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5. Heuristics, Utility theory (expected utility, expected outcome), certainty equivalent, risk premium and risk preferences, Arrow-Pratt coefficient of risk aversion (absolute, relative, CARA, DARA etc.), different v-N.M. utility functions
HEURISTICS FOR DECISIONS UNDER UNKNOWN PROBABILITIES
▪ Maximin: maximize the minimum payoff
▪ Maximax: maximize the maximum gain
▪ Least regret rule (Savage-Niehans): max regret I could have by choosing this
▪ Hurwicz rule:
EXPECTED UTILITY THEORY
Formula part
Notations
Comments
Actions
𝑎𝑠
to be taken by decision maker (GER: Handlungsalternative), state dependent (𝑠)
Events
𝑒𝑖
occur with Pr 𝑒𝑖 , 𝑖 = 1,2,3 ... states of nature: 𝑎𝑠 = 𝑎 𝑒𝑖
Outcomes
Z 𝑎 𝑒𝑖
combination of event and action, gives consequence based on action
Payoffs
𝐸𝑈 𝑍
decision maker values the pay-offs in terms of utility
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An example-coin toss: heads→100€, tails→0€ 𝑒𝑖 = {h𝑒𝑎𝑑𝑠 𝑡𝑎𝑖𝑙𝑠
• 𝑎 (h𝑒𝑎𝑑𝑠)= 100€, 𝑎(𝑡𝑎𝑖𝑙𝑠)= 0€
• Pr (𝑎 (h𝑒𝑎𝑑𝑠))= Pr (𝑎 (𝑡𝑎𝑖𝑙𝑠))= 0.5
• Wecanwrite:𝐸𝑈(𝑎)= 0.5𝑈(𝑎(h𝑒𝑎𝑑𝑠))+0.5𝑈(𝑎(𝑡𝑎𝑖𝑙𝑠))=0.5𝑈(0)+0.5𝑈(100)
• Note: E(U) model is linear in probabilities
▪ Monetary rewards are represented by a random variable Z 𝑎 𝑒𝑖 = Z
▪ Decision maker decides between risky prospects, has preferences towards these actions:
𝑎1~𝑎2, 𝑎1 ≺ 𝑎2 or 𝑎1 ≻ 𝑎2
▪ Link to observed decisions: 𝑎1 is chosen over 𝑎2 if 𝑎1 ≿ 𝑎2
▪ Decisions affect the distribution of the outcome, depending on the states of nature: Z 𝑎 𝑒𝑖
▪ Payoff-valuation: in terms of utility:
Preferences towards risky choices represented with a utility function 𝑈 𝑍 →typically: von Neumann Morgenstern utility function
▪ Decisions based on expected utility
▪Defined:𝐸𝑈 𝑍 =σ𝑖≥1𝑈 𝑍(𝑎 𝑒𝑖 Pr 𝑎 𝑒𝑖 ,
where E is the expectation operator based on subjective probability distributions of Z(𝑎(𝑒𝑖 )) (random reward)
▪ Rational decision maker: will maximize the expected value of her/his utility: expected utility function EU
▪Thisis :max𝐸[𝑈 𝑍 𝑎 𝑒𝑖 =max σ𝑖≥1𝑈 𝑍(𝑎 𝑒𝑖 Pr 𝑎 𝑒𝑖
Preferences
Relative Risk Premium: 𝑅𝑅𝑅𝑅 = 𝑅𝑅𝑅𝑅 is the proportion (ratio) of the risk premium in relation to expected
(terminal) wealth
Arrow-Pratt coefficient of relative risk aversion
𝐸𝐸(𝑥𝑥)
′′
𝑉𝑉̅ ≡ 𝑉𝑉 � 𝑍𝑍 = − 𝑈𝑈 /𝑈𝑈𝑈 𝑍𝑍
16
RISK PREFERENCES AND LEVEL OF INCOME
r can be constant, increasing or decreasing in Z (changes in r with higher absolute Z):
Constant Absolute Risk AversionCARA Increasing Absolute Risk AversionIARA Decrasing Absolute Risk AversionDARA
Similar for 𝑉𝑉̅ (proportional changes in 𝑉𝑉̅ to Z): CRRA, IRRA, DRRA
RP sign is not enough to understand all complexity of DM beavior. ARA and RRA helps to diffentiate behavior of DM based on Z variance, Z is initial wealth for example
r can be constant, increasing or decreasing in Z (changes in r with higher absolute Z):
Similar for rr (proportional changes in rr to Z):
Type of function
Formula
ARA (r)
RRA(rr)
Linear
U(x)=a+bx,b>0.
r(x)=0 and R(x)=0 → the decision maker is risk neutral.
r(x)=0 and rr(x)=0 → the decision maker is risk neutral.
