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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 10 Due December 2, 2010 (in recitation) 1. A financial parable. An investment bank is managing $1 billion, which it invests in various financial instruments (“assets”) related to the housing market (e.g., the infamous “mortgage backed securities”). Because the bank is investing with borrowed money, its actual assets are only $50 million (5%). Accordingly, if the bank loses more than 5%, it becomes insolvent. (Which means that it will have to be bailed out, and the bankers may need to forgo any huge bonuses for a few months.) (a) The bank considers investing in a single asset, whose gain (over a 1year period, and mea sured in percentage points) is modeled as a normal random variable R , with mean 7 and standard deviation 10. (That is, the asset is expected to yield a 7% profit.) What is the probability that the bank will become insolvent? Would you accept this level of risk? (b) In order to safeguard its position, the bank decides to diversify its investments. It considers investing $50 million in each of twenty different assets, with the i th one having a gain R i , which is again normal with mean 7 and standard deviation 10; the bank’s gain will be ( R 1 + ··· + R 20 ) / 20. These twenty assets are chosen to reﬂect the housing sectors at different states or even countries, and the bank’s rocket scientists choose to model the R i as independent random variables. According to this model, what is the probability that the bank becomes insolvent? bank becomes insolvent?...
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 Fall '10
 Prof.DimitriBertsekas
 Electrical Engineering, Standard Deviation, Probability theory, Probabilistic Systems Analysis, First Coin

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