Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A...
Transcript of Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A...
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Mathematical Behavioural Finance A Mini Course
Mathematical Behavioural FinanceA Mini Course
Xunyu Zhou
January 2013 Winter School @ Lunteren
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Mathematical Behavioural Finance A Mini Course
Overview of This Course
Chapter 1: Introduction
![Page 3: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/3.jpg)
Mathematical Behavioural Finance A Mini Course
Overview of This Course
Chapter 1: Introduction
Chapter 2: Portfolio Choice under RDUT - QuantileFormulation
![Page 4: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/4.jpg)
Mathematical Behavioural Finance A Mini Course
Overview of This Course
Chapter 1: Introduction
Chapter 2: Portfolio Choice under RDUT - QuantileFormulation
Chapter 3: Market Equilibrium and Asset Pricing under RDUT
![Page 5: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/5.jpg)
Mathematical Behavioural Finance A Mini Course
Overview of This Course
Chapter 1: Introduction
Chapter 2: Portfolio Choice under RDUT - QuantileFormulation
Chapter 3: Market Equilibrium and Asset Pricing under RDUT
Chapter 4: Portfolio Choice under CPT
![Page 6: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/6.jpg)
Mathematical Behavioural Finance A Mini Course
Chapter 1: Introduction
![Page 7: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/7.jpg)
Mathematical Behavioural Finance A Mini Course
1 Expected Utility Theory
2 Expected Utility Theory Challenged
3 Alternative Theories for Risky Choice
4 Summary and Further Readings
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Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Section 1
Expected Utility Theory
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Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Evaluation of Future Cash Flow
Future cash flow X : Random variable or lottery or prospecte.g. X = (110, 60%; 90, 40%)
![Page 10: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/10.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Evaluation of Future Cash Flow
Future cash flow X : Random variable or lottery or prospecte.g. X = (110, 60%; 90, 40%)
How to compare random variables?
![Page 11: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/11.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Evaluation of Future Cash Flow
Future cash flow X : Random variable or lottery or prospecte.g. X = (110, 60%; 90, 40%)
How to compare random variables?
Expected value or mean E[X ]: 110× 60% + 90× 40% = 102
![Page 12: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/12.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
St Petersburg Paradox
A fair coin is tossed repeatedly until the first head appears. Youget 2 ducats if the first head appears on the 1st toss, 4 ducats ifthe first head appears on the 2nd toss, and 2n ducats if the firsthead appears on the nth toss
![Page 13: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/13.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
St Petersburg Paradox
A fair coin is tossed repeatedly until the first head appears. Youget 2 ducats if the first head appears on the 1st toss, 4 ducats ifthe first head appears on the 2nd toss, and 2n ducats if the firsthead appears on the nth toss
How much would you be willing to pay to play?
![Page 14: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/14.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
St Petersburg Paradox
A fair coin is tossed repeatedly until the first head appears. Youget 2 ducats if the first head appears on the 1st toss, 4 ducats ifthe first head appears on the 2nd toss, and 2n ducats if the firsthead appears on the nth toss
How much would you be willing to pay to play? 4, 20, or 50ducats?
![Page 15: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/15.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
St Petersburg Paradox
A fair coin is tossed repeatedly until the first head appears. Youget 2 ducats if the first head appears on the 1st toss, 4 ducats ifthe first head appears on the 2nd toss, and 2n ducats if the firsthead appears on the nth toss
How much would you be willing to pay to play? 4, 20, or 50ducats?
The prospect is X = (2, 12 ; 4,14 ; ...; 2
n, 12n ...)
![Page 16: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/16.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
St Petersburg Paradox
A fair coin is tossed repeatedly until the first head appears. Youget 2 ducats if the first head appears on the 1st toss, 4 ducats ifthe first head appears on the 2nd toss, and 2n ducats if the firsthead appears on the nth toss
How much would you be willing to pay to play? 4, 20, or 50ducats?
The prospect is X = (2, 12 ; 4,14 ; ...; 2
n, 12n ...)
The expected payoff
E[X ] =1
2× 2 +
1
4× 4 +
1
8× 8 + · · ·+
1
2n× 2n + · · · = +∞!
![Page 17: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/17.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
St Petersburg Paradox
A fair coin is tossed repeatedly until the first head appears. Youget 2 ducats if the first head appears on the 1st toss, 4 ducats ifthe first head appears on the 2nd toss, and 2n ducats if the firsthead appears on the nth toss
How much would you be willing to pay to play? 4, 20, or 50ducats?
The prospect is X = (2, 12 ; 4,14 ; ...; 2
n, 12n ...)
The expected payoff
E[X ] =1
2× 2 +
1
4× 4 +
1
8× 8 + · · ·+
1
2n× 2n + · · · = +∞!
