test1314_21003_2A.pdf - Erasmus School of Economics FEM21003– Asset Pricing(QF-Variant Written examination General information Date examination

test1314_21003_2A.pdf - Erasmus School of Economics...

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Erasmus School of Economics FEM21003– Asset Pricing (QF-Variant) Written examination General information Date examination: 1/7/2014 Lecturer: E. Kole Duration: 18.30 – 21.30 Number of questions 8 questions Number of pages: 9 (incl. cover page) Instructions You are allowed to use any (including programmable) calculator(s). You are not allowed to use notes. You are not allowed to use books. You are not allowed to use a dictionary. You are not allowed to take the examination papers with you. You are allowed to take home scrap paper that is handed out by the university during this written examination. Additional information The total amount of points you can obtain in this exam is 70. Your number of points divided by 7/10 determines the grade for this exam. Please answer the questions in a concise and to-the-point way. Good luck! Please notice: if you have not registered for this examination, you can only do so on the day of the examination itself, against payment of 50.- in administrative charges at the Information Desk of the ESC (H6-02). If your examination ends after 16.00 or takes place on a Saturday, payment has to take place on the next working day.
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Part I: Theory (39 points) 1. ( 9 points ) We consider an agent with an exponential utility function defined over consumption c t , u ( c t ) = - e - αc t , α > 0 , (1) where α is his coefficient of absolute risk aversion. The agent lives in a two-period economy with four future states of the world. The agent holds an optimal portfolio of (unspecified) risky assets. In this economy, a new risky asset A becomes available. The agent does not change his optimal portfolio after the introduction of asset A. The table below summarizes the economy. The agent’s current consumption is equal to 100, his coefficient of risk aversion is equal to 0.02 and his subjective discount factor δ is equal to 0.99. state probability c t +1 pay-off A 1 0.25 120 11 2 0.25 80 10 3 π 3 130 11 4 π 4 90 10 (a) ( 2 points ) Show that the pricing kernel obeys m t +1 = δ e α ( c t - c t +1 ) , where δ is the agent’s subjective discount factor. The pricing kernel follows from the general form m t +1 = δu 0 ( c t +1 ) /u 0 ( c t ) . The first derivative of the utility function is given by u 0 ( c t ) = α e - αc t . Consequently, m t +1 = δ · α e - αc t +1 / ( α e - αc t ) = δ e - αc t +1 + αc t = δ e α ( c t - c t +1 ) . (b) ( 2 points ) Calculate the values of the pricing kernel. For which value of π 3 is the gross risk-free rate equal to 1? The gross risk-free rate is the inverse of the expectation of the pricing ker- nel. The values of the pricing kernel can be found by plugging the values for consumption in the expression for the pricing kernel, which yields m t +1 = (0 . 6636 , 1 . 4769 , 0 . 5433 , 1 . 2092) 0 . The average of the pricing kernel follows as E[ m t +1 ] = 0 . 25 · 0 . 6636 + 0 . 25 · 1 . 4769 + π 3 · 0 . 5433 + (0 . 5 - π 3 ) · 1 . 2092 = 1 . 1397 - 0 . 6659 π 3 . If the gross risk-free rate is equal to one, the average of the pricing kernel should also be equal to 1. Solving 1 . 1397 - 0 . 6659 π 3 = 1 yields π 3 = (1 - 1 . 1397) / ( - 0 . 6659) = 0 . 2098 .
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