2010 qa exam july .pdf - Erasmus School of Economics FEM21003– Asset Pricing(QF-Variant Written examination General information Date examination

2010 qa exam july .pdf - Erasmus School of Economics...

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Erasmus School of Economics FEM21003– Asset Pricing (QF-Variant) Written examination General information Date examination: 7/7/2010 Lecturer: E. Kole Duration: 9.30 – 12.30 Number of questions 8 questions Number of pages: 9 (incl. cover page) Instructions You are allowed to use a non-graphical calculator during the examination. You are not allowed to use a graphical calculator during the examination. You are not allowed to use notes during the examination. You are not allowed to use books during the examination. You are not allowed to use a dictionary during the examination. It is allowed to take the examination papers with you after the 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 EUR 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 (37 points) 1. ( 10 points ) An investor considers assets A and B with the following pay-off scheme. state probability pay-off A pay-off B 1 0.97 10 10 2 0.01 0 5 3 0.02 20 x B 3 Units of asset A and B trade at the same price of 10. The investor has initial wealth 100. He has a log utility function defined over the value of his consumption, U ( C ) = ln C . His subjective discount factor equals one, so his expected utility over two periods equals U ( C t ) + E[ U ( C t +1 )]. He chooses his allocation to maximize his expected utility. (a) ( 2 points ) For which values of x B 3 has the investor found arbitrage opportuni- ties? For values of x B 3 20 , the pay-off of asset B is equal to the pay-off of asset A in state 1, strictly better in state 2, and better or equal in state 3. This implies an arbitrage opportunity. (b) ( 2 points ) Explain why the investor will never invest only in asset A. If the investor does not invest in asset B, he will have a consumption of zero in state 2. His utility function will diverge to minus infinity because of this, meaning that the investor is bankrupt or dead. (c) ( 2 points ) The investor invests ξ A in asset A and ξ B in asset B. Write down the two conditions that the optimal investments ξ * A and ξ * B should satisfy. The optimal investment maximizes U ( W t - p A ξ A - p B ξ B ) + E[ U ( x A ξ A + x B ξ B )] . The marginal costs and expected marginal gains of investing in assets A and B should be equal. To express marginal costs and gains, we need marginal utility, which equals U 0 ( C ) = 1 /C . The marginal costs of assets A are 1 / ( W t - p A ξ * A - p B ξ * B ) · p A and similar for asset B. Substituting prices and current wealth we find for both 10 / (100 - 10( ξ * A + ξ * B )) . The marginal gain in state s for asset A are x A s / ( x A s ξ * A + x B s ξ * B ) , and similar for asset B. Taking expectations yields π 1 1 x A 1 ξ * A + x B 1 ξ * B x A 1 + π 2 1 x A 2
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