“Few of us would pay even 25 ducats to enter such a game”(R. Martin 2004, The Stanford Encyclopedia of Philosophy)
![Page 18: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/18.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Bernoulli’s Resolution
Daniel Bernoulli (1738): Satisfaction from a payoff x should not beproportional to x, but should be a proper function U of x
![Page 19: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/19.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Bernoulli’s Resolution
Daniel Bernoulli (1738): Satisfaction from a payoff x should not beproportional to x, but should be a proper function U of x
Bernoulli proposed U(x) = log(x) (1 utiles = log(ducats))
![Page 20: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/20.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Bernoulli’s Resolution
Daniel Bernoulli (1738): Satisfaction from a payoff x should not beproportional to x, but should be a proper function U of x
Bernoulli proposed U(x) = log(x) (1 utiles = log(ducats))
Value of the St Petersburg game in utiles
E[U(X)]= 1
2× log(2) + 1
4× log(4) + 1
8× log(8) + · · ·+ 1
2n× log(2n) + · · ·
= log(4)
![Page 21: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/21.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Bernoulli’s Resolution
Daniel Bernoulli (1738): Satisfaction from a payoff x should not beproportional to x, but should be a proper function U of x
Bernoulli proposed U(x) = log(x) (1 utiles = log(ducats))
Value of the St Petersburg game in utiles
E[U(X)]= 1
2× log(2) + 1
4× log(4) + 1
8× log(8) + · · ·+ 1
2n× log(2n) + · · ·
= log(4)
... or 4 ducats
![Page 22: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/22.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Gossen’s First Law: Diminishing Marginal Utility
Herman Gossen (1854) “Laws of Economic Activity”
![Page 23: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/23.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Gossen’s First Law: Diminishing Marginal Utility
Herman Gossen (1854) “Laws of Economic Activity”
U(x) = log(x) is increasing in x
![Page 24: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/24.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Gossen’s First Law: Diminishing Marginal Utility
Herman Gossen (1854) “Laws of Economic Activity”
U(x) = log(x) is increasing in x
Speed of increase is decreasing in x (U ′(x) = 1x), and
negligible when x is sufficiently large - diminishing marginalutility
![Page 25: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/25.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Gossen’s First Law: Diminishing Marginal Utility
Herman Gossen (1854) “Laws of Economic Activity”
U(x) = log(x) is increasing in x
Speed of increase is decreasing in x (U ′(x) = 1x), and
negligible when x is sufficiently large - diminishing marginalutility
Key to Bernoulli’s resolution of St Petersburg paradox
![Page 26: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/26.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Gossen’s First Law: Diminishing Marginal Utility
Herman Gossen (1854) “Laws of Economic Activity”
U(x) = log(x) is increasing in x
Speed of increase is decreasing in x (U ′(x) = 1x), and
negligible when x is sufficiently large - diminishing marginalutility
Key to Bernoulli’s resolution of St Petersburg paradox
Mathematically, diminishing marginal utility equivalent to aconcave utility function
![Page 27: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/27.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Gossen’s First Law: Diminishing Marginal Utility
Herman Gossen (1854) “Laws of Economic Activity”
U(x) = log(x) is increasing in x
Speed of increase is decreasing in x (U ′(x) = 1x), and
negligible when x is sufficiently large - diminishing marginalutility
Key to Bernoulli’s resolution of St Petersburg paradox
Mathematically, diminishing marginal utility equivalent to aconcave utility function
Payoff
Util
ity
![Page 28: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/28.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Gossen’s First Law: Diminishing Marginal Utility
Herman Gossen (1854) “Laws of Economic Activity”
U(x) = log(x) is increasing in x
Speed of increase is decreasing in x (U ′(x) = 1x), and
negligible when x is sufficiently large - diminishing marginalutility
Key to Bernoulli’s resolution of St Petersburg paradox
Mathematically, diminishing marginal utility equivalent to aconcave utility function
Payoff
Util
ity
![Page 29: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/29.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Gossen’s First Law: Diminishing Marginal Utility
Herman Gossen (1854) “Laws of Economic Activity”
U(x) = log(x) is increasing in x
Speed of increase is decreasing in x (U ′(x) = 1x), and
negligible when x is sufficiently large - diminishing marginalutility
Key to Bernoulli’s resolution of St Petersburg paradox
Mathematically, diminishing marginal utility equivalent to aconcave utility function
Payoff
Util
ity
![Page 30: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/30.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Gossen’s First Law: Diminishing Marginal Utility
Herman Gossen (1854) “Laws of Economic Activity”
U(x) = log(x) is increasing in x
Speed of increase is decreasing in x (U ′(x) = 1x), and
negligible when x is sufficiently large - diminishing marginalutility
Key to Bernoulli’s resolution of St Petersburg paradox
Mathematically, diminishing marginal utility equivalent to aconcave utility function
Payoff
Util
ity
![Page 31: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/31.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
![Page 32: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/32.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
![Page 33: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/33.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
A: Win $10,000 with 50% chance and $0 with 50% chance
![Page 34: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/34.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chance
![Page 35: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/35.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceMost people chose B
![Page 36: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/36.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceMost people chose B
Generally, most people choose B in the following
![Page 37: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/37.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceMost people chose B
Generally, most people choose B in the following
A: Win x with 50% chance and y with 50% chance
![Page 38: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/38.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceMost people chose B
Generally, most people choose B in the following
A: Win x with 50% chance and y with 50% chanceB: Win 1
2(x+ y) with 100% chance
![Page 39: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/39.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceMost people chose B
Generally, most people choose B in the following
A: Win x with 50% chance and y with 50% chanceB: Win 1
2(x+ y) with 100% chance
People dislike mean-preserving spread – “risk averse”
![Page 40: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/40.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceMost people chose B
Generally, most people choose B in the following
A: Win x with 50% chance and y with 50% chanceB: Win 1
2(x+ y) with 100% chance
People dislike mean-preserving spread – “risk averse”
U(12x+ 12y) ≥
12U(x) + 1
2U(y)
![Page 41: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/41.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Risk Aversion
A concave utility function, in turn, suggests risk aversion
Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceMost people chose B
Generally, most people choose B in the following
A: Win x with 50% chance and y with 50% chanceB: Win 1
2(x+ y) with 100% chance
People dislike mean-preserving spread – “risk averse”
U(12x+ 12y) ≥
12U(x) + 1
2U(y) - Concave function!
