UEH - Bài giảng Môn quản lý danh mục đầu tư Chương 06 Giới thiệu về quản lý danh mục đầu tư - Sách ReillyBrown

UEH - Bài giảng Môn quản lý danh mục đầu tư Chương 06 Giới thiệu về quản lý danh mục đầu tư- Sách ReillyBrown. Bài giảng tham khảo của đại học kinh tế TPHCM

CHAPTER 6

An Introduction

to Portfolio Management

6.1 Some Background Assumptions

• Investors want to maximize from the total set of investments for a given level of risk

the portfolio is important

individually good investments

6.1.1 Risk Aversion

rates of return, risk-averse investors will select the asset with the lower level of risk

• Evidence

• Many investors purchase insurance purchase various types

of insurance, including life insurance, car insurance, and health insurance

• Yield on bonds increases with risk classifications, which indicates that investors require a higher rate of return to accept higher risk

• Not all investors are risk averse

• It may depend on the amount of money involved

6.1.2 Definition of Risk

• For most investors, risk means the

uncertainty of future outcomes

• An alternative definition may be the

probability of an adverse outcome

6.2 Markowitz Portfolio Theory

(slide 1 of 2)

• The Markowitz model is based on several assumptions regarding investor behavior:

1 Investors consider each investment alternative as being

represented by a probability distribution of potential returns over some holding period

2 Investors maximize one-period expected utility, and their utility

curves demonstrate diminishing marginal utility of wealth

3 Investors estimate the risk of the portfolio on the basis of the

variability of potential returns

4 Investors base decisions solely on expected return and risk, so

their utility curves are a function of expected return and the variance (or standard deviation) of returns only

5 For a given risk level, investors prefer higher returns to lower

returns Similarly, for a given level of expected return, investors

6.2 Markowitz Portfolio Theory

(slide 2 of 2)

• Using these assumptions, a single asset

or portfolio of assets is considered to be

efficient if no other asset or portfolio of

assets offers higher expected return with the same (or lower) risk or lower risk with the same (or higher) expected return

6.2.1 Alternative Measures of Risk

(slide 1 of 2)

• Variance or standard deviation of expected return

• Range of returns

• Returns below expectations

considers deviations below the mean

investors want to minimize the damage from

6.2.1 Alternative Measures of Risk

(slide 2 of 2)

• Advantages of using variance or standard deviation of returns:

asset pricing models

6.2.2 Expected Rates of Return

(slide 1 of 3)

• Expected rate of return

• Equal to the sum of the potential returns multiplied with the corresponding probability of the returns

• Exhibit 6.1

• Equal to the weighted average of the expected rates of return for the individual investments in the portfolio

6.2.2 Expected Rates of Return

(slide 2 of 3)

6.2.2 Expected Rates of Return

(slide 3 of 3)

6.2.3 Variance (Standard Deviation) of Returns for an Individual Investment (slide 1 of 2)

measure of the variation of possible rates of

return:

( ) 2 2

6.2.3 Variance (Standard Deviation) of Returns for an Individual Investment (slide 2 of 2)

6.2.4 Variance (Standard Deviation) of

Returns for a Portfolio (slide 1 of 9)

variables “move together” relative to their

individual mean values over time

of return is defined as:

6.2.4 Variance (Standard Deviation) of

Returns for a Portfolio (slide 2 of 9)

6.2.4 Variance (Standard Deviation) of

Returns for a Portfolio (slide 3 of 9)

6.2.4 Variance (Standard Deviation) of

Returns for a Portfolio (slide 4 of 9)

6.2.4 Variance (Standard Deviation) of

Returns for a Portfolio (slide 5 of 9)

6.2.4 Variance (Standard Deviation) of

Returns for a Portfolio (slide 6 of 9)

6.2.4 Variance (Standard Deviation) of

Returns for a Portfolio (slide 7 of 9)

standardizing (dividing) the covariance by the product of the individual standard deviations

ij ij

i j

Cov r

6.2.4 Variance (Standard Deviation) of

Returns for a Portfolio (slide 8 of 9)

• The coefficient can vary only in the range +1 to −1

• A value of +1 would indicate perfect positive

correlation This means that returns for the two

assets move together in a positively and completely linear manner

• A value of −1 would indicate perfect negative

correlation This means that the returns for two

assets move together in a completely linear manner, but in opposite directions

• Exhibit 6.9

6.2.4 Variance (Standard Deviation) of

Returns for a Portfolio (slide 9 of 9)

6.2.5 Standard Deviation of a Portfolio

σ port = standard deviation of the portfolio

w i = weights of an individual asset in the portfolio; where weights are

determined by the proportion of value in the portfolio

σ 2

i = variance of rates of return for Asset i

Cov ij = covariance between the rates of return for Assets i and j

6.2.5 Standard Deviation of a Portfolio

(slide 2 of 10)

