the risky asset.
Returns on any investment in time are never stable, they can be above or below the expected
return- the concept of risk.

RISK
Is an expectation that the actual outcome of a project may differ from the expected
outcome/mean or return. The risk of financial assets is judged in terms of the risk that the
expected cash-flows will be different from the expected amounts. Risk is inherent in any
investment. Risks may be
 Loss of capital [depreciation in value/price]
 Delay in repayment
 Non-payment of interest
 Variability in returns

At the heart of investment analysis is the observation that the market rewards those willing to
bear risk. An investor purchasing an asset faces two potential sources of risk. The future price at
which the asset can be sold may be unknown, as may the payments received from ownership of
the asset. The magnitude of possible differences/spread or dispersion of a distribution reflects
the size of the risk.

MEASURING RISK
Total risk of an investment is measured by the variance and the standard deviation of its
returns. An asset with a return that never changes has no risk. For this asset the variance of
return is 0. Any asset with a return that does vary will have a variance of return that is positive.
The more risk is the return on an asset the larger is the variance of return.

Semi-variance
This measure considers only deviations below the mean. The argument is that returns above the
average return are desirable. The only returns that disturb an investor are those below average.
A measure of this is the average (overall observations) of the squared deviations below the
mean. Semi-variance measures downside risk relative to a benchmark given by expected return.
It is just one of a number of possible measures of downside risk. More generally, we can
consider returns relative to other benchmarks, including a risk-free return or zero return.

These generalized measures are, in aggregate, referred to as lower partial moments. Yet
another measure of downside risk is the so-called value at risk measure, which is widely used by
banks to measure their exposure to adverse events and to measure the least expected loss
(relative to zero, or relative to wealth) that will be expected with a certain probability [especially
if return distributions are not normally distributed, then standard deviation fails].
It is computed as follows:

Intuitively, these alternative measures of downside risk are reasonable, and some portfolio
theory has been developed using them. However, they are difficult to use when we move from
single assets to portfolios.

Variance of an asset
Is a measure of dispersion of a distribution around the mean or the expected value, how far
above or below the mean?
Standard Deviation of a single asset.
It also measures the dispersion or variability of actual returns around the mean (𝑹̅̅). It is the
square root of variance. In other words, standard deviation is a statistical measure of the
variability of a distribution around its mean.

INTERPRETING STANDARD DEVIATION AND EXPECTED RETURN

The likelihood of getting a return which is more or less than expected is found by utilizing the
standard deviation, which we found to be for example 7% and the expected return of for
example 18%. We do this by making the following calculations:
 There is a 68.3% chance that the actual return will be one standard deviation below or
above 18%, that is the return can range between (18% + 7% ) = 25% and ( 18% - 7% ) =
11%.
 Similarly, there is a 95.5% chance that the return will range between (18% + 7% + 7%) =
32.% and (18% - 7% - 7% ) = 4%. [2 standard deviations]
 Lastly, there is a 99.7% chance that it will range between (18% + 7% + 7% + 7%) = 39%
and (18% - 7% - 7% -7%) = -2%. [3 standard deviations]

RELATIONSHIP BETWEEN ASSET RETURNS.

Correlation coefficient
This is a measure or indicator of movement (or lack of movement) between returns of different
assets. Co-relation coefficient is a standardized statistical measure of the linear relationship
between two variables, i.e. it is a standardized/adjusted co-variance. It realizes that co-variance
or co-movement of the returns of two securities is affected by the standard deviations or
dispersions of returns of those securities.
Correlation coefficient ranges between –1 and 1, with –1 representing perfect negative
correlation, 0 (zero) representing no correlation and 1 representing a perfect positive co-
relation-i.e the two variables are perfectly correlated meaning they have a perfect positive
linear relationship with each other. Combining assets with zero correlation reduces risk of the
portfolio, but provides less diversification than combining negatively correlated assets.

NB when the covariance is positive, then the correlation is positive too. Standard deviation can
never be negative
Correlation coefficient is calculated using the formula:

δA,B = covariance between A & B divided by standard deviation of A multiplied by standard

deviation of B

Coefficient of determination
It is the correlation coefficient raised to the power of two. It shows how much variability in the
returns of one asset can be associated with variability in the returns of the other. If the
correlation coefficient of A & B is 0.5 then the coefficient of determination is 0.25 meaning that
approximately 25% of the variability in the returns of stock A can be explained by the returns
variability of stock B. if it is 1 then in such a case, if you know what will be the changes in A then
you could predict exactly the return of asset B.

