Review of Quantitative Finance and Accounting, 16, 81–94, 2001
2001 Kluwer Academic Publishers. Manufactured in The Netherlands.
Measuring Investment Risk Based on Tail Thickness
G. R. DARGAHI-NOUBARY, Ph.D.
Department of Mathematics and Statistics, Bloomsburg University, Bloomsburg, Pennsylvania
Wm. STEVEN SMITH, Ph.D.
Finance Faculty, College of Business Administration, North Dakota State University, Fargo, North Dakota
Abstract. In recent years, both institutional and individual investors have come to rely heavily upon techniques
for analyzing (deﬁning and measuring) risk. In this respect, the issue that continues to require the attention of
academic researchers and practitioners alike is how to concisely deﬁne investment risk and, more importantly,
how to best measure it. Selecting an appropriate risk deﬁnition involves trade-offs among ease of measurement,
forecast ability, and intuition of individual investors. The purpose of this paper is to present an alternative index
for measuring unconditional (or total) risk. The proposed measure reﬂects behavior in general, and thickness
in particular, of the lower tail of the distribution of returns. We therefore argue it provides a more useful and
reasonable index because, unlike measures frequently used, its estimation depends upon the most relevant data
from the sample distribution. We describe risk analysis based on lower tail behavior and identify its advantages
over existing methods. Finally, using data of weekly returns to the CREF Stock Fund, we provide an empirical
example to illustrate the technique.
Key words: risk probability distribution, threshold, extreme value, tail thickness, generalized pareto, conditional
JEL Classiﬁcation: G11
Since the pioneering work of Rothschild and Stiglitz (1970), academic researchers and
practitioners alike have struggled to develop a more concise deﬁnition of unconditional
investment risk (or simply total risk) that is both theoretically sound and widely applicable.
Broadly deﬁned, total risk is uncertainty caused by the tendency of the nominal return
realized (ex post nominal return) on investment to differ from the nominal return predicted
or expected (ex ante return).
Arguably, the risk deﬁnition provided above is reﬁned by redirecting the focus to the
tendency of the return realized to be less than that expected. The attention to unconditional
“downside” risk (or simply downside risk) is especially relevant given a “representative”
investor in the market who is risk averse and possibly, for a given level of risk, demonstrates
preference for positive skewness (see Elton and Gruber, 1995, pp. 247 and 248). Logically,
this discussion leads to behavior of the lower half of the returns distribution as source of
the relevant portion of total risk for at least a well-diversiﬁed portfolio.
As commonly recognized by many within the academic community, a single parameter
for measuring risk is clearly not sufﬁcient (see, e.g., Rothschild and Stiglitz, 1970). Yet,