• Impact of a New Security in a Portfolio

• Two effects to the portfolio’s standard deviation when

we add a new security to such a portfolio:

• The asset’s own variance of returns

• The covariance between the returns of this new asset and the returns of every other asset that is already in the portfolio

• The relative weight of these numerous covariances is substantially greater than the asset’s unique variance; the more assets in the portfolio, the more this is true

• The important factor to consider when adding an

investment to a portfolio that contains a number of

other investments is not the new security’s own

variance but the average covariance of this asset with all other investments in the portfolio

6.2.5 Standard Deviation of a Portfolio

(slide 3 of 10)

Calculation

described by two characteristics:

• The expected rate of return

• The standard deviation of returns

6.2.5 Standard Deviation of a Portfolio

(slide 4 of 10)

• Equal Risk and Return—Changing Correlations

• The expected return of the portfolio does not change because it

is simply the weighted average of the individual expected returns

• Demonstrates the concept of diversification, whereby the risk of

the portfolio is lower than the risk of either of the assets held in the portfolio

• Risk reduction benefit occurs to some degree any time the

assets combined in a portfolio are not perfectly positively

correlated (that is, whenever r i,j < +1)

• Diversification works because there will be investment periods when a negative return to one asset will be offset by a positive return to the other, thereby reducing the variability of the overall portfolio return

6.2.5 Standard Deviation of a Portfolio

(slide 5 of 10)

• The negative covariance term exactly offsets the individual variance terms, leaving an overall

standard deviation of the portfolio of zero

• This would be a risk-free portfolio, meaning that

the average combined return for the two

securities over time would be a constant value

(that is, have no variability)

• Thus, a pair of completely negatively correlated assets provides the maximum benefits of

diversification by completely eliminating variability from the portfolio

6.2.5 Standard Deviation of a Portfolio

(slide 6 of 10)

6.2.5 Standard Deviation of a Portfolio

(slide 7 of 10)

Returns and Risk

different expected rates of return and

individual standard deviations

standard deviation is not zero

equal weights, but the asset standard

deviations are not equal

•

6.2.5 Standard Deviation of a Portfolio

(slide 8 of 10)

6.2.5 Standard Deviation of a Portfolio

(slide 9 of 10)

Weights

while holding the correlation coefficient

constant, a set of combinations is derived that trace an ellipse

dependent on the correlation between assets

• Exhibit 6.12

6.2.5 Standard Deviation of a Portfolio

(slide 10 of 10)

6.2.6 A Three-Asset Portfolio

• The results presented earlier for the two-asset

portfolio can be extended to a portfolio of n assets

• As more assets are added to the portfolio, more risk will be reduced (everything else being the same)

• The general computing procedure is still the same, but the amount of computation has increase rapidly

• For the three-asset portfolio, the computation has

doubled in comparison with the two-asset portfolio

6.2.7 Estimation Issues (slide 1 of 3)

• Results of portfolio allocation depend on

accurate statistical inputs

• Estimates of

• Among entire set of assets

• With 100 assets, 4,950 correlation estimates

• Estimation risk refers to potential errors

6.2.7 Estimation Issues (slide 2 of 3)

described by a single market model, the number

of correlations required reduces to the number

b i = the slope coefficient that relates the returns for security i to the

returns for the aggregate market

6.2.7 Estimation Issues (slide 3 of 3)

shown that the correlation coefficient between

two securities i and j is given as:

6.3 The Efficient Frontier (slide 1 of 4)

• The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk

or the minimum risk for every level of return

• Every portfolio that lies on the efficient frontier has either

a higher rate of return for the same risk level or lower

risk for an equal rate of return than some portfolio falling below the frontier

• See Exhibit 6.13

• Portfolio A in Exhibit 6.13 dominates Portfolio C because it has

an equal rate of return but substantially less risk

• Portfolio B dominates Portfolio C because it has equal risk but a

6.3 The Efficient Frontier (slide 2 of 4)

6.3 The Efficient Frontier (slide 3 of 4)

• Markowitz defined the basic problem that the investor needs to solve as:

2 2 port

6.3 The Efficient Frontier (slide 4 of 4)

• The general method for solving the formula is called a

constrained optimization procedure because the task

the investor faces is to select the investment weights that will “optimize” the objective (minimize portfolio risk) while also satisfying two restrictions (constraints) on the

investment process:

i The portfolio must produce an expected return at least as large as the

return goal, R; and

ii All of the investment weights must sum to 1.0

• The approach to forming portfolios according to this

equation is often referred to as mean-variance

optimization because it requires the investor to minimize

portfolio risk for a given expected (mean) return goal

6.3.1 The Efficient Frontier: An Example (slide 1 of 3)

• What would be the optimal asset allocation strategy using these five asset classes?

