4) Returning to question 1, let's say you want to consider hedging your airline's exposure to fluctuations in jet fuel prices with futures market hedges in two contracts: heating oil and propane. (After all why stop at one hedging contract? What if you could construct a better hedge with more than one hedging contract?)

(a) How might you calculate the number of contracts required for each asset in the two-asset case? (Hint: consider how the one-asset MVHR is derived -- can you extend this to multiple assets?) Describe your methodology, and then repeat the exercise in question 1: download data for propane futures prices from 1993 to 2009, estimate the hedge ratios for heating oil and propane futures (again using jet fuel spot prices) using the data from 1994 to 2000, and interpret your results in the context of the standard deviations of the assets' price movements and the correlations between them.

(b) Next, calculate the changes in the value of your hedged portfolio at monthly intervals from 2001 to 2008. How does the volatility of this hedged portfolio compare to the volatility of the portfolio that you hedged using only heating oil futures? Why do you think this is? Based on your results, what is the single most important criteria that you should use when selecting a hedging contract?

Question 1 for Reference:

1) Cross-hedging

It is the year 2000, and you work in risk management for an airline. You want to hedge the airlines exposure to fluctuations in the price of jet fuel (your airline buys one million gallons of jet fuel every month), but the companys bylaws restrict you to using exchange-traded futures for hedging, and there is not a liquid market for jet fuel futures contracts. Suppose you choose to hedge using futures in a closely-related asset, heating oil.

a) Download the following data:

(1) Monthly NYMEX futures prices for heating oil (No. 2 heating oil, NY Harbor, Contract 1), 1990-2019, from www.eia.gov

(2) Monthly spot prices for jet fuel (U.S. Gulf Coast Kerosene-Type Jet Fuel), 1990- 2019, from www.eia.gov

Combine your data in Excel, aligning by date. Use the data from 1990 to 2000 to calculate the minimum-variance hedge ratio (MVHR), . Look up the contract size for heating oil futures contracts on the CME Group web site, also called Heating Oil USLD (www.cmegroup.com), and determine the optimal number of heating oil futures contracts to use for hedging (round to the nearest whole number). Specify whether your futures hedge position is long or short.

b) Use the optimal number of heating oil futures contracts you found in (a) to calculate monthly changes in the value of a hedged position over the period from 2001 to 2010. Calculate the volatility of these changes. Repeat these steps using values of 2 and 2 to test whether a hedge using produces the minimum volatility in the changes in value of the position.

c) Produce a frequency histogram comparing the sizes of the fluctuations in your airlines fuel cost (a) with the hedge from part (a) in place, and (b) without a hedge. How do you interpret the results? Was the hedge effective?

d) Explain why the MVHR for an equity portfolio being hedged by stock index futures is given by the portfolio's CAPM beta.