mn30380 - financial marketspeople.bath.ac.uk/mnsrf/mn30380/lecture 01.pdf · mn30380 - financial...
TRANSCRIPT
![Page 1: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/1.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
MN30380 - Financial Markets
Andreas Krause
Lecture 1 - Utility Theory
![Page 2: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/2.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Password
JCH140LD9GBX
![Page 3: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/3.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Structure of this lecture
1 Utility axioms
2 Expected utility
3 Risk aversion
4 Classes of utility functionsCARA utility functionsCRRA utility functionHARA utility functions
5 Outlook
Readings: Handout, Chapter 1
![Page 4: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/4.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
1 Utility axioms
2 Expected utility
3 Risk aversion
4 Classes of utility functionsCARA utility functionsCRRA utility functionHARA utility functions
5 Outlook
![Page 5: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/5.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Preferences
Preferences allow to order any possible choices, i.e. rankthem
In economics a set of axioms have been developed thatachieve this aim
![Page 6: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/6.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Preferences
Preferences allow to order any possible choices, i.e. rankthem
In economics a set of axioms have been developed thatachieve this aim
![Page 7: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/7.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 1 (Reflection)
Any choice is at least as good as itself
a � a
![Page 8: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/8.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 1 (Reflection)
Any choice is at least as good as itself
a � a
![Page 9: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/9.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 2 (Transitivity)
If a � b and b � c , then a � c
In experiments, transitivity of choices has been found tobe frequently violated
![Page 10: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/10.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 2 (Transitivity)
If a � b and b � c , then a � c
In experiments, transitivity of choices has been found tobe frequently violated
![Page 11: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/11.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 3 (Completeness)
All choices can be compared
∀a, b : a � b or b � a
![Page 12: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/12.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 3 (Completeness)
All choices can be compared
∀a, b : a � b or b � a
![Page 13: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/13.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Utility function
A utility function U : S → R assigns a real number toeach choice such that
a � b ⇔ U(a) > U(b)
a ∼ b ⇔ U(a) = U(b)
The absolute value of these numbers is irrelevant, they canalso be negative
![Page 14: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/14.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Utility function
A utility function U : S → R assigns a real number toeach choice such that
a � b ⇔ U(a) > U(b)a ∼ b ⇔ U(a) = U(b)
The absolute value of these numbers is irrelevant, they canalso be negative
![Page 15: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/15.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Utility function
A utility function U : S → R assigns a real number toeach choice such that
a � b ⇔ U(a) > U(b)a ∼ b ⇔ U(a) = U(b)
The absolute value of these numbers is irrelevant, they canalso be negative
![Page 16: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/16.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Utility function
A utility function U : S → R assigns a real number toeach choice such that
a � b ⇔ U(a) > U(b)a ∼ b ⇔ U(a) = U(b)
The absolute value of these numbers is irrelevant, they canalso be negative
![Page 17: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/17.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
1 Utility axioms
2 Expected utility
3 Risk aversion
4 Classes of utility functionsCARA utility functionsCRRA utility functionHARA utility functions
5 Outlook
![Page 18: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/18.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
The problem
When outcomes are uncertain, the simple utility axioms donot hold any more
We can compare individual possible outcomes, but how toaggregate them?
A set of axioms has been developed to deal with thisproblem
![Page 19: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/19.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
The problem
When outcomes are uncertain, the simple utility axioms donot hold any more
We can compare individual possible outcomes, but how toaggregate them?
A set of axioms has been developed to deal with thisproblem
![Page 20: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/20.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
The problem
When outcomes are uncertain, the simple utility axioms donot hold any more
We can compare individual possible outcomes, but how toaggregate them?