PORTFOLIO RISK ANALYSIS

In forming a portfolio investors need to diversify and spread the risk [your portfolio include all
your assets and liabilities]. In order for this to be effective the assets comprising the portfolio
should be of different class, sectors or industries i.e negative correlation coefficient and negative
covariance. This is crucial so that the results of the assets will behave in different directions
which is a requisite in risk spreading.
Before calculating risk for a portfolio we need to understand the concept of diversification,
covariance and coefficient of correlation. Every sports fan knows that a team can be much more
(or less) than the sum of its parts. It is not just the ability of the individual players that matters
but how they combine. The same is true for assets when they are combined into portfolios.

Diversification
Is the process of combining securities in a portfolio with the aim of reducing total risk but
without sacrificing portfolio return. It is buying or holding different securities in one portfolio for
the purposes of spreading risk. For this to be effective, the securities in question should have
different risks, return trade-off characteristics. Thus different classes of securities should be
included in the portfolio in order for it to be well diversified.

Types of risks
Specific risk/unsystematic risk
It is that part of total risk that can be directly identified with a particular project or firm. It is the
variability in return due to factors unique to the individual project or firm. Specific factors that
affect the company’s return like demand, management deficiency equipment failure, and
Research and Development achievements. This specific risk can be reduced through
diversification. As you increase the number of negatively correlated securities, total risk falls. It
is of paramount significance to note that as you increase your asset holding, you can reduce
standard deviation, but you cannot eliminate variability [covariance], i.e your portfolio reach the
market portfolio status, that is diversified away all unsystematic risk but remain with
market/systematic risk, as depicted by the diagram below, because specific risks for projects are
independent of other bad events by stock will be off-set by good events effects on the other.
Market risk can be reduced by including in your portfolio securities with negative betas [β].
By implication, in equilibrium, the market should not pay a premium for specific risk as investors
can simply diversify it away [CAPM]. For the same reason, firms should not undertake a merger
activity in-order to diversify risk. It does not add value as investors can do this on their own
easily and cheaply.

Systematic risk / market risk

It is that part of risk which cannot be diversified in any way because it is caused by factors
common to all economic activities. It is also known as market risk e.g. the general level of
demand in the economy, interest rates, labour costs , exchange rates etc which affect
performance of all firms albeit to different extents [different sensitivities/elasticities]. It
therefore refers to macro-economic factors that cause reactions on stock market and the whole
economy. There is no insulation against such factors by an individual firm.
In a well-diversified portfolio only the covariance between assets counts for portfolio
risk/variance. In other words, the variance of the individual assets can be eliminated by
diversification. The contribution to the portfolio variance of the variance of the individual
securities goes to zero as N gets very large. This follows because of risk pooling/insurance
principle; as more and more uncorrelated risks are pooled together, total risk is reduced.
However, the contribution of the covariance terms approaches the average covariance as N gets
large. The individual risk of securities can be diversified away, but the contribution to the total
risk caused by the covariance terms cannot be diversified away.
The market will pay a premium for this non-diversifiable risk, thus the return on an asset is
influenced by this one factor [the market risk/factor]

Co-variance
When constructing a portfolio it is not just the risk on individual assets that matters but also the
way in which this risk combines across assets to determine the portfolio variance. Two assets
may be individually risky, but if these risks cancel when the assets are combined then a portfolio
composed of the two assets may have very little risk. The risks on the two assets will cancel if a
higher than average return on one of the assets always accompanies a lower than average
return on the other.
Diversification or risk reduction will work if the returns in creating a portfolio behave or move
differently over the same period of time. The term given to such movement measurement is
called covariance. The co-variance of returns of assets that make up a portfolio is a measure of
the extent to which the returns of each one vary with another in different conditions. If an
increase in the rate of return of Asset A is associated with an increase in the rate of return of B
then there is a positive covariance, that is A and B move in the same direction over the same
period, and the reverse is true.
Remember, covariance is a measure of the degree to which two variables move together over
time or the degree to which the returns of two variables move together or in tandem [it
measures how often they move up or down together. If an increase in asset A is associated
which a decrease in asset B then there is a negative covariance. A covariance of zero implies
independence between the movements of the assets in question. Covariance of a variable with
itself is its variance.

