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Rethinking Risk: Aspiration as Pure Risk Greg B Davies University College London
FUR XII
26 June 2006

1. 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
Psychophysics or probability perception
Attitudes to risk itself
Existing theories of risk attitude are inadequate
Not psychologically intuitive
Entirely derived from other effects
Only operational where outcomes are purely numerical
Require restrictive assumptions on EUT
Practical risk measures often ad hoc
More intuitively sensible
But lack firm theoretical grounding

2. Cumulative Prospect Theory Value Function Diminishing Sensitivity
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
Loss Aversion
People are far more sensitive to losses than to gains
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
Utility
Reference Point
Gains (£)
Losses (£)
Loss aversion: Steeper for losses
EU = EB[v(x)]

3. IN RANK DEPENDENT UTILITY THEORIES DECISION WEIGHTS ADD A FURTHER SOURCE OF RISK ATTITUDE
Probability Transformation Function
Principle of Attention
Diminishing sensitivity to probability away from extreme outcomes
Psychological interpretation
Optimism/Hope – Convex function
Pessimism/Fear – Concave Function 1
Underweighting of probability of middle outcomes of gamble
Weighting
“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)
Most sensitive (steepest) at extreme outcomes: probability overweighting 1
Cumulative or Decumulative Probability

4. Where are attitudes to “Pure Risk” itself?
THE BEHAVIOURAL COMPONENTS OF RISK ATTITUDE DO NOT PROVIDE A COMPLETE CHARACTERISATION
“Risk Attitude” arises from four behavioural components
Utility curvature for gains
Utility curvature for losses
Loss aversion
Distortion of probabilities
None capture intuitive concepts of “risk”
First three capture strength of preference for sure outcomes
Last captures effects of attention
All derivative on perceptions of value and probabilities
None capture the effect of introducing risk into a decision
Where are attitudes to “Pure Risk” itself?
“After all the study and all the clever theorizing, we are left with a theory of risk taking that fails to mention risk” - Lopes 1987

5. PRACTICAL RISK MEASURES REFLECT DISSATISFACTION WITH STANDARD CONCEPTS OF RISK
Standard Deviation as a risk measure
Requires either normal returns distributions or quadratic utility functions
Not intuitive: positive deviations as weigh as heavily as negative deviations
Many other measures seem more intuitively plausible
Semi-standard deviation; VaR; probability of loss; higher order stochastic dominance; etc...
But these imply highly restrictive assumptions on utility functions (if they can be reconciled with EUT at all)
Reliance on value functions leaves agents with implausible risk attitudes
We require a way to model attitudes to risk that are not simply the sum of incidental but unrelated psychophysical responses

6. INTRODUCING PURE RISK ATTITUDES: ALLOWING RISK ITSELF TO INFLUENCE UTILITY ATTRIBUTION
Example
Imagine gamble: (50%, 8 apples; 50%, 0 apples)
Risk premium is 1 apple: indifference between gamble and 3 apples for sure
If Strength of Preference for gaining 3 apples when I have 0 = that of gaining 5 apples when I have 3
Diminishing marginal utility
“Risk attitude” entirely explained without recourse to “risk”
Otherwise some aspect of the risk premium must be due to Risky Utility
Postulate that attitude to the introduction of Pure Risk exists
Primitive concept
Psychologically intuitive
Application to all decisions, not just numerical outcomes
Distinct from existing concepts
Traditional measures affect risk attitude but don’t completely describe it
Utilities that represent a “rational” preference ordering must already embody all risk attitudes…

7. PURE RISK NEUTRALITY: DECOMPOSITION OF THE PREFERENCE ORDERING TO ISOLATE PURE RISK ATTITUDE
Given a role for Pure Risk we can ask what it means to be Pure Risk Neutral
Postulate a related complete preference ordering which is neutral with regard to Pure Risk (so excludes any attitudes to Pure Risk)
PR neutral individual will have a different attribution of utilities uN
UN to form substrate for Pure Risk theory
Pure Risk Theory describes the relationship between uN and u c u
Overall preference ordering uN
Pure Risk neutral preference ordering
NB: Pure Risk by definition does not include standard risk conceptions which are included in this portion
Portion of allocation due solely to Pure Risk attitude
Pure Risk preference ordering

