Review of Quantitative Finance and Accounting, 18: 39–58, 2002
2002 Kluwer Academic Publishers. Manufactured in The Netherlands.
Gain, Loss, and Two-State Modeling
University of Wyoming
MICHAEL S. ROZEFF
University of Buffalo, Department of Finance and Managerial Economics, Jacobs Management Center,
Box 604000, Buffalo, NY 14260-4000
Abstract. Gain and loss, calculated from the upside and downside portions of return distributions, play a pivotal
role in the two-state model. A two-state economy possesses a universal gain-loss ratio (G/L) for all assets that
is related to the ratio of state prices and to the familiar risk-neutral probabilities. This paper derives many asset
pricing properties in a two-state context and shows the role of gain and loss. Applied to bonds, for example, risky
debt yields depend directly on both G/L and a bond’s potential loss. Using S&P 500 data over a 72-year period,
the market has priced an Arrow-Debreu security in the gain state at approximately $0.36, while the Arrow-Debreu
security in the loss state has been priced at $0.61. Historically, the S&P 500’s expected gain is about three times
its expected loss.
Key words: gain, loss, state prices, asset pricing
JEL Classiﬁcation: G12
Although gain and loss, which measure the upside and downside portions of return distri-
butions, seem to be natural ways to think of reward and risk, mean and variance dominate
academic conceptions of portfolio theory. Recently, however, Bernardo and Ledoit (2000)
show that gain and loss are valuable in analyzing asset pricing, and O’Connor and Rozeff
(2000) develop a rigorous gain-loss portfolio theory. This paper extends the inquiry into
gain and loss by showing that when there are just two states, gain and loss play an absolutely
central role as exhaustive measures of reward and risk. In a two-state world, gain and loss
are extremely effective concepts in characterizing asset pricing.
Two-state models are not as realistic as multi-state models.
But, following Occam, sim-
pliﬁcation has its beneﬁts.
Such an approach gains the following advantages.
The model is
easy to follow, thereby strengthening economic intuition. Important parameters such as state
prices can be directly estimated. An integration of certain ﬁnance principles occurs. For ex-
ample, the binomial model (which is a two-state model) and asset pricing models such as the
capital asset pricing model (CAPM) are ordinarily placed into separate categories (see, for
example, Grinblatt and Titman, 1998). In a two-state world, the binomial model and CAPM
each can be derived from the other. In the same vein, two-state modeling integrates bond and
stock pricing, although each asset has a distinctive model form, one in terms of returns, the
other in terms of yields.
Furthermore, two-state modeling implies a closed-form expression
for risky yields that provides useful intuition concerning relevant pricing parameters.
As simple as it is, the two-state model gives rise to several new ﬁndings. One is an
economic interpretation of risk-neutral probabilities as functions solely of gain and loss.