Risk in Agriculture
Paul D. Mitchell
AAE 575
Goal




How to make economically optimal
decisions/choices under risk
First: Discuss how to talk about risk
(measuring risk)
Second: How to make economically optimal
choices under risk (risk management)
Like before: First production functions,
Second economics
What is Risk?

Risk is a four-letter word!!!




Risk versus Uncertainty
Risk: outcomes and probabilities are not know



Soybean Rust in USA; Ag Bioterrorist Attack
Uncertainty: outcomes and probabilities are known


“Possibility of a loss”
“Chance of a bad outcome”
CBOT corn price for next fall; Crop yield distributions
A fine distinction that many people ignore and use
the two terms interchangeably
Technically, we will do Uncertainty, but call it Risk
Major Categories of Agricultural Risk
1.
2.
3.
4.
5.
Production and Technical Risk
Market and Price Risk
Financial Risk
Human Resource Risk
Legal and Institutional Risk
Production and Technical Risk
Uncertainty in crop yields or livestock gains
due to numerous factors




Weather: flood, drought, hail, frost, etc.
Pests and Diseases: ECB, CRW, Soybean Aphid,
Soybean Rust, BSE, brucellosis, etc.
New Technologies: new herbicides, hybrids
(transgenics), tillage, planter, harvest
machines, milking facilities, organic, intensive
grazing methods, IPM, soil testing, etc.
Input Shortages: labor, custom machinery or
application, trucking, fungicides (for rust)
Market and Price Risk
Uncertainty in market prices or in ability to
market production




Input price changes: fuel, fertilizer, fungicide,
feed/grain, etc.
Crop and livestock prices vary continuously
(CBOT, CME, etc.)
Market Access: Hurricane Katrina and barge
traffic shut down last fall
Processor/Contractor/Buyer: goes out of
business or changes quality requirements
Financial Risk
Money borrowed or external equity provided
creates risk




Interest rate changes
Change in value of assets used as collateral
Ability to generate income to meet debt
obligations (liquidity and solvency)
Lender’s/investor’s willingness to continue
lending/providing capital
Human Resource Risk
Several people are key to a farm business and
potential for changes creates risk





Employee management problems: retention, turnover,
criminal activity, disputes, etc.
Injury, illness, death of manager/key employee
Key employee, spouse, child: retires, career change,
relocates, etc.
Family disputes, divorces, etc.: personal stress, plus
losses from legal settlements, property diversions,
financial reallocations, etc.
Estate Planning: how are farm assets going to be
transferred between generations? Can create risk
Legal and Institutional Risk
Created by regulations and legal liabilities





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Regulations for manure, chemicals, facility
siting, antibiotic use, carcass disposal, burning
Liability for accidents: machinery, livestock
Labor laws: taxes, worker health and safety,
residency requirements
Contractual obligations: hedge-to-arrive and
futures contracts, contracts with processors
Tax liability: properly file all required forms
Ignorance of law is not a legal excuse
Risk is a Major Topic


We will just barely touch the surface
Our Focus: Production/technical risk




Calculating and interpreting commonly used
measures of risk (How to measure risk)
Applying common criteria for decision making
under uncertainty (How to manage risk)
Common tools to manage different risks
Measurement first, so can describe effect
of risk management
Measuring Risk

Convert “Risk” to “Uncertainty”



Know all outcomes and their probabilities
Gives the “probability density function”
Statistical function describing all possible
outcomes and associated probabilities


Discrete: die roll, coin flip, game
Continuous: tomorrow’s max temperature,
corn price at harvest, earnings next year
Discrete Example: Probability Density
Function of Course Grade for Two Persons
Course Grade
(outcome)
A
Person 1
(probability)
0.70
Person 2
(probability)
0.05
AB
0.15
0.15
B
0.10
0.40
BC
0.05
0.20
C
0.00
0.10
D
0.00
0.05
F
0.00
0.05
0.8
0.7
Probability
0.6
0.5
Person 1
Person 2
0.4
0.3
0.2
0.1
0
A
AB
B
BC
Course Grade
C
D
F
Continuous Distribution
Maximum temperature t has a normal distribution
with a mean of 70 and standard deviation of 10,
so its probability density function is:
 (t  70)2 
t ~ f (t ) 
exp 

