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Exercise 1
You are the CFO of an automotive company. Your company signed a large contract with an engine manufacturing company to purchase sport engines for your new line of race cars. The company requires payments to be made in cryptocurrency, 1000 Ethereum to be exact, and by the end of December 2022. You are naturally worried about the volatility of Ethereum prices and, therefore, want to hedge your position using futures or options.
You can find:
The spot price of Ethereum here: https://www.coindesk.com/price/ethereum/
The futures prices on Ethereum here: https://www.cmegroup.com/markets/cryptocurrencies/ether/ether.quotes.html (Ignore daily settlement for the purpose of this exercise.)
A. Discuss the no-arbitrage relation between futures and spot prices in the context of Ethereum. Does this no-arbitrage relation hold given current prices?
B. Describe the appropriate position in (1) a futures contract and (2) an option contract. For the option contract, focus on (European) options that mature at the end of December 2022 and have a strike price equal to the futures price.
C. Represent in a figure the cash flow (in $) of the automotive company in December 2022 as a function of Ethereum’s spot price when they do not hedge.
D. Present (in the same graph as C.) the total cash flow from a fully hedged position using futures. Also, present the payoffs from the appropriate position in futures, indicating at which ranges of the Ethereum spot price in December 2022 the company makes a loss and at which ranges it makes a profit on the futures position.
E. Present (in the same graph as C. and D.) the payoff at maturity of the appropriate option position as well as the total cash flow from the option-hedged position.
F. Discuss the relative attractiveness of the hedging strategies of D. and E. taking into account the price of Ethereum options. Do you think the price of Ethereum options is high or low relative to options on other assets (such as the S&P500 or oil)?

Exercise 2

You manage a portfolio worth $1 Billion for a US pension fund. The portfolio is equal-weighted over 17 industry indices (see:
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/det_17_ind_port.html ). The monthly returns (including dividends, in excess of the t-bill rate) are given in the attached excel file. Your goal is to analyze the prospects of this portfolio over the next year.
a. Calculate the annualized portfolio return volatility using the last five years of data (i.e., 60 months of returns from 2017-8 to 2022-7).
b. Which industry contributed the least to the portfolio volatility?
c. Calculate the betas of the industries with respect to the market portfolio (reported also in the Excel file) over the last five years. Interpret the numbers you obtain.
d. Assume the expected return (in excess of the risk-free rate) on the market portfolio equals 5% and the risk-free rate equals 1% (both annualized). Calculate the annual expected return for each of the industries and the equal-weighted portfolio. You will use this expected return in the questions that follow.
e. Assume that the portfolio returns are normally distributed. Calculate the probability that the portfolio will incur a loss (in excess return) larger than 25 percent in total over the next year. Similarly, what is the probability that the portfolio will realize a loss of more than 25 percent over the next six months?
f. Calculate the 1-year Value-at-Risk at the 1%, 5%, 10% level. Interpret your results.
g. Calculate the 10% expected shortfall using a simulation. (A bonus point is awarded when this number is verified using the properties of a truncated normal distribution.)
h. You plan to go green and sell your position in the oil industry and replace it with a position in the food industry. Approximate the impact of this sale on the 1-year Value-at-Risk at the 5%-level calculated in f. Discuss how accurate your approximation is when compared to the exact VaR(0.05) of the new portfolio.
i. In all your calculations until now, you have used the last five years of data. This is a simple approach to overweight recent observations when estimating risk. In practice, risk is more commonly estimated using the exponentially-weighted moving average (EWMA) technique. Use a smoothing parameter (lambda) of 0.97 and recalculate the 1-year Value-at-Risk at the 5% level using the most recent volatility estimate from the EWMA model. Compare your results to those obtained in f and discuss how effective our simple approach (of focusing on the last 60 observations) is.

Exercise 3

Suppose you want to hedge the exposure of the EW industry portfolio from Exercise 2 to the Euro (coming from the fact that many US companies have sales and buy input goods in Europe) and to oil futures (because you are worried about climate risk). For this purpose, you consider investing in two futures contracts: US$-Euro and oil. A long position means you buy 1 Euro or 1 barrel of oil at the futures price in US$. Suppose an analyst provides you with the following information on the monthly returns of the futures:

Monthly data
EW industry portfolio US$-Euro futures Oil futures
Exp return 0.53% 0.00% 1.00%
St. Dev. 5.67% 3.00% 5.00%
Covariance matrix
EW industry portfolio 0.0032 0.0007 -0.0009
US$-Euro futures 0.0007 0.0009 0.0008
Oil futures -0.0009 0.0008 0.0025

A. Given the positive expected return on the oil futures contract, is the oil futures market more likely in backwardation or contango?
B. Given the positive correlation between the US$-Euro futures contract and the industry portfolio, is the average US firm more likely to (i) have sales in Europe (for which they receive Euros) or (ii) buy input goods in Europe (for which they pay Euros).
C. What is the variance-minimizing hedge ratio in each of the two futures contracts? (Hint: for a simple regression Y = a + bX + e, we have b = Cov(Y,X)/Var(X). You need to figure out the multiple regression equivalent of this relation.) Which contract(s) do you go long / short to hedge?
D. What is the fraction of the variation in the EW industry portfolio that is hedged using the futures contracts (and which fraction remains unhedged)?

