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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