Measuring covariance;

NB: Covariance between an asset and a portfolio containing that asset, is the value weighted
average of the covariances of the assets in the portfolio. Remember, the covariance of asset A
with itself is its variance. For a two asset portfolio, then the covariance between asset A and a
portfolio containing A and B will be;

MEASURING PORTFOLIO RISK

As opposed to the calculation of expected return which is simply the weighted average expected
return of individual sum in the portfolio, calculation of the standard deviation/ risk is different. It
takes into account not only the weights, variances, and standard deviation of each security but
also the covariance or correlation coefficient between pairs individual securities in the portfolio.
It’s not simply weighted average of risks of individual securities. It depends on;
 Riskness of security constituting the portfolio [variances of securities]
 Relationship among the securities in the portfolio [covariances of assets]
 Weights of assets

For the assets in a portfolio it is not just the variability of the return on each asset that matters
but also the way returns vary across assets. A set of assets that are individually high performers
need not combine well in a portfolio. Just like a sports team the performance of a portfolio is
subtly related to the interaction of the component assets.
For the purposes of this course, we only consider a two-asset portfolio. The measures of risk are
variance and standard deviation. Variance of a two asset portfolio is calculated as follows;
NB The covariance has already been described as an indicator of the tendency of the returns on
two assets to move in the same direction (either up or down) or in opposite directions. Although
the sign of the covariance (whether it is positive or negative) indicates this tendency, the value
of the covariance does not in itself reveal how strong the relationship is. In order to determine
the strength of the relationship it is necessary to measure the covariance relative to the
deviation from the mean experienced by the individual assets. This is achieved by using the
correlation coefficient which relates the standard deviations and covariance.

Relationship between stock returns and market portfolio

When picking the relevant assets to the investment portfolio on the basis of their risk and return
characteristics and the assessment of the relationship of their returns, investor must consider to
the fact that these assets are traded in the market. How could the changes in the market
influence the returns of the assets in the investor’s portfolio? What is the relationship between
the returns on an asset and returns in the whole market (market portfolio)?
The market portfolio at large represents all the securities and assets in an global economy- but
for analysis and statistical purposes all shares and securities listed on stock exchange is used as a
proxy/instead . Thus, in measuring the returns on the market portfolio, the stock market index
is used.

Statistics can be explored to answer the above questions;

 The characteristic line and the beta factor
More often than not, the relationship between the asset return and market portfolio return is
demonstrated and examined using the common stocks as assets, but the same concept can be
used analyzing bonds, or any other assets. With the given historical data about the returns on
the particular common stock (rJ) and market index return (rM) in the same periods of time
investor can draw the stock’s characteristic line

Stock’s characteristic line:

 describes the relationship between the stock and the market;
 shows the return investor expect the stock to produce, given that a particular
rate of return appears for the market;
  helps to assess the risk characteristics of one stock relative to the market.
Stock’s characteristic line as a straight line can be described by its slope and by point in which it
crosses the vertical axis – intercept. The slope of the characteristic line is called the Beta factor.
Beta factor for the stock J and the market can be calculated using following formula:

The Beta factor of the stock is an indicator of the degree to which the stock reacts to the
changes in the returns of the market portfolio. The Beta gives the answer to the investor how
much the stock return will change when the market return will change by 1 percent. Beta of a
stock measures the stock’s responsiveness or variability attributable to the variability of the
market portfolio.

PORTFOLIO BETA
Is the responsiveness of the portfolio returns to market portfolio return movements. It is a
measure of the portfolio’s systematic/market risk. Is measured as the portfolio weighted
average of betas of the securities in the portfolio, thus

NB: As in interpreting security beta, a negative index number shows an inverse relationship
between your portfolio returns and market portfolio and the reverse is true. a beta of 1 means
the portfolio will move perfectly positive with the market [swings/variability from the mean.
beta of less than 1 means your portfolio is less volatile. a beta of 1.3 means your portfolio is 30%
more volatile than the market theoretically, i.e if the market returns increase by 10%, your
portfolio will increase by 13%
SECURITY AND PORTFOLIO SELECTION
Single asset selection
The purpose of all the statistical measures computations is to aid us in selecting the best asset
among others.
 Mean-variance rule- states that; investment A will be preferred to investment B
provided one of the following 2 conditions exist;
a. Either the mean/expected return on A exceeds that of B and the variance of A is equal or
less than that of B [nonsatiation] OR
b. The mean/expected return on A exceed or equal that of B and the variance of A is smaller
than that of B [risk aversion]
  Coefficient of variation
It’s a measure of relative variability that indicates risk per unit of return. Any asset with high risk
per unit of return is a bad pick, that is choose the asset with low coefficient of variation. It is
calculated as follows;
C.V = standard deviation/Expected return
= δ/E(r)

An asset with a high C.V has a high risk per unit of return and should be avoided. With this
forming part of our weaponry, which asset will you choose between A and B in the last
example.

