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An Introduction to Portfolio Management An Introduction to Portfolio Management Table of contents 01 02 Some Background Assumptions Markowitz Portfolio Theory Table of contents Alternative Measures of[.]

An Introduction to

Portfolio Management

Variance (Standard Deviation)

The Efficient Frontier and Investor Utility

Some

Background Assumptions

0

1

Maximize the returns for a given level

of risk

A good portfolio is

not simply a collection of individually good investments

The relationship among the returns for all full spectrum

of investments in the portfolio is important

According to the assumption of

portfolio theory, are most

investors with a large investment

portfolio averse or

Evidence 1

The difference in the required rate of return for different grades of bonds with different degrees of credit risk

Evidence 2

7

Not everybody

buys insurance for

everything

averse regarding all financial

commitments

There is the combination

of risk preference and

risk aversionPeople like to gamble with negative expected returns but buy insurance to protect themselves against large losses Because of choice or

unaffordable insurance

Markowitz

Portfolio

Theory

02

The Markowitz model is based on several

assumptions regarding investor behavior

Investors maximize

one-period expected utility &

their utility curves demonstrate the

diminishing marginal

utility of wealth.

For a given risk level, investors prefer higher returns to lower returns and vice versa.

Utility curves are a function

of expected return and the expected variance (or standard deviation) of

Investors estimate the risk

of the portfolio on the basis

of the variability of expected returns.

01

04

05 02

03

Alternative Measures of Risk

A measure that only considers deviations below the mean is the

specific asset (T-bills, the rate of inflation, or a benchmark)

An extension of the semivariance

measure only computes

We will use the variance or standard deviation of returns because

This measure is somewhat

intuitive

It is a correct and widely recognized risk measure

It has been used in most of the

theoretical asset pricing models.

Alternative

Return

The expected rate of return

for a portfolio of investments is simply the weighted average of the expected rates of return

for the individual investments in the portfolio

The weights are the proportion of total value

for the individual investment.

We can generalize this computation of the expected return for the portfolio E(Rport) as follows:

Variance (Standard Deviation) of

Individual

Investment

2.3

The variance, or standard deviation, is a measure of the variation of possible rates of return Ri from the expected rate of return E(Ri) as follows:

Variance (Standard Deviation) of

Portfolio

2.4

Measure the degree to which two variables move together relative to their individual mean values over time

rij = the correlation coefficient

-1<= rij <= 1

● (+1): a perfect positive linear relationship

● (-1): a perfect negative linear relationship

● 0: the returns had no linear relationship

Standard

a Portfolio

2.5

Markowitz derived the general formula for the standard deviation of a portfolio as follows:

The standard deviation

for a portfolio of assets

consists of not only the

variances of the individual assets but

covariances between all

the pairs of individual

assets in the portfolio

large number of securities , this formula reduces to the sum of

the weighted covariances

Impact of a New Security in a

of this new asset and the returns of every other asset that is already in the portfolio

The important factor to consider when adding an investment to a portfolio that contains other investments is not the new security’s own variance but

the average covariance of this asset with all other investments in the

portfolio

The relative weight of

these many covariances is

far bigger than the asset's

individual variance

Portfolio

● Risk of a portfolio is based

● The investor's utility

function is concave and

increasing, due to their

risk aversion and

Equal Risk and Return—

Changing Correlations

The only value that changes is the correlation between the returns for the two assets

When we eventually reach the combination of perfect negative correlation, risk is eliminated.

Combining assets that are not perfectly correlated (correlation 1) does not affect the expected return of the portfolio, but it does reduce the risk of the portfolio (as

measured by its standard deviation)

1st case, case a), where the returns for the two assets are perfectly positively correlated (correlation equals 1), the standard deviation for the portfolio is, in fact, the weighted

average of the individual standard deviations:

0.2 x 0.5 + 0.2 x 0.5 = 0.2 - There is no diversification

This would be a

The returns for the portfolio are

constant because perfect negative

correlation yields a mean

combined return for the two

securities over time that is equal

to the mean for each of them.

There is no fluctuation in the portfolio's total returns, which means there is no risk, because any returns that are above or below the mean for any one of the assets are entirely offset by the return on the other asset.

Thus, the correlation coefficient is the engine that drives the

whole theory of portfolio diversification

Combining Stocks with Different

Returns and Risk

Recalculating the covariances

Case Correlation Coefficient

A

Three-Asset

Portfolio

Issues

2.7

The ‘estimation risk’: The potential source of error that

arises from these approximations

Where: bi = the slope coefficient that relates the returns for Security i to the returns for the aggregate stock market

Rm = the returns for the aggregate stock market

Where: σi = the standard deviation of ^2 m= the variance of returns for the

aggregate stock market

The Efficient

Frontier

2.8

The efficient frontier is

the envelope curve

containing the best of all

these possible

combinations

The set of portfolios with

the highest rate of

return for any given

level of risk or the

lowest risk for any level

of return is known as

the efficient frontier

Based on your utility function,

which expresses your attitude

point along the efficient

frontier as an investor

any portfolio from dominating

and risk measure , with

expected rates of return that

rise with increased risk.

The Efficient Frontier and

Investor Utility

2.9

As we move upward, the

slope of the efficient

frontier curve gradually

decreases This implies

that the expected return

decreases as we move up

the efficient frontier when

we add equal risk

increments.

The utility curves of an

individual investor outline the risks and returns trade-offs that person is prepared

to make

These utility curves work with the efficient frontier to identify the specific portfolio

on the efficient frontier that

best satisfies a particular investor

The optimal portfolio is the most efficient portfolio with the greatest utility for

a given investor It lies at the point of tangency between the efficient frontier and the U1 curve with the greatest utility

The curves labeled

U1, U2, and U3 are

for a strongly

risk-averse investor

The curves labeled (U3′ , U2′ , U1′ ) characterize a less risk-averse investor

Chapter 7 provides an introduction to portfolio theory, which was developed

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