Bob Farquharson AGRI90016

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Managing Risk
Bob Farquharson
AGRI90016
2020 Semester 2
Week 2
Agricultural Decisions
and Risk
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This lecture …
1. Think about decision making in farming
systems
2. The issue of risk in decision making
3. Important concepts and issues for risk
analysis
4. Some examples
Readings
 Anderson, Dillon and Hardaker (ADH)
(1988), pp. 3-7
 Makeham and Malcolm (1988), pp. 162-
165
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1. Human decision making
 ‘The human estate embodies the privileges
and responsibilities of decision making, for
it is the capacity for rational choice that
principally distinguishes man from the
animals’;
 Anderson Dillon and Hardaker (1988), p.3
Decision problems
 A decision problem exists only when we
feel that the possible consequences are
important and yet we are unsure of what is
the best thing to do;
 When a person is uncertain about the
consequences of his or her decision, we
can say he or she faces a risky choice
 Decision analysis is about the future
 We can’t do anything about the past
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Risky decisions
 The issue for risky decisions is that when we
make a decision we are uncertain about the
outcome
 Because the outcome depends on some
process which has not yet been observed
 And/or that process has uncertain outcomes
 Deterministic = outcomes are known for certain
 Stochastic = relationship not always right on
target, or we miss the target (varying randomly)
 (Greek ‘stokhos’ meaning target or bulls eye)
 Stochasticity is widely observed 7
Probability
 Ways of thinking of how likely an outcome
might be from an uncertain event
 Historically: probability = relative frequency
 What has observed in the past, ‘Objective’
 Modern thinking: probabilities are ‘subjective’
 Degree or strength of conviction an individual has
about a (future) proposition
 The only basis for decision making, provided
 Subjective probabilities agree with the rules
 They are consistent with the degree of belief really
held
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‘Good’ and ‘Right’ decisions
 A good decision is a rational decision in that it
 Incorporates beliefs and probabilities at the time
 Is consistent with the decision maker’s preferences
 But because of uncertainty, there is no
guarantee that a ‘good’ decision turns out to
be the ‘right’ decision
 ‘The best laid plans of mice and men’
 e.g. an unforeseen drought or fall in price
 When the decision maker has little or no control
over the outcome
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2. Components of decision problems
ADH Ch. 1 framework:
1. Acts: or possible actions (decisions) of DM
• Including ‘do nothing’
2. States: possible events or states (states of
nature), e.g. a drought occurs
• If the DM doesn’t know which states of nature
will prevail, the decision problem is said to be
risky
3. Probabilities: attached to the occurrence of
states in the decision problem
• These probabilities are subjective (personal in
nature)
Components ..
4. Consequences: depending on which state
occurs
• Choice of an act (decision) and the resulting
state of nature leads to a consequence,
outcome or payoff
5. Choice criterion: an objective of the DM
(or objective function)
• Closely tied to the measurement of
consequences
• Often maximising income (or wealth)
• The Utility (U) from the outcome
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Definitions of states (and acts)
 Mutually exclusive
 States/acts are defined to be separate from
one another, no overlap
 e.g. rainfall totals on one day <10 mm, 10–19 mm,
20-29 mm, and so on
 Collectively exhaustive
 States/acts must cover all possible outcomes
 e.g. rainfall totals on one day <10 mm, 10–19 mm,
…, 90-99 mm, 100 mm or more
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A simple example (ADH)
 A grain crop has just been harvested, the
decision is how to dispose (sell)
 Possible acts or decisions (ai)
 a1 = sell now
 a2 = store and speculate
 Possible states of nature (sj), the market
condition
 s1 = market normal
 s2 = market in short supply
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Outcomes and probabilities
 Probabilities (subjective) of state outcomes
(P(sj))
 P(s1) = 0.8 (normal market)
 P(s2) = 0.2 (market in short supply)
 These sum to 1.0 (all possibilities included)
 Consequences (U) (prices received, $/t)
 U(sell|normal), U(store|short) or U(a1|s1),
U(a2|s2)
 Expected payoff (see below)
 Sum of Prob x Price for each outcome
 Develop a decision matrix 14
First, Expected Values
 We can account for the probabilities
associated with each outcome
 Calculate a weighted average
 Each outcome is weighted by the probability
of the outcome
 A weighted mean or average, an expected value
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Example
 Example, two possible outcomes, $10 or $20
 Equally likely, each with a probability of 50%
 The average is $15: (10 + 20)/2 = 15
 Actually this is: (10*0.5 + 20*0.5) = 5 + 10 = 15
 A different probability of these 2 outcomes?
