期貨理論與實務英文版課件：Chapter 5 Portfolio Management

1、 Chapter 5 Portfolio Management main content5.1 Risk5.2 Two Securities5.2.1 Risk and Expected Return on a Portfolio5.3 Several Securities5.3.1 Risk and Expected Return on a Portfolio5.3.2 Efficient Frontier5.4 Capital Asset Pricing Model5.4.1 Capital Market Line5.4.2 Beta Factor5.4.3 Security Market

2、 Line5.1 RiskThe return K on a risky investment is a random variable. It is natural to take the expectation E(K) as the reference value. The variance VAR(K) turns out to be a convenient measure of risk. In some circumstances the standard deviation of the return is a more convenient measure of risk.W

3、hat is the Risk?Another natural way to quantify risk would be to use the variance Var(k) (or the standard deviation k) of the logarithmic return k. Var(k(0, n) = Var(k(1) + +Var(k(n)where k(i) is the logarithmic return in time step i = 1, . . . , n and k(0, n) is the logarithmic return over the whol

4、e time interval from 0 to n, provided that the k(i) are independent.Discussion: Compare Risk IndexesWhat are the advantages and disadvantages of the following risk indexes?VAR(K)VAR(k) K kRelationship between Risk and Expected Return of a PortfolioSuppose that the prices of two stocks behave as foll

5、ows:Expected Return : 5%;Risk: 05.2 Two SecuritiesThe return K on a portfolio consisting of two securities is the weighted averageK = w1K1 + w2K2for logarithmic returns，where w1 and w2 are the weights (the allocation of funds).5.2.1 Risk and Expected Return on a PortfolioExpected Return on a portfol

6、io： E(K) = w1E(K1) + w2E(K2).Variance of the return on a portfolio：PropositionThe variance of a portfolio cannot exceed the greater of the variances and of the components, if short sales are not allowed.ConstructionTry to construct the examples that 1 = 2 1 2 1 2 Equation between u and Deduce the eq

7、uation between u and !Exercise: calculate the maximum value of u/ .TheoremFor 1 12 1 the portfolio with minimum variance is attained at If short sales are not allowed, then the smallest variance is attained atTypical (u, ) PlaneCorollarySuppose that 1 2. The following three cases are possible:1) If

8、1 12 1/2, then there is a portfolio without short selling such that 1 (lines 4 and 5 in Figure 5.4);2) If 12 = 1/2, then 1 for each portfolio (line 3 in Figure 5.4);3) If 1/2 12 1, then there is a portfolio with short selling such that 0, and a risk-free security with return rF and standard deviatio

9、n zero depends on the weight w1 of the risky security as follows:= |w1| 15.3 Several SecuritiesThe expected return = E(K) and variance = Var(K) of a portfolio with weights w are given by = mwT 2= wCwTwhere w =w1 w2 wn m =1 2 nExerciseCompute the expected return and standard deviation of a portfolio

10、consisting of three securities with weights w1 = 40%, w2 = 20%,w3 = 80%, given that the securities have expected returns 1 = 8%,2 = 10%, 3 = 6%, standard deviations 1 = 1.5, 2 = 0.5, 3 = 1.2 and correlations 12 = 0.3, 23 = 0.0, 31 = 0.2.Our GoalTo find a portfolio with the smallest variance in the a

11、ttainable set. It will be called the minimum variance portfolio.To find a portfolio with the smallest variance among all portfolios in the attainable set whose expected return is equal to a given number V. The family of such portfolios, parameterized by V, is called the minimum variance line.Minimum

12、 Variance Portfolio whereMinimum Variance LineWhat is the relationship between w and uv?What is the relationship among all w in minimum variance line?Example 5.10Figure of (, ) planeMarkowitz Bullet:What is the figure of minimum variance line?5.3.2 Efficient FrontierDefinition：a security with expect

13、ed return 1 and standard deviation 1 dominates another security with expected return 2 and standard deviation 2 whenever1 2 and 1 2.A portfolio is called Efficient if there is no other portfolio, except itself, that dominates it. The set of efficient portfolios among all attainable portfolios is cal

14、led the Efficient Frontier .Efficient Portfolio has the highest expected return among all attainable portfolios with the same standard deviation (the same risk), and has the lowest standard deviation (the lowest risk) among all attainable portfolios with the same expected return.The efficient fronti

15、er consists of all portfolios on the minimum variance line whose expected return is greater than or equal to the expected return on the minimum variance portfolio.5.4 Capital Asset Pricing ModelAssumptions: Every investor uses the same values of expected returns, standard deviation and correlations

16、for all securities, making investment decisions based only on these values.According to the assumptions of the CAPM, every rational investor will select his or her portfolio on this half-line, called the capital market line.Capital Market LineCapital Market LineCapital Market LineThe capital market

17、line joining the risk-free security and the market portfolio satisfies the equationRisk PremiumMarket PortfolioEveryone will be holding a portfolio called market portfolio with the same relative proportions of risky securities.Where is (M,M)?What is w of (M,M)?How to calculate the w?PropositionThe w

18、eights w of any portfolio belonging to the efficient frontier (except for the minimum variance portfolio) satisfy the conditionwC=mu for some real numbers 0 and .V: the gradient of the tangent line to the efficient frontier: the intercept of this tangent line5.4.2 Beta Factorthe values of KM are mar

19、ked along the x axis and the values of KV along the y axis. The equation of the line of best fit will be y = V x + V .The difference = KV ( + KM) between the actual return KV and the predicted return + KM is called the residual random variable.The condition defining the line of best fit is thatas a function of and should attain its minimum at = V and = V .Exercise