FUR XII 24 June 2006

1. FUR XII 24 June 2006 Use this cover page for internal presentations Dynamic Reference Points: Investors as Consumers of UncertaintyGreg B Daviesg.b.davies.97@cantab.netUniversity College London

2. INTRODUCTION: WHAT HAPPENS WHEN RDU REFERENCE POINTS ARE SHIFTED? • Since the advent of Prospect Theory reference dependent choice models have become very popular • However, there is little discussion of what predictions can be made as the reference point shifts • Notation in which Prospect Theory is presented cannot cope with such issues • “House Money Effect” suggests that reference point updating is not instantaneous • Lack of both theory and empirical evidence examining the response of risk attitudes to past gains and losses THIS PAPER DEVELOPS A VERSION OF CPT WHICH CAN ACCOMMODATE DYNAMIC REFERENCE POINTS…

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. 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)]

5. 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

6. THE INVERSE-S SHAPED DECISION WEIGHTING FUNCTION IS A PDF MULTIPLIER WHICH MAGNIFIES THE TAILS Multiplier for Mixed Distribution with 80% Probability of Gain Tails of distribution over-weighted Break in multiplier at reference point Centre of distribution underweighted

7. FOR MIXED DISTRIBUTIONS WE SPLICE THE MULITIPLIERS FOR GAINS AND LOSSES Multiplier for Gains Only Distribution Multiplier for Mixed Distribution with 80% Probability of Gain Multiplier for Loss Only Distribution

8. STANDARD CPT NOTATION TELLS US NOTHING ABOUT HOW VALUATIONS CHANGE AS THE REFERENCE POINT SHIFTS Standard CPT Notation • Assume : • Wealth: y=100 • Prospect f • Outcomes x coded as gains and losses from y • Absolute outcomes: f=y+x • Changing reference point to z=105 requires recoding all outcomes as x’=x-5 • But, useful to be able to maintain consistent outcome coding Dynamic CPT • Define y as the baseline reference point • All outcomes coded relative to y • We examine what happens as reference point shifts by a=z–x for both absolute and relative prospects • Denote value of original prospect as V0[f,y] • Superscript is distance of current ref point from baseline (ie, a)

9. y y + a IT IS USEFUL TO EXAMINE BOTH ABSOLUTE AND RELATIVE PROSPECTS AFTER SHIFTING REFERENCE POINTS Equivalent Absolute Prospect Equivalent Relative Prospect • Absolute prospect f unchanged • Reference shifted upwards by a • Prospect value: Va[f,y+a] • Prospect shifted so gains and losses from y + a are the same as from y • Prospect value: Va[f+a,y+a] y y + a

10. y y y + a y + a ASSUME SHAPE OF VALUE & DECISION WEIGHTING FUNCTIONS INVARIANT TO REFERENCE CHANGES Equivalent Absolute Prospect Equivalent Relative Prospect All gains & losses identical to those before shift Outcomes that change in sign due to shift • Change in prospect value after shift determined by: • Change in outcome values: x replaced by (x–a) • Change in decision weights for outcomes that change from gain to loss • Prospect value: Va[f+a,y+a] • No change in the value of equivalent relative prospect: Va[f+a,y+a] = V0[f,y] • Relies on invariant perceptual functions

11. INCREASING REFERENCE POINT ALWAYS DECREASES VALUE OF OUTCOMES FOR EQUIV ABSOLUTE PROSPECT Effect of Reference Point Increase on Outcome Values*(Shift reference point up by a=1) Without Loss Aversion With Loss Aversion v(x) – v(x-a) always negative for a>0 Therefore: Va[f,y+a] < V0[f,y] for a>0 Loss Aversion accentuates effect of reference point shift *Value function is CARA in example

12. WITH PERCEPTUAL INVARIANCE DECISION WEIGHTS ONLY CHANGE FOR OUTCOMES THAT CHANGE IN SIGN Effect of Reference Point Increase on Decision Weights(Assume upward shift by a=1 changes probability of loss from 20% to 50%) 20% probability of loss vs 50% probability of loss Decision weights unchanged Decision weights unchanged Decision weights decrease as reference point shifts up THESE CHANGES ALTER THE WEIGHT GIVEN TO OUTCOMES THAT CHANGE SIGN BUT Va[f,y+a] < V0[f,y] for a>0 HOLDS