 If $10 is more likely (75% Probability)
 And $20 is less likely (25% Probability)
 The weighted average: (10*0.75 + 20*0.25) = 11.5
 The expected value (weighted average) accounts for
probabilities of outcomes
 If $10 is more likely than $20, the weighted
(expected) outcome will be lower
 A common sense check of the calculation!
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Now: represent the decision
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* Value associated with the prior optimal act
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3. Probability basics
 Probabilities are a measure of personal
strengths of conviction about the
occurrence of uncertain events
 These strengths of conviction are measured
on a scale from 0 to 1
 If we believe an event is impossible it has a
probability of 0, if certain the prob is 1
 The basic rule
 Probabilities assigned to two or more related
events must add up to 1.0 (100%)
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Tossing dice
 One die, two dice
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An example
 Craps is a dice game in which players bet on
the outcomes of the roll, or a series of rolls,
of a pair of dice
 Street craps or a casino game
 The thrower must roll with one hand and the
dice must bounce off a wall surrounding the
table
 To keep the game fair
 Movie, ‘A Bronx Tale’ shows the street game
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Probabilities (one die)
 If we throw a six-sided die (6 outcomes)
 And the die is fair (equal probabilities)
 The probabilities of each outcome are
 Outcome 1: 1/6 = 0.16666, 0.167 or 17%
 Outcome 2: 1/6
 Outcome 3: 1/6
 Outcome 4: 1/6
 Outcome 5: 1/6
 Outcome 6: 1/6
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Class activity
 Throw a single die 6 times and record your
answers
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My results
 When I conducted this experiment ..
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Probabilities and outcomes
 The probabilities above (1/6 or 17%)
relate to a large number of trials
 A shorter experiment (or game) will not
necessarily come up with those outcomes
 My experience today
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Expected Value (1)
 If N1 = number showing on one die, and N2 =
number showing on the other die, and
 The set of values, written as .,. is a list of
all possible values from throwing a die, then
 N
1 = N2 = 1, 2, 3, 4, 5, 6
 The mean, or average, or expected value (E)
of a single die cast is:
 E(N1) = E(N2) = (1 + 2 + 3 + .. +6)/6 = 3.5
 The average is not even one of the possible
outcomes!
 We must be wary of averages! 25
Probabilities (two dice)
 The measure of interest in Craps is the sum
of the top side of the two dice (Y)
 Y = N1 + N2 is the variable of interest to the
Craps players
 Remember that N1 = 1, 2, 3, 4, 5, 6 = N2
 What is Y = ?, ? ..
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Two dice: totals
 Y = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
 The set of all possible values of Y
 Y = 1 is impossible, why?
 Y = 2: 1+1
 Y = 3: 2+1, 1+2
 Y = 4: 2+2, 1+3, 3+1
 Y = 5: 3+2, 2+3, 4+1, 1+4
 Y = 6: 3+3, 4+2, 2+4, 1+5, 5+1
 And so on ..
