Relations between Prices, Dividends and Returns. Present Value Relations (Ch7inCampbell et al.) Thesimplereturn:

Transcription

1 Present Value Relations (Ch7inCampbell et al.) Consider asset prices instead of returns. Predictability of stock returns at long horizons: There is weak evidence of predictability when the return history is used. The goal is now whether the predictability can be improved, when other variables, such as dividend price ratio or the level of interest rate, are brought into the analysis. An empirical nding is that expected returns are at long horizon roughly consistent with persistent AR(1). Why this is so is yet an open question. We consider the basic methodology to investigate this. Relations between Prices, Dividends and Returns Thesimplereturn: R t+1 = P t+1 + D (1) t+1 1: P t Continuously compounded return: () r t+1 =log( t+1 ) Simple Case: Constant expected return E t [R t+1 ]=R (a constant); where E t denotes the conditional expectation given information up to time point t. Then " # Pt+1 + D () P t = E t+1 t : A simple application of the law of iterated expectations, E t [E t+1 [X]] = E t [X] gives then 1

2 A further simpli ed (but more unrealistic) special case is when E t [D t+i ]=(1+G)E t [D t+i 1 ]=(1+G) i D t ; (4) " K # X ³ 1 i ³ 1 K P t = E t D t+i + E t P t+k : i=1 (Just for fun verify!) Assume that " µ # 1 K P t+k =0: (5) lim E t K!1 Thus we obtain a stock price model, called a constant return present value model, referred to as P Dt µ P t = P Dt = E t 4 1 i (6) D 5 t+i : i=1 i.e., a constant growth model with growth rate G<R. Then P t = E t[d t+1 ] R G = 1+G R G D t: The Gordon growth model. 4

3 Implications of the dividend discount model ² P t is not a martingale,becausee t [P t+1 ] 6= P t. ² However, if all the dividends are reinvested in the stock, giving a portfolio with N t number of shares at time t, where N t+1 = N t à 1+ D t+1 P t+1 The value of the portfolio is M t = N tp t (1 + R) t; which is a martingale. A stochastic process, Y t,isamartingale if E t [Y t+1 ]= Y t.! : 5 ² P t follows a linear process with unit root, if D t follows a linear process with unit root. ² If D t (and P t ) are I(1)) then they are under the model (6) (Prove it!) cointegrated y Above prices and dividends are related by a linear modeled. A more appropriate approach may be by using log prices instead. This will be done in what follows. Y t has a unit root, or is integrated of order one, denoted as Y t» I(1) if Y t = Y t Y t 1 is stationary [is integrated of order 0, denoted as Y t» I(0)] y There exist a real number such that P t + ad t is stationary 6

4 The term P Dt is sometimes called fundamental value, andb t a rational bubble. Rational Bubbles Relax the assumption in the model (4) that discounted end value (5) of the stock converges to zero. Then P t is no more unique. Anysolutioncanbewrittenintheform (7) where (8) P t = P Dt + B t ; Bt+1 B t = E t : 7 Consider the following example by Blanchard and Watson (198) ( B t+1 = ¼ B t + ³ t+1 ; with probability ¼ ³ t+1 with probability 1 ¼: where E t ³ t+1 = 0. This example bubble obeys the restriction (8). The bubble bursts at any period with probability 1 ¼. If it does not burst it grows at the rate (1 + R)=¼ 1, faster than R to compensate the probability of bursting. Several other examples of bubbles can be given. Can rational bubbles exist? The word "bubble" recalls some of the famous episodes in nancial history in which asset prices rose far higher than could be explained by fundamentals, andinwhichinvestorsappearedtobebettingthat other investors would drive prices even higher. "Rationale" is used because B t in the price equation is fully consistent with rational expectations and constant expected returns. 8

5 Approximate Present-Value Relation with Time Varying Expected Returns Empirical evidence that stock returns are predictable at least to some extends implies that expected returns are time-varying rather than constant. As a consequence working with present value relations become much more di±cult, for the relation between prices and returns become nonlinear A popular approach is a log linear approximation for the nonlinearity. Rationale in accounting framework: High prices must be followed eventually by ² high future dividends, ² low future returns, or ² or some combination of the two above. 9 The log linear approximation: r t+1 = log(p t+1 + D t+1 ) log(p t ) = p t+1 p t +log ³ 1 + exp(d t+1 p t+1 ) ; where in the last form it is necessary to assume that D t > 0. The last term is a nonlinear function of the dividend price ratio. Taylor approximation gives (9) r t+1 ¼ k + ½p t+1 +(1 ½)d t+1 p t ; where k and ½ are linearization parameters de ned by 1 ½ = 1 + exp(d p) ; with d p the average of log dividend-price ratio, and k = log ½ (1 ½) log( 1 ½ 1): (Exercise: Verify). If f(x) is a di erentiable function then the rst order Taylor approximation of f at c is f(x) ¼ f(c)+f 0 (c)(x c) 10

