马克维茨投资组合选择(5)

lines; the isovariance

curves are a system of concentric ellipses (see Fig. 2). For example,

if~2 p3 equation 1' can be written in the familiar form X2 = a +

bX1; specifically (1)

Thus the slope of the isomean line associated with E = Eois-(pl-

j~3)/(.~2- p3) its intercept is (Eo- p3)/(p2 - p3). If we change E we

change the intercept but not the slope of the isomean line. This confirms

the contention that the isomean lines form a system of parallel lines.

Similarly, by a somewhat less simple application of analytic geometry,

we can confirm the contention that the isovariance lines form a

family of concentric ellipses. The \system is the point

which minimizes V. We will label this point X. Its

expected return and

variance we will label E and V. Variance increases as you move away

from X. More precisely, if one isovariance curve, C1, lies closer to X

than another, Cz, then C1 is associated with a smaller variance than Cz.

With the aid of the foregoing geometric apparatus let us seek the efficient sets.

X, the center of the system of isovariance ellipses, may fall either

inside or outside the attainable set. Figure 4 illustrates a case in which

Xfalls inside the attainable set. In this case: Xis efficient. For no other

portfolio has a V as low as X; therefore no portfolio can have either

smaller V (with the same or greater E) or greater E with the same or

smaller V. No point (portfolio) with expected return E less than E

is efficient. For we have E >E and V

Consider all points with a given expected return E; i.e., all points on

theisomean line associated with E. The point of the isomean line at

which V takes on its least value is the point at which the isomean line

9. The isomean \when = pz= pa In the

latter case all portfolios have the same expected return and the investor chooses the one with minimum variance.

As to the assumptions implicit in our description of the isovariance curves see footnote 12.

Portfolio Selection 85 A

is tangent to an isovariance curve. We call this point X(E). If we let

hE vary, X(E) traces out a curve.

Algebraic considerations (which we omit here) show us that this curve

is a straight line. We will call it the critical line

I. The critical line passes

through X for this point minimizes V for all points with E(X1, Xz) = E.

As we go along l in either direction from X, V increases. The segment

of the critical line from X to the point where the critical line crosses

*direction of increasing E depends on p,. p:. p3 FIG.2

the boundary of the attainable set is part of the efficient set. The rest of

the efficient set is (in the case illustrated) the segment of the 3 line

from d to b. b is the point of maximum attainable E. In Figure 3, X lies

outside the admissible area but the critical line cuts the admissible

area. The efficient line begins at the attainable point with minimum

variance (in this case on the Z line). It moves toward

b until it intersects

the critical line, moves along the critical line until it intersects a

boundary and finally moves along the boundary to b. The reader may

efficient portfolios

Portfolio Selection 87

wish to construct and examine the following other cases: (1) X lies

outside the attainable set and the critical line does not cut the attainable

set. In this case there is a security which does not enter into any

efficient portfolio. (2) Two securities have the same

pi. In this case the

isomean lines are parallel to a boundary line. It may happen that the

efficient portfolio with maximum E is a diversified portfolio. (3) A case

wherein only one portfolio is efficient.

The efficient set in the 4 security case is, as in the 3 security and also

百度搜索“77cn”或“免费范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，免费范文网，提供经典小说综合文库马克维茨投资组合选择(5)在线全文阅读。