Log
U(Z) = α-β*ln(Z)
where z=x, and Z>0, and β<0
Example: ln(0.5x)
r=1/Z or 1/x
DARA
Example answer: 1/x
rr = 1
CRRA
Example answer: 1, cause 1/x*x =1
willingness to take on risk (or willingness to insure) as a proportion of their wealth remains constant
And even Log with exponenta
In case of
U(Z) = α-β*ln(Z^k)
where k is exponenta
like ln(a*x^k)(with a>0,k>0)
r=1/Z or 1/x
DARA
Example answer (regardless of k): 1/x
rr=1
CRRA
Example answer (regardless of k): 1
willingness to take on risk (or willingness to insure) as a proportion of their wealth remains constant
Power
U(Z) = Z^1-θ/1-θ
where z =x
and Z,θ>0, and θ do not equals 1
Example:U(x) = -x^-1
r=θ/Z
DARA
Example: U(x) = -x^-1=(x^1-2)/(1-2)=x^-1/-1=-x^-1
then theta θ=2
then ARA (r) = 2/x
rr=θ
CRRA
Example: as θ=2
then RRA (rr) = 2
willingness to take on risk (or willingness to insure) as a proportion of their wealth remains constant
- Root as subtype of Power above
U(Z)=Z^0.5
Example: x^0.5
r=θ/Z
Example: theta θ=1/2
r=1/2*1/x=1/2x
rr=θ
Example: as theta θ=1/2, then rr=1/2
Exponential
U(Z) = α-βe^𝜆Z
where z=x, and β,𝜆>0
Example:U(x) = -e^-2x
r=𝜆
CARA
Example: 𝜆=2, then then r=2
rr=𝜆x
IRRA
Example:𝜆=2, then then rr=2x
Implies that the proportion of wealth (z or x) an individual is willing to put at risk decreases as their wealth increases. Wealthier individuals are more averse to a proportional risk; they put a smaller percentage of wealth into risky assets, the empirical evidence is still debated and inconclusive.
Quadratic (parabola)
U(Z) = α+β*Z-γ*Z^2
where
Example: U(x) = 2x - 0.01x^2
r=2γ/(β−2γZ)
IARA
Example: γ=0.01, β=2
r=2*0.01/(2-2*0.01x)=0.02/(2-0.02x)
it is often excluded in practice as "too unrealistic" due to its IARA property, which means individuals become more risk-averse as they get wealthier. I.e. wealthier individual would be willing to pay more to avoid a specific absolute amount of risk (e.g., a $1,000 loss) than a less wealthy individual. This implies that private wealth accumulation and insurance motives behave as complements
rr=(2γ/(β−2γZ))Z=2γZ/(β−2γZ)
IRRA
Example: as r =0.02/(2-0.02x)
rr=(0.02/(2-0.02x))*x =0.02x/(2-0.02x)
Implies that the proportion of wealth (z or x) an individual is willing to put at risk decreases as their wealth increases. Wealthier individuals are more averse to a proportional risk; they put a smaller percentage of wealth into risky assets, the empirical evidence is still debated and inconclusive.
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Example lecture, heristics
Example, taken from Musshow/Hirschauer 2009: Modernes Agrarmanagement. 4. Auflage
maximin = A2 Rye / Rent
maximax = A1 Wheat
Heurits = A2 Rye
What is the preferred option, using the Maximin, Maximax, Hurwicz, Least-regret rules?
