Theory of Finance

- m +1 = 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: 1/2 2+ +11 (4) where 6+1 ~ iidN (0,0%). The time series process of is: 10t+1 = (1 - 0)* + $a4 ++1 (5) Denote a log price of a zero-coupon bond at time t with face value equal to one and remaining maturity m by Pm,t = log(Pm.t), and the corresponding m-period yield by ym.t (a) (20 marks] Derive an expression for the one-period yield, y = -Pre, in terms of c, and model parameters. (b) [20 marks] Derive the recursive solution of the affine coefficients am and born in the model: - Port = a + b milliy (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 E.pm-16. Pans nu] =- (3 - + abon-1) ore (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 at for the properties of the model? Comment on the implied properties of interest rates, their volatility and the risk premium. [Word limit: 300) - m +1 = 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: 1/2 2+ +11 (4) where 6+1 ~ iidN (0,0%). The time series process of is: 10t+1 = (1 - 0)* + $a4 ++1 (5) Denote a log price of a zero-coupon bond at time t with face value equal to one and remaining maturity m by Pm,t = log(Pm.t), and the corresponding m-period yield by ym.t (a) (20 marks] Derive an expression for the one-period yield, y = -Pre, in terms of c, and model parameters. (b) [20 marks] Derive the recursive solution of the affine coefficients am and born in the model: - Port = a + b milliy (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 E.pm-16. Pans nu] =- (3 - + abon-1) ore (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 at for the properties of the model? Comment on the implied properties of interest rates, their volatility and the risk premium. [Word limit: 300)