PORTFOLIO SELECTION
To make a good choice we must first know the full range of alternatives. In finance terms, no
investor wishes to bear unnecessary risk for the return that they are achieving. This implies
being efficient and maximizing return for given risk [mean-variance rule]. Given this, what
remains is to choose the investment strategy that makes the best trade-off between risk and
return. What is necessary is to find the relationship between risk and return as portfolio
composition is changed.
We already know that this relationship must depend on the variances of the asset returns and
the covariance between them. The relationship that we ultimately construct is the efficient
frontier. This is the set of efficient portfolios from which a choice is made. Efficient portfolios is a
set of asset portfolios that result in the lowest possible risk for a given level of return or highest
return given risk.
NB: asset allocation deals with the proportion to be invested in each asset class such as bonds,
stocks and commodities. Security selection focuses on the analysis and picking of specific
security within asset class. Coming up with a portfolio simply involves two processes or stages;
selecting the composition of risky assets and assigning weights between risky portfolio and risk
free asset.

 Minimum variance portfolio- 2 asset case.

A minimum variance portfolio indicates a well-diversified portfolio that consists of individually
risky assets, which are hedged when traded together, resulting in the lowest possible risk for the
rate of expected return.
It is a portfolio of individually risky assets that when taken together result in the lowest possible
risk level for the rate of expected return.
The starting point for investigating the relationship between risk and return is a study of
portfolios composed of just two risky assets with no short-selling- which is illegal in Zimbabwe.
The relationship between risk and return that is constructed is termed the portfolio frontier
[portfolio opportunity set for this two asset portfolio] and the shape of the frontier is shown to
depend primarily upon the coefficient of correlation between the returns on the two assets. The
portfolio opportunity set is the set of all possible portfolios with different weight combinations.

The analysis now considers the two limiting cases of perfect positive correlation and perfect
negative correlation, followed by the intermediate case.

Case 1: ρAB = +1 (Perfect Positive Correlation)

The first case to consider is that of perfect positive correlation where ρAB = +1. This can be
interpreted as the returns on the assets always rising or falling in unison. Setting ρAB = +1, the
standard deviation of the return on the portfolio becomes:
δp = WAδA + WBδB remember weights must add up to 1.

Which is weighted sum of the standard deviations of the returns on the individual assets, where
the weights are the portfolio proportions. The expected return on the portfolio remains the
same – formula unchanged being also a weighted sum of the expected returns on the individual
assets.

Graphing the portfolio returns and risk results to:

As the example illustrates, because the equations for portfolio expected return and standard
deviation are both linear, the relationship between δp and Rp is also linear. This produces a
straight line graph when expected return is plotted against standard deviation. Thus with the
correlation coefficient equal to +1, both risk and return of the portfolio are simply linear
combinations of the risk and return of each security.

NB: In the case of perfectly correlated assets, the return and risk on the portfolio of the two
assets is a weighted average of the return and risk on the individual assets. There is no reduction
in risk from purchasing both assets that’s there is no benefit from diversification [it is not
possible to sacrifice risk without sacrificing some return]

The investment implication of the fact that the frontier is a straight line is that the investor can
trade risk for return at a constant rate. Therefore, when the returns on the assets are perfectly
positively correlated, each extra unit of standard deviation that the investor accepts has the
same reward in terms of additional expected return.

The relationship that we have derived between the standard deviation and the expected return
is called the portfolio frontier. It displays the trade-off that an investor faces between risk and
return as they change the proportions of assets A and B in their portfolio.
The chart displays the location on this frontier of some alternative portfolio proportions of the
two assets. It can be seen that as the proportion of asset B (the asset with the higher standard
deviation) is increased the location moves up along the frontier. It is important to be able to
locate different portfolio compositions on the frontier as this is the basis for understanding
the consequences of changing the structure of the portfolio.