Least regrets - A1 Wheat
R1 Payoff matrix Z s;j
x1: Low
R2 Payoff matrix Z s;j
x2: Medium
R3 Payoff matrix Z s;j
x3: High
R4 Decision Rules
Maximin-
R5 Decision Rules
Maximax-
R6 Decision Rules
Maximax-Hurwicz 𝜆=0.25
R7 Decision Rules
Savage-Niehans-
A1: Wheat
12
20
28
12 (3)
28 (1)
16,0 (2)
4 (1)
A2: Rye
16
18
21
16 (1)
21 (2)
17,3 (1)
7 (2)
A3: Rent it out
16
16
16
16 (1)
16 (3)
16,0 (2)
12 (3)
Probability Pj
?
?
?
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HEURISTICS FOR DECISIONS UNDER UNKNOWN PROBABILITIES - CRITICISM
▪ Only provides decision rules “cooking recipes”
▪ Answers might be inconsistent
▪ No theoretical foundation
▪ Not clear what underlying preferences are
Example lecture, sure income, random reward, including graph CE, RP
▪ Get some intuition why risk utility function represents risk preferences
▪ Stick to risk aversion
Illustration
Consider the following situation with two alternatives:
▪ 𝑃𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝐴𝑐𝑡𝑖𝑜𝑛 1 (𝑎1) :sure income, denote by 𝑍0 ,which gives: 𝐸 𝑍 𝑎1 = 𝑍0
▪ 𝑃𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝐴𝑐𝑡𝑖𝑜𝑛 2 (𝑎2) :income is random reward, you know with 50% probability you get 𝑍+ and 𝑍− , respectively
that is: Pr(𝑍+) = Pr(𝑍−) = 0.5
but same expected value 𝐸 𝑍 𝑎2 = 𝐸 𝑍 𝑎1 (15 euro) = 𝑍0
▪ So for example 15 (a1) = 0.5*10 + 0.5*20 (a2) =5+10 = 15
𝑎1: sure income, denote by 𝑍0, which gives: 𝐸 𝑍 𝑎1 = 𝑍0
𝑎2: random reward, but equally likely
Now:
1.) Take 5min yourself to understand the graphical representation on the next two slides.
2.) Then make groups (all people in a row) – try to explain the graph to each other in the groups.
How many imes you play and what is my goal? + what ius my financial resilience? Can I let myself loose 5 euro?
E[U(z1))=U(Z0) higher?
E[U(Z2)] = U(CE)
Example lecture E(U) on agriculture example Wheat, Rye, Rent out
▪ There are 3 options to be considered by the farmer: grow mainly wheat, mainly rye, or rent out the land
▪ Recall: heuristics delivered no clear result*
▪ Presume exponential risk utility function:
𝑈(𝑍) =𝛼−𝛽∙exp( −𝜆∙𝑍) with 𝛼=5, 𝛽=200 and 𝜆=0.3 (we can find out parameteres for risks for each preferences with survey or games)
▪ To get E(U) = 0.3*U(Zlow)+0.5*U(Zmed)+0.2*U(Zhigh)
Action
precipitation
Payoff matrix
Low Pr=30%
Payoff matrix
Med Pr=50%
Payoff matrix
High Pr=20%
Statistics
Variance
Statistics
Exp.value
Exp. Utility
A1: Wheat
12
20
28
31.36
19.20
3.10
A2: Rye
16
18
21
3.00
18.00
3.98
Rent out
16
16
16
0.00
16.00
3.35
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▪Get CE by using inverse of the utility function: 𝑍 𝑈 = −ln
▪Evaluate inverse at 𝐸 𝑈 : use 𝑍(𝐸(𝑈)) = 𝐶𝐸
▪E.g. for action rye: 𝑍 3.98 = −ln 5−3.98 1 = 17.60
Action
precipitation
Payoff matrix
Low Pr=30%
Payoff matrix
Med Pr=50%
Payoff matrix
High Pr=20%
Statistics
Variance
Statistics
Exp.value
Exp. Utility
CE
A1: Wheat
12
20
28
31.36
19.20
3.10
15.53
A2: Rye
16
18
21
3.00
18.00
3.98
17.60
Rent out
16
16
16
0.00
16.00
3.35
16.00
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▪Get the RP by difference between expected value and certainty equivalent
▪I.e.: 𝑅𝑃 = 𝐸(𝑍) − 𝐶𝐸
▪E.g. 𝑅𝑃_𝑟𝑦𝑒 = 18 − 17.60 = 0.4
Action
precipitation
Payoff matrix
Low Pr=30%
Payoff matrix
Med Pr=50%
Payoff matrix
High Pr=20%
Statistics
Variance
Statistics
