Review of Quantitative Finance and Accounting, 9 (1997): 289–300
© 1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
A Note on the Analytics and Geometry of Limiting
Mean—Variance Investment Opportunity Sets
Department of Finance and Management Science, University of Alberta, Edmonton, Alberta, Canada, T6G
2R6, Phone: (403) 492-2764, Fax: (403) 492-3325, Email: firstname.lastname@example.org
HARRY J. TURTLE
Department of Finance, Insurance and Real Estate, Washington State University, Pullman Washington
Abstract. This paper extends the mathematics developed by Merton (1972) to the limiting investment oppor-
tunity set as smaller risk assets are added. Investment opportunity sets of risky assets are well-known to be
described by hyperbolae in mean-standard deviation space. In practice, the asset classes in portfolios may vary
from high risk common stocks to near cash assets. Low variability assets change the appearance of the invest-
ment opportunity set to the extent that a unique optimum risky asset portfolio disappears. The limiting result is
similar to the investment opportunity set that arises when two assets are perfectly correlated. The location of the
IOS is shown to mathematically depend upon the level of the riskless interest rate and one slope parameter. The
slope parameter is estimable, using a ﬁnite number of assets, and represents a bound on market Sharpe ratios.
Key words: investment opportunity sets, Markowitz, convergence, estimation, equity premium
The fundamental investment-consumption decisions of economic agents are based in part
on the available investment opportunities. This paper demonstrates that there is a well-
deﬁned limit to the investment opportunity set (IOS), as one adds assets including the
many low risk assets available to investors. A proof of this limiting IOS and the resulting
analytics and geometry have not appeared in the literature. Although there is ample
empirical evidence of a relationship between interest rate levels and the expected returns
and risks of other assets, there has been no analytics that establish a relationship between
the level of the riskless interest rate and the IOS.
This paper develops the limiting investment opportunity set due to small risk assets.
Our development begins by reviewing the mathematics of investment opportunity sets in
mean—standard deviation space when covariance matrices are singular or nonsingular.
Then, sequence-of-economies proofs are constructed to give the uniform convergence of
the IOS to its limit, as the number of assets increases. The IOS converges to a portfolio
set bounded by straight lines emanating from the known riskless rate; the straight line
slopes are nonlinearly related to the level of the riskless rate. We call this the “limiting
The limit represents a simpliﬁcation because the slopes of the limiting straight lines are
a function of the riskless rate and two efﬁcient set parameters. In practice, the efﬁcient set
@ats-ss11/data11/kluwer/journals/requ/v9n3art4 COMPOSED: 09/16/97 2:14 pm. PG.POS. 1 SESSION: 9