STOCHASTIC

Note I — Alternative Proof of Separation
Theorem and Its Corrolaries
In this note, I present an alternative proof of the
Separation Theorem and its corrolaries using utility
functions explicity. Some readers may prefer this
form, since it follows traditional theory more closely.
Let y and o-„ be the expected value and variance of
the rate of return on any asset mixture and An be the
amount of the investor's total net investment. Given
the assumptions regarding the market and the investor,
stated in the text, the investor will seek to maximize
the expected utility of a function which can be written
in general form as
(i') E[U(ABy,Aeffv)} = O{A0yt A,^),
subject to his investment opportunities characterized
by the risk-free rate r*, at which he can invest in savings
deposits or borrow any amount he desires, and by the
set of all stock mixes available to him, each of which
in turn is represented by a pair of values (r, , r) values lie
on the part of the boundary associated with values of
f > o, and for which changes in o> and f are positively
associated. This is Markowitz' Efficient Set or "E-V"
Frontier. We may write its equation 7 * as
71
For formal proof of these properties, see Tobin, [21],
pp. 72-77.
71
Specifically, that the amount invested in any stock
in any stock mix is infinitely divisible, that all expected
returns on individual stocks are finite, that all variances are
positive and finite, and that the variance-covariance matrix
is positive-definite.
74
Markowitz [14] has shown that, in general, this closed
curve will be made up of successive hyperbolic segments vrhich
are strictly tangent at points of overlap.
71
Harry Markowitz, [14], chapter VII. The shape of the
boundary follows from the fact that the point corresponding
to any mix (in positive proportions summing to one) of any
two points on the boundary lies to the left of the straight line
joining those two points; and all points on and within the
boundary belong to the set of available (o>, f) pairs because any
such point corresponds to an appropriate combination in
positive proportions of at least one pair of points on the
boundary.
78
Note that the stated conditions on the derivatives in
(2') f -/(o,/w >of rur)