2
2
10
2 10


1
0.045
probability density
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
30
50
70
temperature
90
110
Many Types of Distributions






Many possible pdf’s, discrete and
continuous, some with upper and/or lower
limits, some symmetric or skewed, etc.
Lognormal (prices)
Beta (yields, % losses, rates of gain)
Gamma or Weibull (yields)
Triangular (low information)
Empirical: draw from data (discrete)
Measures of Risk






Goal: have a function (pdf): outcomes and
probabilities
How do you describe this pdf?
Typically focus on measures of
1) Central tendency or location
2) Spread or variability
3) Skewness or “bad” outcomes
Measures of Risk








Mean/Expected Value/Average
Central
Tendency or
Median
Location
Mode
Standard Deviation
Variance
Spread or
Variability
Coefficient of Variation
Confidence Interval
Probability of Key Events (i.e.,  < 0)
Mean



Average or expected outcome
Probability weighted average of random variable
Discrete: If x is a random variable with N possible
outcome values, each with probability pi, then the
mean or expected value of x is
N
E[ x]   pi xi
i 1

Continuous: If x is a random variable with
probability density function f(x), then the mean or
expected value of x is: E[ x]  xf ( x)dx

x
Simple Example




A crop has three possible yields:
 low 50 bu/A, with probability 0.25
 typical 100 bu/A, with probability 0.60
 high 150 bu/A, with probability 0.15
Expected Yield = Mean = m =
0.25 x 50 + 0.60 x 100 + 0.15 x 150 =
12.5 + 60 + 22.5 = 95 bu/A
Mean is a “probability weighted average”
Another Example


Profit for a crop has 4 possible outcomes with
probabilities as reported in table
Mean = 50 + 5400 + 3200 + 1800 = $10,450
Yield Price Probability
Profit
Probability x Profit
lo
lo
0.05
$1,000
$50
lo
hi
0.45
$12,000
$5,400
hi
lo
0.40
$8,000
$3,200
hi
hi
0.10
$18,000
$1,800
Interpreting Means

The mean is not what will happen, but rather, if the
random event occurs several times, the mean is the
average of the outcomes


The mean die roll is 3.5, which does not imply that if you
roll a die you will get a 3.5. Rather if you roll a die several
times, the average of all these rolls will be close to 3.5
If your mean corn yields is 150 bu/ac, this does not imply
that next year you will get 150 bu/ac, rather the average
of your corn yields over the next several years will be
around 150 bu/ac (if you plant the same hybrids)
Central Tendency


Besides Mean, the Median & Mode also
measure a distribution’s Central Tendency
Median = the middle or half way point



Half of the draws will be < the median
Half of the draws will be > the median
Mode = most common or most likely value
Mean-Median-Mode

Symmetric Distribution: Mean = Median



Mean = Median = Mode: Normal Distribution
Mean = Median ≠ Mode: Uniform (die roll)
Skewed/Asymmetric Distribution


Mean ≠ Median ≠ Mode
Gamma, Beta, Lognormal, Weibull, etc.
probability density
Mean ≠ Median ≠ Mode
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
0.2
0.4
0.6
0.8
1
% Survival per Spray
Beta distribution for % survival per spray of ECB larvae in
sweet corn after 1, 3, and 5 sprays of Capture
Measures of Spread or Variability



Probability weighted measures of
variability or spread of possible outcomes
All begin with the deviation of the possible
outcomes from the mean
If xi is an outcome and E[x] = mean, then
Deviation Di = xi – E[x] = xi – m
Variance




Variance =
s2
N
N
i 1
i 1
2
2
p
D

p
(
x

E[
x
])
= i i  i i
Variance = probability weighted average of the
squared deviation of each outcome from the mean
Variance = sort of the average squared deviation
Why squared deviation?


Converts all deviations to be positive
Negative and positive deviations of same size have
same value: –32 = 9 and 32 = 9
Standard Deviation


Standard Deviation = square root of the
2
s

s
Variance:
Why take the square root of the variance?





Variance units are squared unit of Mean
Standard Deviation units are same as Mean’s
If yield has Mean of bu/a, then the Variance is bu2/A2
and the Standard Deviation is bu/A
Think of it as the “typical deviation from mean”
Technically “square root of the mean squared deviation”
Calculating Variance and St. Dev.

