2007 a exam december.pdf - This exam consists of 8 numbered...

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This exam consists of 8 numbered pages Erasmus University Rotterdam Master Quantitative Finance Exam FEW2614 “Asset Pricing” Thursday December 21, 2007, 9h30–12.30h This exam consists of 9 questions. Please formulate your answers in a clear, compact, to-the-point matter (no more than 5 sentences or 100 words per essay question). You are allowed to use a calculator. Avoid speculating or including details that are beyond the point. The total points of the exam equals 70. For each question, it is indicated how many points a good answer produces. The sum of your points divided by 7 determines your grade. You need at least 39 points to pass the exam. Good Luck! Part I: Theory (35 points) 1. ( 8 points ) Consider an agent with a concave utility function u ( c ). His sub- jective discount factor is equal to 1. The economy has two periods, and can be in four states of the world in the future. The probabilities of the different states to occur are (0.3, 0.3, 0.2, 0.2). The agent has determined his optimal consumption and investment accordingly. His marginal utility in the different future states of the world is given by (105, 95, 120, 70). (a) ( 1 point ) Rank the states from good to bad. As the utility function is concave, marginal utility is low if utility is high. Consequently, the rank should be 4, 2, 1, 3. 1
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(b) ( 3 points ) Write the risk-free rate as a function of his current marginal utility. Is the risk-free rate increasing or decreasing in current consump- tion? The pricing kernel for each state s is given by u 0 ( c s ) /u 0 ( c 0 ) . The risk-free rate is equal to 1 / E[ m ] , and E[ m ] = 4 s =1 m s π s = 4 s =1 u 0 ( c s ) π s /u 0 ( c 0 ) , so R f = u 0 ( c 0 ) / 4 s =1 u 0 ( c s ) π s . Substitution yields R f = u 0 ( c 0 ) / 98 . If cur- rent consumption goes up, it’s marginal utility goes down, so the risk-free rate is decreasing in current consumption. (c) ( 2 points ) Assume the risk-free rate is equal to 1.05. Value an asset A with pay-off (20, 10, 25, 15). What is the expected return on this asset? (If you cannot find the values of the pricing kernel for different states of the world, assume they are given by (1.03, 0.9, 1.1998, 0.6671).) If the risk-free rate is 1.05, current marginal utility is 1 . 05 × 98 = 102 . 90 . The different values for the pricing kernel can subsequently be found as (1.0204, 0.9232, 1.1662, 0.6803). Contingent claims prices can be found as p c s = m s π s , which yields (0.3061, 0.2770, 0.2332, 0.1361). The asset’s price can be found as p A = 4 s =1 p c s x A s = 16 . 76 . The expected pay-off is equal to 17. The expected return is hence 17 / 16 . 76 = 1 . 014 . Under the alternatively given pricing kernel, the contingent claim prices would be (0.309, 0.27, 0.23996, 0.13342), and the price of the asset would be 16.88. The return would be 17 / 16 . 88 = 1 . 007 . (d) ( 2 points ) Under the same assumption, value an asset B with pay-off (40, 25, 60, 15). What is the expected return on this asset? Which asset is judged on returns the best hedge against future variation in consump- tion?
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