 The total number of dice combinations is 36 27
Two dice: probabilities
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Expected Value (2)
 For our 2-dice variable Y
 When we calculate the expected value it
must be weighted by the probabilities
 E(Y) = ∑Y.py = 2x.028 + 3x.056 + .. = 7
 Notation or shorthand
 ∑ is a Greek letter ‘Sigma’ used to
represent ‘the sum of’
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Summary statistics
 Arithmetic Mean (average or expected value)
 Sum of values divided by number of values
 Maximum
 Largest value in a series of numbers
 Minimum
 Smallest value in a series of numbers
 Standard Deviation
 Measures the amount or variation or dispersion in
a series of numbers from the mean
 Variance
 Square root of the SD 30
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4. Probability graphs
 We can express probability data graphically
 To help in decision making
Probability density for Craps
 For each possible outcome Y
 Graph Y against the number of outcomes
 Symmetrical
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The Normal Distribution
 Other types of distributions of data can be
observed
 e.g. the Normal (‘Bell Curve’), but others too
 Picture from Wikipedia
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µ = mean, σ = standard deviation
Another example
 Skewed distributions and effect on
expected values
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Graphically
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Probability Density Functions
 The Craps outcome graph above and the
Normal Distribution are examples of
Probability Density Functions (PDF)
 Different series of random numbers can be
grouped in this way
 A way of summarising and visualising the
data
 Counting the number of times the random
number is within various numerical ranges
 Plotting the counts for each random value
range against the range
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An example PDF
 Hypothetical data
 1,000 data points
 Randomly between $0 and $1,000
 Count the number of data points in income
ranges
 $0 – 99, $100 – 199 and so on
 Plot the number of occurrences (counts) for
each income range
 As a column graph or smoothed line
 Probability Density Function (PDF)
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PDF: a picture
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0
20
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80
100
120
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Numbeer of accorrences
Income range ($)
PDF
Interpretation of the PDF
 How many outcomes from the random
process are within any range?
 19% of the random outcomes are less
than $200
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Cumulative Distribution Functions
 Another way of presenting these data is by
a Cumulative Distribution Function (CDF)
 Count the number of data occurrences
cumulatively for increasing levels of income
 The CDF shows the probability that the
random variable ($ outcomes) will take a
value less than or equal to any value
between 0 and 1,000
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From PDF to CDF
 We can plot the PDF and CDF on one graph
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A less-than CDF
 Probability on Y-axis from 0 to 1
 Prob < $0 = 0
 Prob < $1,000 = 1.0 (100%)
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Interpreting the CDF
 Prob of outcome < $200 = 19%
 Same information as from the PDF
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5. Which decision is better?
 If we have an idea of the distributions of
possible outcomes for two management
decisions (A and B)
 We can compare the distributions to help
make our decision
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PDFs of 2 decisions
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CDFs of 2 decisions
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Stochastic dominance
 If one PDF (A) is, in a probabilistic sense,
always better than another PDF (B)
 Then A stochastically dominates B
 For any level of income, e.g. $400
 If the Prob (outcome<$400) for decision A
 Is less than Prob (outcome<$400) for decision B
 Then decision A stochastically dominates
decision B
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CDF A dominates CDF B
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Not always smooth
 In the real world the distributions are
never as smooth as we see here
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Simulated rice yields in Sri Lanka
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 CDFs of simulated rice yields in the Yala Season
 Yields under a fixed sowing date are generally
better, in a probabilistic sense, than other sowing
date strategies
An important assumption
 The above uses historical climate data as
a basis for prediction of future outcomes
in decision analysis
 The (important) implicit assumption is that
past variability is a good guide to future
variability
 We must be wary of such an assumption
 Even apart from climate change possibilities
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What if CDFs cross?
 It is possible that CDFs can cross over
 The probability of a lower outcome some of
the time
 What will the decision makers do?
 Depends on their attitudes to risk
 How will they trade off a higher expected
income with a small chance of a worse
outcome?
 A large area of study, not for us in this
subject
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A difficult decision Problem
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6. In summary
 Risk analysis incorporates the probabilities
of different outcomes from a decision
 Decision analysis is based on personal
(subjective) probabilities
 A good decision relies on incorporating
probabilities (rational choice)
 But doesn’t ensure the outcome is ‘right’
 Using historical data as a basis for risky
decisions about the future assumes that
the past will be the same as the future
 Concepts of stochastic comparison can help
decision analysis
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Any questions
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