Exercise 4

MBFS is a European non-profit association that organizes study trips to the United States. In particular, MBFS pays students to go on a six-month visit to a US college of their choice. This price includes everything from plain tickets, housing, food, tuition, and so on. Since MBFS pays these costs in US$, while donations to the association are mostly in €, MBFS is worried about changes in the exchange rate.

A. Is MBFS worried about an appreciation or a depreciation of the US$?

MBFS wants to perform a detailed scenario analysis for the total cost of their study abroad program depending on the total number of contracts (students going to the US) and using various hedging strategies. For this scenario analysis, assume that
• the US$ cost per student is 5000$ and
• there are three possible exchange rates:

0.9€ for 1$ Strong Euro
1€ for 1$ The current exchange rate
1.1€ for 1$ Weak Euro

B. Consider first the expected scenario of 10000 contracts. What is the expected cost of the study abroad program and how does it compare to the best and worst case scenarios when the exchange rate changes?
C. Assume the forward price is 1€ (for 1US$). If MBFS decides to hedge exchange rate risk for all 10000 contracts using forwards, what is the total cost in the three exchange rate scenarios? Is MBFS going long or short the forward?
D. Assume options with a strike price of 1€ are traded in the market. If MBFS decides to hedge exchange rate risk for all 10000 contracts using options, what is the total cost in the three exchange rate scenarios? In your answer, use that the options have a premium that is 5% of the strike price. Is MBFS going long put or call options?
E. Discuss the relative advantages and disadvantages of the two hedging strategies in C and D. In your discussion, assume that MBFS does not intend to speculate on changes in the exchange rate.

Now suppose there is volume risk, meaning that the total number of contracts may vary: it can either be low (5000 contracts) or high (15000 contracts). Assume that MBFS will only know the number of contracts after it takes any hedging decisions. Therefore, it will always use the expected number of contracts of 10000 to determine the position in forwards and options.

F. Redo your calculations from C and D for the low and high volume scenarios. Discuss important differences with the expected scenario of 10000 contracts. Also, discuss the relative advantages and disadvantages of hedging with forwards versus options when there is volume risk. In your answer, (i) take all the evidence from C, D, E, and F into account and (ii) assume that MBFS is most worried about the low volume case, because this indicates a decreasing interest in their study trips.

Exercise 5

A. Look up (Henry Hub) natural gas futures for all expirations up to December 2023 (https://www.cmegroup.com/markets/energy/natural-gas/natural-gas.quotes.html) as well as risk-free interest rates (see, e.g., https://www.treasury.gov/resource-center/data-chart-center/interest-rates/pages/textview.aspx?data=yield). What do these futures prices and risk-free interest rates together imply about the convenience yield and storage cost priced into the futures?
B. Look up the prices of March 2023 puts and calls on the above (Henry Hub) natural gas contract with a strike price of 5900. Do these prices satisfy the put-call parity relation (as defined in Chapter 18.4 of the Hull book)?
C. What does the price of the March 2023 put option imply about the market’s expectation of natural gas volatility? Present a calculation and discuss briefly why option-implied estimates of forward-looking volatility may be useful in practice.

Exercise 6

Assume today is September 30, 2022. A US company is expecting to mine a total of 100,000 ounces of gold over the next three months, i.e., until 31 December 2022. The current spot price of gold is 1640$ per ounce. The company is worried about a large drop in the gold price, because the company will face significant financial distress costs if the average price they receive for gold drops below 1400$ per ounce. (Ignore daily settlement for the purpose of this exercise.)
a. Assuming that the annualized volatility of gold is 16%, what is the probability that the firm will have to pay financial distress costs until December 2022? Assume that gold returns follow a normal distribution and that the expected change in the spot price of gold is 0%.
b. Using the same assumptions, what is the probability that the company will receive a per ounce price that is lower than the current spot price, but larger than 1400$?
c. Suppose that the company wants to hedge using futures and is considering three hedge ratios: 0, 0.5, and 1. Plot the hedged cash flows at the end of December for the three possible hedge ratios and interpret. Use that the December 2022 futures price is 1554$.
d. Calculate the smallest hedge ratio so that the probability of having to pay distress costs is ≤2%. Continue to assume the expected change in the spot price of gold is 0%.
e. Suppose the company is expecting to produce 100,000 ounces of gold for the next 2 years. The company is considering two hedging strategies:
(i) short 8 futures contracts (for 100,000 ounces each) with maturities equal to 3, 6, 9, … and 24 months (expiring, respectively, in December 2022, March 2023, June 2023, September 2024); or,
(ii) short only the 3 month contract today (the contract expiring in December 2022) and 3 months from now, roll over into the new 3 month contract (expiring in March 2023). This rollover strategy will never be invested in futures contracts with a maturity larger than 3 months.
Discuss the relative advantages and disadvantages of strategy (i) vs (ii). In your discussion, take a stance on whether the gold futures market is currently in backwardation or contango.