15.
481x – FINANCIAL MARKET
DYNAMICS AND HUMAN
BEHAVIOR
Utility &
Expected Utility
PREFERENCES
Preference Relation:
• A binary relation ≽ over a set of choices.
• E.g., if you like pizza more than salad, you would write
Pizza ≽ Salad
Common Assumptions on ≽ :
• Complete: Given two alternatives A and B, you can always decide
which one you prefer to the other.
• Reflexive: You are indifferent between something and itself: A ≽ 𝐴.
• Transitive: if you prefer A to B, and B to C, then you should prefer A
to C.
Are these assumptions realistic?
UTILITY FUNCTIONS
It’s hard to deal with preferences… How about we assign a “score” to
each choice we have?
Utility Functions:
• A function 𝑈 ⋅ does exactly that!
• If I like Pizza more than Salad, then the score of Pizza should be
higher than the score of Salad:
𝑈 𝑃𝑖𝑧𝑧𝑎 ≥ 𝑈(𝑆𝑎𝑙𝑎𝑑)
• E.g., I could assign the scores:
𝑈 𝑃𝑖𝑧𝑧𝑎 = 10 utility points
𝑈 𝑆𝑎𝑙𝑎𝑑 = 5 utility points
Does the magnitude of the score really matter?
PROPERTIES OF UTILITY FUNCTIONS
How should utility functions look like?
Properties:
The function 𝑈 ⋅ should (ideally!) be:
• Strictly increasing: 𝐴 ≻ 𝐵 ⟹ 𝑈 𝐴 > 𝑈(𝐵) 𝑈! ⋅ > 0
• Concave: you prefer $1,000$ to $0, and $101,000 to $100,000, but
when is your preference stronger? 𝑈 !! ⋅ < 0
“Diminishing Marginal Utility”
è
BUT WHEN LIFE IS UNCERTAIN…
Expected Utility:
• Given your utility function 𝑈 𝑋 …
… but X is random!
• Then your expected utility is:
E 𝑈 𝑋 =𝑈 𝑎 ×𝑃 𝑋 =𝑎 +𝑈 𝑏 ×𝑃 𝑋 =𝑏 + …
• Example: 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = log 100 + 𝑊𝑒𝑎𝑙𝑡ℎ
100$, 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 25%,
and 𝑊𝑒𝑎𝑙𝑡ℎ = E
−20$, 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 75%,
𝐸 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = log 100 + 100 × 0.25 + log 100 − 20 × 0.75
= 0.25 × log 200 + 0.75 × log 80
= 4.61 utility points
15.481x – FINANCIAL MARKET
DYNAMICS AND HUMAN
BEHAVIOR
Risk Aversion
EXAMPLE 1:
Example 1:
You have a utility: 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = log 100 + 𝑊𝑒𝑎𝑙𝑡ℎ
Choice 1: you get 10$ for sure
Choice 2: you get 100$ with prob. 25%, and lose 20$ with prob. 75%
1. 𝐸 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = log 100 + 10 × 1 = 4.70 utility points
2. 𝐸 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = log 100 + 100 × 0.25 + log 100 − 20 × 0.75
= 0.25 × log 200 + 0.75 × log 80
= 4.61 utility points
So which choice do you pick?
Note: 10$ = 0.25 x 100 + 0.75 x -20, so… why aren’t you indifferent?
EXAMPLE 2:
Example 2:
You have a utility: 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = 100 + 𝑊𝑒𝑎𝑙𝑡ℎ
Choice 1: you get 10$ for sure
Choice 2: you get 100$ with prob. 25%, and lose 20$ with prob. 75%
1. 𝐸 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = 100 + 10 × 1 = 110 utility points
2. 𝐸 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = 100 + 100 × 0.25 + 100 − 20 × 0.75
= 0.25 × 200 + 0.75 × 80
= 110 utility points
Why are you indifferent now?
The utility function you pick accounts for your risk-aversion!
VISUALIZING RISK AVERSION
Recall:
• In example 1, you had a utility: 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = log 100 + 𝑊𝑒𝑎𝑙𝑡ℎ
• In example 2, you had a utility: 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = 100 + 𝑊𝑒𝑎𝑙𝑡ℎ
Choice 1: you get 10$ for sure.
