Please advise with the following questions. Please refer to attached pictures for the questions
Yeshiva University Financial Economics (E CON 5112?) Spring 2018 - Midterm Exam Always show your work Partial credit will be given This is a take-home exam. No communication with anyone else but me shall occur while you take the exam, and no discussion of the exam shall take place before Tue March 13 after class Question 1 (70%): Beyond Shylock (ll in blanks, provide answers as asked.) Many borrowers do not repay their debts unless they have the incentives to do so. This simple observation justies the institution of secured debt borrowers must provide a collateral in exchange for a loan. In the following stylized model of secured debt, there are two time periods, t = 0 and t = 1, and S > 1 states of uncertainty tomorrow There is one physical commodity (wheat). Traders are expected utility maximizers with Bernoulli utility us which is [continuously differentiable and) strictly increasing, concave, and have endowments w: in state 3 = O, , S Traders have access to K nancial assets, with prices the, k- = 1, ,K and (real) payoffs ak E Rf. In order to borrow (i e , to set Zk' 0 units of asset k as collateral per unit of asset lc' sold, so that the demand for assets must be restricted to satisfy a collateral restriction, K max(0, 3k) 2 Z 23%,? min(0, 21:1], for all R: in K (00) k':1 (a) Prove that if an asset has Erma = 1 then 2!: Z 0 (no short sales} As traders are never required to deliver more than the value of their col- lateral, and traders are utility-maximizing, deliveries must satisfy the following restrictions, which are equations dening the vector (is : (of; .., (if) of eectwe deliveries given requirements i : (Zk\"k)k'.k=1.....Kl d'; = min(af, Sakai}, all Is, all s > o. (D) kl Hereafter, assume that the ags are such that a solution (is > 0 exists for each s > 0. Let 2;" : max(0,zk) and z; : min(0, zk) be the long and short positions in asset Ft, respectively With this change of variable in place, we can write the sequential budget and collateral constraints as (insert your answer here): (b) We derive the implications of the absence of arbitrage for asset prices and payoffs. Let R be the (1+5) x Kdimensional matrix of prices and payoffs, and 231,1 21x NI N Zch: ZK,1 2K,K be the matrix of collateral requirements. With this notation, there is no 3 arbitrage portfolio it system (complete the system below} ?'<:f with strict inequality corresponding to the s-th row of matrix r s g has no solution. there is arbitrage if for all prove that collateral constraints holcl eective delivery rates are set according equations and then cannot be net borrowing today let a convent: cone containing zero thus portfolio only it multipliers us> 0 (1") (why? Be explicit about the math result you use) so that Rxsa '13s rs+(,\\s #5 VS)' 1,72 0,1 (g) Show that this implies that there is A\"! > U and u, u 2 0 such that. for all k, U n : 2nd: + a (1) 5>0 and file = 2 A31]: + :Hk'zk' Vic (2) 5)0 k' Interpreting the no arbitrage equations (1) and {2), the rst equation says that no arbitrage implies that the price of an asset k is the sum of two components: a payoff value, 2550 Aadk and a collateral value, 'uk. 557 b.) Assuming K = 2 = S, asset is = 2 is in positive net supply, 21,1 2 51,2 2 0 while 52,2 2 232,1 2 1, while 0