![Page 42: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/42.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
A Few Good Axioms
Expected Utility Theory (EUT): To evaluate gambles (randomvariables, lotteries) and form preference
![Page 43: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/43.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
A Few Good Axioms
Expected Utility Theory (EUT): To evaluate gambles (randomvariables, lotteries) and form preference
Foundation laid by von Neumann and Morgenstern (1947)
![Page 44: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/44.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
A Few Good Axioms
Expected Utility Theory (EUT): To evaluate gambles (randomvariables, lotteries) and form preference
Foundation laid by von Neumann and Morgenstern (1947)
Axiomatic approach: completeness, transivity, continuity andindependence
![Page 45: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/45.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
A Few Good Axioms
Expected Utility Theory (EUT): To evaluate gambles (randomvariables, lotteries) and form preference
Foundation laid by von Neumann and Morgenstern (1947)
Axiomatic approach: completeness, transivity, continuity andindependence
Behaviour of a rational agent necessarily coincides with thatof an agent who values uncertain payoffs using expectedconcave utility
![Page 46: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/46.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Human Judgement Implied by Expected Utility Theory
EUT: Dominant model for decision making under uncertainty,including financial asset allocation
![Page 47: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/47.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Human Judgement Implied by Expected Utility Theory
EUT: Dominant model for decision making under uncertainty,including financial asset allocation
Basic tenets of human judgement implied by EUT in thecontext of asset allocation:
![Page 48: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/48.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Human Judgement Implied by Expected Utility Theory
EUT: Dominant model for decision making under uncertainty,including financial asset allocation
Basic tenets of human judgement implied by EUT in thecontext of asset allocation:
Frame of problem: Investors’ preference is independent ofhow problem is stated (described, or framed)
![Page 49: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/49.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Human Judgement Implied by Expected Utility Theory
EUT: Dominant model for decision making under uncertainty,including financial asset allocation
Basic tenets of human judgement implied by EUT in thecontext of asset allocation:
Frame of problem: Investors’ preference is independent ofhow problem is stated (described, or framed)Source of satisfaction: Investors evaluate assets according tofinal asset positions
![Page 50: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/50.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Human Judgement Implied by Expected Utility Theory
EUT: Dominant model for decision making under uncertainty,including financial asset allocation
Basic tenets of human judgement implied by EUT in thecontext of asset allocation:
Frame of problem: Investors’ preference is independent ofhow problem is stated (described, or framed)Source of satisfaction: Investors evaluate assets according tofinal asset positionsAttitude towards risk: Investors are always risk averse(concave utility)
![Page 51: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/51.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Human Judgement Implied by Expected Utility Theory
EUT: Dominant model for decision making under uncertainty,including financial asset allocation
Basic tenets of human judgement implied by EUT in thecontext of asset allocation:
Frame of problem: Investors’ preference is independent ofhow problem is stated (described, or framed)Source of satisfaction: Investors evaluate assets according tofinal asset positionsAttitude towards risk: Investors are always risk averse(concave utility)Beliefs about future: Investors are able to objectivelyevaluate probabilities of future returns
![Page 52: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/52.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Human Judgement Implied by Expected Utility Theory
EUT: Dominant model for decision making under uncertainty,including financial asset allocation
Basic tenets of human judgement implied by EUT in thecontext of asset allocation:
Frame of problem: Investors’ preference is independent ofhow problem is stated (described, or framed)Source of satisfaction: Investors evaluate assets according tofinal asset positionsAttitude towards risk: Investors are always risk averse(concave utility)Beliefs about future: Investors are able to objectivelyevaluate probabilities of future returns
Neoclassical economics
![Page 53: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/53.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Market Is Always Right
Efficient market hypothesis (Eugene Fama 1960s): Financialmarkets “informationally efficient”, or “prices are right”
![Page 54: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/54.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Market Is Always Right
Efficient market hypothesis (Eugene Fama 1960s): Financialmarkets “informationally efficient”, or “prices are right”
Chicago school (Milton Friedman 1912-2006): regulation andother government intervention always inefficient compared toa free market
![Page 55: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/55.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory
Market Is Always Right
Efficient market hypothesis (Eugene Fama 1960s): Financialmarkets “informationally efficient”, or “prices are right”
Chicago school (Milton Friedman 1912-2006): regulation andother government intervention always inefficient compared toa free market
Reaganomics: “Only by reducing the growth of government,can we increase the growth of the economy”
![Page 56: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/56.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Section 2
Expected Utility Theory Challenged
![Page 57: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/57.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Paradoxes/Puzzles with EUT
EUT is systematically violated via experimental work, andchallenged by many paradoxes and puzzles
Allais paradox: Allais (1953)
Ellesberg paradox: Ellesberg (1961)
Friedman and Savage puzzle: Friedman and Savage (1948)
Equity premium puzzle: Mehra and Prescott (1985)
Risk-free rate puzzle: Weil (1989)
![Page 58: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/58.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Independence
Frame: the form used to describe a decision problem
![Page 59: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/59.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Independence
Frame: the form used to describe a decision problem
Frame independence: form is irrelevant to behaviour
![Page 60: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/60.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Independence
Frame: the form used to describe a decision problem
Frame independence: form is irrelevant to behaviour
People can see through all the different ways cash flows mightbe described
![Page 61: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/61.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Independence
Frame: the form used to describe a decision problem
Frame independence: form is irrelevant to behaviour
People can see through all the different ways cash flows mightbe described
Frame independence: the foundation of neoclassicaleconomics/finance
![Page 62: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/62.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Independence
Frame: the form used to describe a decision problem
Frame independence: form is irrelevant to behaviour
People can see through all the different ways cash flows mightbe described
Frame independence: the foundation of neoclassicaleconomics/finance
Merton Miller: “If you transfer a dollar from your right pocketto your left pocket, you are no wealthier. Franco (Modigliani)and I proved that rigorously”
![Page 63: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/63.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
![Page 64: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/64.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
In HK, the penalty charge notice (PCN) read:
![Page 65: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/65.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
In HK, the penalty charge notice (PCN) read:
A penalty HK$400 is now payable and must be paid in 14 days
![Page 66: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/66.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
In HK, the penalty charge notice (PCN) read:
A penalty HK$400 is now payable and must be paid in 14 daysIf you pay after 14 days there is a surcharge of an additionalHK$400
![Page 67: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/67.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
In HK, the penalty charge notice (PCN) read:
A penalty HK$400 is now payable and must be paid in 14 daysIf you pay after 14 days there is a surcharge of an additionalHK$400
I paid reluctantly, on the last day
![Page 68: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/68.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
In HK, the penalty charge notice (PCN) read:
A penalty HK$400 is now payable and must be paid in 14 daysIf you pay after 14 days there is a surcharge of an additionalHK$400
I paid reluctantly, on the last day
The PCN in UK said:
![Page 69: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/69.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
In HK, the penalty charge notice (PCN) read:
A penalty HK$400 is now payable and must be paid in 14 daysIf you pay after 14 days there is a surcharge of an additionalHK$400
I paid reluctantly, on the last day
The PCN in UK said:
A penalty £70 is now payable and must be paid in 28 days
![Page 70: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/70.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
In HK, the penalty charge notice (PCN) read:
A penalty HK$400 is now payable and must be paid in 14 daysIf you pay after 14 days there is a surcharge of an additionalHK$400
I paid reluctantly, on the last day
The PCN in UK said:
A penalty £70 is now payable and must be paid in 28 daysBut ... if you pay in 14 days there is a discount of 50% to £35
![Page 71: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/71.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
In HK, the penalty charge notice (PCN) read:
A penalty HK$400 is now payable and must be paid in 14 daysIf you pay after 14 days there is a surcharge of an additionalHK$400
I paid reluctantly, on the last day
The PCN in UK said:
A penalty £70 is now payable and must be paid in 28 daysBut ... if you pay in 14 days there is a discount of 50% to £35
I paid immediately ...
