Economics 202A Lecture Outline October 28 30 2008 version 1 3 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 dt and the stock s ex dividend real price i e in terms of output or more generally in terms of the CPI basket is qt Then in a risk neutral world we would have the arbitrage condition 1 r Et dt 1 qt 1 qt 1 which equates the gross return on bonds to that on stocks dividends capital gains This works for a time dependent interest rate rt as well do that case as an exercise To see how the preceding return relationship translates into a theory of stock pricing write qt Et Et Et Et dt 1 qt 1 1 r dt 1 Et 1 r dt 1 Et 1 r dt 1 Et 1 r qt 1 1 r 1 dt 2 qt 2 Et 1 1 r 1 r qt 2 dt 2 Et 2 1 r 1 r 2 Here I have used the law of iterated conditional expectations Et fEt 1 xt 2 g Et fxt 2 g One can continue the iterative substitution procedure above inde nitely successively substituting the versions of eq 1 for dates t 2 t 3 etc 1 The result is 1 X qt i i i 1 1 r i 1 n o qt i What to make of the term limi 1 Et 1 r This term represents a i potential speculative bubble of one particular rational kind in the stock price it captures the idea of a self ful lling frenzy in the asset price More on this later for now let s assume there is no bubble In that case 1 X dt i qt E t 2 i 1 r i 1 qt Et dt i 1 r i lim Et 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 re ect news Suppose that within a particular trading instant people change their expected dividend stream to be E0t fdt 1 g Then the stock price will jump by the amount 1 1 X dt i X dt i qt0 qt E0t Et i i 1 r 1 r i 1 i 1 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 1 1 X dt 1 i X dt i qt 1 qt Et 1 Et i i 1 r 1 r i 1 i 1 Let dividends follow the AR 1 process dt 1 dt t 1 where Et t 1 0 Then qt 1 r 2 dt and qt 1 qt dt 1 1 r 1 r dt 1 dt 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 We get at the essence of the e cient markets hypothesis by examining the ex post excess return et 1 dt 1 qt 1 qt 1 r Our arbitrage condition guarantees that this is uncorrelated with date t information In our particular AR 1 example dt 1 qt 1 qt 1 r dt 1 dt 1 1 r 1 r dt 1 r 1 r dt 1 1 r dt 1 r dt t 1 1 r dt t 1 dt n o xdtt Et t 1 0 For any random variable xt realized as of date t Et t 1 x dt t The excess return is unpredictable Note Even if there is a rational bubble in the stock price the preceding implication of unpredictable excess returns will hold That is because the result follows entirely from eq 1 rather than from eq 2 Summers s Critique on the Interpretation of E ciency Tests Some nancial economists argued that if one fails to nd lagged variables helping to predict excess returns et one can infer that the PDV formula 2 3 for a stock s price is valid stocks are priced according to their fundamentals Larry Summers o ers a persuasive critique of this inference in his paper Does the Stock Market Rationally Re ect Fundamental Values on the reading list Let qt temporarily for this section denote the PDV price given in equation 2 and imagine that perhaps do to fads in investment preferences or the like the actual stock price qt is given by qt q t e ut where the log discrepancy ut follows an autoregressive process ut u t 1 vt j j 1 where the innovation vt is uncorrelated with all economic variables at all leads and lags It is a pure sunspot In this alternative model stock prices can di er from fundamental values due to a slow moving pricing error that can be expected to diminish over time if j j 1 The question Summers asks is will standard tests of excess return predictability disclose the presence of this possibly large pricing error His answer is no Let s see why De ne the e cient excess return as et 1 dt 1 qt 1 qt 1 r and following Summers de ne the actual excess return as zt 1 dt 1 qt 1 qt 1 r Let s adopt the approximations qt 1qt qt log qt 1 log qt and e ut 1 ut The latter is not going to be a great approximation unless ut is relatively small but I am getting closer to the right answer than Summers does He dt 1 assumes that dqt 1 which amounts to the very bad approximation qt t ut e 1 I worry about Summers s approximation because it is only good when qt qt whereas the whole point of this exercise is to argue that the 4 two q s can diverge widely Then we may write zt 1 dt 1 e qt log qt 1 log qt log qt 1 log qt ut 1 et 1 ut 1 ut r ut dt 1 ut qt dt 1 qt r dt 1 ut qt ut To nish up imagine that dividends follow the AR 1 process dt 1 dt t 1 To make life easier let us take 1 As per our earlier result we have qt dt r and so dt 1 dt t 1 ut ut ut qt qt qt assuming that t 1 ut qt is small So zt 1 et 1 ut 1 ut rut et 1 vt 1 r Using this approximation and the fact that the variance of z …