Exp.value
Exp. Utility
CE
RP
A1: Wheat
12
20
28
31.36
19.20
3.10
15.53
3.67
A2: Rye
16
18
21
3.00
18.00
3.98
17.60
0.40
Rent out
16
16
16
0.00
16.00
3.35
16.00
0.00
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Trial exam 2025 Q1.1 + Q3 + Q4
Given:
o True
o False
o We cannot say it depends on the certainty equivalent
If the risk premium = 0, the decision maker is risk neutral, linear
If the risk premium > 0, the decision maker is risk averse, convex
If the risk premium < 0, the decision maker is risk loving, concave
What needs to be answered here:
Visualize task with scheme, picture, short mathematical writing or table:
Formulas to use:
Solution:
If the ARA or r or risk premium > 0, the decision maker is risk averse, convex
If the ARA or r or risk premium < 0, the decision maker is risk loving, concave
Given:
What needs to be answered here:
Visualize task with scheme, picture, short mathematical writing or table:
Solution:
Trial exam 2025 Q6
Trial exam 2025 Q7
Practical 3, A1: Drought risk in crop production
Practical 3, A2: Start-up investment
Practical 3, A3: Coefficients of risk aversion
Practical 3, A4: Certainty equivalent
Charlie owns a house worth 300,000€. There is a probability equal to 0.05 that the house will burn
down completely in a fire. He can insure his house against a loss from this fire. The premium for
full insurance is 17,000€. Additionally Charlie has wealth equal to 100,000€ in non- risky assets.
1. If he had v.N-M utility function 𝑢(𝑥) = √𝑥 ,would he buy this full insurance policy? Which is
the maximum premium that he is willing to pay for full insurance?
2. If, instead, he had v.N-M utility function 𝑢(𝑥) = 𝑙𝑛(𝑥), which would be the maximum
premium that he would be willing to pay for full insurance?
3. How do these findings correspond with your findings in Q2?
→ Without insurance: 𝐸(𝑥) = 400,000 * 0.95 + 100,000 * 0.05 = 385,000
→ With insurance: 𝐸(𝑥) = 400,000 − 17,000 = 383,000 →From having the insurance, the
expected wealth is reduced by 2000€. Thus, this is not a ‘fair’ premium. If the
decision maker would not be risk averse, she would not buy the insurance!
thus 𝑈(𝐸(𝑥)) = √385000 = 620,48
The maximum premium m is:
𝐸(𝑈(𝑥)) = 616.64 and 𝑢(𝑥) = √𝑥
thus 𝑥 = 616.642 = 380,244.89
𝑅𝑃 = 385,000 − 380,244.89 = 4,755.11
As this additional value given by the risk averse decision maker is larger than the
‘price’ in terms of reduced wealth, she would buy the insurance.
2. → 𝐸(𝑈(𝑥)) = 0.95𝑙𝑛(400000) + 0.05𝑙𝑛(100000) = 12.83 , and 𝑈(𝐸(𝑥)) = 𝑙𝑛(385000) = 12.860 he would also buy that insurance policy when 𝑢(𝑥) = 𝑙𝑛(𝑥)
The maximum premium m is:
𝐸𝑈(𝑥) = 12.83 and 𝑢(𝑥) = 𝑙𝑛(𝑥) thus 𝑥 = 𝑒12.83 = 373248.61
𝑅𝑃 = 385,000 – 373,248.61 = 11,751.39 The individual is willing to pay more for full insurance
when 𝑢(𝑥) = 𝑙𝑛(𝑥).
3. An individual with 𝑢(𝑥) = 𝑙𝑛𝑥 is more risk averse than an individual with and 𝑢(𝑥) = √𝑥, as
ARA[ln(𝑥)] =
1
𝑥
>
1
2𝑥 =ARA(√𝑥), and, therefore, is willing to pay more for full insurance (the risk
premium of the risky alternative is greater for an individual with 𝑢(𝑥) = √𝑥 than for an
individual with 𝑢(𝑥) = 𝑙𝑛(𝑥)).
Given:
What needs to be answered here:
Visualize task with scheme, picture, short mathematical writing or table:
Formulas to use:
where:
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