Advances in Pure Mathematics, 2011, 1, 28-29
doi:10.4236/apm.2011.12007 Published Online March 2011 (
Copyright © 2011 SciRes. APM
Endogenous Risk Measures
Moawia Alghalith
The University of the West Indies, St. Augustine, Trinidad
Received January 11, 2011; revised February 10, 2011; accepted February 28, 2011
We present a methodology that allows endogenous derivation of the moments of the probability distributions.
In doing so, we, present an alternative objective function and alternative concept of risk aversion. In addition,
we show that the risk measure depends on the preferences. Moreover, we show that a higher level of risk
aversion yields higher values of the risk measure.
Keywords: Risk, Risk Measures, Uncertainty
1. Introduction
In the absence of risk neutrality, probability measures
have been subjectively determined by the individual. The
risk-averse or risk-loving individual is assumed to sub-
jectively determine the moments and the distribution of
the risky variable. In addition, these distributions are
assumed exogenous. That is, they are determined outside
the stochastic model. Consequently, these results are ad
hoc empirical and theoretical results. Examples include
numerous theoretical and empirical results in the ex-
pected utility theory, arbitrage pricing theory, and sto-
chastic finance.
For example, according to the expected utility theory,
the probability distributions are still determined exoge-
nously and independently of the individual’s preferences
(attitude towards risk), since preferences are determined
by the form of the utility function [1]. That is, prefer-
ences determine the quantity of the decision variable but
not the probability measure [2], among others). This ap-
pears to be counter-intuitive, since preferences should
play a formal role in deriving the probability distribu-
tions. That is, it should be endogenously determined.
Other exogenous risk measures are extensively used in
finance. The most prominent of these measures is the
value at risk (VaR). The limitations of VaR are well-
documented and hence are needless to discuss (for ex-
ample, see [3] and [4], among many others). Coherent
risk measures and deviations measures are developed as
an alternative to VaR. However, these measures are still
exogenous and subjective measures. Even as exogenous
measures, they have limitations (see [5] and [4] among
In this note, we develop a methodology that enables us
to endogenously derive the moments of the probability
distributions as the decision variables of the model. In so
doing, we formally link the functional form of the objec-
tive function (attitude towards risk) to the derivation of
the moments of the probability distributions. We use a
model of decision-making under uncertainty as an exam-
ple. Though the method is applicable to many other
models, moreover, we present a more general and flexi-
ble model of decision-making under uncertainty, com-
pared to the expected utility models (see [6]).
2. The Model
The conventional theory of the firm under uncertainty
assumes that the firm maximizes the expected utility of
the profit. However, the expected-utility-maximization
objective does not describe the behavior of all firms.
There is empirical evidence that suggests that the agent's
behavior is inconsistent with the expected utility theory
(see, for example, [7]). In the real world, the risk averse
firm’s objective is a mixture of profit-maxi- mization and
risk-minimization. This is particularly true in the inter-
mediate/long run. Consequently, we assume that the firm
maximizes the function
ππ πfEuVaru
 with respect to the mo-
ments of the distribution
max πd
where u is a bounded utility function
0u, 0
and 0
are parameters representing the firms mo-
tive/preferences. A higher value of
indicates a higher
Copyright © 2011 SciRes. APM
level of risk-aversion and a more inclination to minimize
risk, as opposed to profit maximizing. The opposite is
true for
. If the investor is extremely risk averse,
. Clearly, the EU theory is a special
case of this theory when 0
. It is also a generaliza-
tion of the mean-variance framework since our model
does not require quadratic preferences. It is worth notic-
ing, according to this model, 0u does not necessar-
ily imply risk aversion; the converse is also true (risk
aversion does not necessarily imply
0u .
The profit function is specified by
pyc y
 ,
where y is output, c is the cost, p is the random
output price so that 0pp
 , where
is the
second moment, pEp and
is random (see [6]),
among others). The assumption 0p guarantees that
0p and 0
. The output range is given by
Y where Y is the full-capacity output. For
each fixed value of output ˆ
The solution yields
π2ππ 0,EuEu u
 
π2ππ0,EuEu u
 
 Thus, since we have two equa-
tions, optimal solutions can be obtained such that p
arg maxπf
. That is, the moments of the pro-
bability distribution can be obtained for each possible
value of y. However, since we are we interested in
measuring risk, we can modify the objective function to
have the risk as the only decision variables
max πd
and the solution is
π2ππ 0.EuEu u
 
3. A Numerical Example
Under the assumptions of normality, exponential utility
 (the usual assumption in the literature; see
for example, [4]) and 1
(other assumptions
can also be used such as different utility functions or
values of ,
). Under the assumption of
πππ,πEfEfcov p yEf
 
. Given
10, 1, 9ypcy
, we obtain 0.146
We used the assumptions of normality and exponential
utility to simplify the calculations. Other functional
forms and distributions are feasible using computer pro-
4. References
[1] M. Yaari, “The Dual Theory of Choice under Risk,”
Econometrica, Vol. 55, No. 1, 1987, pp. 95-115.
[2] A. Sandmo, “On the Theory of the Competitive Firm
Under Price Uncertainty,” American Economic Review,
Vol. 61, No. 1, 1971, pp. 65-73.
[3] R. E. Bailey, “The Economics of Financial Markets,”
Cambridge University Press, Cambridge, 2005.
[4] R. Elliott and E. Kopp, “Mathematics of Financial Mar-
kets,” Springer, New York, 2005.
[5] R. Elliott and T. Siu, “On Risk Minimizing Portfolios
under a Markovian Regime-Switching Black-Scholes
Economy,” Annals of Operations Research, forthcoming
2010. doi:10.1007/s10479-008-0448-5
[6] M. Alghalith, “New Economics of Risk and Uncertainty:
Theory and Applications,” Nova Science Publishers Inc.,
New York, 2007.
[7] B. J. Cohen, “Is Expected Utility Theory Normative for
Medical Decision Making?” Medical Decision Making,
Vol. 16, 1996, pp. 1-6.