![Page 72: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/72.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Frame Dependence: My Parking Ticket
I got parking tickets in both HK and UK, and needed to pay
In HK, the penalty charge notice (PCN) read:
A penalty HK$400 is now payable and must be paid in 14 daysIf you pay after 14 days there is a surcharge of an additionalHK$400
I paid reluctantly, on the last day
The PCN in UK said:
A penalty £70 is now payable and must be paid in 28 daysBut ... if you pay in 14 days there is a discount of 50% to £35
I paid immediately ... filled with gratitude and joy
![Page 73: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/73.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames
Game 1: Choose between
![Page 74: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/74.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames
Game 1: Choose between
A: 25% chance to gain $10,000, 75% chance to gain nothing
![Page 75: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/75.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames
Game 1: Choose between
A: 25% chance to gain $10,000, 75% chance to gain nothingB: gain $2,400 for sure
![Page 76: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/76.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames
Game 1: Choose between
A: 25% chance to gain $10,000, 75% chance to gain nothingB: gain $2,400 for sureB was more popular
![Page 77: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/77.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames
Game 1: Choose between
A: 25% chance to gain $10,000, 75% chance to gain nothingB: gain $2,400 for sureB was more popular
Game 2: Choose between
![Page 78: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/78.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames
Game 1: Choose between
A: 25% chance to gain $10,000, 75% chance to gain nothingB: gain $2,400 for sureB was more popular
Game 2: Choose between
C: 75% chance to lose $10,000, 25% chance to lose nothing
![Page 79: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/79.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames
Game 1: Choose between
A: 25% chance to gain $10,000, 75% chance to gain nothingB: gain $2,400 for sureB was more popular
Game 2: Choose between
C: 75% chance to lose $10,000, 25% chance to lose nothingD: lose $7,500 for sure
![Page 80: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/80.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames
Game 1: Choose between
A: 25% chance to gain $10,000, 75% chance to gain nothingB: gain $2,400 for sureB was more popular
Game 2: Choose between
C: 75% chance to lose $10,000, 25% chance to lose nothingD: lose $7,500 for sureC was more popular
![Page 81: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/81.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames
Game 1: Choose between
A: 25% chance to gain $10,000, 75% chance to gain nothingB: gain $2,400 for sureB was more popular
Game 2: Choose between
C: 75% chance to lose $10,000, 25% chance to lose nothingD: lose $7,500 for sureC was more popular
“B + C > A+D”
![Page 82: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/82.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames (Cont’d)
Game 3: Choose between
![Page 83: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/83.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames (Cont’d)
Game 3: Choose between
E: 75% chance to lose $7,600, 25% chance to gain $2,400
![Page 84: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/84.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames (Cont’d)
Game 3: Choose between
E: 75% chance to lose $7,600, 25% chance to gain $2,400F: 75% chance to lose $7,500, 25% chance to gain $2,500
![Page 85: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/85.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames (Cont’d)
Game 3: Choose between
E: 75% chance to lose $7,600, 25% chance to gain $2,400F: 75% chance to lose $7,500, 25% chance to gain $2,500
“F = E + $100 > E”
![Page 86: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/86.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames (Cont’d)
Game 3: Choose between
E: 75% chance to lose $7,600, 25% chance to gain $2,400F: 75% chance to lose $7,500, 25% chance to gain $2,500
“F = E + $100 > E”
However: B + C = E, A+D = F !
![Page 87: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/87.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Decisions Depend on Frames (Cont’d)
Game 3: Choose between
E: 75% chance to lose $7,600, 25% chance to gain $2,400F: 75% chance to lose $7,500, 25% chance to gain $2,500
“F = E + $100 > E”
However: B + C = E, A+D = F !
Frame dependence: frames are not transparent, but opaque
![Page 88: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/88.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Reference Point: Tough Jobs
Alan Greenspan “The Age of Turbulence” (2007): Choose betweenthe following two job offers
![Page 89: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/89.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Reference Point: Tough Jobs
Alan Greenspan “The Age of Turbulence” (2007): Choose betweenthe following two job offers
A: Earn $105,000/year while all your colleagues earn at least$210,000/year
![Page 90: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/90.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Reference Point: Tough Jobs
Alan Greenspan “The Age of Turbulence” (2007): Choose betweenthe following two job offers
A: Earn $105,000/year while all your colleagues earn at least$210,000/year
B: Earn $100,000/year while all your colleagues earn at most$50,000/year
![Page 91: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/91.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Reference Point: Tough Jobs
Alan Greenspan “The Age of Turbulence” (2007): Choose betweenthe following two job offers
A: Earn $105,000/year while all your colleagues earn at least$210,000/year
B: Earn $100,000/year while all your colleagues earn at most$50,000/year
B was more popular
![Page 92: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/92.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Reference Point: Tough Jobs
Alan Greenspan “The Age of Turbulence” (2007): Choose betweenthe following two job offers
A: Earn $105,000/year while all your colleagues earn at least$210,000/year
B: Earn $100,000/year while all your colleagues earn at most$50,000/year
B was more popular
Reference point: what matters is deviation of wealth fromcertain benchmark, not wealth itself
![Page 93: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/93.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Risk Aversion vs. Risk Seeking
Experiment 1: Choose between
![Page 94: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/94.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Risk Aversion vs. Risk Seeking
Experiment 1: Choose between
A: Win $10,000 with 50% chance and $0 with 50% chance
![Page 95: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/95.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Risk Aversion vs. Risk Seeking
Experiment 1: Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chance
![Page 96: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/96.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Risk Aversion vs. Risk Seeking
Experiment 1: Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceB was more popular
![Page 97: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/97.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Risk Aversion vs. Risk Seeking
Experiment 1: Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceB was more popular
Experiment 2: Choose between
![Page 98: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/98.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Risk Aversion vs. Risk Seeking
Experiment 1: Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceB was more popular
Experiment 2: Choose between
A: Lose $10,000 with 50% chance and $0 with 50% chance