8. OUR INTUITIONS OF “RISK” ARE RELATED TO THE PROBABILITY OF NOT ACHIEVING ASPIRATIONS
Aspirations as Pure Risk
Minimise probability of bad outcomes
Evidence from psychology*, applied finance**, economics***
Analogous to VaR in finance
However this is an incomplete theory, which relies on an arbitrary choice of  
* Lopes 1987, Lopes and Oden 1999, Payne 1980, 1981, 2004
** Coombs 1975, Roy 1952, Mao 1970
*** Diecidue & van de Ven 2004, Camerer et al. 1997
Pure Risk Neutral Utility (uN)

9. GENERALISING ASPIRATION POINTS: MULTIPLE POINTS
Multiple Aspiration points
Survival threshold
Peer group references
Status quo
Options ranked by cumulative probability at each point
Lower Aspiration levels of greater importance?
How to generalise aspiration notion to arrive at single concept of Pure Risk? 1
 2
Pure Risk Neutral Utility (uN)

10. GENERALISING THE ASPIRATION CONCEPT: ASPIRATION WEIGHTING FUNCTION
Take two gambles f and g
If, for every possible aspiration point we have:
Pure Risk can thus be represented by a non-decreasing function over uN
Thus u=(uN)
Function governs importance of the downside uN values in choice
If importance of aspiration levels decreases with uN,  is concave
If Pure Risk decreases when a positive constant is added to every outcome,  is negative exponential
then we must prefer f to g
But this is exactly the requirement of first-order stochastic dominance applied to uN so

11. IGNORING PURE RISK ATTITUDES COULD MEAN EXISTING MODELS AND PREDICTIONS ARE BIASED
Hitherto value function v(x) used to proxy true EUT utilities u
But v(x) does not include attitudes to Pure Risk
Therefore
Estimates of v(x) parameters mis-specified
Predictions from outcomes based on v(x) inaccurate
Pure Risk Neutral Utilities
Utilities from Full Preferences
Numerical Outcomes x v uN u
Traditional functions have been trying to force v(x) to accomplish both psychophysical and Pure Risk effects

12. PURE RISK ATTITUDE CAN BE USED TO PROVIDE A MORE COMPLETE ACCOUNT OF AGENTS PREFERENCES
Instead use v(x) as proxy for uN, not u:
Thus v(x) is transformed through concave (v(x)): EU = E[(v(x))]
v(x) includes strength of preference as a source of risk attitude
Other non-Pure Risk aspects can be incorporated into uN
Social Preferences
Regret
Dynamics (eg, reference point adjustment, house money effect)
Aspiration
Decision weighting can be included when arriving at uN
Pure Risk Neutral Utilities
Utilities from Full Preferences
Numerical Outcomes x v uN u
Value function (reference point, curvature, loss aversion)
Missing components of value function?
Pure Risk Attitude

13. PURE RISK INDUCES LOSS AVERSION AND DIFFERENTIAL CURVATURE WITHOUT ASSUMING EITHER…
Adding Pure Risk Attitudes to Cumulative Prospect Theory Can Explain Observed Choice Effects With Only Two Parameters
CPT function (1 parameter)
Equal curvature for gains and losses
No loss aversion
Pure Risk and CPT (2 Params)
Losses more linear than gains
Displays loss aversion
Achieving these effects in CPT requires 4 parameters

14. SUMMARY: PURE RISK ATTITUDE AS AN ADDITIONAL COMPONENT OF COMPREHENSIVE AGENT PREFERENCES
Risk attitude now arises from
Pure Risk attitude (primitive)
Utility curvature (derived - diminishing sensitivity to monetary amounts)
Loss Aversion (derived - differential attitudes to gains vs. losses)
Probability transformation (derived - attention drawn to extreme outcomes)
Pure Risk attitude derives from a generalisation of Aspiration levels
Aspiration weighting function transforms Pure Risk neutral utilities
Function is non-decreasing, concave, negative exponential
Traditional value function may be used to proxy Pure Risk neutral utilities: EU = E[(v(x)]