A set of axioms has been developed to deal with thisproblem
![Page 21: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/21.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Notation
A choice ai has a set of potential outcomes{ci1, ci2, . . . , ciM}These outcomes corresponds to the different states ofassets
Each outcome (state) has a certain probability ofoccurring, pij
A choice we also write as ai = [pi1ci1, . . . , piMciM ]
![Page 22: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/22.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Notation
A choice ai has a set of potential outcomes{ci1, ci2, . . . , ciM}These outcomes corresponds to the different states ofassets
Each outcome (state) has a certain probability ofoccurring, pij
A choice we also write as ai = [pi1ci1, . . . , piMciM ]
![Page 23: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/23.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Notation
A choice ai has a set of potential outcomes{ci1, ci2, . . . , ciM}These outcomes corresponds to the different states ofassets
Each outcome (state) has a certain probability ofoccurring, pij
A choice we also write as ai = [pi1ci1, . . . , piMciM ]
![Page 24: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/24.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Notation
A choice ai has a set of potential outcomes{ci1, ci2, . . . , ciM}These outcomes corresponds to the different states ofassets
Each outcome (state) has a certain probability ofoccurring, pij
A choice we also write as ai = [pi1ci1, . . . , piMciM ]
![Page 25: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/25.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 1 (Completeness)
All choices can be compared
∀ai , aj : ai � aj or aj � ai
![Page 26: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/26.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 1 (Completeness)
All choices can be compared
∀ai , aj : ai � aj or aj � ai
![Page 27: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/27.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 2 (Decomposition)
Let ai = [pi1bi1, . . . , piMbiM ] with bij = [qij1c1, . . . , qijLbL]
If we define p∗ik =∑M
l=1 pilqilk then ai ∼ [p∗i1c1, . . . , p∗iLcL]
If the possible outcomes of a choice are themselvesrandom, we can generate an equivalent choice based onlyon final payoffs
![Page 28: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/28.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 2 (Decomposition)
Let ai = [pi1bi1, . . . , piMbiM ] with bij = [qij1c1, . . . , qijLbL]
If we define p∗ik =∑M
l=1 pilqilk then ai ∼ [p∗i1c1, . . . , p∗iLcL]
If the possible outcomes of a choice are themselvesrandom, we can generate an equivalent choice based onlyon final payoffs
![Page 29: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/29.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 2 (Decomposition)
Let ai = [pi1bi1, . . . , piMbiM ] with bij = [qij1c1, . . . , qijLbL]
If we define p∗ik =∑M
l=1 pilqilk then ai ∼ [p∗i1c1, . . . , p∗iLcL]
If the possible outcomes of a choice are themselvesrandom, we can generate an equivalent choice based onlyon final payoffs
![Page 30: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/30.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 3 (Composition)
Let ai = [pi1bi1, . . . , piMbiM ] and bij ∼ [qij1c1, . . . , qijLbL]
Then ai ∼ [p∗i1c1, . . . , pij [qij1c1, . . . , qijLbL] , . . . , p∗iLcL]
We can generate more complex outcomes, but they do notaffect the preferences
![Page 31: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/31.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 3 (Composition)
Let ai = [pi1bi1, . . . , piMbiM ] and bij ∼ [qij1c1, . . . , qijLbL]
Then ai ∼ [p∗i1c1, . . . , pij [qij1c1, . . . , qijLbL] , . . . , p∗iLcL]
We can generate more complex outcomes, but they do notaffect the preferences
![Page 32: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/32.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 3 (Composition)
Let ai = [pi1bi1, . . . , piMbiM ] and bij ∼ [qij1c1, . . . , qijLbL]
Then ai ∼ [p∗i1c1, . . . , pij [qij1c1, . . . , qijLbL] , . . . , p∗iLcL]
We can generate more complex outcomes, but they do notaffect the preferences
![Page 33: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/33.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 4 (Monotonicity)
Let ai = [pi1c1, pi2c2] and bi = [qi1c1, qi2c2] with c1 � c2
If pi1 > qi1 then ai � bi
Given the same possible payoffs, the choice with thehigher probability on the better payoff is preferred
![Page 34: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/34.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 4 (Monotonicity)
Let ai = [pi1c1, pi2c2] and bi = [qi1c1, qi2c2] with c1 � c2
If pi1 > qi1 then ai � bi
Given the same possible payoffs, the choice with thehigher probability on the better payoff is preferred
![Page 35: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/35.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 4 (Monotonicity)
Let ai = [pi1c1, pi2c2] and bi = [qi1c1, qi2c2] with c1 � c2
If pi1 > qi1 then ai � bi
Given the same possible payoffs, the choice with thehigher probability on the better payoff is preferred
![Page 36: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/36.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 5 (Continuity)
If ai � bi and bi � ci then there exists a di = [p1ai , p2ci ]such that di ∼ bi
This axiom ensures that we always can generateintermediate outcomes from its extremes.
