Exercise 4: We consider two risky assets with returns Y1,2,Y2,t for period (t - 1,t) and a riskfree asset with zero riskfree rate. The vector of returns satisfy the following V AR(1) model: unt Y = Y1.t-1 Y2,1-1 U2. (M :)-(53) 3( )+( where the () [CO). )] are i.i.d. N 4.1 What is the conditional distribution of Y4 given Y4-2? 4.2 What is the unconditional distribution of Y4 ? Compare the matrices of volatility-covolatility at horizon 1, horizon 2 and infinite horizon. 4.4 Three investors have at date t the same information Y, the same horizon 1, the same absolute risk aversion A. They differ by the risky assets included in their portfolios. Investor 1 includes the riskfree asset and the two risky assets; Investor 2 includes the riskfree asset and risky asset 1, Investor 3 includes the riskfree asset and risky asset 2. Describe the mean-variance efficient allocations in risky assets of these three investors. 4.5" What are the characteristics, that are the conditional mean and condi- tional variance, of the associated portfolio returns ? Compare these three different portfolio managements. We consider a market with 3 financial assets, i.e. a riskfree asset with zero riskfree rate and two risky assets with risky returns yit, Y2t. These returns satisfy the autoregressive dynamics : t $1(41,0-1 + 42,1-1) + Uits Yat 42(41,t-1 + 42,t-1) + U2t, N(0,0), U2t ~ where ult, uz are independent Gaussian variables, uit N(0, 0). i) Write system (*) under a vector form: Yt = Vyt-1 + ut, where y = (91t, Y2t)', ut = (uit, U2t)'. ii) Show that the process Yt = yt + y2t follows a one-dimensional AR(1) process. What is the associated autoregressive coefficient ? iii) Find the portfolio allocations (01, a2), such that the portfolio return Q1 Ynt + 02 Y2t is a white noise. iv) What are the eigenvalues and eigenvectors of matrix 0 ? Interprete the result. Deduce the stationarity condition for process (yt). v) Compute the power oh. vi) Derive the mean-variance portfolio allocation at horizon h= 1? At horizon h = 2 ? At long term horizon h = o ? Exercise 4: We consider two risky assets with returns Y1,2,Y2,t for period (t - 1,t) and a riskfree asset with zero riskfree rate. The vector of returns satisfy the following V AR(1) model: unt Y = Y1.t-1 Y2,1-1 U2. (M :)-(53) 3( )+( where the () [CO). )] are i.i.d. N 4.1 What is the conditional distribution of Y4 given Y4-2? 4.2 What is the unconditional distribution of Y4 ? Compare the matrices of volatility-covolatility at horizon 1, horizon 2 and infinite horizon. 4.4 Three investors have at date t the same information Y, the same horizon 1, the same absolute risk aversion A. They differ by the risky assets included in their portfolios. Investor 1 includes the riskfree asset and the two risky assets; Investor 2 includes the riskfree asset and risky asset 1, Investor 3 includes the riskfree asset and risky asset 2. Describe the mean-variance efficient allocations in risky assets of these three investors. 4.5" What are the characteristics, that are the conditional mean and condi- tional variance, of the associated portfolio returns ? Compare these three different portfolio managements. We consider a market with 3 financial assets, i.e. a riskfree asset with zero riskfree rate and two risky assets with risky returns yit, Y2t. These returns satisfy the autoregressive dynamics : t $1(41,0-1 + 42,1-1) + Uits Yat 42(41,t-1 + 42,t-1) + U2t, N(0,0), U2t ~ where ult, uz are independent Gaussian variables, uit N(0, 0). i) Write system (*) under a vector form: Yt = Vyt-1 + ut, where y = (91t, Y2t)', ut = (uit, U2t)'. ii) Show that the process Yt = yt + y2t follows a one-dimensional AR(1) process. What is the associated autoregressive coefficient ? iii) Find the portfolio allocations (01, a2), such that the portfolio return Q1 Ynt + 02 Y2t is a white noise. iv) What are the eigenvalues and eigenvectors of matrix 0 ? Interprete the result. Deduce the stationarity condition for process (yt). v) Compute the power oh. vi) Derive the mean-variance portfolio allocation at horizon h= 1? At horizon h = 2 ? At long term horizon h = o