Risk Aversion and Individual Preferences
Modelling
Prof. Constantin ANGHELACHE PhD.
Bucharest University of Economic Studies, “Artifex” University Bucharest
Lecturer Mădălina Gabriela ANGHEL PhD.
Assoc. prof. Aurelian DIACONU PhD.
Artifex” University Bucharest
Abstract
Confronting risky situations is a feature of any decision making, in
general, but, especially for specialized decision making contexts. Risk aversion
concentrates the problems of modeling individual preferences. Models can
grasp with accuracy and largely enough the fundamental human tendencies.
As consequence, understanding economic behavior confronted to risk might
be understood. Modeling market forecasting activities is possible based on
“marginal consumer”. Thus, the vague use of mathematical instruments is
necessary, but not enough as condition to develop an economic analysis.
A historical approach of risk analysis might bring an idea regarding the
development of the concept instruments used today when analyzing “risky
choices”
Key words: risk theory, diversification, risk aversion, risk premium,
absolute risk aversion, relative risk aversion, utility function, marginal
consumer, marginal utility, DARA, CARA, CRRA.
A short presentation
In 1738, Daniel Bernoulli published “Specimen theoriae novae
de mensura sortis”, a paper in Latin, or “Exposition of a new theory on
measurement of risk”. The English translation made in 1854 seemed to be
non-technical and focused on the way two people confronted to the same
risky situation might choose depending on psychological differences. While
presenting his hypothesis, Bernoulli used three examples. In this sense, “Sankt
Petersburg paradox” is quite famous and still present today in academic
circles, and shows the importance of psychological aspects with regard to
risky choices. Unfortunately, the celebrity of this paradox has overshadowed
the other two examples. They show that the value of a risky situation is not
equal to its mathematical expectation. One is known as “Sempronius” and it
anticipates in an exemplary manner the contributions brought 230 years later
in the risk theory by Arrow, Pratt et alias.
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Quoting Bernoulli, Sempronius owns goods at home worth a total of
4 000 ducats and in addition he possesses 8 000 ducats worth commodities
in foreign countries from where they can be transported only by sea. Our
common experience shows that from two ships one perishes.”
Sempronius is confronted with a risk on his wealth. x̃ represents his
wealth on a value of 4 000 ducats with probability
(if ship is sunk), or 12
000 ducats with probability . We denote this lottery as x̃ being distributed as
Its mathematical expectation is given by:
(4000,
Ex̃
4000 +
12000 = 8000 ducats.
Sempronius comes with a big idea. Instead of trusting all 8 000 ducats
of goods to one ship, he “entrusts equal portions of his commodities to two
ships”. Assuming that the ships follow independent but equally dangerous
routes, Sempronius now faces a more diversified lottery ỹ distributed as
(4000, ; 8000, ; 12000, )
In case Sempronius looses all his ships, he still has 4000 ducats wealth.
AS the two risks are independent, their probability equals the result of the
individual events, exactly ( )2 = .Thus, there is the probability of that the
two ships to be back home, and thus the final wealth of Sempronius is 12000
ducats. Evidently, there is also the other chance for him to have only one
ship back with the final result of half of the previous wealth. The final wealth
would be of 8000 ducats. The probability of this event is of completing the
other two events with probability of .
The common experience leads to the idea that diversification is good,
so that we can expect the value attached to the event ỹ to exceed the one
attached to the event x̃ . Even so, calculating the expected profit, we may
obtain:
Eỹ = ¼ 4 000+ ½ 8 000+ ¼ 12 000 = 8 000 ducats,
which means the same value as for Ex̃. Thus, diversification seems
useless!
Using intuition, Bernoulli’s demonstration will lead to the conclusion
that lottery ỹ might bring a satisfaction value larger than lottery x̃, thus the
satisfaction level generated by lottery ỹ will make Sempronius choose the
first of them. Thus, Bernoulli shows that diversification generates a transfer
conserving means to gain wealth, from extremes to the middle. Thus, transferring
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probability from x=4 000 to x=8 000 a, increase of expected utility is obtained.
