Shauna Phillips
School of Economics
Quantifying uncertainty(II)
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AREC3005 Agricultural Finance & Risk
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Dr Shauna Phillips (Unit Coordinator) Phone: 93517892
R479 Merewether Building
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Recall the disease outbreak problem
› Decision tree
Subjective probabilities
don’t insure
no outbreak (P=0.94) $500
bans only (P=0.5)
P(outbreak & bans=0.03) $490
don’t insure
outbreak (P=0.06)
› E(net asset | insured)= 0.94*492.8+0.06*492.8=$492.8
› E(net asset | uninsured)= 0.94*500+0.03*490+0.03*490=$493.7
$300 Slaughter (P=0.5)
P(outbreak & slaughter=0.03)
Simple risk decision example (cont)
› Repeat the CE elicitation exercise to further simplify the tree (at the first decision node)
› CE must be between $500,000 and $384,000
– If the CE is less than $492,800 → take out insurance – If the CE is more than $492,800 → take the gamble
Elicit CE for this gamble
Recall the disease outbreak problem
› Decision tree
no outbreak (P=0.94) $500
don’t insure
outbreak (P=0.06)
› E(net asset | insured)= 0.94*492.8+0.06*492.8=$492.8 › E(net asset | uninsured)= 0.94*500+0.06*384=$493.04
Using CE version of the same problem
Subjective probabilities
don’t insure
no outbreak
A decision based on EMV indicates not insuring is optimal. But what if the farmer is uncertain about own priors (subjective) beliefs (probabilities)? What if a forecast is obtainable for disease outbreaks?
Accounting for new information
› Usually, we are presented with situations in which the initial information about uncertain events is incomplete and we might be able to gather more information:
– Purposeful experimentation, obtaining forecasts, recording more events as they occur
– How should we revise our prior probabilities to account for new information?
Probability updating with Bayes’ theorem
› Often called Bayesian Inference
– A standard method for updating probabilities as information is acquired
› New information may be summarised as a new set of probabilities – Say, data from an experiment
– These data may be used to make some projections of future outcomes
Probability updating with Bayes’ theorem
Bayes’ theorem
Example: Foot and mouth disease [HHA2004 pp.25]
› Continuation of example from earlier in book, which involved the farmer making a choice of whether or not to take out insurance against losses related to FMD outbreak
› Let’s say the farmer initially believes the probabilities are: – Outbreak: 6%
– No outbreak: 94%
› These are called prior probabilities
– That is, those elicited before subsequent information is acquired
Example: Foot and mouth disease [HHA2004 pp.25]
Example: Foot and mouth disease [HHA2004 pp.25]
› The accuracy of the group’s predictions can be judged by evaluating past performance of predictions against actual events
› These are (Zk|Si):
Example: Likelihood probabilities
Example: Probability tree
Example: Joint probabilities
Example: Joint probabilities
Likelihood Joint
Example: Marginal probabilities
Example: Posterior probabilities
Example: ‘Flipped’ probability tree
Example: ‘Flipped’ probability tree
Example: ‘Flipped’ probability tree
Example: ‘Flipped’ probability tree
We really care about these probabilities
Example: Posterior probabilities
Example: Posterior probabilities
Event | Prediction
Marginal probability
Joint probability
Posterior probability
Example: Posterior probabilities
Bayesian updating process
Bayesian updating process
When the forecast is made-comparison
Subjective probabilities
Posterior Probs – outbreak possible
don’t insure
don’t insure
Posterior Probs – outbreak unlikely
don’t insure
P(S|Z ) P(Si)i 1
Posterior Probs – outbreak probable
don’t insure
P(PS(S) |Z ) ii2
P(S|Z ) ii3
What drives these results?
Comparison (using CE values from last week’s lecture)
forecast unlikely
Subjective probabilities
Posterior Probs – outbreak unlikely
don’t insure
don’t insure
ak0.989474
forecast possible
forecast probable
Posterior Probs – outbreak possible
Posterior Probs – outbreak probable
P(S|Z ) P(Si)i 2
don’t insure
P(S) P(Si i|Z3)
don’t insure
ak0.921569
ak0.758065
What drives these results?
› If a forecast that the outbreak is unlikely, don’t insure.
› If a forecast that the outbreak is possible or probable, insure.
› Although priors are strong, results driven by the accuracy of the forecast tool- likelihood probabilities.
› We can used these revised probabilities in decision making analysis.
› We can also determine a value of incorporating this information. How?
– Essentially calculate the expected value/utility of the problem before incorporating the information (ie the problem based on prior probabilities). Then, the value of the information is the difference between this value/utility amount and the amount of expected value/utility from an optimum strategy that following from incorporating the information. If this is positive, the forecast is worth buying at a cost up to the value/utility of this difference in terms of $.
– NB if calculated in terms of utility then it is necessary to back transform the utility value to $ amount.
Computation of forecast value
› Expected value before the forecast= $493.04K (1)
› Expected value with forecast, need to incorporate the probabilities of
getting the forecasts (P(Zi)):
› P(Z1)*489.7789+P(Z2)*492.8+P(Z3)*492.8
› =0.57*489.7789+0.306*492.8+0.124*492.8
› =$496.208K (2)
› Expected value from incorporating forecast information: (2)-(1)=
› =$496.208K -$493.04K = $3.168K
› This value is an upper bound on what the DM should be prepared to pay for the forecast.
› [HHA2004] Hardaker, Huirne and Anderson (2004) Coping with Risk in Agriculture, CAB International.
– Chapter 3, ‘Accounting for new information’, at the end of the chapter
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