Case 2: ρAB = -1 (Perfect Negative Correlation)

Perfect negative correlation occurs when an increase in the return on one asset is met with by a
reduction in the return on the other asset. With ρAB = -1 the standard deviation of the portfolio
becomes the square root of:
Thus, a portfolio with standard deviation of zero can be constructed from two risky assets is a
general property when there is perfect negative correlation. To find the portfolio with a
standard deviation of zero, the correlation coefficient must be -1, the proportion of each
investment in each security must be calculated as;

The most important observation to be made about the figure above is that for each portfolio on
the downward sloping section there is a portfolio on the upward sloping section with the same
standard deviation but a higher return. Those on the upward sloping section therefore dominate
in terms of offering a higher return for a given amount of risk.

NB: reduction in risk is greater when the correlation is perfectly negative than perfectly positive.

Case 3: -1 < ρAB < +1 [correlation coefficient lies between the extreme bounds]
For intermediate values of the correlation coefficient the frontier must lie between that for the
two extremes of ρAB = -1 and ρAB = 1. It will have a curved shape that links the positions of the
two assets.
It can be seen that there is no portfolio with a standard deviation of zero, but there is a portfolio
that minimizes the standard deviation. This is termed the minimum variance portfolio and is
the portfolio located at the point furthest to the left on the portfolio frontier. The composition
of the minimum variance portfolio is implicitly defined by its location on the frontier.

The observation that there is a minimum variance portfolio is an important one for investment
analysis. It can be seen in the chart above that portfolios with a lower expected return than the
minimum variance portfolio are all located on the downward-sloping section of the portfolio
frontier [and inefficient].

As was the case for perfect negative correlation, for each portfolio on the downward sloping
section there is a portfolio on the upward-sloping section with a higher excepted return but
the same standard deviation. Conversely, all portfolios with a higher expected return than the
minimum variance portfolio are located on the upward sloping section of the frontier. This leads
to the simple rule that every efficient portfolio has an expected return at least as large as the
minimum variance portfolio.

NB In the case where the correlation coefficient is zero, the risk of the portfolio is less than the
risk of either of the individual securities. That is, if the return patterns of two assets are
independent so that the correlation coefficient and covariance are zero, a portfolio can be found
that has a lower variance than either of the assets by themselves.

For such a zero coefficient case, the minimum variance portfolio can be found by assigning
correct weights. In our two asset case the weightings are calculated as follows:
When the correlation ranges between 0 and -1 [negative] there is a possibility of minimizing
the total risk by combing the two assets. The percentage of investment in security A can be
ascertained using the following equation:

In summary:
 With a perfect positive correlation the portfolio frontier is upward sloping and
describes a linear trade-off of risk for return.
 At the opposite extreme of perfect negative correlation, the frontier has a
downward sloping section and an upward-sloping section which meet at a
portfolio with minimum variance. For any portfolio on the downward-sloping
section there is a portfolio on the upward-sloping section with the same
standard deviation but a higher return.
 Intermediate values of the correlation coefficient produce a frontier that lies
between these extremes. For all the intermediate values, the frontier has a
smoothly-rounded concave shape. The minimum variance portfolio separates
inefficient portfolios from efficient portfolios.
This can be diagrammatically presented as follows;
Efficient Frontier
It is the set of optimal portfolios that offers the highest expected return for a defined level of
risk or the lowest risk for a given level of expected return.
The portfolios that lie below the efficient frontier are sub-optimal because they do not
provide enough return for the level of risk.
The important role of the minimum variance portfolio has already been described. Every point
on the portfolio frontier with a lower expected return than the minimum variance portfolio is
dominated by others which have the same standard deviation but a higher return. It is from
among those assets with a higher return than the minimum variance portfolio that an investor
will ultimately make a choice. The minimum variance portfolio separates efficient portfolios
that may potentially be purchased from inefficient ones that should never be purchased.

The set of portfolios with returns equal to, or higher than, the minimum variance portfolio is
termed the efficient frontier. The efficient frontier is the upward section of the portfolio
frontier and is the set from which a portfolio will actually be selected. From the previous
sections, how to come up with a minimum variance portfolio was covered.
On the efficient frontier, there is a portfolio with the minimum risk, as measured by the
variance of its returns — hence, it is called the minimum variance portfolio — that also has
a minimum return, and a maximum return portfolio with a concomitant maximum risk.
Portfolios below the efficient frontier offer lower returns for the same risk, so a wise
investor would not choose such portfolios.
MORE THAN TWO RISKY ASSETS
The first consequence of the introduction of additional assets is that it allows the formation of
many more portfolios. The definition of the efficient frontier remains that of the set of portfolios
with the highest return for a given standard deviation. It is that part of the portfolio frontier that
begins with the minimum variance portfolio and includes all those on the portfolio frontier with
return greater than or equal to that of the minimum variance portfolio.
But, rather than being found just by varying the proportions of two assets, it is now constructed
by considering all possible combinations of assets and combinations of portfolios. The process of
studying these combinations of assets and portfolios is eased by making use of the following
observation: a portfolio can always be treated as if it were a single asset with an expected return
and standard deviation. Constructing a portfolio by combining two other portfolios is therefore
not analytically different from combining two assets. So, when portfolios are combined, the
relationship between the expected return and the standard deviation as the proportions are
varied generates a curve with the form discussed above. The shape of this curve will again be
dependent upon the coefficient of correlation between the returns on the portfolios.

