Endogenous Risk Measures

doi:10.4236/apm.2011.12007

Paper Menu >>

Journal Menu >>

Advances in Pure Mathematics, 2011, 1, 28-29

doi:10.4236/apm.2011.12007 Published Online March 2011 (http://www.SciRP.org/journal/apm)

Endogenous Risk Measures

Moawia Alghalith

The University of the West Indies, St. Augustine, Trinidad

E-mail: [email protected]

Received January 11, 2011; revised February 10, 2011; accepted February 28, 2011

Abstract

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

others).

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

M. ALGHALITH

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,



 or



. 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







where



2πEu

 

 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



πeu





 (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 ,





and



). Under the assumption of

normality













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-

gramming.

4. References

[1] M. Yaari, “The Dual Theory of Choice under Risk,”

Econometrica, Vol. 55, No. 1, 1987, pp. 95-115.

doi:10.2307/1911158

[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.

doi:10.1177/0272989X9601600101