Topic 4: Incorporating attitudes to risk, Part A
Shauna Phillips
School of Economics
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AREC3005 Agricultural Finance & Risk
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Dr Shauna Phillips (Unit Coordinator) Phone: 93517892
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› Definition of a “good” decision – not one that turns out to be the “right call” given the states of nature that eventuate, but one that is consistent with DM’s beliefs about probabilities and their preferences.
› How can we represent preferences in decision theory?
› Preferences can be mapped to utilities: utility functions are a compact way to represent
preferences over outcomes. What must we assume about preferences for utilities to work?
› Utility functions should assign utilities such that rules of decision making ensure manager is acting in accordance with own wishes ie maximising utility should lead to actions consistent with preferences.
› Use utility concept in this context first of arose with Bernoulli 1780ish in response to the St Petersburg paradox.
St Petersburg Paradox
› For any gamble (risky decision), eg 50/50 chance of winning $100 or losing $50, people should be willing to pay the expected value of the gamble in order to participate: 0.5*100 – 0.5*50 = $75
Consider a fair coin would be tossed continuously until tails occurs. If tails occurs on the nth toss, you win $2n i.e. tails on the 10th toss, earns $210 = $1042.
What should someone be willing to pay, to participate?
1738 , a mathematician, proposed an explanation.
St Petersburg Paradox & Bernoullian response
Expected utility hypothesis
› To sum up, first developed by Bernoulli 1738, refined by von Neumann and Morgenstern 1940s
› An agent will prefer one risky action a1, over another, a2, if the former has a greater expected utility.
› Utility functions are analytical methods for expressing an agent’s preferences-translate outcomes of risky actions to a real number index of their desirability-serve as basis for decision making.
Expected utility hypothesis- preference axioms
› 1.Ordering and transitivity. For any 2 risky prospects, a1 and a2 that belong to a, DM prefers a1 to a2 or a2 to a1 or is indifferent between them. Transitivity is a logical extension for systems with more than 2 prospects eg a1 , a2 and a3 . If DM prefers a1 to a2 (or is indifferent) and prefers a2 to a3 (or is indifferent), then DM will prefer a1 to a3 (or is indifferent).
› 2.Independence. For any 3 risky prospects, a1,a2 and a3 if the DM prefers a1to a2 , the DM will prefer a lottery made up of a1 and a3 as possible outcomes compared to a lottery made up of a2 and a3 as outcomes, when the probabilities of a1 and a2 are equal.
Expected utility hypothesis hypothesis- preference axioms
› 3.Continuity. For any 3 risky prospects, a1,a2 and a3 if DM prefers a1to a2 , and if DM prefers a2to a3 , then there exists a probability other than 0 or 1, which will make the DM indifferent between a2 for certain, and a lottery composed of a1 (with P=p) and a3 (with P=1-p).
Violations of the axioms
› Violations of the any of the axiom possible
› Arguably most problematic violations is of the independence axiom
› The most famous violation (AKA common consequence effect) is the Allais paradox(Allais, 1953).
Alias Paradox (Source Econport)
› Subjects are asked to choose between the following 2 gambles, i.e. which one they would like to participate in if they could:
: A 100% chance of receiving $1 million.
: A 10% chance of receiving $5 million, an 89% chance of receiving $1 million, and a 1% chance of receiving nothing.
After they have made their choice, they are presented with another 2 gambles and asked to choose between them:
: An 11% chance of receiving $1 million, and an 89% chance of receiving nothing.
: A 10% chance of receiving $5 million, and a 90% chance of receiving nothing.
Alias Paradox (Source Econport)
› Most people prefer A to B, and D to C. So why is this a paradox?
Expected value of A is $1 million, while the expected value of B is $1.39 million. Preferring A to B, people are maximizing expected utility, not expected value. By preferring A to B, we have the following expected utility relationship:
u(1) > 0.1 * u(5) + 0.89 * u(1) + 0.01 * u(0), i.e. 0.11 * u(1) > 0.1 * u(5) + 0.01 * u(0)
Adding 0.89 * u(0) to each side, we get:
0.11 * u(1) + 0.89 * u(0) > 0.1 * u(5) + 0.90 * u(0)
This implies that an expected utility maximizer prefers C to D. The expected value of C is $110,000, the expected value of D is $500,000, so if people were maximizing expected value, they should prefer D to C. However, their choice in the first stage is inconsistent with their choice in the second stage, hence, the paradox.
› Expected utility theory is a theory seeks to explain optimal decisions under risk- EUH applied to “rational” agents.
› Violations of EUH have been shown in empirical applications: e.g.Alias Paradox.
› -Use SEU-flawed responses? SEU maximisation is best as a normative tool, but a poor descriptive tool.
› Alternatives:
– Prospect Theory (how preferences of individuals are inconsistent among same choices, depending on how those choices are presented- see the work of Khanemann and Tversky ).
– Generalised Expected Utility Theory ( )
Expected utility hypothesis
› If the DM’s preferences are consistent with these axioms then a utility function defined on risky prospects exists that can assign a single real number utility value for each prospect.
