北大光华 资产定价讲义 lect4capm.

2. Mathematics of the Portfolio Frontier
In this section we minimize the variance of a portfolio’s returns subject to a given level of expected return. Of course, this approach is equivalent to maximizing the expected return subject to a given variance, the pricing implications of which will be discussed in the next section. However, richer results concerning the characteristics of the portfolio frontier can be generated using the minimization approach. This section outlines the solution to the minimization problem and derives a number of interesting results concerning the portfolio frontier.
A = e V −1e
B = e V −11
C = 1 VBiblioteka Baidu−11
D = AC − B2
Pre-multiply equation (4) by e to get
e x = λe V −1e + γe V −11
Using equation (2) and our defined scalars this reduces to
E[Rp] = λA + γB
(5)
Pre-multiply equation (4) by 1 to get
Let e be a vector of expected returns of assets and V be the co-variance matrix of the returns on these assets. Suppose investors choose portfolio weights x to minimize variance x V x subject to a given expected return on the portfolio, E[Rp] = x e and the constraint on the portfolio weights x 1 = 1. Our problem becomes
min
x
1 2
x
V
x
+
λ(E
[Rp]
−
x
e)
+
γ
(1
−
x
1)
which has the following necessary and sufficient first order conditions:
V x = λe + γ1
(1)
E[Rp] = x e
(2)
1=x1
(3)
1
Solving equation (1) for x yields
Doctoral Seminar in Asset Pricing (Four) Mean-Variance Analysis and the Capital Asset Pricing Model
1. Mean-variance Analysis
Suppose that you, as a risk averse investor, want a simple rule for choosing between various investment alternatives. One rule that you might consider is to select the investment that delivers the highest expected return for a given level of variance. That is, you might decide that you want to maximize expected return and minimize variance. Even if you had only a passing familiarity with economic theory, you would probably agree that this approach sounds quite sensible. Later on, we will see that the mean-variance rule of portfolio selection is fully consistent with expected utility maximization only under special circumstances. For the moment, however, we want to consider how an investor might behave under circumstances where the mean-variance approach is optimal.
x = λV −1e + γV −11
(4)
Thus, we see that portfolios on the efficient frontier are linear combinations of just two portfolios (V −1e and V −11).
Let’s solve for the Lagrange multipliers λ and γ so that we have an exact expression for the frontier portfolio x. For ease of exposition, we introduce the following notations.