大三下合作課lecture

1、Lecture OutlineSome Background Assumptions (Risk Aversion, Definition of Risk) Markowitz Portfolio TheoryAlternative Measures of RiskExpected Rates of Return (Portfolio)Covariance of ReturnsCovariance and CorrelationStandard Deviation of a Portfolio Estimation IssuesThe Efficient Frontier and Invest

2、or Utility1-1Some Background AssumptionsAs an investor you want to maximize the returns for a given level of risk.Your portfolio includes all of your assets and liabilities.The relationship between the returns for assets in the portfolio is important.A good portfolio is not simply a collection of in

3、dividually good investments. 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-2Some Background AssumptionsRisk Aversion Given a choice between two assets with equal rates of return, risk-averse invest

4、ors will select the asset with the lower level of riskEvidenceMany investors purchase insurance for: Life, Automobile, Health, and Disability e. Yield on bonds increases with risk classifications from AAA to AA to A, etc.Not all Investors are risk averseIt may depends on the amount of money involved

5、: Risking small amounts, but insuring large losses 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-3Some Background Assumptions and Markowitz Portfolio TheoryDefinition of RiskUncertainty: Risk means

6、 the uncertainty of future es. For instance, the future value of an investment in Googles stock is uncertain; so the investment is risky. On the other hand, the purchase of a six-month CD has a certain future value; the investment is not risky.Probability: Risk is measured by the probability of an a

7、dverse e. For instance, there is 40% chance you will receive a return less than 8%.Main Results of Markowitz Portfolio TheoryQuantifies riskDerives the expected rate of return for a portfolio of assets and an expected risk measureShows that the variance of the rate of return is a meaningful measure

8、of portfolio riskDerives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-4Markowitz Portfolio T

9、heoryAssumptions for Investors Consider investments as probability distributions of expected returns over some holding periodMaximize one-period expected utility, which demonstrate diminishing marginal utility of wealthEstimate the risk of the portfolio on the basis of the variability of expected re

10、turnsBase decisions solely on expected return and riskPrefer higher returns for a given risk level. Similarly, for a given level of expected returns, investors prefer less risk to more riskUsing these five assumptions, a single asset or portfolio of assets is considered to be efficient if no other a

11、sset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return. 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-5Alte

12、rnative Measures of RiskVariance or standard deviation of expected returnRange of returnsReturns below expectationsSemivariance a measure that only considers deviations below the meanThese measures of risk implicitly assume that investors want to minimize the damage from returns less than some targe

13、t rateThe Advantages of Using Standard Deviation of ReturnsThis measure is somewhat intuitiveIt is a correct and widely recognized risk measure It has been used in most of the theoretical asset pricing models 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or poste

14、d to a publicly accessible website, in whole or in part.7-6Expected Rates of ReturnFor An Individual AssetIt is equal to the sum of the potential returns multiplied with the corresponding probability of the returns 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or

15、 posted to a publicly accessible website, in whole or in part.7-7Expected Rates of ReturnFor A Portfolio of InvestmentsIt is equal to the weighted average of the expected rates of return for the individual investments in the portfolio 2012 Cengage Learning. All Rights Reserved. May not scanned, copi

16、ed or duplicated, or posted to a publicly accessible website, in whole or in part.7-8Individual Investment Risk MeasureVarianceIt is a measure of the variation of possible rates of return Ri, from the expected rate of return E(Ri) 2012 Cengage Learning. All Rights Reserved. May not scanned, copied o

17、r duplicated, or posted to a publicly accessible website, in whole or in part.7-9where Pi is the probability of the possible rate of return, RiStandard Deviation () It is simply the square root of the varianceIndividual Investment Risk Measure 2012 Cengage Learning. All Rights Reserved. May not scan

18、ned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-10Exhibit 7.3Variance ( 2) = 0.000451Standard Deviation ( ) = 0.021237=2.1237%Covariance of ReturnsA measure of the degree to which two variables “move together” relative to their individual mean values over

19、 timeFor two assets, i and j, the covariance of rates of return is defined as: Covij = ERi - E(Ri) Rj - E(Rj)Example The Wilshire 5000 Stock Index and Barclays Capital Treasury Bond Index in 2010See Exhibits 7.4 and 7.7 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicate

20、d, or posted to a publicly accessible website, in whole or in part.7-11Exhibit 7.4 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-12Exhibit 7.7 2012 Cengage Learning. All Rights Reserved. May not sc

21、anned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-13Covariance and Correlation 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-14The correlation coefficie

22、nt is obtained by standardizing (dividing) the covariance by the product of the individual standard deviationsComputing correlation from covariance Correlation CoefficientThe coefficient can vary in the range +1 to -1. A value of +1 would indicate perfect positive correlation. This means that return

23、s 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. 2012 Cengage Learning. All Rights Reserved. M

24、ay not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-15Standard Deviation of a Portfolio 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-16The Formu

25、la Standard Deviation of a Portfolio Computations with A Two-Stock Portfolio Any asset of a portfolio may be described by two characteristics:The expected rate of returnThe expected standard deviations of returnsThe correlation, measured by covariance, affects the portfolio standard deviationLow cor

26、relation reduces portfolio risk while not affecting the expected return 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-17Standard Deviation of a Portfolio Two Stocks with Different Returns and Risk

27、2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-181 .10 .50 .0049 .07 2 .20 .50 .0100 .10 W)E(R Asset ii2ii Case Correlation Coefficient Covariance a +1.00 .0070 b +0.50 .0035 c 0.00 .0000 d -0.50 -.

28、0035 e -1.00 -.0070Standard Deviation of a Portfolio Assets may differ in expected rates of return and individual standard deviationsNegative correlation reduces portfolio riskCombining two assets with +1.0 correlation will not reduces the portfolio standard deviationCombining two assets with -1.0 c

29、orrelation may reduces the portfolio standard deviation to zero 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-19Standard Deviation of a Portfolio 2012 Cengage Learning. All Rights Reserved. May not

30、 scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-20Constant Correlation with Changing WeightsAssume the correlation is 0 in the earlier example and let the weight vary as shown below. Portfolio return and risk are (See Exhibit 7.13):Exhibit 7.13 2012

31、Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-21Standard Deviation of a Portfolio A Three-Asset Portfolio The results presented earlier for the two-asset portfolio can extended to a portfolio of n asset

32、sAs more assets are added to the portfolio, more risk will be reduced everything else being the sameThe general computing procedure is still the same, but the amount of computation has increase rapidlyFor the three-asset portfolio, the computation has doubled in comparison with the two-asset portfol

33、io 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-22Estimation IssuesResults of portfolio allocation depend on accurate statistical inputsEstimates ofExpected returns Standard deviationCorrelation c

34、oefficient Among entire set of assetsWith 100 assets, 4,950 correlation estimatesEstimation risk refers to potential errorsWith the assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assetsSingle index market model: 2

35、012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-23Estimation IssuesWith the assumption that stock returns can be described by a single market model, the number of correlations required reduces to the

36、number of assetsSingle index market model: 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-24bi = the slope coefficient that relates the returns for security i to the returns for the aggregate market

37、Rm = the returns for the aggregate stock marketThe Efficient FrontierThe 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 returnEfficient frontier are portfolios of investments rather than indiv

38、idual securities except the assets with the highest return and the asset with the lowest risk 2012 Cengage Learning. All Rights Reserved. May not scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.7-25The efficient frontier curves:Exhibit 7.14 shows the process of deriving the efficient frontier curveExhibit 7.15 shows the final efficient frontier curve 2012 Cengage Learning. All Rights Reserved. May not scanne