Question 3 (Recourse function, (20 marks)). Let ? be some set of scenarios and X C R" be an arbitrary set. Consider the risk-neutral stochastic program IEX min c z + E [Q(x,w)] where the recourse function Q: X x 0 - R is given by HEY Q(x,w) = min fo(y,w) s.t. A(y,w) + g.(x, w) Shi(w), i= 1,....m, where Y is a convex set and for every we ? and i = 1, ..., m, g,(,w) is a convex function of r. 1. Suppose that for every w E ? and i = 0, ..., m, f,(.,w) is a convex function of y. Prove that Q(.,w) is a convex function of r for every w e !. (6 marks) 2. Suppose that for every w E ? and i = 0, .... m, /,(, w) is a concave function of y. Also assume that Y is a polytope, which is a convex set with finitely many extreme points. Let the extreme points of Y be the vectors {y', ...."} for some finite integer K. Prove that for every A 2 0, the recourse function can be lower bounded by the function d: X x R" x 0 - R which is defined as (14 marks) (x, A, w) := -Ah,(w) + ) dig (x, w) + min folg", w) + AS.(g*, w). i-1 Also argue that this lower bound is a convex function of r for every A 2 0 and we !. An extreme point is a point that cannot be written as a convex combination of two other points in the set.Question 4 (Computational Exercise, (30 marks)). Consider the second question from tutorial 4. 1. Batch the data and create a table giving the geometric means pit and standard deviation if for the 8 indices (two 4 x 8 tables, one for mean, one for std.dev). (5 marks) 2. Solve the risk-neutral stochastic program as a Linear Program. (5 marks) Use the ten scenarios given below, and we assume for simplicity each of these scenarios repre- sents the scenarios for all assets i and quarters t, Scenario Return rate rit(w;) Probability p(w}) WI Hit - 80i,t 0.10 Pit - 30it 0.04 Wa Hit - 201,t 0.07 WA Hit - 1.501,t 0.12 Hit - Gift 0.20 Hi,t 0.15 WT Hit + Fit 0.05 Hit + 1.501,t 0.13 wg Hit + 201,t 0.08 Mir + 301,t 0.06 The investment cost c for each index is Index | S&P100 S&P500 S&P600 Dow NASDAQ Russell 2000 Barron's Wilshire Ci 0.45 1.15 0.65 0.8 1.25 1.1 0.9 0.7 and the penalty cost by for the four quarters are, respectively, 1.3, 2.5, 1.75, 3.25. Target return rates for the different quarters are R, (0.5) where for A E [0, 1] we define Ri(X) := min Air + A max Hit - min.. 3. Vary target returns as Re(X) for AC {0, 0.05, 0.1, 0.15, ..., 1.0} and using the optimal portfolios for the above stochastic program, plot the composition of portfolios. (5 marks) 4. Formulate a Linear Program for the risk-averse stochastic program using CVaR as the risk measure, where the term sCVQR,[Q(r, w)] is added to the objective. (5 marks) 5. Solve the risk-averse problem and plot the portfolio composition by varying R, (A) as before and 8 = 90% and & e {1, 10, 50}. (5 marks) 6. Comment on how your plots compare for the risk-neutral vs. risk-averse problems