This process of forming combinations can be continued until all possible portfolios of the
underlying assets have been constructed. As already described, every combination of portfolios
generates a curve with the shape of a portfolio frontier. The portfolio frontier itself is the upper
envelope of the curves found by combining portfolios.

Graphically, it is the curve that lies outside all other frontiers and inherits the general shape of
the individual curves. Hence, the portfolio frontier is always concave.

In total, the portfolio frontier and the portfolios located in the interior are called the portfolio
set or feasibility set.

REMEMBER: Systematic Risk

Systematic risk, also known as market or non-diversifiable risk, is the variability of return on
stocks or portfolios associated with changes in return on the market as a whole. It cannot be
eliminated no matter how one diversifies.
In other words market risk measures the sensitivity of an individual’s securities return to
movements in the market portfolio. This sensitivity measure is called Beta (β) and can be said
to measure the marginal contribution of a security to the total risk of the market portfolio.
Beta is a metric that shows the extent to which a given stock’s returns move up and down with
the stock market. Beta thus measures market risk.

Beta coefficient can be any figure. A beta of zero means the security is not sensitive to the
movements in the market portfolio. Thus Treasury Bills have a beta of zero as they are said to be
riskless and have no correlation with the market portfolio. A beta of one means the individual
security moves in line with the market portfolio, i.e. if the market or average return increases or
decreases by X% the return on that security is expected to change by the same margin, and
because of this the market portfolio has a beta of one.

MARKET EFFICIENCY THEORY

The role of markets in a competitive economy is to allocate scarce resources between competing
ends in a way that leads to the scarce resources being used most productively. When this occurs
markets are said to be allocatively efficient.
A market is said to be operationally/transactionally efficient when the transaction costs of operating
in the market are determined competitively- includes also speed of execution and accuracy. A strict
definition of operational efficiency implies that the transaction costs of making a market are zero.
A market is said to be informationally efficient if the current price ‘instantaneously and fully reflect
all relevant available information’. Informational efficiency is a measure of how quickly and
accurately the market reacts to new information. EMH is centered on this type of efficiency.
If a market is simultaneously allocationally, operationally and informationally efficient, it is said to be
perfectly efficient. Professor Eugene Fama, who coined the phrase “efficient markets”, defined
market efficiency as follows:
"In an efficient market, competition among the many intelligent participants leads to a situation
where, at any point in time, actual prices of individual securities already reflect the effects of
information based both on events that have already occurred and on events which, as of now, the
market expects to take place in the future. In other words, in an efficient market at any point in time
the actual price of a security will be a good estimate of its intrinsic value."

EFFICIENT MARKET HYPOTHESIS [EMH]

The origins of the EMH can be traced back to the work of two individuals in the 1960s: Eugene F.
Fama and Paul A. Samuelson. There is an old joke, widely told among economists, about an
economist strolling down the street with a companion. They come upon a $100 bill lying on the
ground, and as the companion reaches down to pick it up, the economist says, ‘Don’t bother – if it
were a genuine $100 bill, someone would have already picked it up’. This humorous example of
economic logic gone awry is a fairly accurate rendition of the efficient markets hypothesis (EMH),
one of the most hotly contested propositions in all the social sciences. Emh is the hypothesis that
prices of securities fully reflect available information about securities.

EMH AND RANDOM WALK.

If EMH holds, it means that the market prices of securities will always equal the fair/fundamental or
economic/intrinsic value of those securities i.e they- prices-are rational. If discrepancies exists, they
will be sufficiently small that given transaction cost this differences cannot be exploited profitably.

