FUR XII 24 June 2006

1. FUR XII 24 June 2006 Use this cover page for internal presentations The Behavioural Components Of Risk Aversion Greg B Daviesg.b.davies.97@cantab.netUniversity College London

2. 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

3. Utility Increases in Utility get slower as wealth increases EXPECTED UTILITY THEORY: THE “RATIONAL” STANDARD Value Function • Individuals always choose the option with the highest expected utility: EU = E[v(x)] • Assumes utility is a function of wealth • Often diminishing marginal returns (implied risk aversion) • Underlying function is stable • Options can be evaluated independently • Individuals accurately use subjective assessments of probability Total Wealth (£)

4. Value Function Utility Utility of EV Utility of Gamble EV of Gamble Total Wealth (£) 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)] • Risk premium always positive for a concave value function • Positive risk premium indicates Risk Aversion • Requires gamble outcomes to be defined on single numerical scale Risk Premium

5. 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 RESULTS FROM EXPERIMENTAL PSYCHOLOGY SUGGEST A VERY DIFFERENT VALUE FUNCTION Cumulative Prospect Theory Value Function Utility Reference Point Gains (£) Losses (£) Loss aversion: steeper for losses V[f] = EB[v(x)]

6. Probability Transformation Function 1 Weighting 0 1 Cumulative or Decumulative Probability IN RANK DEPENDENT UTILITY THEORIES DECISION WEIGHTS ADD A FURTHER SOURCE OF RISK ATTITUDE • Principle of Attention • Diminishing sensitivity to probability away from extreme outcomes • Psychological interpretation • Optimism/Hope – Convex function • Pessimism/Fear – Concave Function Underweighting of probability of middle outcomes of gamble “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

7. 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

8. 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

9. THE CPT RISK PREMIUM IS MADE UP OF TWO COMPONENTS REPRESENTING CURVATURE AND LOSS AVERSION • rCPT is made up of two terms: • 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 • 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

10. AN EXAMPLE USING S&P 500 RETURNS ILLUSTRATES THE EFFECT OF CPT PARAMETERS ON THE RISK PREMIUM rCPT for Different Decision Weighting and Loss Aversion rCPT for Different Curvatures of Gain and Loss Value Functions Gain Concavity Loss Convexity Loss Aversion Decision Weighting 1 – no weighting <1 – Inverse-S >1 – S-Shaped

11. 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 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 Inverse-S shaped decision weighting curve: People believe themselves to be more risk averse than they actually behave Behavioural Risk Premium Decision Weighting Parameter