![Page 99: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/99.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Risk Aversion vs. Risk Seeking
Experiment 1: Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceB was more popular
Experiment 2: Choose between
A: Lose $10,000 with 50% chance and $0 with 50% chanceB: Lose $5,000 with 100% chance
![Page 100: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/100.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Risk Aversion vs. Risk Seeking
Experiment 1: Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceB was more popular
Experiment 2: Choose between
A: Lose $10,000 with 50% chance and $0 with 50% chanceB: Lose $5,000 with 100% chanceThis time: A was more popular
![Page 101: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/101.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Risk Aversion vs. Risk Seeking
Experiment 1: Choose between
A: Win $10,000 with 50% chance and $0 with 50% chanceB: Win $5,000 with 100% chanceB was more popular
Experiment 2: Choose between
A: Lose $10,000 with 50% chance and $0 with 50% chanceB: Lose $5,000 with 100% chanceThis time: A was more popular
Risk averse on gains, risk seeking on losses
![Page 102: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/102.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Loss Aversion: Losses Matter More
Paul Samuelson (1963): Choose between
![Page 103: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/103.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Loss Aversion: Losses Matter More
Paul Samuelson (1963): Choose between
A: Win $100,000 with 50% chance and lose $50,000 with50% chance
![Page 104: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/104.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Loss Aversion: Losses Matter More
Paul Samuelson (1963): Choose between
A: Win $100,000 with 50% chance and lose $50,000 with50% chance
B: Don’t take this bet
![Page 105: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/105.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Loss Aversion: Losses Matter More
Paul Samuelson (1963): Choose between
A: Win $100,000 with 50% chance and lose $50,000 with50% chance
B: Don’t take this bet
B was more popular
![Page 106: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/106.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Loss Aversion: Losses Matter More
Paul Samuelson (1963): Choose between
A: Win $100,000 with 50% chance and lose $50,000 with50% chance
B: Don’t take this bet
B was more popular
Loss aversion: pain from a loss is more than joy from a gain ofthe same magnitude
![Page 107: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/107.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Probability Distortion (Weighting): Lottery Ticket andInsurance
Experiment 3: Choose between
![Page 108: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/108.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Probability Distortion (Weighting): Lottery Ticket andInsurance
Experiment 3: Choose between
A: Win $50,000 with 0.1% chance
![Page 109: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/109.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Probability Distortion (Weighting): Lottery Ticket andInsurance
Experiment 3: Choose between
A: Win $50,000 with 0.1% chanceB: Win $50 with 100% chance
![Page 110: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/110.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Probability Distortion (Weighting): Lottery Ticket andInsurance
Experiment 3: Choose between
A: Win $50,000 with 0.1% chanceB: Win $50 with 100% chanceA was more popular
![Page 111: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/111.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Probability Distortion (Weighting): Lottery Ticket andInsurance
Experiment 3: Choose between
A: Win $50,000 with 0.1% chanceB: Win $50 with 100% chanceA was more popular
Experiment 4: Choose between
![Page 112: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/112.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Probability Distortion (Weighting): Lottery Ticket andInsurance
Experiment 3: Choose between
A: Win $50,000 with 0.1% chanceB: Win $50 with 100% chanceA was more popular
Experiment 4: Choose between
A: Lose $50,000 with 0.1% chance
![Page 113: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/113.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Probability Distortion (Weighting): Lottery Ticket andInsurance
Experiment 3: Choose between
A: Win $50,000 with 0.1% chanceB: Win $50 with 100% chanceA was more popular
Experiment 4: Choose between
A: Lose $50,000 with 0.1% chanceB: Lose $50 with 100% chance
![Page 114: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/114.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Probability Distortion (Weighting): Lottery Ticket andInsurance
Experiment 3: Choose between
A: Win $50,000 with 0.1% chanceB: Win $50 with 100% chanceA was more popular
Experiment 4: Choose between
A: Lose $50,000 with 0.1% chanceB: Lose $50 with 100% chanceThis time: B was more popular
![Page 115: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/115.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Probability Distortion (Weighting): Lottery Ticket andInsurance
Experiment 3: Choose between
A: Win $50,000 with 0.1% chanceB: Win $50 with 100% chanceA was more popular
Experiment 4: Choose between
A: Lose $50,000 with 0.1% chanceB: Lose $50 with 100% chanceThis time: B was more popular
Probability weighting (distortion): People tend to exaggerate,intentionally or unintentionally, small probabilities of bothwinning big and losing big
![Page 116: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/116.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Equity Premium and Risk-Free Rate Puzzles
Equity premium puzzle (Mehra and Prescott 1985): observedequity premium is too high to be explainable by classicalconsumption-based capital asset pricing model (CCAPM)
![Page 117: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/117.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Equity Premium and Risk-Free Rate Puzzles
Equity premium puzzle (Mehra and Prescott 1985): observedequity premium is too high to be explainable by classicalconsumption-based capital asset pricing model (CCAPM)
Mehra and Prescott found historical equity premium of S&P500 for 1889–1978 to be 6.18%, much higher than could bepredicted by EUT-based CCAPM
![Page 118: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/118.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Equity Premium and Risk-Free Rate Puzzles
Equity premium puzzle (Mehra and Prescott 1985): observedequity premium is too high to be explainable by classicalconsumption-based capital asset pricing model (CCAPM)
Mehra and Prescott found historical equity premium of S&P500 for 1889–1978 to be 6.18%, much higher than could bepredicted by EUT-based CCAPMSubsequent empirical studies have confirmed that this puzzle isrobust across different time periods and different countries
![Page 119: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/119.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Equity Premium and Risk-Free Rate Puzzles
Equity premium puzzle (Mehra and Prescott 1985): observedequity premium is too high to be explainable by classicalconsumption-based capital asset pricing model (CCAPM)
Mehra and Prescott found historical equity premium of S&P500 for 1889–1978 to be 6.18%, much higher than could bepredicted by EUT-based CCAPMSubsequent empirical studies have confirmed that this puzzle isrobust across different time periods and different countries
Risk-free rate puzzle (Weil 1989): observed risk-free rate istoo low to be explainable by classical CCAPM
![Page 120: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/120.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Economic Data 1889–1978 (Mehra and Prescott 1985)
Consumption growth riskless return equity premium S&P 500 return
Periods Mean S.D. Mean S.D. Mean S.D. Mean S.D.
