1. In what follows, we consider non-constrained optimized portfolios. (a) Define the diversification ratio DR (x) by considering a general risk measure R (x). How can one interpret this measure from a risk allocation perspective? (b) We assume that the weights of the portfolio are positive. Show that DR (x) ? 1 for all risk measures satisfying the Euler allocation principle. Find an upper bound of DR (x). (c) We now consider the volatility risk measure. Calculate the upper bound of DR (x). (d) What is the most diversified portfolio (or MDP)? In which case does it correspond to the tangency portfolio? Deduce the analytical expression of the MDP and calculate its volatility. (e) Demonstrate then that the weights of the MDP are in some sense proportional to ??1?. 2. We suppose that the return of asset i satisfies the CAPM: Ri = ?iRm + ?i where Rm is the return of the market portfolio and ?i is the specific risk. We note ? = (?1, . . . , ?n) and ? = (?1, . . . , ?n). We assume that Rm ? ?, var (Rm) = ? 2 m and cov (?) = D = diag

AntiquesNow is a start-up company that is considering building a flag- ship brick-and-mortar store as well as an online e-business to sell an- tiques. It plans to use the flagship store as a warehouse from which to fill orders from the e-business. DCF analysis using an appropriate risk- adjusted discount rate places the NPV of the expected future cash flows at $50 million. The annual volatility factor for this payoff is calculated to be 25%. Based on the competition, AntiquesNow estimates that it has three years to make a goo-go decision on this project. The continuous annual risk-free interest rate over this period is 5%. Construction of the flagship store is expected to cost $50 million and the development of the e-business is estimated at $20 million. AntiquesNow can start the e- business project at any time, but the flagship store must be built before the website is launched, thereby creating a parallel compound option. The company is interested in taking advantage of this option in project valuation to make a better investment decision that takes into account the payoff uncertainty. What is the ROV of the project? 8-5. Anti-Fossil Fuels is contemplating building a manufacturing plant to produce specialty fuel cells that generate power without emitting green- house gases. Since this is a novel approach to generating power, there is uncertainty regarding how well it will be received in the marketplace. The payoff for this project is also influenced by the price of oil, which determines the market for alternative sources of energy. Thus, the ulti- mate payoff for this project is affected by two sources of uncertainty: (1) uncertainty in the market for fuel cells and (2) uncertainty in the price of oil. The payoff volatilities due to these two factors are estimated to be 20% and 30%, respectively. Assuming that the two uncertainties are uncorrelated, Anti-Fossil Fuels is interested in valuation of the project using rainbow options. The present value of the expected cash flows from future fuel cell sales is estimated to be $250 million. Plant construc- tion is expected to cost $300 million. What is the value of the option to wait to invest given an option life of two years, over which the continuous annual risk-free rate is expected to be 5%? 8-6. Recalculate the value of the option to wait for Anti-Fossil Fuels assum- ing that the volatility in the payoff due to the uncertainty in the market for fuel cells is 30% and the volatility due to the uncertainty in the price of oil is 20%. Is this value the same as or different than in Problem 8-5? Why or why not? 8-7. Smart Textiles, a specialty textile manufacturer, is considering an invest- ment in a new plant that will manufacture a spill-proof fabric, but there is uncertainty about the market demand for the new fabric. The uncer- tainty will increase at the end of two years, when a regional trade pact goes into effect, resulting in a larger market but an increased number of competitors. Smart Textiles forecasts the present value of the expected future cash flows to be $200 million and the investment cost to be $250 million. The annual volatility of the logarithmic returns for the future cash flows is estimated to be 20% for the next two years and is expected to increase to 30% at that point when the trade pact goes into effect. The annual continuous risk-free rate for the next four years is 5%. What is the ROV of this project?

Assume that an underlying stock has a current price of $50 and pays no dividends. The risk-free rate is 5% per period, and the stock price follows the following binomial tree: $60 $50 $40 A new derivative: if the stock price goes up, payoff of this derivative is $12.38; $1.27 if the stock price goes down What is the current price of this derivative?