notes: use this cover page for internal presentations dynamic reference points: investors as...

Dynamic Reference Points: Investors as Consumers of Uncertainty

Greg B [email protected] University College London

FUR XII24 June 2006

FUR XII 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…


• 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


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] = EB[v(x)]


Cumulative or Decumulative Probability

00 11

11

Wei

gh

tin

g

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


THE INVERSE-S SHAPED DECISION WEIGHTING FUNCTION IS A PDF MULTIPLIER WHICH MAGNIFIES THE TAILS

10.750.50.250

5

4

3

2

1

0

F(x)

Weight

Multiplier for Mixed Distribution with 80% Probability of Gain

Centre of distribution underweighted

Tails of distribution over-weighted

Break in multiplier at reference point


10.750.50.250

5

4

3

2

1

0

F(x)

Weight

FOR MIXED DISTRIBUTIONS WE SPLICE THE MULITIPLIERS FOR GAINS AND LOSSES

10.750.50.250

8

6

4

2

0

Cumulative Probability - F(x)

Pdf Multiplier


Pdf Multiplier

10.750.50.250

10

8

6

4

2

0


Pdf Multiplier


Pdf Multiplier

Multiplier for Gains Only Distribution

Multiplier for Mixed Distribution with 80% Probability of Gain

Multiplier for Loss Only Distribution


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)


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: Va[f,y+a]

Equivalent Relative Prospect

• 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


ASSUME SHAPE OF VALUE & DECISION WEIGHTING FUNCTIONS INVARIANT TO REFERENCE CHANGES


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


• 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

y y + a

Outcomes that change

in sign due to shift

All gains & losses

identical to those before

shift


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: Va[f,y+a] < V0[f,y] for a>0


10.750.50.250

F(x)

Weight

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

Decision weights

unchanged

THESE CHANGES ALTER THE WEIGHT GIVEN TO OUTCOMES THAT CHANGE SIGN BUT Va[f,y+a] < V0[f,y] for a>0 HOLDS


THUS, WHERE PERCEPTUAL FUNCTIONS ARE INVARIANT TO REFERENCE POINT SHIFTS, WE HAVE


y y + 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]

• No change in value:

Va[f+a,y+a] = V0[f,y] for any a


y y + a


THIS CAN PROVIDE AN ACCCOUNT OF THE HOUSE MONEY EFFECT

Original Prospect

y

• Value: V0[f,y]• Assume decision making

experiences a gain of a>0• But reference point does not

immediately adjust to reflect

new wealth

y y + ay

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: V0[f+a,y] = Va[f,y-a]• Value increases from original

prospect• Greater risk taking

• 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]


THE VALUE OF THE BET WILL CHANGE DYNAMICALLY AS THE REFERENCE POINT GRADUALLY ADJUSTS

Val

ue

of

Re

lati

ve P

rosp

ect

Value (Invariant Perceptual Functions)

Baseline valueV0[f,y] = Va[f+a,y+a]

Full “House Money” value V0[f+a,y] = Va[f,y-a]

Gradual absorbtion of

gain into reference point

Time


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


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]


AS BEFORE, THE PATH WILL BE DETERMINED BY EFFECT OF ADDITIONAL WEALTH ON PERCEPTUAL FUNCTIONS

Val

ue

of

Re

lati

ve P

rosp

ect

Possible Value Paths with Variable Perceptual Functions?

Time

Val

ue

of

Re

lati

ve P

rosp

ect

Value Path(Invariant Perceptual Functions)

Baseline valueV0[f,y] = Va[f+a,y+a]

Full “House Money” value

V0[f+a,y] = Va[f,y-a]

Time

Baseline valueV0[f,y] ≠ Va[f+a,y+a]

DARA holds Va[f+a,y+a] > V0[f,y]

Full “House Money” value different from equivalent absolute prospect after shift:V0[f+a,y] ≠ Va[f,y-a]

DARA doesn’t hold


SUMMARY OF REFERENCE POINT SHIFTS

Change in Prospect Evaluation V0[f,y] After

Reference Point Shift of a>0

Perceptual Functions Invariant to Reference

Shifts

Perceptual Functions Change with Reference

Shifts


Va[f+a,y+a]

No Change

Va[f+a,y+a] = V0[f,y]

Indeterminate, but increase in value if DARA

Va[f+a,y+a] > V0[f,y]


Va[f,y+a]

Decrease in Value

Va[f,y+a] < V0[f,y]

Indeterminate, but likely still to decrease except for strong value function change effects

House Money Value

V0[f-a,y]

Identical to Equivalent Absolute Prospect

V0[f-a,y] = Va[f,y+a]

No change, but no longer identical to Equivalent

Absolute Prospect

V0[f-a,y] ≠ Va[f,y+a]


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

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