”Risk Perception and Utility Curves Aggregation”

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Behavioural Finance A 3D model Measure ofθ,λ,γ The Joint Formula Main result

”Risk Perception and Utility Curves Aggregation”

Cyril Lanquetuit

Avril 2019

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1 Behavioural Finance

2 A 3D model

3 Measure of _θ,λ,γ

4 The Joint Formula

Main result

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Behavioural Finance

”We are not Econs” R.H.Thaler (Nobel Prize 2017)

”Prospect Theory” Kahnman & Tversky (1979)

”Psychological Fundamentale Rule” J.M.Keynes (1936)

”Risk, Uncertainty and Profit” F.H.Knight (1921)

”La main invisible” A.Smith (1776)

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Behavioural Finance

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Behavioural Finance

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Behavioural Finance

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Behavioural Finance

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Concave Utility Function

”Back to Bentham” Kahnman, Wakker, Sarin (1787 idea)

The first euro you win makes you happier than the next one Could we aggregate two concave utility curves ?

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Concave Utility Function

”Back to Bentham” Kahnman, Wakker, Sarin (1787 idea) The first euro you win makes you happier than the next one

Could we aggregate two concave utility curves ?

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Concave Utility Function

”Back to Bentham” Kahnman, Wakker, Sarin (1787 idea) The first euro you win makes you happier than the next one Could we aggregate two concave utility curves ?

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Litterature

Allais paradoxe : (1953)

A) 100e100% >B) 150e90% & C) 100e10%<D) 150e9%

Kahneman & Tversky : prospect & cumulative theory (1978, 1992)

prelec formula w(p;_γ)₌ ^p^γ

(p^γ₊₍1₋p)^γ)^γ¹

P.P.Wakker : Curved Value Function (CRRA)

”Explaining the caracteristics of the power utility family” (2008)

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Litterature

A) 100e100% >B) 150e90% & C) 100e10%<D) 150e9%

prelec formula w(p;_γ)= ^p^γ

(p^γ₊₍1₋p)^γ)^γ¹

”Explaining the caracteristics of the power utility family” (2008)

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Litterature

A) 100e100% >B) 150e90% & C) 100e10%<D) 150e9%

prelec formula w(p;_γ)= ^p^γ

(p^γ₊₍1₋p)^γ)^γ¹

”Explaining the caracteristics of the power utility family”

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”We are not Econs”*

Curved Value Function (CRRA, Von Newman Utility)

Loss Aversion : |U(−x)| >U(x) Probability Weighting (Prelec)

Other eluded bias : Narrow Framing, Reference Return

*R.Thaler (2017 Economy Nobel Prize)

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”We are not Econs”*

Curved Value Function (CRRA, Von Newman Utility) Loss Aversion : |U(−x)| >U(x)

Probability Weighting (Prelec)

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”We are not Econs”*

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”We are not Econs”*

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Utility function

Pw(p;_γ)⁽¹⁺^r₁⁾_−θ¹^−θ⁻¹(1₊_λ1r_>0)

Proba Weigt-CRRA-Loss Aversion γ− − − − − − −θ− − − − − − −λ

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Utility function

Pw(p;_γ)⁽¹⁺^r₁⁾_−θ¹^−θ⁻¹(1₊_λ1r_>0) Proba Weigt-CRRA-Loss Aversion

γ− − − − − − −θ− − − − − − −λ

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Utility function

Pw(p;_γ)⁽¹⁺^r₁⁾_−θ¹^−θ⁻¹(1₊_λ1r_>0) Proba Weigt-CRRA-Loss Aversion γ− − − − − − −θ− − − − − − −λ

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How to measure _θ , λ , γ ?

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How to measure _θ , λ , γ ?

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How to measure _θ , λ , γ ?

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3D Comb

θ risk aversion λloss aversion γprobability deformation

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Orthogonality

Trajectory : horizontal Wheel : slowly

decreasing

Histogram : highly decreasing

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Classification

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Theta - Lambda determination

Trajectory, Wheel, Histogram : decreasing slope

Better _θ,λdetermination

Higher risk profile knowledge precision

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Theta - Lambda determination

Trajectory, Wheel, Histogram : decreasing slope Better _θ,λdetermination

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Theta - Lambda determination

Trajectory, Wheel, Histogram : decreasing slope Better _θ,λdetermination

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Risk Aversion : Theta from 3 to 33

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Curves aggregation outlook

What are we looking for ?

How to aggregate two utility curves ?

Given two curves, how would the aggregation looks like ?

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Curves aggregation outlook

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Curves aggregation outlook

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Artsy (but rigorous !) investigation

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The Joint Formula

h(^λ₂)=^f⁽^λ^p⁾⁺^g₂⁽^λ⁽¹⁻^p⁾⁾

h(1−^λ₂)=^f⁽¹^−λ⁽¹⁻^p₂⁾⁾⁺^g⁽¹^−λ^p⁾

with 0≤p≤¹₂≤1−p≤1 and 0≤λ≤1

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The Joint Formula

h(^λ₂)=^f⁽^λ^p⁾⁺^g₂⁽^λ⁽¹⁻^p⁾⁾

h(1−^λ₂)=^f⁽¹^−λ⁽¹⁻^p₂⁾⁾⁺^g⁽¹^−λ^p⁾

with 0≤p≤¹₂≤1−p≤1 and 0≤λ≤1

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The Joint Formula

h(^λ₂)=^f⁽^λ^p⁾⁺^g₂⁽^λ⁽¹⁻^p⁾⁾

h(1−^λ₂)=^f⁽¹^−λ⁽¹⁻^p₂⁾⁾⁺^g⁽¹^−λ^p⁾

with 0≤p≤¹₂≤1−p≤1 and 0≤λ≤1

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Experimentation

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First results

We seem to have_θ₁_≤_θ_couple_≤_θ₂

We have not enough data yet to compute the real ratio In the Joint Formula, ¹₂ looks not so bad

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First results

We have not enough data yet to compute the real ratio

In the Joint Formula, ¹₂ looks not so bad

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First results

We have not enough data yet to compute the real ratio In the Joint Formula, ¹₂ looks not so bad

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Conclusion

The Joint Formula seems to be a good candidate for h₌f&g

Under perfect symetry the ¹₂ ratio suits well

In reality weighted centro¨ıd instead of iso-centro¨ıd : 0_≤_λ_≤1

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Conclusion

The Joint Formula seems to be a good candidate for h₌f&g Under perfect symetry the ¹₂ ratio suits well

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Conclusion

The Joint Formula seems to be a good candidate for h₌f&g Under perfect symetry the ¹₂ ratio suits well

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Profil aggregation

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Questions et Critiques constructives

MERCI Questions ? ? ?

et apr`es... ?

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Curves aggregation

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https ://videotheque.u-

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”Risk Perception and Utility Curves Aggregation”