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4. Consider a market where there are only one, two and three-period zero- coupon treasury bonds. Under the stochastic discount factor model, if we assume that the log prices of 2-period and 3-period zero coupon bonds and the log stochastic discount factor are jointly normally distributed, the pricing equation in terms of the log prices of the bonds is given by: Pnx = E, (m.: + Pn-x+1)+(1/2)ar, (m. 1 + Pu-k+d) where Pw, is the log price of an n period zero coupon bond in period t and m, is the log stochastic discount factor in period t+1. (a) What is the equivalent expression for the one period (risk free) bond? [10 marks] (b) Show that the pricing equation for the excess return for the two risky, 2 and 3-period bonds takes the form: E pn-1+1 - Pnx + Py = (1/2)Var/(m +1 + PM-) - Cov, (m.:1+Pm-10+1) [15 marks) (c) Give these two pricing equations in terms of the yields to maturity of the two bonds (ie solve for R.,). [10 marks] (d) Given your answer to (c), show what would be required for the yield curve to slope upwards. [15 marks] 4. Consider a market where there are only one, two and three-period zero- coupon treasury bonds. Under the stochastic discount factor model, if we assume that the log prices of 2-period and 3-period zero coupon bonds and the log stochastic discount factor are jointly normally distributed, the pricing equation in terms of the log prices of the bonds is given by: Pnx = E, (m.: + Pn-x+1)+(1/2)ar, (m. 1 + Pu-k+d) where Pw, is the log price of an n period zero coupon bond in period t and m, is the log stochastic discount factor in period t+1. (a) What is the equivalent expression for the one period (risk free) bond? [10 marks] (b) Show that the pricing equation for the excess return for the two risky, 2 and 3-period bonds takes the form: E pn-1+1 - Pnx + Py = (1/2)Var/(m +1 + PM-) - Cov, (m.:1+Pm-10+1) [15 marks) (c) Give these two pricing equations in terms of the yields to maturity of the two bonds (ie solve for R.,). [10 marks] (d) Given your answer to (c), show what would be required for the yield curve to slope upwards. [15 marks]

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