![Page 37: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/37.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Axiom 5 (Continuity)
If ai � bi and bi � ci then there exists a di = [p1ai , p2ci ]such that di ∼ bi
This axiom ensures that we always can generateintermediate outcomes from its extremes.
![Page 38: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/38.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
The proof
Let ai = [pi1ci1, . . . , piMciM ] with c1 � c2 � . . . � cM
Such an order exists by axiom 1
![Page 39: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/39.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
The proof
Let ai = [pi1ci1, . . . , piMciM ] with c1 � c2 � . . . � cM
Such an order exists by axiom 1
![Page 40: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/40.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Using the extreme outcomes
We can rewrite ci as ci ∼ [uic1, (1− ui )cM ] ≡ c∗iThis uses axiom 5
![Page 41: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/41.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Using the extreme outcomes
We can rewrite ci as ci ∼ [uic1, (1− ui )cM ] ≡ c∗iThis uses axiom 5
![Page 42: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/42.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Substitute in the choice
ai = [pi1c∗1 , . . . , piMc∗M ]
This uses axiom 3 and replaces the outcomes withstochastic outcomes
Each c∗i consists of the same two possible outcomes, c1
and cM , only with different probabilities
![Page 43: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/43.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Substitute in the choice
ai = [pi1c∗1 , . . . , piMc∗M ]
This uses axiom 3 and replaces the outcomes withstochastic outcomes
Each c∗i consists of the same two possible outcomes, c1
and cM , only with different probabilities
![Page 44: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/44.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Substitute in the choice
ai = [pi1c∗1 , . . . , piMc∗M ]
This uses axiom 3 and replaces the outcomes withstochastic outcomes
Each c∗i consists of the same two possible outcomes, c1
and cM , only with different probabilities
![Page 45: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/45.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Simplifying the outcomes
ai ∼ [pic1, (1− pi )cM ]
pi =∑M
j=1 uijpij
pi = E [ui ]
The choice has been reduced to two possible outcomes
![Page 46: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/46.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Simplifying the outcomes
ai ∼ [pic1, (1− pi )cM ]
pi =∑M
j=1 uijpij
pi = E [ui ]
The choice has been reduced to two possible outcomes
![Page 47: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/47.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Simplifying the outcomes
ai ∼ [pic1, (1− pi )cM ]
pi =∑M
j=1 uijpij
pi = E [ui ]
The choice has been reduced to two possible outcomes
![Page 48: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/48.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Simplifying the outcomes
ai ∼ [pic1, (1− pi )cM ]
pi =∑M
j=1 uijpij
pi = E [ui ]
The choice has been reduced to two possible outcomes
![Page 49: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/49.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Comparing outcomes
For another choice we get equivalentlyaj ∼ [pjc1, (1− pj)cM ] with pi = E [ui ]
If pi > pj we find ai � aj using axiom 4
Or E [ui ] > E [uj ]
![Page 50: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/50.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Comparing outcomes
For another choice we get equivalentlyaj ∼ [pjc1, (1− pj)cM ] with pi = E [ui ]
If pi > pj we find ai � aj using axiom 4
Or E [ui ] > E [uj ]
![Page 51: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/51.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Comparing outcomes
For another choice we get equivalentlyaj ∼ [pjc1, (1− pj)cM ] with pi = E [ui ]
If pi > pj we find ai � aj using axiom 4
Or E [ui ] > E [uj ]
![Page 52: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/52.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Interpreting ui
From our definition we know from axiom 4 thatci � cj ⇔ ui > uj
The ordering of ui therefore reflects the order if thepreferences
This is exactly the same as the definition of the utilityfunction
ui can be interpreted as utility
![Page 53: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/53.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Interpreting ui
From our definition we know from axiom 4 thatci � cj ⇔ ui > uj
