2. Consider a one-factor square-root model, which is a discrete-time version of the Cox, Ingersoll and Ross (1985) model. Denote log stochastic discount factor by me = log(M.) and assume that it is defined as: - m.t1 = + x26-1 (4) where foundN(0,7). The time series process of :, is: 2+1 = (1 - 0) +or+*12+1 (5) Denote a log price of a zero-coupon bond at time with face value equal to one and remaining maturity m by Paul = log(P), and the corresponding m-period yield by ./ - - Po (a) [20 marks] Derive an expression for the one period yield, yu = - , in terms of , and model parameters. (b) (20 marks] Derive the recursive solution of the affine coefficients and D... in the model: (6) Show the intermediate steps in the derivation. (c) [20 marks] Show that the expected log excess return on an m-period bond over one period rate is given by En pom-1+1 Pous - 2) =-(-a) on (7) Show the intermediate steps. (d) (20 marks) From Eq.(7), what sign should we expect a to have? Why? Explain. (Word limit: 300] (e) (20 marks] What are the consequences of the square-root assumption of the state variable... for the properties of the model? Comment on the implied properties of interest rates, their volatility and the risk premium, [Word limit: 300) 2. Consider a one-factor square-root model, which is a discrete-time version of the Cox, Ingersoll and Ross (1985) model. Denote log stochastic discount factor by me = log(M.) and assume that it is defined as: - m.t1 = + x26-1 (4) where foundN(0,7). The time series process of :, is: 2+1 = (1 - 0) +or+*12+1 (5) Denote a log price of a zero-coupon bond at time with face value equal to one and remaining maturity m by Paul = log(P), and the corresponding m-period yield by ./ - - Po (a) [20 marks] Derive an expression for the one period yield, yu = - , in terms of , and model parameters. (b) (20 marks] Derive the recursive solution of the affine coefficients and D... in the model: (6) Show the intermediate steps in the derivation. (c) [20 marks] Show that the expected log excess return on an m-period bond over one period rate is given by En pom-1+1 Pous - 2) =-(-a) on (7) Show the intermediate steps. (d) (20 marks) From Eq.(7), what sign should we expect a to have? Why? Explain. (Word limit: 300] (e) (20 marks] What are the consequences of the square-root assumption of the state variable... for the properties of the model? Comment on the implied properties of interest rates, their volatility and the risk premium, [Word limit: 300)
