Question 1 : Risk Parity Portfolio (20 points) Risk Parity Portfolio (RPP) aims at equalizing the risk contribution from the invested assets in the global portfolio risk. Given a portfolio wRN and the return covariance matrix , the portfolio volatility is (w)=wTw The relative risk contribution (RCC) form the ith asset is defined as RRCi=2(w)wi(w)i=ci, where c is the given risk budget vector. One RPP formulation is given by xminimizesubjectto21xxclog(x)x0 We can recover the portfolio by w=x/(1x). (a) Prove the problem is convex. (b) Use log-returns from "2015-01-06" to "2015-05-29" to find W and the corresponding relative risk contribution (RRC) for following c : c1=[0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1]c2=[0.2,0.2,0.2,0.05,0.05,0.05,0.05,0.05,0.05,0.1]c3=[0.15,0.25,0.1,0.1,0.02,0.02,0.02,0.02,0.02,0.3] (c) Plot capital allocation distribution as the format of the following figure. (d) Plot RRC distribution as the same format. Question 1 : Risk Parity Portfolio (20 points) Risk Parity Portfolio (RPP) aims at equalizing the risk contribution from the invested assets in the global portfolio risk. Given a portfolio wRN and the return covariance matrix , the portfolio volatility is (w)=wTw The relative risk contribution (RCC) form the ith asset is defined as RRCi=2(w)wi(w)i=ci, where c is the given risk budget vector. One RPP formulation is given by xminimizesubjectto21xxclog(x)x0 We can recover the portfolio by w=x/(1x). (a) Prove the problem is convex. (b) Use log-returns from "2015-01-06" to "2015-05-29" to find W and the corresponding relative risk contribution (RRC) for following c : c1=[0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1]c2=[0.2,0.2,0.2,0.05,0.05,0.05,0.05,0.05,0.05,0.1]c3=[0.15,0.25,0.1,0.1,0.02,0.02,0.02,0.02,0.02,0.3] (c) Plot capital allocation distribution as the format of the following figure. (d) Plot RRC distribution as the same format