The ordering of ui therefore reflects the order if thepreferences
This is exactly the same as the definition of the utilityfunction
ui can be interpreted as utility
![Page 54: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/54.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Interpreting ui
From our definition we know from axiom 4 thatci � cj ⇔ ui > uj
The ordering of ui therefore reflects the order if thepreferences
This is exactly the same as the definition of the utilityfunction
ui can be interpreted as utility
![Page 55: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/55.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Interpreting ui
From our definition we know from axiom 4 thatci � cj ⇔ ui > uj
The ordering of ui therefore reflects the order if thepreferences
This is exactly the same as the definition of the utilityfunction
ui can be interpreted as utility
![Page 56: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/56.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Expected utility principle
Choices should be made according to the expected utilityof the choices
ai � aj ⇔ E [U(ai )] > E [U(aj)]
Determine the utility of each possible outcome and thencalculate the expected value
![Page 57: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/57.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Expected utility principle
Choices should be made according to the expected utilityof the choices
ai � aj ⇔ E [U(ai )] > E [U(aj)]
Determine the utility of each possible outcome and thencalculate the expected value
![Page 58: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/58.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Expected utility principle
Choices should be made according to the expected utilityof the choices
ai � aj ⇔ E [U(ai )] > E [U(aj)]
Determine the utility of each possible outcome and thencalculate the expected value
![Page 59: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/59.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
1 Utility axioms
2 Expected utility
3 Risk aversion
4 Classes of utility functionsCARA utility functionsCRRA utility functionHARA utility functions
5 Outlook
![Page 60: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/60.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Definition
Individuals are risk averse if they always prefer to receive afixed payment to a random payment of equal expected value.
![Page 61: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/61.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Certainty equivalent
In order to be indifferent between a fixed and randompayment, the fixed payment must be lower by an amountof π
E [U(x)] = E [U (E [x ]− π)] = U(E [x ]− π)
E [x ]− π is called the cash equivalent or certaintyequivalent of x
π is called the risk premium of x
![Page 62: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/62.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Certainty equivalent
In order to be indifferent between a fixed and randompayment, the fixed payment must be lower by an amountof π
E [U(x)] = E [U (E [x ]− π)] = U(E [x ]− π)
E [x ]− π is called the cash equivalent or certaintyequivalent of x
π is called the risk premium of x
![Page 63: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/63.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Certainty equivalent
In order to be indifferent between a fixed and randompayment, the fixed payment must be lower by an amountof π
E [U(x)] = E [U (E [x ]− π)] = U(E [x ]− π)
E [x ]− π is called the cash equivalent or certaintyequivalent of x
π is called the risk premium of x
![Page 64: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/64.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Certainty equivalent
In order to be indifferent between a fixed and randompayment, the fixed payment must be lower by an amountof π
E [U(x)] = E [U (E [x ]− π)] = U(E [x ]− π)
E [x ]− π is called the cash equivalent or certaintyequivalent of x
π is called the risk premium of x
![Page 65: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/65.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Approximating the random payoff
Using a 2nd order Taylor series expansion around E [x ] weget
E [U(x)] = E [U(E [x ]) + U ′(E [x ])(x − E [x ])+1
2U ′′(E [x ])(x − E [x ])2]
= U(E [x ]) + 12U ′′(E [x ])Var [x ]
![Page 66: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/66.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Approximating the random payoff
Using a 2nd order Taylor series expansion around E [x ] weget
E [U(x)] = E [U(E [x ]) + U ′(E [x ])(x − E [x ])+1