Previous Example: three possible yields:








Low 50, Typical 100, High 150
Probabilities 0.25, 0.60, and 0.15
Mean is 95 bu/A
D12 = (50 – 95)2 = 452 = 2025
D22 = (100 – 95)2 = 52 = 25
D32 = (150 – 95)2 = 552 = 3025
Var. = 2025 + 25 + 3025 = 5075 bu2/A2
St. Dev. = sqrt(5075) = 71.24 bu/A
Variance and St. Dev. Example




Previous Table: 4 possible profits, Mean = $10,450
Variance = 4,465,125 + 1,081,125 + 2,401,000 + 5,700,250
Variance = 13,647,500 squared dollars
Standard Deviation = sqrt(13,647,500) = $3,694
Squared Prob. X Squared
Yield Price Prob.
Profit Deviation
Deviation
lo
lo
0.05
$1,000 89,302,500
lo
hi
0.45 $12,000
2,402,500
1,081,125
hi
lo
0.40
6,002,500
2,401,000
hi
hi
0.10 $18,000 57,002,500
5,700,250
$8,000
4,465,125
Normal Density with mean
of 70 and st. dev. of
10 (red)
15 (green)
20 (blue)
0.040
probability density
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0
10
20
30
40
50
60
70
80
90 100 110 120 130 140
Normal Density with st.
dev. of 15 and means of
50 (red)
70 (green)
90 (blue)
0.030
probability density
0.025
0.020
0.015
0.010
0.005
0.000
0
10
20
30
40
50
60
70
80
90 100 110 120 130 140
Beta Density with mean of
0.30 and st. dev. of
0.10 (red)
0.15 (green)
0.20 (blue)
4.0
probability density
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Beta Density with st. dev.
of 0.15 and means of
0.25 (red)
0.30 (green)
0.35 (blue)
3.0
probability density
2.5
2.0
1.5
1.0
0.5
0.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Coefficient of Variation




Coefficient of Variation (CV) is s/m (st.
dev./mean), expressed as a percent
If standard deviation is 60 and mean 200,
then CV = 60/200 = 0.30 or 30%
CV Normalizes the standard deviation by
the mean, thus expressing the standard
deviation as a percentage of the mean
CV of 30% implies the standard deviation
is 30% of the mean
Coefficient of Variation (CV)





CV is a relative measure of risk
CV “corrects for” difference in mean
Corn price and yield (Coble, Heifner and Zuniga 2000)
Price variability the same everywhere (CV 21%-23%)
Yield variability varies greatly by location (CV 15%-40%)
-------- Price --------
-------- Yield --------
County
mean
st. dev.
CV
mean
st. dev.
CV
Iroquois, IL
2.24
0.49
22%
143.2
28.6
20%
Douglas, KS
2.33
0.49
21%
90.48
30.8
34%
Lincoln, NE
2.20
0.46
21%
150.3
22.5
15%
Pitt, NC
2.52
0.58
23%
87.7
35.1
40%
How Actually Done

Discrete pdf: 50 bu Pr = 0.25, 100 bu
Pr = 0.60, 150 bu Pr = 0.15

Calculate
Continuous pdf: don’t do integrals
yourself, use formulas
 People get confused by normal pdf
where parameters = mean & st dev
 Most pdfs are not this easy

Some pdfs



Lognormal, Beta, Gamma, Weibull
Look up the pdf and formula's for the
mean and st dev etc. in books, Wikipedia,
Wolfram alpha, Google, etc.
Main Point: lots of pdfs out there, mean
and variance are not the parameters, but
functions of the parameters
Lognormal pdf






 (ln( y )  m ) 2 
exp 
Pdf: y ~ f ( y | m , s ) 

2
2
2s
y 2s


1
Min = 0
Max = +infinity
Mean = exp(m + ½s2)
Variance = exp(2m)exp(s2)(exp(s2) – 1)
If y ~ lognormal, then ln(y) ~ normal with
mean m and st dev s
Beta pdf






ya 1 (1  y)w 1
Pdf: y ~ f ( y | a , w ) 
(a , w )
Min = 0
Max = 1
Mean = a/(a + w)
Variance = aw/[(a + w)2(a + w + 1)]
Can re-scale to be between other upper
and lower limits
Beta pdf

ya 1 (1  y )w 1
Pdf: y ~ f ( y | a , w ) 
(a , w )
1

a 1
w 1

(
a
,
w
)

u
(1

u
)
du
Beta Function:

(a )(w )
(a , w ) 
(a  w )

0

Gamma Function: ( z )   exp(u )u z 1du
0

Note: Programs have the gamma function
Rescaled Beta pdf






y ~ beta(a,w) with min 0 and max 1
z = L + (U – L)y also has a beta
distribution, but with
Min = L
Max = U
Mean = L + (U – L)(a/(a + w))
Variance = (U – L)2(aw/[(a+w)2(a+w+1)])
Rescaled Beta pdf

If y ~ beta(a,w), then z = L + (U – L)y
has a pdf
( z  L)a 1 (U  z )w 1
z ~ f ( z | a , w , L, U ) 
a w 1
(a , w )(U  L)
( z  L)a 1 (U  z )w 1 (a )(w )
z ~ f ( z | a , w , L, U ) 
a w 1
(a  w )(U  L)
Gamma pdf





( y / b )l 1 exp( y / b )
y ~ f ( y | b , l) 
b  (l )
Pdf:
Min = 0
Max = +infinity
Mean = bl
Variance = b2l
Weibull pdf
l 1





  y l 
exp     
 b  


ly
y ~ f ( y | b , l) 
Pdf:
bl
Min = 0
Max = +infinity
Mean = b[(l + 1)/l]
Variance =
b2[(l + 2)/l] – b2([(l + 1)/l])2
Confidence Interval



Confidence Interval: the limits between
which the random variable will be with a
predefined probability
Example: 95% confidence interval for corn
yield: Corn Yield is between 100 bu/ac
and 195 bu/ac with 95% probability
Another measure of variability: a wider
confidence interval implies greater
variability
Confidence Interval Rule of Thumb






95% confidence interval is approximately the mean
plus and minus 2 standard deviation
65% confidence interval is approximately the mean
plus and minus 1 standard deviation
Example: Suppose returns have a mean of $100/ac
and a standard deviation of $25/ac, then
Returns in range $50-$150/ac with 95% probability
Returns in range $75-$125/ac with 65% probability
Approximation close for most random variables
Confidence Intervals




Use the inverse of the cumulative
distribution function (CDF) to calculate
confidence intervals
If y ~ f(y), then F(z) = Pr(y ≤ z)
f(y) is the pdf and F(y) is the cdf
95% CI


F(zlo) = 0.025, so F-1(0.025) = zlo
F(zhi) = 0.975, so F-1(0.975) = zhi
Probability of Key Events


Sometimes key events important
What’s the probability that:




Yield is less than 80 bu/ac?
Price is less than $1.90/bu?
Returns will exceed $5/ac?
Break Even Probability: probability that



Recover the investment cost?
Get the cost of insecticide back in saved yield?
Per acre returns will be positive?
Probability of Key Events:
If y ~ f(y), then Pr(y ≤ z) = F(z)

What’s the probability that:

Yield is less than 80 bu/ac?


Price is less than $1.90/bu?


p ~ f(p), then Pr(p < 1.90) = F(1.90)
Returns will exceed $5/ac?


y ~ f(y), then Pr(y < 80) = F(80)
r ~ f(r), then Pr(r > 5) = 1 – F(5)
Break Even Probability: probability that

Per acre returns will be positive?

 ~ f(), then Pr( > 0) = 1 – F(0)
Probabilities




In special cases, can calculate probability of key
events, but usually need numerical simulations
(Monte Carlo analysis)
Confidence Interval: pick the probability and
then derive the limits
Probability of Key Events: pick the limit or limits
and the derive the probability
Value at risk (VAR): pick the probability and the
limit, find the portfolio allocation to get them
Cumulative Distribution
Functions and Their Inverses


If z ~ f(z) and F(z) is CDF, a = Pr(z ≤ Z) = F(Z)
Standard Normal CDF: a = F(z)


If ~ N(m,s) then CDF: a  F((Z – m)/s)


Excel NORMDIST(Z, m, s, true)
Inverse Standard Normal CDF z = F-1(a)