6.3.1 The Efficient Frontier: An Example (slide 2 of 3)

6.3.1 The Efficient Frontier: An Example (slide 3 of 3)

6.3.2 The Efficient Frontier and Investor Utility (slide 1 of 3)

trade-offs he is willing to make between

expected return and risk

decreases steadily as you move upward

determine the particular portfolio selected by an individual investor

given investor

6.3.2 The Efficient Frontier and Investor Utility (slide 2 of 3)

between the efficient frontier and the utility

curve with the highest possible utility

6.3.2 The Efficient Frontier and Investor Utility (slide 3 of 3)

6.4 Capital Market Theory: An Overview

portfolio theory we have just developed by

extending the Markowitz efficient frontier into a model for valuing all risky assets

implications for how portfolios are managed in practice

the existence of a risk-free asset, which in turn will lead to the designation of the market portfolio

6.4.1 Background for Capital Market

Theory (slide 1 of 2)

1 All investors are Markowitz efficient investors

who want to target points on the efficient frontier

2 Investors can borrow or lend any amount of

money at the risk-free rate of return (RFR)

3 All investors have homogeneous expectations;

that is, they estimate identical probability

distributions for future rates of return

4 All investors have the same one-period time

horizon such as one month, six months, or one year

6.4.1 Background for Capital Market

Theory (slide 2 of 2)

5 All investments are infinitely divisible, which

means that it is possible to buy or sell fractional shares of any asset or portfolio

6 There are no taxes or transaction costs involved

in buying or selling assets

7 There is no inflation or any change in interest

rates, or inflation is fully anticipated

8 Capital markets are in equilibrium, which implies

that all investments are properly priced in line

6.4.2 Developing the Capital Market Line (slide 1 of 8)

• A risky asset is one for which future

returns are uncertain

• Uncertainty is measured by the standard deviation of expected returns

• Because the expected return on a risk-free asset is entirely certain, the standard

deviation of its expected return is zero

• The rate of return earned on such an asset should be the risk-free

6.4.2 Developing the Capital Market Line (slide 2 of 8)

• Covariance with a Risk-Free Asset

• Covariance between two sets of returns is

• Similarly, the correlation between any risky asset and

6.4.2 Developing the Capital Market Line (slide 3 of 8)

• Combining a Risk-Free Asset with a Risky Portfolio

w w



= −

6.4.2 Developing the Capital Market Line (slide 4 of 8)

riskless security and the risky Portfolio M can expect a return equal to the risk-free rate plus compensation for the number of risk units

6.4.2 Developing the Capital Market Line (slide 5 of 8)

• The Capital Market Line

• The risk–return relationship holds for every

combination of the risk-free asset with any

collection of risky assets

• This relationship holds for every combination of

the risk-free asset with any collection of risky

assets

• However, when the risky portfolio, M, is the market portfolio containing all risky assets held anywhere

in the marketplace, this linear relationship is called

the Capital Market Line (CML)

6.4.2 Developing the Capital Market Line (slide 6 of 8)

• Risk–Return Possibilities with Leverage

• One can attain a higher expected return than is available

at point M

• One can invest along the efficient frontier beyond point M, such as point D

• With the risk-free asset, one can add leverage to the

portfolio by borrowing money at the risk-free rate and

investing in the risky portfolio at point M to achieve a point like E

• Clearly, point E dominates point D

• Similarly, one can reduce the investment risk by lending money at the risk-free asset to reach points like C

6.4.2 Developing the Capital Market Line (slide 7 of 8)

6.4.2 Developing the Capital Market Line (slide 8 of 8)

6.4.3 Risk, Diversification, and the Market Portfolio (slide 1 of 8)

• Because portfolio M lies at the point of tangency,

it has the highest portfolio possibility line

• Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML

• It must include all risky assets

• Because the market is in equilibrium, all assets in this portfolio are in proportion to their market

values

• Because it contains all risky assets, it is a

completely diversified portfolio, which means that all the unique risk of individual assets

(unsystematic risk) is diversified away

6.4.3 Risk, Diversification, and the Market Portfolio (slide 2 of 8)

• Only systematic risk remains in the market portfolio

• Systematic risk can be measured by the standard

deviation of returns of the market portfolio and can

change over time

• Systematic risk is the variability in all risky assets

caused by macroeconomic variables:

• Variability in growth of money supply

• Interest rate volatility

• Variability in factors like industrial production, corporate