2U ′′(E [x ])(x − E [x ])2]
= U(E [x ]) + 12U ′′(E [x ])Var [x ]
![Page 67: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/67.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Approximating the random payoff
Using a 2nd order Taylor series expansion around E [x ] weget
E [U(x)] = E [U(E [x ]) + U ′(E [x ])(x − E [x ])+1
2U ′′(E [x ])(x − E [x ])2]
= U(E [x ]) + 12U ′′(E [x ])Var [x ]
![Page 68: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/68.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Approximating the random payoff
Using a 2nd order Taylor series expansion around E [x ] weget
E [U(x)] = E [U(E [x ]) + U ′(E [x ])(x − E [x ])+1
2U ′′(E [x ])(x − E [x ])2]
= U(E [x ]) + 12U ′′(E [x ])Var [x ]
![Page 69: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/69.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Approximating the fixed payoff
Using a 1st order Taylor series expansion around E [x ] weget
U(E [x ]− π) = U(E [x ]) + U ′(E [x ])π
![Page 70: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/70.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Approximating the fixed payoff
Using a 1st order Taylor series expansion around E [x ] weget
U(E [x ]− π) = U(E [x ]) + U ′(E [x ])π
![Page 71: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/71.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Risk premium
Combining the two equations we get
π = 12
(−U′′(E [x])
U′(E [x])
)Var [x ]
z = −U′′(E [x])U′(E [x]) is known as the absolute risk aversion
The more concave the utility function is, the more riskaverse the individual
The relative risk aversion is defined as zV
![Page 72: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/72.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Risk premium
Combining the two equations we get
π = 12
(−U′′(E [x])
U′(E [x])
)Var [x ]
z = −U′′(E [x])U′(E [x]) is known as the absolute risk aversion
The more concave the utility function is, the more riskaverse the individual
The relative risk aversion is defined as zV
![Page 73: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/73.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Risk premium
Combining the two equations we get
π = 12
(−U′′(E [x])
U′(E [x])
)Var [x ]
z = −U′′(E [x])U′(E [x]) is known as the absolute risk aversion
The more concave the utility function is, the more riskaverse the individual
The relative risk aversion is defined as zV
![Page 74: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/74.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Risk premium
Combining the two equations we get
π = 12
(−U′′(E [x])
U′(E [x])
)Var [x ]
z = −U′′(E [x])U′(E [x]) is known as the absolute risk aversion
The more concave the utility function is, the more riskaverse the individual
The relative risk aversion is defined as zV
![Page 75: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/75.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Risk aversion and expected utility
6
-
����������
x1 x2E [x]E [x]− π
π�-
U(x2)
U(x1)
U(E [x])
E [U(x)] =U(E [x]− π)
U(x)
U(x)
x
![Page 76: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/76.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
1 Utility axioms
2 Expected utility
3 Risk aversion
4 Classes of utility functionsCARA utility functionsCRRA utility functionHARA utility functions
5 Outlook
![Page 77: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/77.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
The idea
Relative utility level should depend only on the differenceof wealth associated with two choicesU(V )U(V ′) = f (V − V ′)
![Page 78: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/78.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
The idea
Relative utility level should depend only on the differenceof wealth associated with two choicesU(V )U(V ′) = f (V − V ′)
![Page 79: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/79.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function
U′(V )U(V ′) = f ′(V − V ′)
U′(V ′)U(V ′) = f ′(0) ≡ −z
[ln U(V ′)]′ = −z
ln U(V ′) = −zV ′ + c
U(V ) = Ce−zV = −e−zV
Set C = −1 to ensure U ′(V ) ≥ 0
![Page 80: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/80.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function
U′(V )U(V ′) = f ′(V − V ′)
U′(V ′)U(V ′) = f ′(0) ≡ −z
[ln U(V ′)]′ = −z
ln U(V ′) = −zV ′ + c
U(V ) = Ce−zV = −e−zV
Set C = −1 to ensure U ′(V ) ≥ 0
![Page 81: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/81.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function
U′(V )U(V ′) = f ′(V − V ′)
U′(V ′)U(V ′) = f ′(0) ≡ −z
[ln U(V ′)]′ = −z
ln U(V ′) = −zV ′ + c