› Such a function has the following properties:
› 1.If a1 is prefers to a2 , the U (a1) > U(a2 ) and vice versa
› 2.The utility of a risky prospect (aj) is equal to the expected utility of its outcome: U(aj) = E[U(aj)]
› 3.The utility value of each risky prospect is assigned an arbitrary origin and unit of scale.
Expected utility hypothesis
› In essence, the EUH establishes a basis for comparing risky prospects in a way that’s consistent with a DM’s preferences.
› Theorem implies that agents select risky prospects so as to maximise expected utility.
› Flaws in the EUH – many experiments have shown that people routinely violate the behavioural axioms (more on this later). Interested students should see the Allais Paradox and the Ellsberg Paradox.
Utility functions
› In order to identify the attitudes towards risk, we must start with the individual’s utility function over payoffs
– We don’t need to go into the methods of eliciting utility functions, because our interest is in the implications for decision making
– If you are interested in these methods, then the first few pages of HHA2004 Chapter 6 provide a good overview
› Utility is just the expression of an individual’s preferences over stuff
– In our case, we will be interested in utility over payoffs, which we can think of as
monetary payments w
› Utility is a unit-less concept
– We can say outcomes with a higher utility are preferred to those with lower utility, but not by how much
– Usually, we will express utility as values between zero and one, with higher values preferred to lower values
Given the existence of the utility function,
› What does it look like?
› Often expressed as function of wealth, or income.
› Utility of given action is then considered in the context of the resulting increase in wealth or income.
› To represent certain classes of economic behaviour and risk preferences we need to understand the shape and curvature of the function
Utility function
Utility function
Utility function
› The shape of the utility function over payoffs reflects the preferences of the decision-maker
› Upward sloping curves suggest higher payoffs are preferred to lower payoffs
– Usually the case for money
– Not so much the case for stuff where some degree of satiation is possible
(e.g. food, beer)
– We might expect to see an initial upward slope, followed by a downward sloping section
EU utility function
DM will choose 1 as it always yields at least as much as 2.
EU utility function example (Rasmussen)
V(450) = 0 .7 v(2300) + 0.3 v(-600) = 0 .7 *1+ 0.3 *0= 0.7
Utility function
› We can formally describe these features (ie the way shape represents preferences) with respect to the derivatives of the utility function, for utility U as a function of payoffs (or wealth) w
– The first derivative of an upward sloping function would be: U = f (w)
– That is, higher payoffs are associated with higher utility
Utility function
Utility function
› We can go further and use the second derivative to describe the curvature of the utility function:
– Convex-increasing:
– Concave-increasing:
– Something in-between:
d U = 0 dw2
Utility function
Utility and risk attitudes
› The second derivative is used to tell us about the preferences towards risk:
– Convex-increasing suggests risk loving:
– Concave-increasing suggests risk aversion:
– Something in-between suggests risk indifference:
Why does curvature indicate attitude towards risk?
› Consider a situation in which we have a 50:50 gamble, with payoffs of $5 and $25:
EV = 0.5 × $5 + 0.5 × $25 = $15
› Let’s say, initially, we also have a utility function which is concave- increasing in payoffs
Why does curvature indicate attitude towards risk?
Expected utility
Why does curvature indicate attitude towards risk?
› Point A is the utility of the $5, Point B is the utility of the $25
– So, point C is the expected utility of the gamble, which is approximately 0.8
– Note that it is not equal to the utility of the expected [money] value of the gamble, which is $15
› The question is, what certain payoff would yield equivalent utility to the expected utility from the gamble?
– We can think of this payoff as the certainty equivalent
Why does curvature indicate attitude towards risk?
Why does curvature indicate attitude towards risk?
› The utility of the certainty equivalent (point D) should be equal to the expected utility of the gamble (point C)
– If the certainty equivalent (CE) (point D) payment is less than the expected value (EV) (payoff corresponding to point C) of the gamble, the DM is risk-averse
› In general:
– If CE < EV, they are risk-averse - If CE > EV, they are risk-loving – If CE = EV, they are risk-neutral
CE > EV with convex-increasing utility
Risk-premium
› We often call the gap between the certainty equivalent (CE) and the expected value (EV) the value of the risk-premium (RP)
– Risk-averse DM has a positive RP: RP = EV – CE > 0 – Risk-loving DM has a negative RP: RP = EV – CE < 0 - Risk-neutral DM has a zero RP: RP = EV – CE = 0
› The value of the risk premium tells us the minimum amount of money the individual needs to be paid to take on the gamble.
