1. Notes: Use this cover page for internal presentations The Behavioural Components Of Risk Aversion Greg B Davies g.b.davies.97@cantab.net University College London FUR XII 24 June 2006

2. FUR XII Notes: Page 1 Copyright © 2006 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. FUR XII Notes: Page 2 Copyright © 2006 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 (£) Utility Value Function Increases in Utility get slower as wealth increases EXPECTED UTILITY THEORY: THE “RATIONAL” STANDARD

4. FUR XII Notes: Page 3 Copyright © 2006 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 ATTITUDE MAY BE MEASURED BY THE RISK PREMIUM Total Wealth (£) Utility Value Function EV of Gamble Utility of Gamble Utility of EV Risk Premium

5. FUR XII Notes: Page 4 Copyright © 2006 Cumulative Prospect Theory Value Function Losses (£) Utility Loss aversion: steeper for losses Reference Point Gains (£) RESULTS FROM EXPERIMENTAL PSYCHOLOGY SUGGEST A VERY DIFFERENT 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 V[f] = E B [v(x)]

6. FUR XII Notes: Page 5 Copyright © 2006 Cumulative or Decumulative Probability 01 1 Weighting Probability Transformation Function 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 “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) Underweighting of probability of middle outcomes of gamble Most sensitive (steepest) at extreme outcomes: probability overweighting

7. FUR XII Notes: Page 6 Copyright © 2006 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 r CPT : –v(E[f] - r CPT ) = 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 r B : –v(E B [f] - r B ) = V[f] –E B [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 THE RISK PREMIUM MAY BE ANALYSED IN THE FRAMWORK PROVIDED BY CPT…

8. FUR XII Notes: Page 7 Copyright © 2006 Pratt-Arrow risk premium –Shows how local risk attitude is affected by the curvature of the EUT value function –rp = -σ 2 v’’(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 WE MAY APPROXIMATE THE DEGREE OF LOCAL RISK AVERSION USING PRATT’S METHODOLOGY

9. FUR XII Notes: Page 8 Copyright © 2006 r CPT is made up of two terms: 1.Curvature component: analogous to Pratt-Arrow, but numerator is a weighted average of σ 2 v’’(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 THE CPT RISK PREMIUM IS MADE UP OF TWO COMPONENTS REPRESENTING CURVATURE AND LOSS AVERSION

10. FUR XII Notes: Page 9 Copyright © 2006 AN EXAMPLE USING S&P 500 RETURNS ILLUSTRATES THE EFFECT OF CPT PARAMETERS ON THE RISK PREMIUM r CPT for Different Curvatures of Gain and Loss Value Functions Loss Convexity Gain Concavity r CPT for Different Decision Weighting and Loss Aversion Decision Weighting 1 – no weighting <1 – Inverse-S >1 – S-Shaped Loss Aversion

11. FUR XII Notes: Page 10 Copyright © 2006 The difference between the CPT risk premium and the behavioural risk premium is the Attitudinal Premium (AP) AP = r CPT – r B = E[f] – E B [f] AP shows the difference between individuals’ beliefs of their own risk aversion, and the risk aversion imputed from their behaviour INDIVIDUALS MAY BELIEVE THEMSELVES TO BE RISK AVERSE BUT YET BEHAVE AS A RISK SEEKER CPT vs Behavioural Risk Aversion (Illustrative Example: S&P 500 Returns) CPT Risk Premium Behavioural Risk Premium Decision Weighting Parameter Inverse-S shaped decision weighting curve: People believe themselves to be more risk averse than they actually behave S shaped decision weighting curve: People believe themselves to be less risk averse than they actually behave People think they are risk seeking, but are actually risk averse…