1889–1978 1.83 3.57 0.80 5.67 6.18 16.67 6.98 16.54
1889–1898 2.30 4.90 5.80 3.23 1.78 11.57 7.58 10.021899–1908 2.55 5.31 2.62 2.59 5.08 16.86 7.71 17.211909–1918 0.44 3.07 -1.63 9.02 1.49 9.18 -0.14 12.811919–1928 3.00 3.97 4.30 6.61 14.64 15.94 18.94 16.181929–1938 -0.25 5.28 2.39 6.50 0.18 31.63 2.56 27.901939–1948 2.19 2.52 -5.82 4.05 8.89 14.23 3.07 14.671949–1958 1.48 1.00 -0.81 1.89 18.30 13.20 17.49 13.081959–1968 2.37 1.00 1.07 0.64 4.50 10.17 5.58 10.591969–1978 2.41 1.40 -0.72 2.06 0.75 11.64 0.03 13.11
![Page 121: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/121.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
EUT Based Theories
Recall EUT based formulae (single period)
r − rf ≈ αCov(g, r),
1 + rf ≈1+αgβ
where α: relative risk aversion index, g: consumption growthrate, r: equity return rate, rf : risk-free rate, β: discount rate
![Page 122: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/122.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
EUT Based Theories
Recall EUT based formulae (single period)
r − rf ≈ αCov(g, r),
1 + rf ≈1+αgβ
where α: relative risk aversion index, g: consumption growthrate, r: equity return rate, rf : risk-free rate, β: discount rate
Noting β ≤ 1, we have upper bound α ≤rfg if g > 0
![Page 123: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/123.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
EUT Based Theories
Recall EUT based formulae (single period)
r − rf ≈ αCov(g, r),
1 + rf ≈1+αgβ
where α: relative risk aversion index, g: consumption growthrate, r: equity return rate, rf : risk-free rate, β: discount rate
Noting β ≤ 1, we have upper bound α ≤rfg if g > 0
For 1889–1978, g = 1.83%, rf = 0.80%
![Page 124: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/124.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
EUT Based Theories
Recall EUT based formulae (single period)
r − rf ≈ αCov(g, r),
1 + rf ≈1+αgβ
where α: relative risk aversion index, g: consumption growthrate, r: equity return rate, rf : risk-free rate, β: discount rate
Noting β ≤ 1, we have upper bound α ≤rfg if g > 0
For 1889–1978, g = 1.83%, rf = 0.80%So α ≤ 0.80
1.83= 0.44
![Page 125: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/125.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
EUT Based Theories
Recall EUT based formulae (single period)
r − rf ≈ αCov(g, r),
1 + rf ≈1+αgβ
where α: relative risk aversion index, g: consumption growthrate, r: equity return rate, rf : risk-free rate, β: discount rate
Noting β ≤ 1, we have upper bound α ≤rfg if g > 0
For 1889–1978, g = 1.83%, rf = 0.80%So α ≤ 0.80
1.83= 0.44
On the other hand, we have lower bound α ≥r−rfσgσr
![Page 126: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/126.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
EUT Based Theories
Recall EUT based formulae (single period)
r − rf ≈ αCov(g, r),
1 + rf ≈1+αgβ
where α: relative risk aversion index, g: consumption growthrate, r: equity return rate, rf : risk-free rate, β: discount rate
Noting β ≤ 1, we have upper bound α ≤rfg if g > 0
For 1889–1978, g = 1.83%, rf = 0.80%So α ≤ 0.80
1.83= 0.44
On the other hand, we have lower bound α ≥r−rfσgσr
For 1889–1978, r = 6.98%, σg = 3.57%, σr = 16.54%
![Page 127: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/127.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
EUT Based Theories
Recall EUT based formulae (single period)
r − rf ≈ αCov(g, r),
1 + rf ≈1+αgβ
where α: relative risk aversion index, g: consumption growthrate, r: equity return rate, rf : risk-free rate, β: discount rate
Noting β ≤ 1, we have upper bound α ≤rfg if g > 0
For 1889–1978, g = 1.83%, rf = 0.80%So α ≤ 0.80
1.83= 0.44
On the other hand, we have lower bound α ≥r−rfσgσr
For 1889–1978, r = 6.98%, σg = 3.57%, σr = 16.54%
So α ≥ 6.98%−0.80%3.57%×16.54%
= 10.47
![Page 128: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/128.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Puzzles under EUT
Large gap between upper bound of 0.44 and lower bound of10.47: a significant inconsistency between EUT basedCCAPM and empirical findings of a low risk-free rate and ahigh equity premium
![Page 129: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/129.jpg)
Mathematical Behavioural Finance A Mini Course
Expected Utility Theory Challenged
Puzzles under EUT
Large gap between upper bound of 0.44 and lower bound of10.47: a significant inconsistency between EUT basedCCAPM and empirical findings of a low risk-free rate and ahigh equity premium
Under EUT, a puzzle thus arises: the solution simultaneouslyrequires a small relative risk aversion to account for the lowrisk-free rate and a large relative risk aversion to account forthe high equity premium
![Page 130: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/130.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Section 3
Alternative Theories for Risky Choice
![Page 131: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/131.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Yaari’s Dual Theory
Preference on random payoff X ≥ 0 represented by (Yaari 1987)
V (X) =
∫
Xd(w ◦ P) :=
∫
∞
0
w(P(X > x))dx
where probability weighting w : [0, 1] → [0, 1], ↑, w(0) = 0,w(1) = 1
![Page 132: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/132.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Risk Preference Reflected by Weighting
Assuming w is differentiable:V (X) =
∫∞
0 xd[−w(1 − FX(x))] =∫∞
0 xw′(1− FX(x))dFX (x)
where FX is CDF of X
Risk averse when w(·) is convex (overweighting small payoffand underweighting large payoff)
![Page 133: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/133.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Risk Preference Reflected by Weighting
Assuming w is differentiable:V (X) =
∫∞
0 xd[−w(1 − FX(x))] =∫∞
0 xw′(1− FX(x))dFX (x)
where FX is CDF of X
Risk averse when w(·) is convex (overweighting small payoffand underweighting large payoff)
Risk seeking when w(·) is concave
![Page 134: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/134.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Risk Preference Reflected by Weighting
Assuming w is differentiable:V (X) =
∫∞
0 xd[−w(1 − FX(x))] =∫∞
0 xw′(1− FX(x))dFX (x)
where FX is CDF of X
Risk averse when w(·) is convex (overweighting small payoffand underweighting large payoff)
Risk seeking when w(·) is concave
Simultaneous risk averse and risk seeking when w(·) isinverse-S shaped
![Page 135: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/135.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Probability Weighting Functions
