handoutassetpricingsp08 - Asset Pricing 1 Lucas (1978,...

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Asset Pricing 1 Lucas (1978, Econometrica) Question: What is the equilibrium price of a stock? Defn: a stock is a claim to a future stream of dividends. 1.1 Environment Tastes: E " X t =0 β t U ( c t ) # Technology: Each project or tree bears uncertain dividends (stochastic endowments of fruit): y t (= d t ) D a f nite set in R ++ with Π ( y t +1 | y t )= prob ( y t +1 = y 0 | y t = y ) . 1.2 Equilibrium Ownership of projects or trees is determined each period in a competitive stock market. Each tree has one outstanding perfectly divisible equity share. A share (denoted s t ) entitles its owner at the beginning of period t to all of the tree’s output in period t . Shares are traded after payment of dividends at price p t . Household sequence problem max { c t ,s t +1 } E " X t =0 β t U ( c t ) # (1) subject to c t + p t s t +1 = s t ( y t + p t ) Def. An equilibrium is a sequence t =0 , 1 , ... of functions: a price func- tion p t = P ( y t ) and an allocation of consumption and stocks c t ( s t ,y t ) and s t +1 ( s t t ) which satis f es household optimization (1), stock market clearing s t =1 , and goods market clearing c t = y t . The f rst order conditions characterizing the optimal choice of stocks sat- is f es p t U 0 ( c t βE t [ U 0 ( c t +1 )( y t +1 + p t +1 )] (2) which is the standard (asset pricing) Euler equation. 1
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Rearranging (2), substituting the market clearing condition c t = y t ,and iterating forward one period yields p t +1 = βE t +1 U 0 ( y t +2 ) U 0 ( y t +1 ) ( y t +2 + p t +2 ) ¸ into (2) yields p t = t U 0 ( y t +1 ) U 0 ( y t ) ½ y t +1 + t +1 U 0 ( y t +2 ) U 0 ( y t +1 ) ( y t +2 + p t +2 ) ¸¾¸ = E t 2 X j =1 β j U 0 ( y t + j ) U 0 ( y t ) y t + j + β 2 U 0 ( y t +2 ) U 0 ( y t ) p t +2 where we have used the “law of iterated expectations", that E t [ E t +1 [ e x ]] = E t [ e x ] . Successive forward iterations yields p t = E t T X j =1 β j U 0 ( y t + j ) U 0 ( y t ) y t + j + β T U 0 ( y t + T ) U 0 ( y t ) p t + T If there exists a bounded price function, then lim T →∞ p t = E t X j =1 β j U 0 ( y t + j ) U 0 ( y t ) y t + j which says that the stock price is the present discounted value of fu- ture dividends, where the “stochastic discount factor" or “pricing kernel” m t + j ³ β j U 0 ( y t + j ) U 0 ( y t ) ´ for any date t + j is just equal to the marginal rate of substitution in consumption. Notice that if U is linear (i.e. households are risk neutral), then lim T →∞ p t = X j =1 β j E t [ y t + j ] Recall that if x and z are random variables, then E [ x · z ]= E [ x ] E [ z ]+ cov ( x, z ) . Then lim T →∞ p t = E t X j =1 E t [ m t + j ] E t [ y t + j cov t ( m t + j ,y t + j ) 2
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If U ( c t )= c 1 ψ t 1 1 ψ , then m t + j = β j ³ y t + j y t ´ ψ , in which case cov t ( m t + j ,y t + j cov μ ³ y t y t +1 ´ ψ t +1 < 0 . That is, dividends on this asset go up just when the marginal utility of consumption is lowest and dividends go down just when you value extra utility. Thus, the asset doesn’t provide a good hedge against consumption risk. In that case, its price will be lower.
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handoutassetpricingsp08 - Asset Pricing 1 Lucas (1978,...

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