Each of 1,000 identical corporations issues one bond worth $100 and maturing next year. The bond pays $125 if the corporations survives, and 0 if it defaults. Corporate defaults partially depend on the business cycle. With probability 20%, there will be a recession next year. With probability 80%, there will be an expansion. In a recession, firms default with probability 40%, independently of each other. In an expansion, firms default with probability 15%, independently of each other. 1. What is the ex-ante default probability of an firm? 2. Show that bonds are fairly priced for risk-neutral investors. 3. A financial institution buys all the bonds, and issues 1,000 notes, each with a claim to a fraction 1/1,000 of each of the bonds. You may assume that the sample is large enough that the law of large numbers holds so, conditional on being in a recession (expansion) exactly 40% (15%) of the corporations default. (a) What is the payoff distribution of the note? That is: What are the possible payoffs of the note and the respective probabilities? (b) Whats the fair price of the note? (c) Does diversification fully remove risk? Why or why not?