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Efficient_Markets,_with_Answers

Efficient_Markets,_with_Answers - Efficient Markets...

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Efficient Markets Financial Economics Bruce C. Dieffenbach Question 1 Consider a stock for which the dividend D t follows a random walk with a positive upward trend: D t + 1 = D t ( 1 + m + e t + 1 ) . Here t denotes the time period. The error term e t + 1 has mean zero and is uncorrelated for different periods; the error term represents the “news” about the dividend. At time t , the expected future dividend is E ( D t + n ) = D t ( 1 + m ) n . In market equilibrium, the price of the stock is the present value of ex- pected dividends, P t = E ( D t + 1 ) 1 + R + E ( D t + 2 ) ( 1 + R ) 2 + E ( D t + 3 ) ( 1 + R ) 3 + ··· . The interest rate R is constant, R > m . a) Find the stock price P t as a function of the dividend D t , by calculating the present value of expected dividends. b) The rate of return from time t to time t + 1 is D t + 1 + P t + 1 P t P t . By substituting your formula from (a) into this expression for the rate of return, show that the efficient-market theory applies: show that expected rate of return equals R , that the rate of return is uncorrelated at different times, and that the news determines the unexpected rate of return. Answer 1 a) We have E ( D t + n ) = D t ( 1 + m ) n , for any n . Computing the present value of expected dividends, we have P t = E ( D t + 1 ) 1 + R + E ( D t + 2 ) ( 1 + R ) 2 + E ( D t + 3 ) ( 1 + R ) 3 + ··· = D t ( 1 + m ) 1 + R + D t ( 1 + m ) 2 ( 1 + R ) 2 + D t ( 1 + m ) 3 ( 1 + R ) 3 + ··· (an infinite geometric sum) = D t ( 1 + m ) 1 + R 1 1 + m 1 + R = D t ( 1 + m ) R m . b) The rate of return comes from the dividend plus the capital gain, rate of return = D t + 1 +( P t + 1 P t ) P t . Using the relationship P t = D t ( 1 + m ) / ( R m ) , we evaluate rate of return = D t + 1 + P t + 1 P t 1 = D t + 1 + D t + 1 ( 1 + m ) R m D t ( 1 + m ) R m 1 = D t + 1 ( R m )+ D t + 1 ( 1 + m ) D t ( 1 + m ) 1 = ( 1 + R ) D t + 1 D t ( 1 + m ) 1 = ( 1 + R ) D t ( 1 + m + e t + 1 ) D t ( 1 + m ) 1 = R +( 1 + R ) e t + 1 ( 1 + m ) .
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The expected rate of return is the first term, R , as the second term is zero on average. The unexpected rate of return is the second term, ( 1 + R ) e t + 1 ( 1 + m ) .
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