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
Trang 1An Introduction to Portfolio
Management
Trang 26.1 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
Trang 36.1.1 Risk Aversion
rates of return, risk-averse investors will select the asset with the lower level of risk
• 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
Trang 56.2 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
Trang 66.2 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
Trang 76.2.1 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
Trang 86.2.1 Alternative Measures of Risk(slide 2 of 2)
•Advantages of using variance or standard deviation of returns:
asset pricing models
Trang 96.2.2 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
Trang 106.2.2 Expected Rates of Return(slide 2 of 3)
Trang 116.2.2 Expected Rates of Return(slide 3 of 3)
Trang 126.2.3 Variance (Standard Deviation) of Returns for an Individual Investment (slide 1 of 2)
measure of the variation of possible rates of return:
( )22
Trang 136.2.3 Variance (Standard Deviation) of Returns for an Individual Investment (slide 2 of 2)
Trang 146.2.4 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:
Trang 156.2.4 Variance (Standard Deviation) of Returns for a Portfolio (slide 2 of 9)
Trang 166.2.4 Variance (Standard Deviation) of Returns for a Portfolio (slide 3 of 9)
Trang 176.2.4 Variance (Standard Deviation) of Returns for a Portfolio (slide 4 of 9)
Trang 186.2.4 Variance (Standard Deviation) of Returns for a Portfolio (slide 5 of 9)
Trang 196.2.4 Variance (Standard Deviation) of Returns for a Portfolio (slide 6 of 9)
Trang 206.2.4 Variance (Standard Deviation) of Returns for a Portfolio (slide 7 of 9)
standardizing (dividing) the covariance by the product of the individual standard deviations
Covr
Trang 216.2.4 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
Trang 226.2.4 Variance (Standard Deviation) of Returns for a Portfolio (slide 9 of 9)
Trang 236.2.5 Standard Deviation of a Portfolio (slide 1 of 10)
σport= standard deviation of the portfolio
wi= weights of an individual asset in the portfolio; where weights are determined by the proportion of value in the portfolio
i= variance of rates of return for Asset i
Covij= covariance between the rates of return for Assets i and j
Trang 246.2.5 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
Trang 256.2.5 Standard Deviation of a Portfolio (slide 3 of 10)
described by two characteristics:
• The expected rate of return
• The standard deviation of returns
Trang 266.2.5 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 ri,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
Trang 276.2.5 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
Trang 286.2.5 Standard Deviation of a Portfolio (slide 6 of 10)
Trang 296.2.5 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
•
Trang 306.2.5 Standard Deviation of a Portfolio (slide 8 of 10)
Trang 316.2.5 Standard Deviation of a Portfolio (slide 9 of 10)
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
Trang 326.2.5 Standard Deviation of a Portfolio (slide 10 of 10)
Trang 336.2.6 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
Trang 346.2.7 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
Trang 356.2.7 Estimation Issues (slide 2 of 3)
described by a single market model, the number of correlations required reduces to the number of assets
R= +ab R+
bi= the slope coefficient that relates the returns for security i to the
returns for the aggregate market
Trang 366.2.7 Estimation Issues (slide 3 of 3)
shown that the correlation coefficient between
two securities i and j is given as:
Trang 376.3 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
Trang 386.3 The Efficient Frontier (slide 2 of 4)
Trang 396.3 The Efficient Frontier (slide 3 of 4)
•Markowitz defined the basic problem that the investor needs to solve as:
22port
Trang 406.3 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
Trang 416.3.1 The Efficient Frontier: An Example (slide 1 of 3)
•What would be the optimal asset allocation strategy using these five asset classes?
Trang 426.3.1 The Efficient Frontier: An Example (slide 2 of 3)
Trang 436.3.1 The Efficient Frontier: An Example (slide 3 of 3)
Trang 446.3.2 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
Trang 456.3.2 The Efficient Frontier and Investor Utility (slide 2 of 3)
between the efficient frontier and the utility curve with the highest possible utility
•
Trang 466.3.2 The Efficient Frontier and Investor Utility (slide 3 of 3)
Trang 476.4 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
Trang 486.4.1 Background for Capital Market Theory (slide 1 of 2)
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
Trang 496.4.1 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
Trang 506.4.2 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
Trang 516.4.2 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
Trang 526.4.2 Developing the Capital Market Line (slide 3 of 8)
•Combining a Risk-Free Asset with a Risky Portfolio
= −
Trang 536.4.2 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
E RRFRE RRFR
Trang 546.4.2 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
•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)
Trang 556.4.2 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
Trang 566.4.2 Developing the Capital Market Line (slide 7 of 8)
Trang 576.4.2 Developing the Capital Market Line (slide 8 of 8)
Trang 586.4.3 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
•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
Trang 596.4.3 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
Trang 606.4.3 Risk, Diversification, and the Market Portfolio (slide 3 of 8)
•Diversification and the Elimination of Unsystematic Risk
•The purpose of diversification is to reduce the standard deviation of the total portfolio
•This assumes that imperfect correlations exist among securities
•As you add securities, you expect the average covariance for the portfolio to decline
•How many securities must you add to obtain a completely diversified portfolio?
•Exhibit 6.19
Trang 616.4.3 Risk, Diversification, and the Market Portfolio (slide 4 of 8)
Trang 626.4.3 Risk, Diversification, and the Market Portfolio (slide 5 of 8)
•The CML and the Separation Theorem
•The CML leads all investors to invest in the M portfolio
•Individual investors should differ in position on the CML depending on risk preferences
•How an investor gets to a point on the CML is based on financing decisions
•Risk averse investors will lend at the risk-free rate, while investors preferring more risk might borrow funds at the RFR and invest in the market portfolio
•The investment decision of choosing the point on CML is separate from the financing decision of reaching there through either lending or borrowing