# This question needs to be solved without using the optimization software. The world of the problem..

This question needs to be solved without using the optimization software. The world of the problem is described by risk-factors. The question consists of 5 parts. You run an oil company, which can invest in three projects numbered 1,2,3. The projectsreturns depend on oil prices, F, and also project-specific risks, _1, _2, _3. Payoffs of the projects are: r1 = 0.02 + _1 r2 = 0.06 + 1 3 F + _2 r3 = 0.04 + F + _3 Covariance matrix of _1, _2, _3, is: 2 4 2 _,1 _,1,2 _,1,3 _,2,1 2 _,2 _,2,3 _,3,1 _,3,2 2 _,3 3 5 = 2 4 0.01 0.01 0.02 0.01 0.08 0.06 0.02 0.06 0.16 3 5 F is
This question needs to be solved without using the optimization software. The world of the problem is described by risk-factors. The question consists of 5 parts. You run an oil company, which can invest in three projects numbered 1,2,3. The projectsreturns depend on oil prices, F, and also project-specific risks, _1, _2, _3. Payoffs of the projects are: r1 = 0.02 + _1 r2 = 0.06 + 1 3 F + _2 r3 = 0.04 + F + _3 Covariance matrix of _1, _2, _3, is: 2 4 2 _,1 _,1,2 _,1,3 _,2,1 2 _,2 _,2,3 _,3,1 _,3,2 2 _,3 3 5 = 2 4 0.01 0.01 0.02 0.01 0.08 0.06 0.02 0.06 0.16 3 5 F is independent of all _1, _2, _3, and has mean 0 and standard deviation F = 0.3. The risk-free rate is rf = 1%. 1. Calculate the covariance matrix between r1, r2, r3. 2. Suppose the firm owner is a mean-variance optimizer, whose only options for investmentare 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 owners optimal portfolio problem, by calculating the tangency portfolio weights on assets 1,2, and 3. What is the Sharpe ratio of the tangency portfolio? 3. Now, suppose you have a security h whose payoff is: rh = rf + 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? 4. Using your answer to part C, construct 3hedged 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 acombination of one unit of 3 and some number of units of h. Write expressions for the returns of these hedged assets. These should express ra, rb, rc in terms of _1, _2, _3, _h and numbers. 5. Suppose _his uncorrelated with _1, _2, _3 and has 0 mean and a standard deviation h = 0.4. What is the variance – covariance matrix of ra, rb, rc? What is the tangency portfolio of these new assets, and what is its Sharpe ratio?

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