Kahneman and Tversky (1992) weighting
w(p) =pγ
(pγ + (1− p)γ)1/γ,
Tversky and Fox (1995) weighting
w(p) =δpγ
δpγ + (1− p)γ,
Prelec (1998) weighting
w(p) = e−δ(− ln p)γ
Jin and Zhou (2008) weighting
w(z) =
yb−a0 keaµ+
(aσ)2
2 Φ(
Φ−1(z)− aσ)
z ≤ 1− z0,
C + kebµ+(bσ)2
2 Φ(
Φ−1(z)− bσ)
z ≥ 1− z0
![Page 136: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/136.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Inverse-S Shaped Functions
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p
w(p
)
Kahneman−Tversky’s distortionTversky−Fox’s distortionPrelec’s distortionJin−Zhou’s distortion
![Page 137: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/137.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Quiggin’s Rank-Dependent Utility Theory
Rank-dependent utility theory (RDUT): Quiggin (1982),Schmeidler (1989)
![Page 138: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/138.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Quiggin’s Rank-Dependent Utility Theory
Rank-dependent utility theory (RDUT): Quiggin (1982),Schmeidler (1989)
Preference dictated by an RDUT pair (u,w)
∫
u(X)d(w ◦ P) ≡
∫ ∞
0w(
P(
u(X) > x)
)
dx
![Page 139: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/139.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Quiggin’s Rank-Dependent Utility Theory
Rank-dependent utility theory (RDUT): Quiggin (1982),Schmeidler (1989)
Preference dictated by an RDUT pair (u,w)
∫
u(X)d(w ◦ P) ≡
∫ ∞
0w(
P(
u(X) > x)
)
dx
Two components
![Page 140: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/140.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Quiggin’s Rank-Dependent Utility Theory
Rank-dependent utility theory (RDUT): Quiggin (1982),Schmeidler (1989)
Preference dictated by an RDUT pair (u,w)
∫
u(X)d(w ◦ P) ≡
∫ ∞
0w(
P(
u(X) > x)
)
dx
Two components
A concave (outcome) utility function: individuals dislikemean-preserving spread
![Page 141: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/141.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Quiggin’s Rank-Dependent Utility Theory
Rank-dependent utility theory (RDUT): Quiggin (1982),Schmeidler (1989)
Preference dictated by an RDUT pair (u,w)
∫
u(X)d(w ◦ P) ≡
∫ ∞
0w(
P(
u(X) > x)
)
dx
Two components
A concave (outcome) utility function: individuals dislikemean-preserving spreadA (usually assumed) inverse-S shaped (probability) weightingfunction: individuals overweight tails
![Page 142: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/142.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Lopes’ SP/A Theory
Security-Potential/Aspiration (SP/A) theory: Lopes (1987)
![Page 143: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/143.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Lopes’ SP/A Theory
Security-Potential/Aspiration (SP/A) theory: Lopes (1987)
A dispositional factor and a situational factor to explain riskychoices
![Page 144: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/144.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Lopes’ SP/A Theory
Security-Potential/Aspiration (SP/A) theory: Lopes (1987)
A dispositional factor and a situational factor to explain riskychoices
Dispositional factor describes people’s natural tendency toachieving security and exploiting potential
![Page 145: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/145.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Lopes’ SP/A Theory
Security-Potential/Aspiration (SP/A) theory: Lopes (1987)
A dispositional factor and a situational factor to explain riskychoices
Dispositional factor describes people’s natural tendency toachieving security and exploiting potentialSituational factor describes people’s responses to specific,immediate needs and opportunities
![Page 146: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/146.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Dispositional Factor
Risk-averse motivated by a desire for security
![Page 147: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/147.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Dispositional Factor
Risk-averse motivated by a desire for security
Risk-seeking motivated by a desire for potential
![Page 148: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/148.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Dispositional Factor
Risk-averse motivated by a desire for security
Risk-seeking motivated by a desire for potential
Lopes applies Yaari’s dual theory to model the dispositionalfactor
![Page 149: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/149.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Dispositional Factor
Risk-averse motivated by a desire for security
Risk-seeking motivated by a desire for potential
Lopes applies Yaari’s dual theory to model the dispositionalfactor
V (X) =∫∞
0 w(P(X > x))dx where
w(p) := νpqs+1 + (1− ν)[1− (1− p)qp+1]
with qs, qp > 0 and 0 < ν < 1
![Page 150: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/150.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Dispositional Factor
Risk-averse motivated by a desire for security
Risk-seeking motivated by a desire for potential
Lopes applies Yaari’s dual theory to model the dispositionalfactor
V (X) =∫∞
0 w(P(X > x))dx where
w(p) := νpqs+1 + (1− ν)[1− (1− p)qp+1]
with qs, qp > 0 and 0 < ν < 1
The nonlinear transformation zqs+1 reflects the security and1− (1− z)qp+1 reflects the potential
![Page 151: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/151.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Situational Factor
Aspiration level is a situational variable that reflects individualcircumstances, opportunities at hand as well as constraintsimposed by the environment
![Page 152: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/152.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Situational Factor
Aspiration level is a situational variable that reflects individualcircumstances, opportunities at hand as well as constraintsimposed by the environment
Situational factor turns into the constraint
P(X ≥ A) ≥ α
![Page 153: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/153.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Situational Factor
Aspiration level is a situational variable that reflects individualcircumstances, opportunities at hand as well as constraintsimposed by the environment
Situational factor turns into the constraint
P(X ≥ A) ≥ α
A is the aspiration level, 0 < α < 1
![Page 154: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/154.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Kahneman and Tversky’s Cumulative Prospect Theory