6 Thus the approximation replaces the log of the sum of the price and dividend by the weighted average. Accuracy of the approximation: If the dividend-price is constant the approximation relation holds exactly (Exercise: prove it). Generally it has proven that the approximation is reasonably accurate, especially for investigating the asset price dynamics. Implication for Prices: Imposing the (terminal) condition we obtain p t = k 1 ½ + 1 X lim j!1 ½j p t+j =0 (Exercise: Verify) ½ j [(1 ½)d t+1+j r t+1+j ]: 11 The relation holds also for ex ante, fore t [p t ]= p t, consequently p t = k 1 ½ +E t 4 ½ j [(1 ½)d t+1+j r t+1+j ] 5 : Interpretation of the model: A high stock price implies expected high future dividends and low future return values, and vice versa. Campbell and Shiller call the model as dynamic Gordon growth model or the dividend-ratio model. Oncethedividendgrowthratesandthediscount rates are constants the DGGM reduces to GGM (Again verify.) Campbell, J. and R. Chiller (1988). The dividendprice ratio and expectations on future dividends and discount factors. Review of Financial Studies, 1, 195{7. Campbell, J. and R. Chiller (1988). Stock prices, earnings and expected dividends. Journal of Finance, 4, 661{676. 1

7 De ning p dt =(1 ½)E t 4 ½ j d 5 t+1+j and p rt = E t 4 ½ j r 5 t+1+j we can rewrite the model p t = k 1 ½ + p dt p rt ; The log dividend-price ratio can be written in terms of the model d t p t = k 1 ½ +E t 4 ½ j [ d t+1+j + r t+1+j ] 5 : Hence, the dividend-price ratio is high if the dividends are expected to grow slowly or the stock returns are expected to be high. Note again that if d t» I(1) and p t» I(1), then d t,andp t are cointegrated with a cointegration coe±cient equal to one. 1 Consider next the return series. Using the above relations we can write (work this out yourself) r t+1 E t [r t+1 ] = E t+1 P 1 ½j d t+1+j Et P 1 ½j d t+1+j or (10) E t+1 P 1 ½j r t+1+j Et P 1 ½j r t+1+j : r t+1 E t [r t+1 ]= t+1 = d;t+1 r;t+1 ; where t+1 is the unexpected stock return, d;t+1 is the change in expectations of future dividends, and r;t+1 is the change in expectations of future returns. Hence, returns must be associated with changes in expectations of future dividends or real returns. 14

8 A Simpli ed Example Suppose (11) where (1) E t [r t+1 ]=r + x t x t+1 = Áx t +» t+1 ; 1 <Á<1 a stationary zero-mean AR(1) process. If Á is close to one the process is highly persistent (approaches to the random walk). Denoting ¾ x =Var(x t )and¾» =Var(» t), we obtain Then (1) p rt = E t 4 (Exercise: Verify!) ¾ x = ¾» 1 Á : ½ j r t+1+j 5 = r 1 ½ + x t 1 ½Á : Consequently the more persistent the expected return is the greater is the e ect on stock price. Since ½ is close to one, a 1% increase in return today decreases the stock pricebyabout%ifá =0:5, by about 4% if Á =0:75, and by about 10% if Á =0:9. Note that the variability of expected stock returns are measured by the standard deviation of x t. Consequently it is tempting to think that if ¾ x is small, then changing expected returns have little in uence on stock prices, i.e. the variability in p rt. However, this may not be so, because the variability in p rt comes through x t =(1 ½Á), and if Á is close to one, then (1 ½Á) isclose to zero, and hence already small changes in x t may impose large changes in p rt

9 In terms of the above model the one-period return becomes (14) r t+1 = r + x t + d;t+1 ½» t+1 1 ½Á : Consequently " Var[r t+1 ]=¾d 1+½ # ½Á +¾ x (1 Á½) ¼ ¾d + ¾ x 1 Á ; where the approximate equality holds when Á ½, and½ is close to one. Variability is the larger the higher is the persistence (provided ¾» > 0). Equations (1) and (14) can be used to show that realized stock returns follow an ARMA(1,1) process. (Exercise: Show this and calculate the autocorrelation function. What are the implications? Assume Á<½.) 17