Log-linear approximate present-value models FINA790C Empirical Finance HKUST Spring 2006

Motivation What are the sources of changes in stock prices over time? –Can we quantify their impact? –How persistent is their impact? Background: J. Campbell and R. Shiller, “The dividend-price ratio and expectations of future dividends and discount factors” RFS 1(3), Fall 1988; J. Campbell and R. Shiller, “Stock prices, earnings and expected dividends” JF 43(3), July 1988.

Stock prices and dividend yields To study these issues we need to be able to relate prices to underlying “fundamentals” ln(R t+1 ) = ln(P t+1 + D t+1 ) – ln(P t ) = ln(P t+1 ) – ln(P t ) + ln( 1 + DY t+1 ) Or r t+1 = p t+1 + ln( 1 + exp(  t+1 ) ) - p t where  t+1 = ln(DY t+1 ) and DY t+1 is the dividend yield at t+1

Returns, prices and yields: an approximate relation Suppose  t is a stationary stochastic process with a constant mean. Then we can expand f(  ) = ln(1+exp(  )) in a Taylor series around its mean (call it  * = d* - p*) This gives r t+1 ≈ k +  p t+1 + (1-  )d t+1 – p t where k = -ln(  )-(1-  )ln( (1/  )-1 ) and  = 1/(1+exp(  *))

Static, constant growth case Suppose dividend growth is constant and return is constant: D t+1 /D t = exp(g) = P t+1 /P t, (P t+1 +D t+1 )/P t = exp(r) Note exp(g-r) = P t+1 /(P t+1 +D t+1 ) constant and (1/exp(g-r))-1 = D t+1 /P t+1 so DY is constant Since k +  p t+1 + (1-  )d t+1 – p t = k + (1-  )(d t+1 – p t+1 ) + p t+1 – p t in this case the approximation is exact

Discounting formula Or p t = k +  p t+1 + (1-  )d t+1 – r t Recursively substitute for p t+1 to get p t = k(1+  +  2 +… + (1-  )d t+1 +  (1-  )d t+2 +  2 (1-  )d t+3 + … - r t+1 -  r t+2 -  2 r t+3 - … Or p t = {k/(1-  )} +  j {(1-  )d t+1+j - r t+1+j } (assuming lim  j p t+j = 0 as j →∞)

Loglinear approximate present value relation Take conditional expectations p t = {k/(1-  )} + E t  j {(1-  )d t+1+j - r t+1+j } The log dividend-price ratio  t = -{k/(1-  )} + E t  j {-Δd t+1+j + r t+1+j }

What moves stock prices? Unexpected stock returns are given by r t+1 - E t r t+1 = (E t+1 - E t )  j Δd t+1+j - (E t+1 - E t )  j r t+1+j Or  rt+1 =  dt+1 -  rt+1

Example Suppose expected returns are given by E t r t+1 = r* + x t where x t is an observable zero mean variable that follows an AR(1) process x t+1 =  x t + u t+1 (-1≤  ≤+1) In this case  rt+1 =  u t+1 /(1-  ) The importance of movements in expected returns for stock price volatility is var(  rt+1 )/var(  rt+1 ) = (1-  2 )(  /(1-  )) 2 (R 2 /(1-R 2 )) where R 2 is the fraction of the variance of return that is predictable

Excess returns If the log riskfree rate is r ft+1 then excess log returns are e t+1 = r t+1 - r ft+1 Substituting for r t+1 gives e t+1 - E t e t+1 = (E t+1 - E t )  j Δd t+1+j - (E t+1 - E t )  j r ft+1+j - (E t+1 - E t )  j e t+1+j or  et+1 =  dt+1 -  ft+1 -  et+1

Empirical implementation Vector autoregression (VAR) approach Description and variance decompositions Testing models for intertemporal behavior of expected returns Testing models for cross-sectional behavior of expected returns (see J. Campbell (1996),”Understanding risk and return”, Journal of Political Economy 104(2), April, )

VAR approach Define k-element vector z t+1 that includes as its first element r t+1. The other variables are potential predictors of returns (such as  t+1, Δd t+1 ). Estimate a vector autoregression for z t+1 as follows z t+1 = Az t + w t+1 Note that E t z t+k = A k z t and in particular E t r t+1+j = e1’A j+1 z t where e1 is k-element vector with first element 1 and others 0.

Return variance decomposition So  rt+1 = (E t+1 - E t )  j r t+1+j = e1’  j A j w t+1 =e1’  A(I -  A) -1 w t+1 = ’w t+1 Since r t+1 - E t r t+1 =  rt+1 = e1’ w t+1 =  dt+1 -  rt+1 this gives  dt+1 = (e1’ + ’)w t+1

Persistence measure How long do shocks to expected returns persist? Shock to one-period ahead expected return = (E t+1 - E t )r t+2 = u t+1 = e1’Aw t+1 Define P r = σ(  rt+1 )/ σ(u t+1 ) = σ( ’w t+1 )/ σ(e1’Aw t+1 )

Estimation Estimate –VAR coefficients A –Variance matrix of innovations var(w t+1 ) Calculate (nonlinear) functions of A and estimate standard errors by delta method

Testing expected return models Suppose we have a theory that specifies the time series behavior of E t r t+1 = E t  t+1 For example –E t r t+1 = constant  –E t r t+1 =  E t ΔlnC t+1 –E t r t+1 =  E t r t+1 2 We can see what this implies for the behavior of the VAR

Example: Constant expected return If expected returns are constant then the log dividend yield is  t = -{k/(1-  )} + E t  j {-Δd t+1+j +  } = (  -k)/(1-  )} + E t  j {-Δd t+1+j } If z t = [  t –Δd t … ]’ then e1’ z t = e2’Az t + e2’  A 2 z t + e2’  2 A 3 z t … = e2’A(I -  A) -1 z t To hold for all z t we must have e1’ = e2’A(I -  A) -1