Cumulative Prospect Theory (CPT): Kahneman and Tversky(1979), Tversky and Kahneman (1992), Nobel wining 2002
![Page 155: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/155.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Kahneman and Tversky’s Cumulative Prospect Theory
Cumulative Prospect Theory (CPT): Kahneman and Tversky(1979), Tversky and Kahneman (1992), Nobel wining 2002
Key ingredients
![Page 156: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/156.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Kahneman and Tversky’s Cumulative Prospect Theory
Cumulative Prospect Theory (CPT): Kahneman and Tversky(1979), Tversky and Kahneman (1992), Nobel wining 2002
Key ingredients
Reference point or customary wealth (Markowitz 1952)
![Page 157: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/157.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Kahneman and Tversky’s Cumulative Prospect Theory
Cumulative Prospect Theory (CPT): Kahneman and Tversky(1979), Tversky and Kahneman (1992), Nobel wining 2002
Key ingredients
Reference point or customary wealth (Markowitz 1952)S-shaped value (utility) function (risk-averse on gains,risk-seeking on losses), steeper on losses than on gains (lossaversion)
![Page 158: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/158.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
Kahneman and Tversky’s Cumulative Prospect Theory
Cumulative Prospect Theory (CPT): Kahneman and Tversky(1979), Tversky and Kahneman (1992), Nobel wining 2002
Key ingredients
Reference point or customary wealth (Markowitz 1952)S-shaped value (utility) function (risk-averse on gains,risk-seeking on losses), steeper on losses than on gains (lossaversion)Probability weighting
![Page 159: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/159.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
CPT Preference Function
V (X) =∫∞
0 w+
(
P(
u+
(
(X − B)+)
> x))
dx
−∫∞
0 w−
(
P(
u−
(
(X − B)−)
> x))
dx
where
B: reference point in wealth (possibly random)
![Page 160: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/160.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
CPT Preference Function
V (X) =∫∞
0 w+
(
P(
u+
(
(X − B)+)
> x))
dx
−∫∞
0 w−
(
P(
u−
(
(X − B)−)
> x))
dx
where
B: reference point in wealth (possibly random)
X: random payoff
![Page 161: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/161.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
CPT Preference Function
V (X) =∫∞
0 w+
(
P(
u+
(
(X − B)+)
> x))
dx
−∫∞
0 w−
(
P(
u−
(
(X − B)−)
> x))
dx
where
B: reference point in wealth (possibly random)
X: random payoff
w± : [0, 1] → [0, 1] probability weightings
![Page 162: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/162.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
CPT Preference Function
V (X) =∫∞
0 w+
(
P(
u+
(
(X − B)+)
> x))
dx
−∫∞
0 w−
(
P(
u−
(
(X − B)−)
> x))
dx
where
B: reference point in wealth (possibly random)
X: random payoff
w± : [0, 1] → [0, 1] probability weightings
u+(x)1x≥0 − u−(x)1x<0: overall value function
![Page 163: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/163.jpg)
Mathematical Behavioural Finance A Mini Course
Alternative Theories for Risky Choice
CPT Preference Function
V (X) =∫∞
0 w+
(
P(
u+
(
(X − B)+)
> x))
dx
−∫∞
0 w−
(
P(
u−
(
(X − B)−)
> x))
dx
where
B: reference point in wealth (possibly random)
X: random payoff
w± : [0, 1] → [0, 1] probability weightings
u+(x)1x≥0 − u−(x)1x<0: overall value function
Note: Tversky and Kahneman (1992) used discrete randomvariables
![Page 164: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/164.jpg)
Mathematical Behavioural Finance A Mini Course
Summary and Further Readings
Section 4
Summary and Further Readings
![Page 165: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/165.jpg)
Mathematical Behavioural Finance A Mini Course
Summary and Further Readings
Summary
Rationality – foundation of neoclassical economics
![Page 166: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/166.jpg)
Mathematical Behavioural Finance A Mini Course
Summary and Further Readings
Summary
Rationality – foundation of neoclassical economics
Dominant in economics theory and practice
![Page 167: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/167.jpg)
Mathematical Behavioural Finance A Mini Course
Summary and Further Readings
Summary
Rationality – foundation of neoclassical economics
Dominant in economics theory and practice
Rationality seriously challenged by paradoxes, experiments,empirical findings, and financial crises
![Page 168: Mathematical Behavioural Finance A Mini Course · 2014-07-14 · Mathematical Behavioural Finance A Mini Course Expected Utility Theory St Petersburg Paradox A fair coin is tossed](https://reader034.fdocuments.us/reader034/viewer/2022042101/5e7e2b9f75cd5157010f0e14/html5/thumbnails/168.jpg)
Mathematical Behavioural Finance A Mini Course
Summary and Further Readings
Summary
Rationality – foundation of neoclassical economics
Dominant in economics theory and practice
Rationality seriously challenged by paradoxes, experiments,empirical findings, and financial crises
Behavioural theories with new risk preferences have emerged
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Mathematical Behavioural Finance A Mini Course
Summary and Further Readings
Further Readings
D. Ariely. Predictably Irrational, HarperCollins, New York, 2008.
D. Kahneman and A. Tversky. Prospect theory: An analysis of decision under risk, Econometrica,47:263–291, 1979.
L. L. Lopes. Between hope and fear: The psychology of risk, Advances in Experimental Social Psychology,20:255–295, 1987.
H. Markowitz. The utility of wealth, Journal of Political Economy, 60:151–158, 1952.
R. Mehra and E.C. Prescott. The equity premium: A puzzle, Journal of Monetary Economics, 15:145–161,1985.
J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior, Princeton UniversityPress, Princeton, 1944.
J. Quiggin. A Theory of anticipated utility, Journal of Economic and Behavioral Organization, 3:323–343,1982.
A. Tversky and D. Kahneman. Advances in prospect theory: Cumulative representation of uncertainty,Journal of Risk and Uncertainty, 5:297–323, 1992.
P. Weil. The Equity premium puzzle and the risk free rate puzzle, Journal of Monetary Economics,24:401–421, 1989.
M. E. Yaari. The dual theory of choice under risk, Econometrica, 55(1):95–115, 1987.
X. Zhou. Mathematicalising behavioural finance, Proceedings of the International Congress ofMathematicians, Hyderabad, India, 2010; available at http://people.maths.ox.ac.uk/˜zhouxy/download/img-PDF342002.pdf