232 Performance Measurement in Finance
of the GMM estimation and testing of the overidentifying restrictions are pre-
sented in section 9.5. Finally, in section 9.6, we offer concluding comments.
9.2 INVESTMENT OPPORTUNITY SETS WITH CONTINUOUS
RISK STRUCTURES
This section presents the conditional investment opportunity set that reflects
the continuum of volatilities available on traded financial assets. We begin
by presenting the familiar conditional IOS hyperbola as described in Merton
(1972). Next we discuss the conditional IOS in the context of a continuous risk
structure. The resultant IOS represents a considerable simplification because
it can be described by a bivariate process governing the IOS vertex and slope.
An investment opportunity set represents the risk and return possibilities
from a set of risky financial assets held over a fixed time interval, (t, t
1
], such
as one month. For convenience, the time subscript t is omitted in all notation
and it is implicit that an opportunity set is conditional on information at a
given point in time, and for a given interval length.
Following Merton (1972), the conditional investment opportunity set is
described by a hyperbola in mean-standard deviation space,
f(σ
p
) = μ
p
=
b
c
±

a
b
2
c
σ
2
p
1
c
1/2
(9.1a)
where μ
p
= X
p
μ is the conditional mean of the portfolio determined by the
(n × 1) vector of portfolio weights, X
p
,andthe(n × 1) vector of asset means,
μ; σ
p
=
X
p
X
p
is the conditional standard deviation of the portfolio deter-
mined by the portfolio weight vector, X
p
,andthe(n × n) covariance matrix
of asset returns, ;anda = μ
1
μ, b = e
1
μ and c = e
1
e are the
efficient set constants determined by the mean vector, μ, covariance matrix
inverse, and the (n × 1) vector of ones, e.
4
The hyperbola’s asymptote equations are described by,
g(σ
p
) =
b
c
±
a
b
2
c
1/2
σ
p
(9.1b)
and the mean and standard deviation of the least risky or vertex portfolio are
μ
o
= b/c and σ
o
= 1/
c, respectively.
4
For tractability in our empirical work, we assume a non-singular covariance matrix of asset returns.
More general results can be shown using generalized inverses when the covariance matrix is of
less than full rank (c.f. Graybill (1969), Buser (1977) and Ross (1977)).
The intertemporal performance of investment opportunity sets 233
A restricted set of risky assets, such as common stocks, is a coarse approxi-
mation to the actual investment opportunity set faced by economic agents. As
demonstrated by Stambaugh (1982) and Kandel (1984), asset omissions almost
surely cause a misspecification of the opportunity set. Here, we are interested
in the omission of risky assets with risks arbitrarily close to zero.
5
To pro-
ceed, we define a continuous risk structure opportunity set in terms of the
volatilities on available traded assets. The development and proof of the con-
tinuous risk structure investment opportunity set, described by equation (9.2),
is available in Korkie and Turtle (1997).
Definition: A continuous risk structure investment opportunity set is a
feasible set of expected returns and risks from financial assets with a sequence
of volatilities arbitrarily close to zero, for all bounded time intervals. The
continuous risk structure IOS can then be described by the mean equation,
μ
p
= r
f
± slσ
p
(9.2)
where r
f
represents the known riskless rate for the interval (t, t
1
], and sl
denotes the finite IOS slope.
Because the vertex mean, b/c, has converged to the known riskless rate, r
f
determines the minimum or zero risk portfolio’s return and the IOS slope, sl,
can be estimated using
%
a b
2
/c or
%
a br
f
.
The investment opportunity set depends mathematically upon the current
level of interest rates through both the IOS vertex and the IOS slope. This is
not surprising given an abundance of early empirical research on the subject.
For example, Fama and Schwert (1977) find that the spread between stock
and bond returns is related to the interest rate level. In addition, they find that
stock returns are negatively related to anticipated inflation rates, which may be
proxied by the Treasury bill rate (Fama and Gibbons, 1984). Geske and Roll
(1983) explain the causality in the linkage between interest rates and stock
returns, which is supported in Solnik (1984) and James, Koreisha and Partch
(1985). In Fama (1984), term premia predict future spot rates of interest. In
Chen, Roll and Ross (1986), the Treasury bill rate is in the term structure shift
factor. In Keim and Stambaugh (1986), term structure spreads are important
determinants of conditional expected returns. Ferson (1989) found that the
information contained in one month Treasury bill rates implies time variation
5
Stambaugh (1982) and Kandel (1984) describe the robustness of mean-variance parameters used
in asset pricing tests to omitted assets. Although they were not concerned specifically with the
omission of nearly riskless assets, in general they find that the omission of assets causes potentially
severe measurement error in the parameters of the investment opportunity set.

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