U(V ) = Ce−zV = −e−zV
Set C = −1 to ensure U ′(V ) ≥ 0
![Page 82: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/82.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function
U′(V )U(V ′) = f ′(V − V ′)
U′(V ′)U(V ′) = f ′(0) ≡ −z
[ln U(V ′)]′ = −z
ln U(V ′) = −zV ′ + c
U(V ) = Ce−zV = −e−zV
Set C = −1 to ensure U ′(V ) ≥ 0
![Page 83: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/83.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function
U′(V )U(V ′) = f ′(V − V ′)
U′(V ′)U(V ′) = f ′(0) ≡ −z
[ln U(V ′)]′ = −z
ln U(V ′) = −zV ′ + c
U(V ) = Ce−zV = −e−zV
Set C = −1 to ensure U ′(V ) ≥ 0
![Page 84: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/84.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function
U′(V )U(V ′) = f ′(V − V ′)
U′(V ′)U(V ′) = f ′(0) ≡ −z
[ln U(V ′)]′ = −z
ln U(V ′) = −zV ′ + c
U(V ) = Ce−zV = −e−zV
Set C = −1 to ensure U ′(V ) ≥ 0
![Page 85: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/85.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Risk aversion
Using U(V ) = −e−zV we can show that −U′′(V )U′(V ) = z
Hence the utility function has Constant Absolute RiskAversion
![Page 86: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/86.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Risk aversion
Using U(V ) = −e−zV we can show that −U′′(V )U′(V ) = z
Hence the utility function has Constant Absolute RiskAversion
![Page 87: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/87.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
The idea
Relative utility levels of two choices depend on the relativechange in wealth the choices produceU(V )U(V ′) = f
(VV ′
)
![Page 88: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/88.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
The idea
Relative utility levels of two choices depend on the relativechange in wealth the choices produceU(V )U(V ′) = f
(VV ′
)
![Page 89: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/89.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function
U(V )U(V ′) = U(exp(ln V ))
U(exp(ln V ′)) = U(ln V )
U(ln V ′)
= f(
VV ′
)= f
(exp(
(ln V
V ′
))= f (ln V − ln V ′)
Log-levels of wealth follow a CARA utility function
![Page 90: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/90.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function
U(V )U(V ′) = U(exp(ln V ))
U(exp(ln V ′)) = U(ln V )
U(ln V ′)
= f(
VV ′
)= f
(exp(
(ln V
V ′
))= f (ln V − ln V ′)
Log-levels of wealth follow a CARA utility function
![Page 91: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/91.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function
U(V )U(V ′) = U(exp(ln V ))
U(exp(ln V ′)) = U(ln V )
U(ln V ′)
= f(
VV ′
)= f
(exp(
(ln V
V ′
))= f (ln V − ln V ′)
Log-levels of wealth follow a CARA utility function
![Page 92: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/92.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function (ctd.)
U(ln V ) = U(V ) = Ce−z ln V = CV−z =V−z
−z
z = U′′(V )U′(V ) = (z+1)
V or z = zV − 1
More commonly written with γ = z + 1 as U(V ) = V 1−γ
1−γ
Hence γ = zV which is the relative risk aversion
Utility function has Constant Relative Risk Aversion
for γ = 1 (z = 0) this becomes U(V ) = ln V
![Page 93: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/93.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function (ctd.)
U(ln V ) = U(V ) = Ce−z ln V = CV−z =V−z
−z
z = U′′(V )U′(V ) = (z+1)
V or z = zV − 1
More commonly written with γ = z + 1 as U(V ) = V 1−γ
1−γ
Hence γ = zV which is the relative risk aversion
Utility function has Constant Relative Risk Aversion
for γ = 1 (z = 0) this becomes U(V ) = ln V
![Page 94: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/94.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function (ctd.)
U(ln V ) = U(V ) = Ce−z ln V = CV−z =V−z
−z
z = U′′(V )U′(V ) = (z+1)
V or z = zV − 1
More commonly written with γ = z + 1 as U(V ) = V 1−γ
1−γ
Hence γ = zV which is the relative risk aversion
Utility function has Constant Relative Risk Aversion
for γ = 1 (z = 0) this becomes U(V ) = ln V
![Page 95: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/95.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function (ctd.)
U(ln V ) = U(V ) = Ce−z ln V = CV−z =V−z
−z
z = U′′(V )U′(V ) = (z+1)
V or z = zV − 1
More commonly written with γ = z + 1 as U(V ) = V 1−γ
1−γ
Hence γ = zV which is the relative risk aversion
Utility function has Constant Relative Risk Aversion
for γ = 1 (z = 0) this becomes U(V ) = ln V
![Page 96: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/96.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function (ctd.)