Each transferred probability unit generates an increase of expected utility equal
to u(8000) - u(4000). On the contrary, transferring a certain probability from x
= 12 000 to x = 8 000, it will reduce the expected probability. Each transferred
probability unit will generate a reduction of the expected utility equal to u (12
000) – u (8 000). But concavity of u implies that
u (8 000) - u (4 000) u (12 000) - u (8 000),
(1)
With the positive aspect that these transfers combined should dominate
the negative effect.
Definition and characteristics of risk aversion in investments
Supposing agents live only one investment cycle, investing their
wealth purchasing and consuming goods and services. The final wealth is
formed of the initial wealth w plus the outcome of any risk borne during the
period.
Definition 1: An agent has risk aversion if, at any level of wealth w
he dislikes every risk with expected payoff of zero. Thus, he always prefers
receiving the expected outcome of a risky decision with certainty, rather than
the risk itself. For an expected-utility maximize with a utility function u, this
implies that, for any risky situation z̃ and for any initial wealth w,
E u ( w + z̃ ) u ( w + E z̃).
(2)
In Sempronius’ case, his risk aversion is obvious. This is true anytime
the utility function is concave. If marginal utility decreases, the possible
loss reduces more the utility generated by potential gain. The preference
for diversification is intrinsically equivalent to risk aversion, at least under
the model of expected utility of Bernoulli. Reversely, if u is convex, the
inequality will be reversed. That is why the agent will prefer risky decision
to any mathematical expectation revealing his inclination to taking risks. This
behavior is known as loving risk behavior.
Finally, u is linear, then the welfare Eu is linear in the expected payoff
of lottery. If u(x) = a + b x for all x, then we have
Eu ( w +z̃) = E [a + b (e + z̃ )] = a + b ( w + Ez̃) = u ( w + Ez̃).
This implies that the agent would rank the lotteries according to their
expected outcome. This behavior is called risk-neutral.
A decision maker is risk averse if inequality (2) holds for all w and z̃ if
and only if u is concave. This proposition is in fact a rewriting of the famous
Jensen inequality with the conclusion that the decrease of the marginal utility
means an increase of the income. Lately, many researchers consider that risk
aversion is generated by marginal utility decrease. Other authors consider that
there is no connection between the two concepts.
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Risk premium and the equivalent of certainty
An agent is risk averse in case he dislikes zero-mean risks. The degree
of risk aversion might be evaluated by asking him how much he is inclined
to pay in order to avoid a zero-mean risk z̃. The answer to this question will
be defined as risk premium associated to that risk. The agent will end up with
the same wealth either he will accept the risk or he will pay the risk premium.
When risk z̃ has a expectation other than zero, we usually use the concept of
the certainty equivalent. This is the sure increase in wealth that has the same
effect on welfare as having to bear the risk z̃.
The cost of risk, as measured by the risk premium, is proportional to
the variance of payoffs. Thus, the variance seems to be a good measure of the
degree of riskiness of lottery. This is why many authors used mean-variance
in order to model the behavior under risk. The validity of these models is
accurate only when the risk is small. In such cases, mean-variance approach
can be seen as a special case of the expected- utility theory. The mean-variance
played a very important role in the development of the finance theory.
The degree of risk aversion
Considering the small risk only, we can see that agents with a larger
absolute risk aversion are more reluctant to accept small risks. In their cases,
the minimum expected payoff is larger. Technically, the measure of the degree
of risk aversion is a measure of concavity of the utility function. It measures
the speed at which marginal utility is decreasing. In small risk cases, we need
to know to determine whether a risk is desirable is the degree of concavity
of u locally at the current wealth level w. For larger risks, we need to know
much more in order to make a decision, meaning needing to know the degree
of concavity of u at all wealth levels. So, the degree of concavity must be
increased at all levels of wealth to guarantee that a change in u makes the
decision maker more reluctant to accept risks.
