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Chapter3_Solutions

Chapter3_Solutions - Chapter 3 Assigned Problems 1 12 15 16...

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Chapter 3 Assigned Problems: 1, 12, 15, 16, 19 1. Perfect or Fully-Informative Information System Current Financial Statement Information GN BN State of nature (future profitability) High 1 0 Low 0 1 Here, each state produces a different message with probability 1. Thus, if the state is H, the financial statements will show GN for certain, and so forth. Prior probabilities of the states of nature are: P(H) = 0.30 P(L) = 0.70 (any other set of prior probabilities with P(H) > 0 would do) Suppose that GN is observed. Then, by Bayes’ theorem: 00 . 1 ) 0 70 . 0 ( ) 00 . 1 30 . 0 ( 00 . 1 30 . 0 ) / ( ) ( ) / ( ) ( ) / ( ) ( ) / ( = × + × × = + = L GH P L P H GN P H P H GN P H P GN H P Page  1  of  9
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0 ) 0 70 . 0 ( ) 00 . 1 30 . 0 ( 0 70 . 0 ) / ( ) ( ) / ( ) ( ) / ( ) ( ) / ( = × + × × = + = L GN P L P H GN P H P L GN P L P GN L P Thus, with a perfect information system, the information perfectly reveals the true state of nature. If BN is observed, similar calculations give P(H/BN) = 0, P(L/BN) = 1. Non-Informative Information System Current Financial Statement Information GN BN State of nature (future profitability) High 0.8 0.2 Low 0.8 0.2 Here, both rows of the information system are the same. Any system with both row probabilities the same would do. Suppose that GN is observed. Then, by Bayes’ theorem: ( 29 ( 29 30 . 0 80 . 0 70 . 0 80 . 0 30 . 0 80 . 0 30 . 0 ) / ( ) ( ) / ( ) ( ) / ( ) ( ) / ( = × + × × = + = L GN P L P H GN P H P H GN P H P GN H P Page  2  of  9
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70 . 0 ) 80 . 0 70 . 0 ( ) 80 . 0 30 . 0 ( 80 . 0 70 . 0 ) / ( ) ( ) / ( ) ( ) / ( ) ( ) / ( = × + × × = + = L GN P L P H GN P H P L GN P L P GN L P The posterior probabilities are the same as the prior probabilities in this case, which is what we would expect if the information system is non-informative. That is, regardless of which is the true state, the state probabilities are the same after the financial statements as before. In effect, the information cannot discriminate between states. Thus, it is non-informative, or useless.
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