U(ln V ) = U(V ) = Ce−z ln V = CV−z =V−z
−z
z = U′′(V )U′(V ) = (z+1)
V or z = zV − 1
More commonly written with γ = z + 1 as U(V ) = V 1−γ
1−γ
Hence γ = zV which is the relative risk aversion
Utility function has Constant Relative Risk Aversion
for γ = 1 (z = 0) this becomes U(V ) = ln V
![Page 97: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/97.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Deriving the utility function (ctd.)
U(ln V ) = U(V ) = Ce−z ln V = CV−z =V−z
−z
z = U′′(V )U′(V ) = (z+1)
V or z = zV − 1
More commonly written with γ = z + 1 as U(V ) = V 1−γ
1−γ
Hence γ = zV which is the relative risk aversion
Utility function has Constant Relative Risk Aversion
for γ = 1 (z = 0) this becomes U(V ) = ln V
![Page 98: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/98.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Definition
CARA and CRRA utility functions are special cases of theHyperbolic Absolute Risk Aversion utility function
U(V ) = (V+V0)1−γ
1−γ
Relative risk aversion isγ = −U′′(V )
U′(V ) V = γ VV+V0
= γ(
1 + V0V
)
![Page 99: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/99.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Definition
CARA and CRRA utility functions are special cases of theHyperbolic Absolute Risk Aversion utility function
U(V ) = (V+V0)1−γ
1−γ
Relative risk aversion isγ = −U′′(V )
U′(V ) V = γ VV+V0
= γ(
1 + V0V
)
![Page 100: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/100.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Definition
CARA and CRRA utility functions are special cases of theHyperbolic Absolute Risk Aversion utility function
U(V ) = (V+V0)1−γ
1−γ
Relative risk aversion isγ = −U′′(V )
U′(V ) V = γ VV+V0
= γ(
1 + V0V
)
![Page 101: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/101.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Properties
If V0 = 0 we have γ = γ and HARA collapses to CARA
If we set γ = +∞ and γ = z HARA becomes a CARAutility function
γ is called the local risk aversion at a risk free investmentof V
![Page 102: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/102.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Properties
If V0 = 0 we have γ = γ and HARA collapses to CARA
If we set γ = +∞ and γ = z HARA becomes a CARAutility function
γ is called the local risk aversion at a risk free investmentof V
![Page 103: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/103.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Properties
If V0 = 0 we have γ = γ and HARA collapses to CARA
If we set γ = +∞ and γ = z HARA becomes a CARAutility function
γ is called the local risk aversion at a risk free investmentof V
![Page 104: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/104.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Different representation
Solving for V0 we get V0 = V(
γγ − 1
)U(V ) = V 1−γ
1−γ
(γγ
)1−γ
γ > γ
![Page 105: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/105.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Different representation
Solving for V0 we get V0 = V(
γγ − 1
)U(V ) = V 1−γ
1−γ
(γγ
)1−γ
γ > γ
![Page 106: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/106.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Different representation
Solving for V0 we get V0 = V(
γγ − 1
)U(V ) = V 1−γ
1−γ
(γγ
)1−γ
γ > γ
![Page 107: MN30380 - Financial Marketspeople.bath.ac.uk/mnsrf/MN30380/Lecture 01.pdf · MN30380 - Financial Markets Andreas Krause Lecture 1 - Utility Theory. Financial Markets Lecture 1 Utility](https://reader030.vdocuments.site/reader030/viewer/2022040701/5d5c597f88c99395318b8446/html5/thumbnails/107.jpg)
FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
1 Utility axioms
2 Expected utility
3 Risk aversion
4 Classes of utility functionsCARA utility functionsCRRA utility functionHARA utility functions
5 Outlook
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FinancialMarkets
Lecture 1
Utility axioms
Expectedutility
Risk aversion
Utilityfunctions
CARA
CRRA
HARA
Outlook
Outlook
We will use utility functions widely to determine the optimalportfolio as of the coming lecture.