Decreasing absolute risk aversion and prudence
As we could see, risk aversion is driven by the fact that one’s marginal
utility is decreasing with wealth.
Arrow considers that wealthier people are generally less willing to pay
in order to eliminate fixed risk. Thus, the risk premium associated to any risk
is decreasing in wealth if and only if absolute risk aversion is decreasing; or
if and only if prudence is uniformly larger than absolute risk aversion. Notice
that Decreasing Absolute Risk Aversion (DARA), a very intuitive condition,
requires the necessary condition that the utility function to be positive, or that
marginal utility to be convex.
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Relative risk aversion
Absolute risk aversion is the rate of decay for marginal utility. More
particularly, absolute risk aversion measures the rate at which marginal utility
decreases when wealth is increased by one monetary unit. The index of
absolute risk aversion is not unit free, as it is measured per monetary units.
Economists prefer unit-free measurements of sensitivity. Defining the
index of relative risk aversion R as a rate at which marginal utility decreases when
wealth is increased by one per cent. In terms of standard economic theory, this
measure is simply the wealth-elasticity of marginal utility. Finally, the measure of
relative risk aversion is simply the product of wealth and absolute risk aversion.
The relative risk premium is equal to half of the variance of the
proportional risk times the index of relative risk aversion. This can be used to
establish a range for acceptable degrees of risk aversion.
There is no argument for or against decreasing relative risk aversion.
Arrow considered that relative risk aversion is likely to be constant or perhaps
increasing, but the intuition is not clear as was for decreasing absolute risk
aversion. Under the intuitive DARA assumption, becoming wealthier also
means becoming less risk-averse. This effect tends to reduce risk premium.
On the other hand, becoming wealthier means facing larger absolute risks, that
tends to raise risk premium. It is not clear if the first effect or the second will
dominate, as well as there is no a priori reason to believe that the dominant
effect will not change over various wealth levels.
Some classical utility functions
The expected utility (EU) theory is not accepted by everyone in
economic theory. Some researchers consider EU criterion satisfying for those
who find expected utility too restrictive. Economics and finance researchers
consider EU theory as an acceptable paradigm for decision making under
uncertainty. EU theory has a long “career” and a prominent place in the
development of decision making under uncertainty. Even those who reject EU
theory use it as a standard by which to compare alternative theories.
Some researchers often restrict the EU criterion by considering a
specific subset of utility functions in order to obtain solutions to different
problems. Of course, particular utility functions have different implications.
WE have to note that utility is unique only up to a linear transformation.
We note that during the 1960s the theory of finance considered the
subset of utility functions that are quadratic of the form
u (w) = aw - w2,
for w a.
u should satisfy the requirement of not being decreasing, which is true
only when w is smaller than a.
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Above wealth level a, marginal utility becomes negative. Since
quadratic utility is decreasing in wealth for w a, many people might feel
this is not appropriate as a utility function. However, it is important that we
try to obtain models of human behavior with mathematical models. Quadratic
utility functions exhibit increasing absolute risk aversion and this is why they
are not in fashion anymore.
The second set of classical functions is the so-called constant-absoluterisk-aversion (CARA) utility functions, which are exponential functions,
whose domain is the real life. They exhibit constant absolute risk aversion
with A (w) =a for all w. The fact that risk aversion is constant is often useful in
analyzing choices among several alternatives. This assumption eliminates the
income effect when dealing with decisions to be made about a risk whose size
is invariant to changes in wealth. This is a reason for criticizing this function,
since absolute risk aversion is constant than decreasing.
Finally, the constant-relative risk-aversion (CRRA) class of preferences
is a set of power utility functions which eliminate any income effects when
making decisions about risk whose size is proportional to one’s level of wealth.
The assumption that relative risk aversion is constant simplifies enormously
many of the problems encountered in macroeconomics and finance.
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