13. Equivalent Relative Prospect y y + a y y + a THUS, WHERE PERCEPTUAL FUNCTIONS ARE INVARIANT TO REFERENCE POINT SHIFTS, WE HAVE Equivalent Absolute Prospect • No change in value: Va[f+a,y+a] = V0[f,y] for any a • Prospect value decreases for a>0 and increases for a<0 • Same as equivalent reduction of absolute outcomes: Va[f,y+a] = V0[f-a,y]

14. y y THIS CAN PROVIDE AN ACCCOUNT OF THE HOUSE MONEY EFFECT After Reference Point Adjusts to Gain Same Relative Bet after Gain of a>0 Original Prospect Prospect shifts up by a y y + a • Gain absorbed into new reference point • Value: Va[f+a,y+a] • This is equivalent relative bet • Va[f+a,y+a] = V0[f,y] • Gain not absorbed into reference point • Outcomes perceived as f+a • Value: V0[f+a,y] = Va[f,y-a] • Value increases from original prospect • Greater risk taking • Value: V0[f,y] • Assume decision making experiences a gain of a>0 • But reference point does not immediately adjust to reflect new wealth

15. THE VALUE OF THE BET WILL CHANGE DYNAMICALLY AS THE REFERENCE POINT GRADUALLY ADJUSTS Value (Invariant Perceptual Functions) Full “House Money” valueV0[f+a,y] = Va[f,y-a] Gradual absorbtion of gain into reference point Value of Relative Prospect Baseline valueV0[f,y] = Va[f+a,y+a] Time

16. BUT THE PERCEPTUAL FUNCTIONS WILL CHANGE WITH WEALTH, THE OVERALL EFFECT IS INDETERMINATE • Both the value and decision weighting functions may be expected to change as the reference point shifts: • Equivalent relative prospects will not be valued identically after reference shifts • Assuming Decreasing Absolute Risk Aversion (DARA) implies that relative prospects should increase with wealth: “An individual’s attitude to money, say, could be described by a book, where each page presents the value function for changes at a particular asset position. Clearly the value functions described on different pages are not identical: they are likely to become more linear with increases in assets” Kahneman & Tversky 1992 Va[f+a,y+a] > V0[f,y]for a>0

17. WHERE THE PERCEPTUAL FUNCTIONS CHANGE WITH WEALTH, THE OVERALL EFFECT IS INDETERMINATE • Hypotheses about the effects of increasing wealth: • Value function becomes more linear (decreasing diminishing sensitivity) • Decision weights more linear (less distortion of attention due to hope and fear) • Lower loss aversion • Only lower loss aversion necessarily increases the value of all prospects • Effect of perceptual function changes on equivalent absolute prospects: • If changes in shape are slight, upward reference point shift will still decrease prospect value: Va[f,y+a] < V0[f,y]for a>0 • But very strong wealth induced changes in value function could reverse this • Increase in reference point no longer equivalent to reduction in absolute outcomes: Va[f,y+a] ≠ V0[f-a,y]

18. AS BEFORE, THE PATH WILL BE DETERMINED BY EFFECT OF ADDITIONAL WEALTH ON PERCEPTUAL FUNCTIONS Possible Value Paths with Variable Perceptual Functions? Value Path(Invariant Perceptual Functions) Full “House Money” valueV0[f+a,y] = Va[f,y-a] Full “House Money” value different from equivalent absolute prospect after shift:V0[f+a,y] ≠ Va[f,y-a] DARA holdsVa[f+a,y+a] > V0[f,y] Value of Relative Prospect Value of Relative Prospect Baseline valueV0[f,y] ≠ Va[f+a,y+a] DARA doesn’t hold Baseline valueV0[f,y] = Va[f+a,y+a] Time Time

19. SUMMARY OF REFERENCE POINT SHIFTS

20. CONCLUSIONS… • CPT notation has been expanded to be capable of accounting for the effects of reference point shifts • Need distinction between equivalent absolute prospects and equivalent relative prospects • The House Money effect is analysable in this dynamic CPT framework • Prospect values change due to • Changes in outcome values and decision weights in evaluation • Wealth dependent changes in the functions themselves • Second half of the paper is not presented here • Combines this dynamic CPT framework with riskless consumer theory • Uses evidence from empirically observed endowment effects to place initial restrictions on changes in perceptual functions