In short, if the EMH is true, securities markets will be in continuous stochastic equilibrium. Any
change in the fundamental values will be reflected immediately in market prices. The only thing that
would cause fundamental value to change would be new information. New information/news is, by
definition, unpredictable- otherwise its not news.
Hence, we expect the return on securities to change in response to new information in a direction
and by an amount that is also unpredictable and random- that is securities prices follows a random
walk because of the random nature of news.
Don’t confuse randomness in price changes with irrationality in the level of prices. If prices are
determined rationally, then only new information will cause them to change. Therefore, a random
walk would be 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.

VERSIONS/DEGREES OF EMH
These arise in trying to answer the questions; what is all available information and what does it
mean to reflect all available information?
A market is efficient with respect to an information set if it is impossible to earn consistent abnormal
profits by trading on the basis of the information set [financial transactions at market prices using
the information set are zero NPV activities.

1. Weak form efficiency.

Says that the current prices instantaneously and fully reflect all information contained in the past
history of security prices. The current price is a fair one that takes into account any information
contained in the past price data. Under this version, all available information is past prices.
Future prices cannot be predicted by analyzing prices from the past. In other words, past prices [or
trend analysis eg buy a stock if it has gone up three days in a row and reverse] provides no
information about future prices that would allow an investor to earn excess returns [over a passive
by-hold strategy] from using active trading rules based on historical prices- its like gnawing white old
bones, no matter the zeal, effort or resources used no flesh can be brought forth.
According to this EMH version, how a stock arrived at its current/ initial trading price is irrelevant. If
a security market is efficient in weak form, technical analysts/chartists are wasting time and
resources because markets do not have memory; neither do exams.
Thus Pt = Pt-1 + Er + random error i.e current price is equal to the last observed price plus the risk
compensation plus the error term due to new information.
NB: at least each and every stock market should be weak-form efficient because historical
information is readily available for all [easy and cheap for all to discover] and no one should profit
consistently from such information

TESTING WEAK-FORM EFFICIENCY.

These are the tests which are/can be carried out by researchers in a quest to find out whether the
market in question is weak form compliance i.e no investor can earn abnormal returns consistently
from the use of historic data.

Simulation tests- these tests generate random series of numbers as returns and compare them with
the actual price changes in the market. The similarity between the two establishes the relevance of
technical analysis as a stock market predictor since random numbers can be generated to know the
future movement of prices.
 Serial correlation tests- is the correlation between current return on a security and the
return on the same security over the last period. It involves one stock and looks for
independence between subsequent price movements using the correlation coefficient.
Highly correlated coefficients indicate dependence on the past data and suggest that past
data can be used to predict the future price behaviour of securities. If the EMH holds at
weak-form level, the coefficients should be close to zero [equal chance of increase and
decrease in price]
A positive serial correlation coefficient indicates tendency towards continuation- higher return is
likely to be followed by other higher than average return. A negative coefficient indicates tendency
towards reversal- lower price followed by higher return and the reverse is true. Presence of positive
or negative serial correlation coefficient indicates market inefficient i.e we can use today’s returns to
predict future prices.
 Runs tests- examines direction of movement of securities prices and not the quantum of
movement. Stock prices when they move at random [random walk theory] will not have any
dominating runs
 Filter rules- trading strategies can be tested for their efficiency using filter rules. Returns
from specific trading strategy can be computed and compared with the returns that an
investor can get out of holding the security over the entire trading period. Abnormality of
returns gives the inference that stock prices can be predicted using such filter rules hence,
make abnormal profits.

2. SEMI-STRONG FORM

States that security prices reflect/ impounds all relevant publicly available information. This
information is likely to include annual reports, patents, announcements, economic reports,
elections, economic policies. Eg the business news headlines- US$ weakens based on poor US
economic data.
Once information becomes public knowledge, prices adjust instantaneously and its virtually
impossible to profit from such news. Some authors likened an efficient market and arrival of new
information to the arrival of lamb chop to a school of flesh-eating piranha. The instant the lamb chop
hits the water, there is turmoil as the fish [investors] devour the meat. Very soon the meat is gone
leaving only the worthless bone behind and the water returns to normalcy. Similarly, when new
information reaches a competitive market, there is much turmoil as investors trade securities in
response, leading to price changes. Once price adjust, all that is left is the worthless bone and no
level of gnawing on the bone will yield any more meat (no further study will provide valuable
intelligence). From this smiley, past information about the stock and trading volumes are the bones
left by the investors and no amount of intelligence can produce any return to the analyst.
It implies that there are no learning lags in the dissemination of publicly available information that
can give rise to profitable trading rules. If news does not lead to any change in the security prices,
then we can infer that the news contained no relevant information/ price sensitive information- if
semi-strong EMH is true.
Tests for semi strong version
i. Event studies- deals with whether the information contained in company reports leads to
significant changes in security prices following the public announcement. No trading rule based on
the announcements can lead to excess economic returns after adjusting for risk and transaction
costs if semi-strong holds. Event study looks at whether there is any abnormal returns on the day of
announcement –eg on dividend declaration date. If semi-strong holds, it will be too late for an
investor to wait until the announcement is made public for him to make abnormal economic returns.
ii. Leads and lags securities- looks at whether there exist some securities which over react or under
react after the day of announcement. If such securities exist, then the stock market in question
might not be efficient in the semi-strong form.
iii. Records of mutual funds- these rely on mostly public information- if the market is efficient in the
semi-strong form then these institutional investors cannot score above the average investor. On this
same note, returns on actively managed funds can be compared to returns on passively managed
funds like index funds. If the market is efficient, then returns should not be significantly different and
fundamental analysts and chartists are wasting resources and time.

