1. Risk in Agriculture Paul D. Mitchell AAE 575

2. 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

3. 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

4. Major Categories of Agricultural Risk 1. 1.Production and Technical Risk 2. 2.Market and Price Risk 3. 3.Financial Risk 4. 4.Human Resource Risk 5. 5.Legal and Institutional Risk

5. 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)

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. Discrete Example: Probability Density Function of Course Grade for Two Persons Course Grade (outcome) Person 1 (probability) Person 2 (probability) A0.700.05 AB0.15 B0.100.40 BC0.050.20 C0.000.10 D0.000.05 F0.000.05

14. 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:

15. 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)

16. 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

17. Measures of Risk Mean/Expected Value/Average Median Mode Standard Deviation Variance Coefficient of Variation Confidence Interval Probability of Key Events (i.e.,  < 0) Central Tendency or Location Spread or Variability

18. 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 p i, 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:

19. 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 =  = 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”

20. Another Example Profit for a crop has 4 possible outcomes with probabilities as reported in table Mean = 50 + 5400 + 3200 + 1800 = $10,450 YieldPriceProbabilityProfitProbability x Profit lo 0.05$1,000$50 lohi0.45$12,000$5,400 hilo0.40$8,000$3,200 hi 0.10$18,000$1,800

21. 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)

22. 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

23. 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.

24. Mean ≠ Median ≠ Mode Beta distribution for % survival per spray of ECB larvae in sweet corn after 1, 3, and 5 sprays of Capture

25. 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 x i is an outcome and E[x] = mean, then Deviation D i = x i – E[x] = x i – 

26. Variance Variance =  2 = 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: – 3 2 = 9 and 3 2 = 9

27. Standard Deviation Standard Deviation = square root of the 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 bu 2 /A 2 and the Standard Deviation is bu/A Think of it as the “typical deviation from mean” Technically “square root of the mean squared deviation”

28. 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 D 1 2 = (50 – 95) 2 = 45 2 = 2025 D 2 2 = (100 – 95) 2 = 5 2 = 25 D 3 2 = (150 – 95) 2 = 55 2 = 3025 Var. = 2025 + 25 + 3025 = 5075 bu 2 /A 2 St. Dev. = sqrt(5075) = 71.24 bu/A

29. 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 YieldPriceProb.Profit Squared Deviation Prob. X Squared Deviation lo 0.05$1,00089,302,5004,465,125 lohi0.45$12,0002,402,5001,081,125 hilo0.40$8,0006,002,5002,401,000 hi 0.10$18,00057,002,5005,700,250

30. 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)

31. Beta Density with mean of 0.30 and st. dev. of 0.10 (red) 0.15 (green) 0.20 (blue) Beta Density with st. dev. of 0.15 and means of 0.25 (red) 0.30 (green) 0.35 (blue)

32. Coefficient of Variation Coefficient of Variation (CV) is  /  (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

33. 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 -------- Countymeanst. dev.CVmeanst. dev.CV Iroquois, IL2.240.4922%143.228.620% Douglas, KS2.330.4921%90.4830.834% Lincoln, NE2.200.4621%150.322.515% Pitt, NC2.520.5823%87.735.140%

34. 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

35. 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

36. Lognormal pdf Pdf: Min = 0 Max = +infinity Mean = exp(  + ½  2 ) Variance = exp(2  )exp(  2 )(exp(  2 ) – 1) If y ~ lognormal, then ln(y) ~ normal with mean  and st dev 

37. Beta pdf Pdf: Min = 0 Max = 1 Mean =  /(  +  ) Variance =  /[(  +  ) 2 (  +  + 1)] Can re-scale to be between other upper and lower limits

38. Beta pdf Pdf: Beta Function: Gamma Function: Note: Programs have the gamma function

39. Rescaled Beta pdf y ~ beta( ,  ) 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)  /(  +  )) Variance = (U – L) 2 (  /[(  +  ) 2 (  +  +1)])

40. Rescaled Beta pdf If y ~ beta( ,  ), then z = L + (U – L)y has a pdf

41. Gamma pdf Pdf: Min = 0 Max = +infinity Mean =  Variance =  2

42. Weibull pdf Pdf: Min = 0 Max = +infinity Mean =  [( + 1)/ ] Variance =  2  [( + 2)/ ] –  2 (  [( + 1)/ ]) 2

43. 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

44. 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

45. 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(z lo ) = 0.025, so F -1 (0.025) = z lo F(z hi ) = 0.975, so F -1 (0.975) = z hi

46. 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?

47. 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?  ~ f(  ), then Pr(  > 0) = 1 – F(0)

48. 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

49. Cumulative Distribution Functions and Their Inverses If z ~ f(z) and F(z) is CDF,  = Pr(z ≤ Z) = F(Z) Standard Normal CDF:  =  (z) Excel NORMDIST(Z, 0, 1, true) If ~ N( ,  ) then CDF:  ((Z –  )/  ) Excel NORMDIST(Z, , , true) Inverse Standard Normal CDF z =  -1 (  ) Excel NORMSINV( ,0,1) If ~ N( ,  ) then inverse CDF z =  -1 (  )  Excel NORMSINV( , ,  )

50. Normal CDF:  = 30,  = 10  = Pr(z ≤ Z) = F(Z)  =  (Z) Non-standard normal  =  ((Z –  )/  )  = NORMDIST(Z, , ,true) z =  -1 (  ) Non-standard normal z =  -1 (  )  z = NORMINV( , ,  )

51. 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) =  {(ln(z)-  )/  } Lognormal inverse CDF: F -1 (  ) = exp[  -1 (  )  +  ] Weibull CDF: F(z) = 1 – exp[–(z/  ) ] Weibull inverse CDF: F -1 (z) =  ln(1/(1-  ))]  1- Beta CDF: incomplete beta function (now Excel) Gamma CDF: ????

52. 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

53. 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

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

55. Bt HallNo Bt HallBt WINo Bt WI Mean168.2155.6130.2124.4 Median172.1158.9133.5127.2 Mode*190165155150 Variance1140.61144.61531.91448.3 St Dev33.7733.8339.1438.06 CV20.1%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

56. Bt HallNo Bt HallBt WINo Bt WI 50% lo147.1133.9103.798.6 50% hi193.8180.8160.6153.1 70% lo131.6118.786.281.8 70% hi203.4190.9172.6165.7 90% lo105.594.659.656.5 90% hi216.8206.0188.9182.2 Different confidence intervals for Bt and non-Bt corn in both Hall County and Wisconsin See graphics to better understand differences

57. 50% Confidence Interval: Upper limit further from mean 70% Confidence Interval: Limits approximately symmetric around mean 90% Confidence Interval: Lower limit further from mean Pattern holds for Bt and non-Bt in both locations

58. 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