1. Notes: Use this cover page for internal presentations Dynamic Reference Points: Investors as Consumers of Uncertainty 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: 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. 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 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)]

5. FUR XII Notes: Page 4 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

6. FUR XII Notes: Page 5 Copyright © 2006 THE INVERSE-S SHAPED DECISION WEIGHTING FUNCTION IS A PDF MULTIPLIER WHICH MAGNIFIES THE TAILS Multiplier for Mixed Distribution with 80% Probability of Gain Centre of distribution underweighted Tails of distribution over- weighted Break in multiplier at reference point

7. FUR XII Notes: Page 6 Copyright © 2006 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. FUR XII Notes: Page 7 Copyright © 2006 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 V 0 [f,y] Superscript is distance of current ref point from baseline (ie, a)

9. FUR XII Notes: Page 8 Copyright © 2006 IT IS USEFUL TO EXAMINE BOTH ABSOLUTE AND RELATIVE PROSPECTS AFTER SHIFTING REFERENCE POINTS Equivalent Absolute Prospect y y + a Absolute prospect f unchanged Reference shifted upwards by a Prospect value: V a [f,y+a] Equivalent Relative Prospect Prospect shifted so gains and losses from y + a are the same as from y Prospect value: V a [f+a,y+a] y y + a

10. FUR XII Notes: Page 9 Copyright © 2006 ASSUME SHAPE OF VALUE & DECISION WEIGHTING FUNCTIONS INVARIANT TO REFERENCE CHANGES Equivalent Absolute Prospect y y + a 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 Equivalent Relative Prospect Prospect value: V a [f+a,y+a] No change in the value of equivalent relative prospect: V a [f+a,y+a] = V 0 [f,y] Relies on invariant perceptual functions y y + a Outcomes that change in sign due to shift All gains & losses identical to those before shift

11. FUR XII Notes: Page 10 Copyright © 2006 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) With Loss Aversion Without Loss Aversion Loss Aversion accentuates effect of reference point shift *Value function is CARA in example v(x) – v(x-a) always negative for a>0 Therefore: V a [f,y+a] 0

12. FUR XII Notes: Page 11 Copyright © 2006 20% probability of loss vs 50% probability of loss 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%) Decision weights decrease as reference point shifts up Decision weights unchanged THESE CHANGES ALTER THE WEIGHT GIVEN TO OUTCOMES THAT CHANGE SIGN BUT V a [f,y+a] 0 HOLDS

13. FUR XII Notes: Page 12 Copyright © 2006 THUS, WHERE PERCEPTUAL FUNCTIONS ARE INVARIANT TO REFERENCE POINT SHIFTS, WE HAVE Equivalent Absolute Prospect y y + a Prospect value decreases for a>0 and increases for a<0 Same as equivalent reduction of absolute outcomes: V a [f,y+a] = V 0 [f-a,y] No change in value: V a [f+a,y+a] = V 0 [f,y] for any a Equivalent Relative Prospect y y + a

14. FUR XII Notes: Page 13 Copyright © 2006 THIS CAN PROVIDE AN ACCCOUNT OF THE HOUSE MONEY EFFECT Original Prospect y Value: V 0 [f,y] Assume decision making experiences a gain of a>0 But reference point does not immediately adjust to reflect new wealth y y + a y Same Relative Bet after Gain of a>0 After Reference Point Adjusts to Gain Prospect shifts up by a Gain not absorbed into reference point Outcomes perceived as f+a Value: V 0 [f+a,y] = V a [f,y-a] Value increases from original prospect Greater risk taking Gain absorbed into new reference point Value: V a [f+a,y+a] This is equivalent relative bet V a [f+a,y+a] = V 0 [f,y]

15. FUR XII Notes: Page 14 Copyright © 2006 THE VALUE OF THE BET WILL CHANGE DYNAMICALLY AS THE REFERENCE POINT GRADUALLY ADJUSTS Value of Relative Prospect Value (Invariant Perceptual Functions) Baseline value V 0 [f,y] = V a [f+a,y+a] Full “House Money” value V 0 [f+a,y] = V a [f,y-a] Gradual absorbtion of gain into reference point Time

16. FUR XII Notes: Page 15 Copyright © 2006 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 V a [f+a,y+a] > V 0 [f,y] for a>0

17. FUR XII Notes: Page 16 Copyright © 2006 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: V a [f,y+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: V a [f,y+a] ≠ V 0 [f-a,y]

18. FUR XII Notes: Page 17 Copyright © 2006 AS BEFORE, THE PATH WILL BE DETERMINED BY EFFECT OF ADDITIONAL WEALTH ON PERCEPTUAL FUNCTIONS Value of Relative Prospect Possible Value Paths with Variable Perceptual Functions? Time Value of Relative Prospect Value Path (Invariant Perceptual Functions) Baseline value V 0 [f,y] = V a [f+a,y+a] Full “House Money” value V 0 [f+a,y] = V a [f,y-a] Time Baseline value V 0 [f,y] ≠ V a [f+a,y+a] DARA holds V a [f+a,y+a] > V 0 [f,y] Full “House Money” value different from equivalent absolute prospect after shift: V 0 [f+a,y] ≠ V a [f,y-a] DARA doesn’t hold

19. FUR XII Notes: Page 18 Copyright © 2006 SUMMARY OF REFERENCE POINT SHIFTS Change in Prospect Evaluation V 0 [f,y] After Reference Point Shift of a>0 Perceptual Functions Invariant to Reference Shifts Perceptual Functions Change with Reference Shifts Equivalent Relative Prospect V a [f+a,y+a] No Change V a [f+a,y+a] = V 0 [f,y] Indeterminate, but increase in value if DARA V a [f+a,y+a] > V 0 [f,y] Equivalent Absolute Prospect V a [f,y+a] Decrease in Value V a [f,y+a] < V 0 [f,y] Indeterminate, but likely still to decrease except for strong value function change effects House Money Value V 0 [f-a,y] Identical to Equivalent Absolute Prospect V 0 [f-a,y] = V a [f,y+a] No change, but no longer identical to Equivalent Absolute Prospect V 0 [f-a,y] ≠ V a [f,y+a]

20. FUR XII Notes: Page 19 Copyright © 2006 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