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1、Mean-variance theoryPhilip H. DybvigWashington University in Saint Louis Decision theory Mean-variance theory Means, variances, and covariances The value of diversification Optimal portfolio choice previous lecturenext lecture Copyright Philip H. Dybvig 1997, 2000Decision TheoryDecision Theory gives
2、 us a conceptual framework for formalizing optimal choice. This framework is almost indispensible if we want to solve for an optimal choice, and it is also useful for thinking about a choice problem even if we are going to use a combination of intuition and informal analysis to make the final decisi
3、on.The essential pieces of a choice problem are the choice variables, the objective function, and the constraints. We may also have parameters, which are inputs to the choice problem that can be varied.A Simple Consumers Decision ProblemChoose c1, c2, and c3 tomaximize U(c1,c2,c3) subject top1 c1 +
4、p2 c2 + p3 c3 = W,c1 = 0,c2 = 0, andc3 = 0.The choice variables are c1, c2, and c3, which are expenditures on three classes of consumption goods. The objective function is U(c1,c2,c3), which represents the consumers preferences for different consumption patterns. There are four constraints, the budg
5、et constraint and three positivity constraints. Some parameters of the problem include the prices p1, p2, and p3, and wealth W. There may be additional parameters describing the features of the utility function. The parameters are not chosen as part of the decision; we leave them flexible to allow u
6、s to study, for example, the sensitivity of the consumption of good one to its price.A Simple Investments ProblemChoose q1, q2, and q3 tomaximize Eu(w1) subject toq1 + q2 + q3 = 1 andw1 = w0(q1(1+r1) + q2(1+r2) + q3(1+r3).The choice variables are q1, q2, and q3, which are proportional investments in
7、 three different securities. The objective function is Eu(w1), which represents the investors preferences over different random payoffs. The first constraint is the budget constraint that proportions in the securities sum to one, and the second constraint defines the payoff w1 given the amounts inve
8、sted and the random returns r1, r2, and r3 on the three securities. To complete this specification, we would have to specify the particular utility function u and the joint probability distribution of the returns.An Even Simpler Investments ProblemSuppose there are two assets, a riskless asset 1 wit
9、h return 10% and a risky asset 2 with equally probable returns -10% and +50%. Assuming that the utility function is u(w1) = w1-.004w12 and the initial wealth is 100, we can substitute in the constraints (e.g. q1 = 100%-q2) and use elementary algebra to reduce the previous choice problem to the follo
10、wing.Choose q2 to 2maximize 61.6 + 1.2 q2 - 4.0 q2The solution is q2 = .15, i.e. 15% of wealth should be in the risky asset and 85% in the riskless asset. (One way of proving this is by writing the objective function as 61.69 - 4.0 (q2 - 0.15)2.) This is a very conservative strategy; a less conserva
11、tive solution would arise if we replaced .004 with a smaller number, representing a smaller aversion to risk-taking.An Even Simpler Investments Problem (Algebra Details)Eu(w1) = 0.5u(w1u)+0.5u(w1d) = 0.5(u(100(1.1q1+1.5q2)+u(100(1.1q1+.9q2) = 0.5(u(100(1.1(1-q2)+1.5q2)+u(100(1.1(1-q2)+.9q2) = 0.5(u(
12、110+40q2)+u(110-20q2) = 0.5(110+40q2)-.004(110+40q2)2 +(110-20q2)-.004(110-20q2)2) = 0.5(110+40q_2-48.4-35.2q_2-6.4q_22 +110-20q_2-48.4+17.6q_2-1.6q_22) = 61.6+1.2q_2-4q_22 = 61.69-4(.15-q_2)2 =61.69(since the square cannot be negative), which is the value when q2=0Types of portfolio problemsFor an
13、individuals portfolio choice, the choice problem in the previous slides is a good start. In most institutional settings, there are at least two levels of management. At the highest level, the plan sponsor (the representative of the beneficiaries) must choose proportions of the portfolio to be alloca
14、ted to different asset classes and more specifically how to allocate funds within each asset class to different managers (who may or may not be in-house). We call this selection of broad asset classes asset allocation. The problem of the specific manager (who may manage a portfolio of equities or go
15、vernment bonds or convertible bonds), we refer to as subportfolio management.Traditionally, academic finance has not looked at subportfolio problems, which are significantly different from asset allocation problems. Subportfolio managers are typically judged relative a benchmark appropriate to the asset class and are constrained directly and indirectly in how far they can deviate from the benchmark. The traditional tools for asset allocation can be modified in a natural way to study subportfolio management.Mean-variance theoryMean-variance theory is an important mo
