McGraw-Hill

Long-Short Portfolio Mariagement , 357
Let Y, represent the return on the risk-free security, and let Ri
represent the expected return on the ith risky security. The expected
return on the investor's total portfolio is
N
r, =h,r, + zhiRi.
i=l
Substituting the expression derived above for h, into this equation
gives the total portfolio return as the sum of a risk-free return component
and a risky return component, expressed as rr, = T, + r,.
The risky return component is
N
r, =xhiri (14.2~)
i=l
where ri = Ri - r, is the expected return on the ith risky security in ex
cess of the risk-free rate. The risky return component can be ex- also
pressed in matrix notation as .
r,=hTr
where h = [h,, h, . . I hNIT and r = [r,, . .,
(14.2b)
rNIT. It can be shown
that the variance of the risky return component, D:, is
D: =hTQh (14.3)
where Q is the covariance matrix of the risky securities' returns. The
2
variance of the overall portfolio is D; =Q R .
With these expressions, the utility function in Eq. (14.1) can be
expressed in terms of controllable variables. We determine the optimal
portfolio by maximization of the utility function through appropriate
choice of these variables. This maximization is performed
subject to the appropriate constraints. A minimal set of appropriate
constraints consists of (1) the Regulation T margin requirement and
(2) the requirement that all the wealth allocated to the risky securities
is fully utilized. The solution (providing Q is nonsingular) gives
the optimal risky portfolio as
h= z Q"~ (14.4)
where Q" is the inverse of the covariance matrix. We refer to the
portfolio in Eq. (14.4) as the minimally constrained portfolio.