FIN 450
Spring 2019

  1. Cross-hedging

You work in risk management for an airline. You want to hedge the airline’s exposure to fluctuations in the price of jet fuel, but the company’s 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), 1990-2018, from

(2)Monthly spot prices for jet fuel, 1990-2019, from

Combine your data in Excel, aligning by date. Use the data from 1990 to 2000 to calculate the minimum-variance hedge ratio (MVHR), ℎ∗. Look up the contract size for heating oil futures contracts on the CME Group web site, also called Heating Oil USLD (, 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. (5 points)

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 ℎ∗ and 2ℎ∗ to test whether a hedge

using ℎ∗ produces the minimum volatility in the changes in value of the position. (3 points)

c)Explain why the MVHR for an equity portfolio being hedged by stock index futures is given by the portfolio’s CAPM beta. (1 point)

  1. Forward exchange rates under no-arbitrage

a)Find the five-year forward AUD/JPY exchange rate under no-arbitrage if the spot exchange rate is 80 yen per Australian dollar, and the five-year risk-free interest rates in Australia and Japan are 4% and 6% per annum, respectively. (1 point)

b)Choose a forward exchange rate that is greater than the no-arbitrage exchange rate you found in (a), and describe the arbitrage strategy you would use to exploit this situation. Calculate your profits from arbitrage, assuming you’re able to borrow 1,000 units of any given currency. (1 point)

c)Choose a forward exchange rate that is lower than the no-arbitrage exchange rate you found in (a), and repeat the steps in (b). (1 point)

d)Forwards and futures exist on cryptocurrencies like Bitcoin. Would you calculate forward prices on these contracts as if Bitcoin was a stock, or forward exchange rates (as if Bitcoin was a currency)? Why? Can you support your answer with research? (2 points)

  1. Forward prices and convergence

a)Show that, at time = ,   =   must be equal to   =   under no-arbitrage forward prices for a non-dividend-paying stock. (1 point)

b)Assume that 1 and 2 are the futures prices of two contracts on the same consumption commodity, with times to maturity 1and 2, 2 > 1. Show that 2 ≤ 1   (  2−  1).

(1 point)

c)At time = 0, a speculator takes a long position in a futures contract that will expire at time . The present value of this contract to the speculator is given by:
− 0  −     +  [  ]  −

Where is the risk-free rate, and is the investor’s required rate of return on the investment. Assume no-arbitrage pricing. Show analytically that if the return from the underlying is positively correlated with the overall return on the stock market, then the futures market must be in backwardation. (2 points)

d)Download the following data series:

(1)Monthly NYMEX futures prices for crude oil (light-sweet, Cushing, OK, Contract 1), 1983-2018, from

(2)Monthly spot prices for crude oil (WTI – Cushing, OK), 1986-2019, from

(3)1-month treasury bill rates (“1-Month Treasury Constant Maturity Rate”), monthly average, Jan. 2010 to Dec. 2018, from

Combine the data series in a spreadsheet (make sure dates are aligned) for the years 2010 to 2018. Assume that futures prices are determined under no-arbitrage, and that storage costs are negligible. Derive an estimate for the monthly convenience yield, and generate a scatter plot showing these estimates over time. (Label your axes and include a title.)

Based on your scatter plot, from the end of 2014 to the end of 2017, was the crude oil market in contango or in backwardation? How do you know? (6 points)

Additional question for FIN 550 students only:

4)Returning to question 1, let’s say you want to hedge your airline’s exposure to fluctuations in jet fuel prices with futures market hedges in two contracts: heating oil and propane. 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 prop