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lecture15 - Economics 202A Lecture Outline November 6-8...

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Economics 202A Lecture Outline, November 6-8, 2007 (version 1.2) Maurice Obstfeld Stock Prices as Present Values The most basic theory of the stock market is that a stock°s price is the present value of expected future dividends. Suppose the real interest rate is r , and is constant. Suppose the stocks real dividend in period t is d t and the stock°s ex dividend real price (i.e., in terms of output, or more generally, in terms of the CPI basket), is q t . Then in a risk-neutral world, we would have the arbitrage condition 1 + r = E t ° d t +1 + q t +1 q t ± ; (1) which equates the gross return on bonds to that on stocks (dividends + capital gains). This works for a time-dependent interest rate r t as well ± do that case as an exercise. To see how the preceding return relationship translates into a theory of stock pricing, write q t = E t ° d t +1 + q t +1 1 + r ± = E t ° d t +1 1 + r ± + E t ° q t +1 1 + r ± = E t ° d t +1 1 + r ± + E t ° 1 1 + r E t +1 ° d t +2 + q t +2 1 + r ±± = E t ° d t +1 1 + r ± + E t ( d t +2 (1 + r ) 2 ) + E t ( q t +2 (1 + r ) 2 ) : Here, I have used the law of iterated conditional expectations, E t f E t +1 x t +2 g = E t f x t +2 g : One can continue the iterative substitution procedure above inde²nitely, successively substituting the versions of eq. (1) for dates t + 2 ; t + 3 , etc. The result is q t = 1 X i =1 E t ( d t + i (1 + r ) i ) + lim i !1 E t ( q i (1 + r ) i ) : What to make of the term lim i !1 E t n q i (1+ r ) i o ? This term represents a potential speculative bubble (of one particular "rational" kind) in the stock price: it captures the idea of a self-ful²lling frenzy in the asset price. More on this later; for now let°s assume there is no bubble. In that case q t = E t ( 1 X i =1 d t + i (1 + r ) i ) ; (2) 1
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and the stock°s price is the expected present value of future dividends. An important implication of this formula is that changes in stock prices re³ect news . Suppose that, within a particular trading instant, people change their ex- pected dividend stream to be E 0 t f d t +1 g : Then the stock price will jump by the amount q 0 t ° q t = E 0 t ( 1 X i =1 d t + i (1 + r ) i ) ° E t ( 1 X i =1 d t + i (1 + r ) i ) ; : where this change is uncorrelated with any information available before the revision in market expectations. This is the basic idea of the "random walk" theory of stock prices, or, more broadly, the "e¢ cient markets" view. As another application, consider the behavior of the stock price from period to period. We have q t +1 ° q t = E t +1 ( 1 X i =1 d t +1+ i (1 + r ) i ) ° E t ( 1 X i =1 d t + i (1 + r ) i ) : Let dividends follow the AR(1) process d t +1 = °d t + " t +1 ; where E t " t +1 = 0 : Then q t = ° 1 + r ° ° d t and q t +1 ° q t = ° 1 + r ° ° ( d t +1 ° d t ) = ° 1 + r ° ° [( ° ° 1) d t + " t +1 ] : Changes in stock prices are proportional to changes in dividends (as in Shiller°s excess volatility tests). Also, for ° near 1, or for a very small time interval, the change in the stock price is essentially proportional to the "news" " t +1 ± the innovation in dividends.
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