› So, a risk-averse DM would accept a riskless amount that’s lower than an Expected amount from a risky project (CE
› Where c is known as the coefficient of absolute risk aversion
› This functional form has the property of constant absolute risk aversion (CARA)
Utility function options
› First and second derivatives of the utility function are then:
dU = ce−cw d U = −c2e−cw
R (w)=−−c2e−cwce−cw=c
Rr (w) = w Ra (w) = wc
Utility function options
› We can also have logarithmic and power functional forms, for w > 0 U = ln(w)
1 U =1−rw
– Where r is some value between 0 and 1
› These functional forms have the property of constant relative risk aversion
Exponent utility function
Empirical application (Kuu & Weber, 2012) )
A crucial determinant of the demand for insurance is risk aversion. A risk averse individual values a risky prospect at less than its expected value and is thus willing to pay for insurance that grants certainty of income. Risk aversion is a property of the utility function that can be measured by calculating the income elasticity of marginal utility:
R(Y)=−dU'(Y) Y =−YU”(Y) dY U'(Y) U'(Y)
Risk premium (ρ) = amount of income one is willing to give up in to be indifferent between the certain insured income and the expected uninsured income.
Via a Taylor series expansion, ρ can be related to R(Y) :
−12 YU”(Y)=1R2 Y 2Y2 U'(Y) 2 Y
An empirical model is based on the equation above:
› Model farmers’ willingness to pay for insurance that protects against wheat yield variability is modelled.
› Hail insurance has the effect of guaranteeing a certain yield in t/ha for the farmer’s crop in the event of damage.
› The model estimates the coefficient of relative risk aversion and the impact that the Federal Government’s ECIRS program has on willingness to pay for hail insurance.
› Dependentvariable:hailinsurancepremiumasapercentageofmeancropvalue.
› Risk is measured as the squared coefficient of variation of wheat yield in t/ha,. The estimated value of
the slope parameter b2 gives an estimate of the coefficient of relative risk aversion R. ›
=b +bECIRS+b2 +u 012Y
Results (NB R2=0.25 and 15 d.f.)
h = 0.39 − 0.22ECIRS + 1.3Y2 Y (0.14) (0.12) (0.47)
WTP for crop insurance depends on yield variability.
B2 estimate implies an R of 2.6-moderately strongly risk aversion
Revisit risk decision example [HHA2004 pp.25/122]
Revisit risk decision example [HHA2004 pp.25/122]
Farmer’s utility function
› Let’s say that the farmer’s utility function is represented by the expression:
Where c = 0.003658
› Taking to payoffs at the terminal nodes in the decision tree above, we can now calculate the corresponding utilities
Farmer’s payoffs as utilities
U =1−e−0.003658w
500.0 0.8394 492.8 0.8351 490.0 0.8334 300.0 0.6663
Replace payoffs with calculated utilities
We also have some information about the likelihoods of outcomes. For example, if there is an outbreak, there is then probability of 0.5 of only bans arising and 0.5 of mandated slaughter.
Rationalise decision tree by calculating expected utility
› Step 2: We work from outcome nodes and rationalise backwards
› To start, if we have an outbreak, there is a 50:50 chance of ‘ban only’ or
‘mandated slaughter’:
E (U ) = 0 . 5 U ( B a n s ) + 0 . 5 U ( S l a u g h t e r ) E(U )= 0.50.8334+ 0.50.6663= 0.7499
Rationalise decision tree by calculating expected utility
And, then:
Rationalise decision tree by calculating expected utility
› Given that we have worked backward to our initial decision node, we can make a judgement based on the relevant expected utilities of one action over another
– So, we would take out insurance
› We can calculate the certainty equivalent of each option by
rearranging the utility function:
U =1−e−0.003658w
w= ln(1−U) − 0.003658
Rationalise decision tree by calculating expected utility
ln(1−U*) − 0.003658
› So, the certainty equivalent for the ‘don’t insure’ option is then:
w = ln (1 − 0 . 8 3 4 0 ) = 4 9 0 . 9 2
dont_insure
− 0.003658 = 492.8
Process summary
1. Assign probabilities to each event branch, making sure these are coherent with probability principles
2. Calculate money payoffs for each terminal node
3. Convert payoffs to utility values using decision-maker’s utility function
4. Working backwards, rationalise the branches by calculating the expected utilities
5. Resolve decision forks by selecting the option with the highest expected utility or certainty equivalent value
› Bardsley, P. & M. Harris (1987). “An Approach To The Econometric Estimation Of Attitudes To Risk In Agriculture,” Australian Journal of Agricultural and Resource Economics, Australian Agricultural and Resource Economics Society, vol. 31(2), pages 112-126, August.
› Dornan,M.andF.Jotzo(2013).RenewableTechnologiesandRiskMitigationinSmallIslandDeveloping States: Fiji’s Electricity Sector.
› Econport.(2020).Decisionsunderuncertainty.Available:http://econport.org/content/handbook/decisions- uncertainty.html
› Hardaker,HuirneandAnderson(2004)CopingwithRiskinAgriculture,CABInternational.
– Chapter 5
› Khuu, A. & Weber, E.J. (2012) How Australian farmers deal with risk. Discussion paper 12.07. Business School University of Western Australia
› Rasmussen, S. Optimisation of Production Under Uncertainty. State-contingent Approach. Chapter 3.
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