The Behavioural Components Of Risk
Aversion
FUR XII
24 June 2006
Greg B Davies
g.b.davies.97@cantab.net
University College London
INTRODUCTION: THERE IS MORE TO RISK ATTITUDE THAN
DIMINISHING MARGINAL UTILITY
•
Traditional economic theory has had a particularly simple view of risk attitude
– Based on Expected Utility Theory
– Identified with diminishing marginal utility for wealth
•
No recognition of psychology
– Psychophysics of value: different curves for gains and losses
– Loss aversion
– Psychophysics of probability perception
•
We analyse the risk premium in a CPT framework and break overall risk attitude
down into the underlying behavioural components
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Page 1
EXPECTED UTILITY THEORY: THE “RATIONAL” STANDARD
• Individuals always choose the option
with the highest expected utility:
EU = E[v(x)]
Value Function
Utility
• Assumes utility is a function of
wealth
– Often diminishing marginal
returns (implied risk aversion)
– Underlying function is stable
– Options can be evaluated
independently
Increases in
Utility get slower
as wealth
increases
• Individuals accurately use subjective
assessments of probability
FUR XII
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Total Wealth (£)
Page 2
RISK ATTITUDE MAY BE MEASURED BY THE RISK PREMIUM
• Risk premium: difference in utility
between holding the gamble, and
holding the EV of the gamble for
sure:
v(E[x] - rp) = E[v(x)]
Value Function
Utility
Utility of
EV
• Risk premium always positive for a
concave value function
• Positive risk premium indicates Risk
Aversion
Risk
Premium
Utility of
Gamble
• Requires gamble outcomes to be
defined on single numerical scale
EV of Gamble
FUR XII
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Total Wealth (£)
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RESULTS FROM EXPERIMENTAL PSYCHOLOGY SUGGEST A
VERY DIFFERENT VALUE FUNCTION
Cumulative Prospect Theory
Value Function
Reference Points
• People evaluate utility as gains or
losses from a reference point not
relative to total wealth
Utility
Loss Aversion
• People are far more sensitive to
losses than to gains
Reference
Point
Gains (£)
Losses (£)
Diminishing Sensitivity
• Weber/Fechner law away from
reference point
• Risk seeking behaviour for losses
Status Quo Bias/Endowment Effect
• People demand more to give up an
object than they are willing to pay
FUR XII
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Loss
aversion:
steeper
for losses
V[f] = EB[v(x)]
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IN RANK DEPENDENT UTILITY THEORIES DECISION
WEIGHTS ADD A FURTHER SOURCE OF RISK ATTITUDE
Probability Transformation
Function
1
Underweighting of
probability of middle
outcomes of gamble
Weighting
• Principle of Attention
– Diminishing sensitivity to
probability away from extreme
outcomes
• Psychological interpretation
– Optimism/Hope – Convex
function
– Pessimism/Fear – Concave
Function
“The attention given to an outcome
depends not only on the probability
of the outcomes but also on the
favourability of the outcome in
comparison to the other possible
outcomes” - Diecidue and Wakker
(2001)
FUR XII
Most sensitive
(steepest) at extreme
outcomes: probability
overweighting
0
Copyright © 2006
Cumulative or Decumulative
Probability
1
Page 5
THE RISK PREMIUM MAY BE ANALYSED IN THE FRAMWORK
PROVIDED BY CPT…
• The concept of risk premium may be applied to the CPT framework
– CPT valuation of prospect f is given by V[f]
– CPT value function given by v(x)
• Standard CPT Risk Premium rCPT:
– v(E[f] - rCPT) = V[f]
– Certain amount that would make the decision maker indifferent between the
prospect and the expected value minus the risk premium
– Shows the degree of risk aversion individuals believe themselves to have
• Behavioural Risk Premium rB:
– v(EB[f] - rB) = V[f]
– EB[f] is the Behavioural Expected Value that takes decision weight distortions
into account
– Shows the degree of risk aversion individuals will demonstrate by their
behaviour
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WE MAY APPROXIMATE THE DEGREE OF LOCAL RISK
AVERSION USING PRATT’S METHODOLOGY
• Pratt-Arrow risk premium
– Shows how local risk attitude is affected by the curvature of the EUT value
function
– rp = -σ2v’’(x)/2v’(x)
• We use Pratt’s methodology to get local approximations for the CPT risk premia at
the reference point with no decision weights (at first)
• Standard Pratt-Arrow risk premium is a special case of CPT risk premium at
reference point under three conditions
– Slope of value function at reference point the same for gains and losses
– Curvature at reference point the same for gains and losses
– Outcome distribution is symmetrical at reference point
• Away from the reference point the CPT risk premium is the same as Pratt-Arrow
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THE CPT RISK PREMIUM IS MADE UP OF TWO COMPONENTS
REPRESENTING CURVATURE AND LOSS AVERSION
• rCPT is made up of two terms:
1. Curvature component: analogous to Pratt-Arrow, but numerator is a weighted
average of σ2v’’(x) taken above and below the reference point, where the
weights are probability of a loss and of a gain
2. Loss Aversion Component: first order effect of loss aversion always
increases risk aversion
• Concavity of both gains and losses is not necessary to ensure risk aversion:
convexity for losses is consistent with risk aversion as long as the value function
for gains is sufficiently concave
• Loss aversion has second order effect through affecting the slope of the loss
value function – if it gets too high this can dominate and reduce risk aversion
• Adding decision weights makes the two components much more complicated but
does not add an additional component
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Page 8
AN EXAMPLE USING S&P 500 RETURNS ILLUSTRATES THE
EFFECT OF CPT PARAMETERS ON THE RISK PREMIUM
rCPT for Different Curvatures of
Gain and Loss Value Functions
Loss
Convexity
rCPT for Different Decision
Weighting and Loss Aversion
Gain
Concavity
Decision
Weighting
Loss Aversion
1 – no weighting
<1 – Inverse-S
>1 – S-Shaped
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INDIVIDUALS MAY BELIEVE THEMSELVES TO BE RISK AVERSE
BUT YET BEHAVE AS A RISK SEEKER
• The difference between the CPT risk premium and the behavioural risk premium
is the Attitudinal Premium (AP)
AP = rCPT – rB = E[f] – EB[f]
• AP shows the difference between individuals’ beliefs of their own risk aversion,
and the risk aversion imputed from their behaviour
CPT vs Behavioural Risk Aversion (Illustrative Example: S&P 500 Returns)
CPT Risk Premium
Inverse-S shaped
decision weighting
curve: People
believe themselves
to be more risk
averse than they
actually behave
Behavioural Risk
Premium
People think they
are risk seeking,
but are actually risk
averse…
S shaped decision
weighting curve:
People believe
themselves to be
less risk averse
than they actually
behave
Decision Weighting Parameter
FUR XII
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