Choice 2: you get 100$ with prob. 25%,
and lose 20$ with prob. 75%.
Why are you indifferent between the gamble and the sure choice in
Example 2 but not in Example 1?
VISUALIZING RISK AVERSION
Choice 1: you get 10$ for sure
Choice 2: you get 100$ with prob. 25%,
and lose 20$ with prob. 75%
𝑼 𝑾𝒆𝒂𝒍𝒕𝒉 = 𝟏𝟎𝟎 + 𝑾𝒆𝒂𝒍𝒕𝒉 𝑼 𝑾𝒆𝒂𝒍𝒕𝒉 = 𝐥𝐨𝐠 𝟏𝟎𝟎 + 𝑾𝒆𝒂𝒍𝒕𝒉
Wealth
Wealth
Your expected utility is lower than
Risk Premium!
the utility of the expected outcome!
CERTAINTY EQUIVALENT
Choice 1: you get 10$ for sure
Choice 2: you get 100$ with prob. 25%,
and lose 20$ with prob. 75%
Example:
• In example 1, you had a utility: 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = log 100 + 𝑊𝑒𝑎𝑙𝑡ℎ
• Choice 2: 𝐸 𝑈 𝑊𝑒𝑎𝑙𝑡ℎ = 4.61 utility points
Choice 3: I give you x$ for sure.
How should I pick x to make sure that you are indifferent between Choice 2
and Choice 3?
Solve: log 100 + 𝑥 = 4.61 ⟹ 𝑥 = 0.59
So this gamble is equivalent to me giving you 0.59$ for sure!
For you, it’s worth much less than 10$, so you don’t want the gamble.
ABSOLUTE VS RELATIVE RISK AVERSION
Utility Function: 𝑈(𝑥)
Absolute Risk Aversion Coefficient:
𝑈′′(𝑥)
𝐴(𝑥) = −
𝑈′(𝑥)
Relative Risk Aversion Coefficient:
𝑈 NN 𝑥
𝑅(𝑥) = − N × 𝑥
𝑈 𝑥
=𝐴 𝑥 × 𝑥
CONSTANT ABSOLUTE RISK AVERSION:
CARA UTILITY
Utility: 𝑈 𝑥 = −𝑒 OP⋅R , for 𝑎 > 0 .
𝐴 𝑥 = 𝑎 is your Absolute Risk Aversion coefficient!
Proof: 𝑈′′(𝑥)
𝐴(𝑥) = −
𝑈 ! 𝑥 = 𝑎 𝑒 "#⋅% 𝑈′(𝑥)
𝑈 !! 𝑥 = −𝑎& 𝑒 "#⋅% 𝑅 𝑥 =𝐴 𝑥 ×𝑥
'!! % "#" ( #$⋅&
⟹𝐴 𝑥 = − ! = − =𝑎
' % # ( #$⋅&
⟹ The absolute risk aversion coefficient is constant!
CONSTANT RELATIVE RISK AVERSION:
CRRA UTILITY
𝑈′′(𝑥)
𝐴(𝑥) = −
log 𝑥 , 𝑖𝑓 𝛾 = 1, 𝑈′(𝑥)
Utility: 𝑈 𝑥 = M \ \O] , 𝑖𝑓 𝛾 > 1. 𝑅 𝑥 =𝐴 𝑥 ×𝑥
\O]
⋅ 𝑥
R 𝑥 = 𝛾 is your Reative Risk Aversion coefficient!
Proof: (case of 𝛾 > 1) Proof: (case of 𝜸 = 𝟏)
"
𝑈 ! 𝑥 = 𝑥 ") 𝑈! 𝑥 = #
"
𝑈 !! 𝑥 = −𝛾 𝑥 ")"* 𝑈 !! 𝑥 = − # !
'!! % ") %#'#( ) $ "" # %"/# ! "
⟹𝐴 𝑥 = − ! =− = ⟹𝐴 𝑥 =− =− =
$" # "/# #
' % %#' % "
) ⟹𝑅 𝑥 =𝐴 𝑥 ×𝑥 = ×𝑥 =1 =𝛾
⟹𝑅 𝑥 =𝐴 𝑥 ×𝑥 = ×𝑥 =𝛾 #
%
⟹ The relative risk aversion coefficient is constant!
