Risk in Agriculture

Paul D. MitchellAAE 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!!!

“Possibility of a loss” “Chance of a bad outcome”

Risk versus Uncertainty Risk: outcomes and probabilities are not

know Soybean Rust in USA; Ag Bioterrorist Attack

Uncertainty: outcomes and probabilities are known 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. Production and Technical Risk2. Market and Price Risk3. Financial Risk4. Human Resource Risk5. 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 RiskUncertainty 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 RiskSeveral 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 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)

Person 1(probability)

Person 2(probability)

A 0.70 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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

A AB B BC C D F

Course Grade

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Person 1

Person 2

Continuous DistributionMaximum temperature t has a normal

distribution with a mean of 70 and standard deviation of 10, so its probability density function is:

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0.005

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0.040

0.045

30 50 70 90 110

temperature

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2

22

1 ( 70)~ ( ) exp

102 10

tt f t

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 Median Mode Standard Deviation Variance Coefficient of Variation Confidence Interval Probability of Key Events (i.e., p < 0)

Central Tendency or Location

Spread or Variability

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

Continuous: If x is a random variable with probability density function f(x), then the mean or expected value of x is:

1

E[ ]N

i ii

x p x

E[ ] ( )x

x xf x dx

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 =

$10,450Yiel

dPric

eProbabili

ty Profit Probability x Profit

lo lo 0.05 $1,000 $50

lo hi 0.45$12,00

0 $5,400

hi lo 0.40 $8,000 $3,200

hi hi 0.10$18,00

0 $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.

Mean ≠ Median ≠ Mode

0.00.51.01.52.02.53.03.5

0 0.2 0.4 0.6 0.8 1

% Survival per Spray

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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 =

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

2 2

1 1

( E[ ])N N

i i i ii i

p D p x x

Standard Deviation

Standard Deviation = square root of theVariance:

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”

2

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

Yield

Price

Prob. Profit

Squared Deviation

Prob. X Squared

Deviation

lo lo 0.05 $1,00089,302,5

00 4,465,125

lo hi 0.45$12,00

02,402,50

0 1,081,125

hi lo 0.40 $8,0006,002,50

0 2,401,000

hi hi 0.10$18,00

057,002,5

00 5,700,250

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0.005

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0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

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Normal Density with st. dev. of 15 and means of 50 (red)70 (green)90 (blue)

Normal Density with mean of 70 and st. dev. of 10 (red)15 (green)20 (blue)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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yBeta Density with mean of 0.30 and st. dev. of 0.10 (red)0.15 (green)0.20 (blue)

0.0

0.5

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2.0

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Beta Density with st. dev. of 0.15 and means of 0.25 (red)0.30 (green)0.35 (blue)

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 meanst.

dev. CV meanst.

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

Pdf: 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

2

22

1 (ln( ) )~ ( | , ) exp

22

yy f y

y

Beta pdf

Pdf: 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

1 1(1 )~ ( | , )

( , )

y yy f y

Beta pdf

Pdf:

Beta Function:

Gamma Function:

Note: Programs have the gamma function

1 1(1 )~ ( | , )

( , )

y yy f y

1

1 1

0

( , ) (1 )u u du ( ) ( )

( , )( )

1

0

( ) exp( ) zz u u du

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

1 1

1

( ) ( )~ ( | , , , )

( , )( )

z L U zz f z L U

U L

1 1

1

( ) ( ) ( ) ( )~ ( | , , , )

( )( )

z L U zz f z L U

U L

Gamma pdf

Pdf: Min = 0 Max = +infinity Mean = bl Variance = b2l

1( / ) exp( / )~ ( | , )

( )

y yy f y

Weibull pdf

Pdf: Min = 0 Max = +infinity Mean = bG[(l + 1)/l] Variance =

b2G[(l + 2)/l] – b2(G[(l + 1)/l])2

1

~ ( | , ) expy y

y f y

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? y ~ f(y), then Pr(y < 80) = F(80)

Price is less than $1.90/bu? p ~ f(p), then Pr(p < 1.90) = F(1.90)

Returns will exceed $5/ac? r ~ f(r), then Pr(r > 5) = 1 – F(5)

Break Even Probability: probability that Per acre returns will be positive?

p ~ f(p), then Pr(p > 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)

Excel NORMDIST(Z, 0, 1, true) 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 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 = 10a = Pr(z ≤ Z) = F(Z)a = F(Z)Non-standard normala = F((Z – m)/s)a = NORMDIST(Z,m,s,true)

z = F-1(a)Non-standard normalz = F-1(a) + s mz = 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

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0 25 50 75 100 125 150 175 200 225 250

Yield bu/ac

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No Bt

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No Bt

Wisconsin

Hall County, NE

Simulated empirical probability density functions of harvested yield with and without Bt corn in Wisconsin and Hall County, NE

Bt HallNo 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

Variance 1140.6 1144.6 1531.9 1448.3

St Dev 33.77 33.83 39.14 38.06

CV 20.1% 21.7% 30.1% 30.6%• 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

Bt HallNo 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

Different confidence intervals for Bt and non-Bt corn in both Hall County and Wisconsin

See graphics to better understand differences

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0 25 50 75 100 125 150 175 200 225 250

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0 25 50 75 100 125 150 175 200 225 250

0.000

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0 25 50 75 100 125 150 175 200 225 250

50% Confidence Interval: Upper limit further from mean

70% Confidence Interval: Limits approximately symmetric around mean90% Confidence Interval: Lower limit further from mean

Pattern holds for Bt and non-Bt in both locations

Example ContinuedBreak-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