Correlation

John Carter was afarmer in NorthernMassachusetts. John,like all of the farmersaround him, grewapples and shipped hisharvest to Boston for sale at the prevailing marketprice. The farm had been in John’s family for threegenerations; from his inheritance and prudentmanagement, John had built his net worth to$300,000. Like many of his neighbors, John was beingpressed by increasing costs and by the failure ofrevenues to keep up with this increase. John worriedthat a really bad year could wipe him out and he mightlose the farm.

John tried to project the amount of harvest and theprice that it would bring at market. This year hebelieved the farm would earn revenues in the vicinity of$310,000. John’s analysis of past costs indicated thatthe farm would incur $220,000 this year in fixed costsand that variable costs would be $0.03 per pound ofapples produced. The tax schedule for farmers meantthat John did not pay any taxes unless he earned over$25,000 in a year. Between $25,000 and $50,000, hewould have to pay a marginal rate of 24 percent, sothat his actual taxes on $50,000 would be 12 percent.Over $50,000, the marginal tax rate increased to45.6 percent on earnings.

During the past few years, John had hedged hisrevenues using forward contracts. When he entered aforward contract, John had to guarantee the delivery ofthe contracted amount of apples at harvest. If John’sown crop fell below the amount of apples that he hadsold forward, he would have to buy enough apples onthe open market to make up the difference. Any applesthat John’s farm produced above the amount specifiedin the contract would be sold at the prevailing marketprice. All transactions, including any payment receivedby John from the forward contract and any selling orbuying of apples on the spot market at harvest wouldbe settled at the same time in the fall

John believed the price at harvest would best beapproximated by a normal distribution with a meanof $0.2079 per pound and a standard deviation of$0.0247 per pound. Forward contracts were onlyavailable in increments of 100 tons, and the current forward rate was $0.2079 per pound. In the past, Johnhad assumed that he could predict with certainty whathis land would produce, and he had usually hedgedroughly half that amount. John sometimes wondered ifthat was the best policy for how much to hedge, givenhis forecast of what his farm would produce.

John Junior, home from college for springvacation, had decided to help his father. He hadproduced a worksheet that modeled the future price ofapples in an @RISK simulation. John Junior initially setup the model so that the harvest was exactly his father’sforecast—743 tons of apples—and he agreed to helphis father figure out what the ideal amount was to sellforward, given the assumption that apple productionwas known with certainty.

However, John Junior was concerned that hisfather’s assumptions about the harvest did not captureall of the risks that the farm faced. After someprodding, John Junior was able to get his father toadmit that his forecasts were not always right, andJohn Senior provided John Junior with data aboutactual market prices and harvest yields in the past(Exhibit 1). John Junior noted with some concern thathis father’s forecast for the upcoming harvest wassimply the mean of the past ten years’harvests. Inaddition to the simple analysis his dad had asked himto do, John Junior decided to model the farm’sprofitability, factoring in uncertainty about how manytons of apples would actually be produced. He decidedthat production quantity was best approximated by anormal distribution with a mean of 743 tons (hisfather’s forecast) and standard deviation of 87 tons.

After completing his basic model for his father andhis“improved”model with the uncertainty about thesize of the harvest, John Junior knew he could helpdecide how many tons of apples to sell forward. Butsomething was nagging him. He suspected there wastypically a relationship between the quantity of applesharvested and the price at market. After all, if his fatherhad a bad year, other farmers might also have bady ears, and this could affect price. As a result, he decided to add a correlation variable to his model to better capture the interaction between the harvest and market price. To explore what correlation did for the decision of how much to sell forward, he decided to analyze how many tons his father should sell forward assuming 0 correlation between price and quantity,þ0.99 correlation, and0.99 correlation. He wondered whether the variables would be that highly correlated.

John Junior concluded, from talking over dinner with his father about risk preference, that his father was decreasingly risk averse and a logarithmic utility function would be an appropriate model of his risk preference. John Junior thus modeled the utility of the profit after tax of each hedging scenario. Based on the results of the different simulations, John Junior hoped to explain to his father what the risks and implications were for the decision about forward contracts.

1.Assume that the output quantity of John Carter’s farm is known with certainty. Is it a good idea to hedge half the production? How much should Carter hedge? Note: There are 2000 pounds in a ton.
2.Now assume that the output quantity is uncertain. How does the correlation between output and prices affect Carter’s decision on how much to hedge? Prepare your analysis showing hedging decisions for correlations of -.99, 0, and .99.

3.Assuming output quantity is uncertain, which correlation would you use to make a hedging decision? How much should Carter hedge assuming that correlation