Read “John Carter: Hedging” beginning on page 789 and analyze the following questions:
John Carter was a farmer in Northern Massachusetts. John, like all of the farmers around him, grew apples and shipped his harvest to Boston for sale at the prevailing market price. The farm had been in John’s family for three generations; from his inheritance and prudent management, John had built his net worth to$300,000. Like many of his neighbors, John was being pressed by increasing costs and by the failure of revenues to keep up with this increase. John worried that a really bad year could wipe him out and he might lose the farm.
John tried to project the amount of harvest and the price that it would bring at market. This year he believed the farm would earn revenues in the vicinity of$310,000. John’s analysis of past costs indicated that the farm would incur $220,000 this year in fixed costs and that variable costs would be $0.03 per pound of apples produced. The tax schedule for farmers meant that John did not pay any taxes unless he earned over$25,000 in a year. Between $25,000 and $50,000, he would have to pay a marginal rate of 24 percent, so that 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 his revenues using forward contracts. When he entered a forward contract, John had to guarantee the delivery of the contracted amount of apples at harvest. If John’s own crop fell below the amount of apples that he had sold forward, he would have to buy enough apples on the open market to make up the difference. Any apples that John’s farm produced above the amount specified in the contract would be sold at the prevailing market price. All transactions, including any payment received by John from the forward contract and any selling or buying of apples on the spot market at harvest would be settled at the same time in the fall
John believed the price at harvest would best be approximated by a normal distribution with a mean of $0.2079 per pound and a standard deviation of$0.0247 per pound. Forward contracts were only available in increments of 100 tons, and the current forward rate was $0.2079 per pound. In the past, John had assumed that he could predict with certainty what his land would produce, and he had usually hedged roughly half that amount. John sometimes wondered if that was the best policy for how much to hedge, given his forecast of what his farm would produce.
John Junior, home from college for spring vacation, had decided to help his father. He had produced a worksheet that modeled the future price of apples in an @RISK simulation. John Junior initially setup the model so that the harvest was exactly his father’s forecast—743 tons of apples—and he agreed to help his father figure out what the ideal amount was to sell forward, given the assumption that apple production was known with certainty.
However, John Junior was concerned that his father’s assumptions about the harvest did not capture all of the risks that the farm faced. After some prodding, John Junior was able to get his father to admit that his forecasts were not always right, and John Senior provided John Junior with data about actual market prices and harvest yields in the past(Exhibit 1). John Junior noted with some concern that his father’s forecast for the upcoming harvest was simply the mean of the past ten years’ harvests. In addition to the simple analysis his dad had asked him to do, John Junior decided to model the farm’s profitability, factoring in uncertainty about how many tons of apples would actually be produced. He decided that production quantity was best approximated by a normal distribution with a mean of 743 tons (his father’s forecast) and standard deviation of 87 tons.
After completing his basic model for his father and his “improved” model with the uncertainty about the size of the harvest, John Junior knew he could help decide how many tons of apples to sell forward. But something was nagging him. He suspected there was typically a relationship between the quantity of apples harvested and the price at market. After all, if his father had a bad year, other farmers might also have bad years, 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
