Question 3 (50 points ). You run an oil company, which can invest in three projects numbered 1, 2, 3. The projects returns depend on oil prices , F, and also project -specific risks , ", , "2. ". Payoffs of the projects are: T, = 0.02 +" T2 = 0.06 + F + 2 Ty = 0.04 + F + 3 The covariance matrix of ",",, Bar is: 0.01 0.01 0.02 ,2,1 0.01 0.08 -0.06 .3.1 .3.2 0.02 - 0.06 0. 16 F is independent of all ",",, ", and has mean o and standard deviation of = 0.3. The risk -free rate is 1 = 1%. a) (10 points ). Calculate the covariance matrix between f, , f2. ;- b) (10 points ). Suppose the firm owner is a mean-variance optimizer, whose only options for investment are the three projects the firm has access to. The firm owner can borrow or lend an infinite amount at the risk -free rate . Solve the firm owner's optimal portfolio problem , by calculating the tangency portfolio weights on assets 1, 2, and 3. What is the Sharpe ratio of the tangency portfolio ? c) (5 points ). Now, suppose you have a security h whose payoff is: The = Ty + 2F + "h For the rest of the question , suppose that each unit of asset 1, asset 2, asset 3 and asset h has a fair market price of $ 100. If you hold one unit of asset 2, how many units of asset h you should long or short in order to neutralize your exposure to oil price risk (i.e. so that you have no exposure to F)? How about using asset h to neutralize one unit of asset 3? d) (5 points ). Using your answer to part C, construct 3 "hedged assets " a, b, and c. Asset a is a combination of one unit of asset 1 and some number of units of h, such that a has no exposure to factor F. Likewise, b is a combination of one unit of 2 and some number of units of h, and c is a combination of one unit of 3 and some number of units of h. Write expressions for the returns of these hedged assets . These should express To , T, f. in terms of ", "2. "3: " and numbers e) (10 points ). Suppose " is uncorrelated with ",. "2 and ", and has a o mean and a standard deviation of o = 0.4. What is the variance-covariance matrix of fast, T,? What is the tangency portfolio of these new assets , and what is its Sharpe ratio ? f) (10 points ). Imagine you are a dealer, this oil company is your client, and you are trying to sell asset h to this company. Building on the answers above, pitch asset h to the company. In particular, how should the company use h? How does using h affect the company's optimal project choice? How does h affect the company's optimal risk and return