NB; shareholders need not worry that they are paying too much for the stock with low dividend etc
because the market has already incorporated such information into price, but still, investors have to
worry about their level of risk exposure.

3. STRONG FORM.

According to this version, current security prices reflect all knowable information about a security
including privately held information- insider information. Under this version, even the corporate
insiders cannot make abnormal economic profits by exploiting their private inside information about
the security/company. Inside information is not a privilege of a few to gain/make abnormal profits or
else there are no secrets- once known by one its public.

Tests for strong form efficiency

 Court cases- this include imprisonments, fines or suspension from trading due to violation of
inside trading laws. The checking will be on the returns based on inside trading to check for
abnormal returns.
 Specialists traders [including mutual funds, investment managers] against a random trading
strategy- checking if these special traders who use legal means to acquire information score
better.

In a nutshell as depicted by the diagram below,

 The circles represent the amount of information that each form of the EMH includes.

 Note that the weak form covers the least amount of information, and the strong form covers
all information.

 Also note that each successive form includes the previous ones.
IMPLICATIONS OF EMH
i. Trust market prices- they reflect the present value- market prices give best estimate value for
projects and firms receive ‘fair’ value for their securities
ii. Only normal returns [risk adjusted] can be earned
iii. Technical and fundamental analysts are doomed to failure

iv. Active management is wasted resources and justify not the costs incurred, hence passive
management/indexing is most appropriate
v. Read into market prices- if market prices reflect all available information we can extract
information from prices
vi. There are no financial illusions- market prices reflects value only from an asset’s payoff and its not
easy to trick the market even with doctored statements.
vii. Values comes from economic rents like superior information, superior skills/abilities, superior
technology, access to cheap resources eg investors who can establish a cost advantage [in
information collection and transaction cost- will be more able to exploit small inefficiencies
viii. No group of investors should be able to consistently beat the market using a common
investment strategy
ix. Negative implications for many investment strategies;
 Equity research and valuations would be a costly task if no benefits ensued hence, at best
the benefits from information collection and research would cover the research cost
 A strategy of randomly diversifying across stocks/indexing to the market carrying little, or no
information cost and minimal execution cost would be superior to any other strategy
 In an efficient market, a strategy of minimizing trading i.e creating a portfolio and not trading
unless cash is needed would be superior to a strategy that required frequent trading
What market efficiency does not imply.
 Stock prices cannot deviate from the true value [but should be random and unbiased]-
markets like people do make mistakes, at times great mistakes, but the true value will
always be found.
 No investor will ‘beat’ the market in any time period
 No group of investors will beat the market in the long term- given the number of investors in
the financial markets, the law of probability would suggest that a fairly large number are
going to beat the market consistently over long periods, not because of their investment
strategies but lucky.

In the same train of thoughts, EMH does not say;

 Prices are uncaused
 Investors are foolish and too stupid to be in the market
 All shares have same expected return
 Investors should throw darts to select stocks- [a blind fold chimp throwing darts on the stock
sheet scoring same returns as professional]
 There is no upward trend in stock prices

But it does say;

 prices reflect underlying value
 financial managers cannot time issues
 managers cannot boost prices via creative accounting

Evidence for and against EMH;

Generally, most academic studies have found that mutual funds do not consistently outperform
their benchmarks, especially after adjusting for risk and fees. Anomalies are unexplained empirical
results that contradict the EMH:
 The Size effect.
 The “Incredible” January Effect.
 P/E Effect.
 Day of the Week (Monday Effect)