Excel NORMDIST(Z, 0, 1, true)
Excel NORMSINV(a,0,1)
If ~ N(m,s) then inverse CDF z = F-1(a)s  m

Excel NORMSINV(a,m,s)
Normal CDF: m = 30, s = 10
a = Pr(z ≤ Z) = F(Z)
a = F(Z)
Non-standard normal
a = F((Z – m)/s)
a = NORMDIST(Z,m,s,true)
z = F-1(a)
Non-standard normal
z = F-1(a)s  m
z = NORMINV(a,m,s)
Cumulative Distribution
Functions and Their Inverses








Many CDF’s require special functions to evaluate
Some software packages have them, some don’t
Lognormal CDF: F(z) = F{(ln(z)-m)/s}
Lognormal inverse CDF: F-1(a) = exp[F-1(a)s+m]
Weibull CDF: F(z) = 1 – exp[–(z/b)l]
Weibull inverse CDF: F-1(z) = b[ln(1/(1-a))] 1-l
Beta CDF: incomplete beta function (now Excel)
Gamma CDF: ????
Some pdfs



Lognormal, Beta, Gamma, Weibull
Look up the cdf and formula's for the
mean and st dev etc. in books, Wikipedia,
Wolfram alpha, Google, etc.
Main Point: lots of pdfs out there, mean
and variance are not the parameters, but
functions of the parameters
Extended Example





Bt Corn Yield Simulations with local ECB
pressure and yield parameters
Wisconsin State Average ECB and Yield
Hall County, NE (irrigation with lots ECB)
Random Variables: yield, ECB pressure,
ECB tunneling, % yield loss
10,000 Monte Carlo random draws
0.014
Wisconsin
Probability Density
0.012
0.010
0.008
Bt
No Bt
0.006
0.004
0.002
0.000
0
25
50
75 100 125 150 175 200 225 250
Yield bu/ac
0.014
Hall County, NE
Probability Density
0.012
0.010
0.008
Bt
No Bt
0.006
0.004
0.002
0.000
0
25
50
75 100 125 150 175 200 225 250
Yield bu/ac
Simulated empirical
probability density
functions of harvested
yield with and without
Bt corn in Wisconsin
and Hall County, NE
Bt Hall
No Bt Hall
Bt WI
No Bt WI
Mean
168.2
155.6
130.2
124.4
Median
172.1
158.9
133.5
127.2
Mode*
190
165
155
150
1140.6
1144.6
1531.9
1448.3
33.77
33.83
39.14
38.06
20.1%
21.7%
30.1%
30.6%
Variance
St Dev
CV
• Bt corn increases mean yield in both locations, more in Hall
County where more pest pressure exists
• Bt corn decreases variance and standard deviation in Hall
County and increases both in WI (irrigated vs dryland)
• Bt corn increases yield CV in both locations, more in Hall
County (risk measure matters)
* Rounded to nearest 5 bu
Different confidence intervals for Bt and non-Bt corn in both
Hall County and Wisconsin
Bt Hall No Bt Hall
Bt WI
No Bt WI
50% lo
147.1
133.9
103.7
98.6
50% hi
193.8
180.8
160.6
153.1
70% lo
131.6
118.7
86.2
81.8
70% hi
203.4
190.9
172.6
165.7
90% lo
105.5
94.6
59.6
56.5
90% hi
216.8
206.0
188.9
182.2
See graphics to better understand differences
0.012
50% Confidence Interval:
Upper limit further from
mean
0.010
0.008
0.006
0.004
0.012
0.002
0.010
0.000
0.008
0
25
50
75
100
125
150
175
200
225
250
70% Confidence Interval:
Limits approximately
symmetric around mean
0.006
0.004
0.012
0.002
0.010
0.000
0.008
0
25
50
75
100
125
150
175
200
225
250
90% Confidence Interval:
Lower limit further from
mean
0.006
Pattern holds for Bt and
non-Bt in both locations
0.004
0.002
0.000
0
25
50
75
100
125
150
175
200
225
250
Example Continued
Break-Even Probability



What’s the probability, given an expected
price and yield, that you will recover the
Bt corn “Tech Fee” in saved yield?
Analyzed this question for WI regions
UWEX bulletin and Spreadsheet on class
web page