Log-linear approximate present-value models

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(Rt+1) = ln(Pt+1 + Dt+1) – ln(Pt) = ln(Pt+1) – ln(Pt) + ln( 1 + DYt+1) • Or rt+1 = pt+1 + ln( 1 + exp(t+1) ) - pt where t+1 = ln(DYt+1) and DYt+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 rt+1≈ k + pt+1 + (1- )dt+1 – pt where k = -ln()-(1- )ln( (1/ )-1 ) and  = 1/(1+exp(*))

Static, constant growth case • Suppose dividend growth is constant and return is constant: Dt+1/Dt = exp(g) = Pt+1/Pt , (Pt+1+Dt+1)/Pt = exp(r) • Note exp(g-r) = Pt+1/(Pt+1+Dt+1) constantand (1/exp(g-r))-1 = Dt+1/Pt+1 so DY is constant • Since k + pt+1 + (1- )dt+1 – pt = k + (1- )(dt+1 – pt+1)+ pt+1– pt in this case the approximation is exact

Discounting formula • Or pt = k + pt+1 + (1- )dt+1 – rt Recursively substitute for pt+1 to get pt = k(1++2+… + (1- )dt+1 + (1- )dt+2 + 2(1- )dt+3 + … - rt+1 -  rt+2 - 2rt+3 - … • Or pt = {k/(1- )} + j{(1- )dt+1+j - rt+1+j} (assuming lim jpt+j = 0 as j →∞)

Loglinear approximate present value relation • Take conditional expectations pt = {k/(1- )} + Etj{(1- )dt+1+j - rt+1+j} • The log dividend-price ratio t = -{k/(1- )} + Etj{-Δdt+1+j + rt+1+j}

What moves stock prices? • Unexpected stock returns are given by rt+1 - Etrt+1 = (Et+1 - Et) jΔdt+1+j - (Et+1 - Et) jrt+1+j • Or rt+1 = dt+1 - rt+1

Example • Suppose expected returns are given by Etrt+1 = r* + xt where xt is an observable zero mean variable that follows an AR(1) process xt+1 = xt + ut+1 (-1≤  ≤+1) • In this case rt+1 = ut+1/(1- ) • The importance of movements in expected returns for stock price volatility is var(rt+1)/var(rt+1) = (1-2)(/(1-))2(R2/(1-R2)) where R2 is the fraction of the variance of return that is predictable

Excess returns • If the log riskfree rate is rft+1 then excess log returns are et+1 = rt+1 - rft+1 • Substituting for rt+1 gives et+1 - Etet+1 = (Et+1 - Et) jΔdt+1+j - (Et+1 - Et) jrft+1+j - (Et+1 - Et) jet+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, 298-345)

VAR approach • Define k-element vector zt+1 that includes as its first element rt+1. The other variables are potential predictors of returns (such as t+1, Δdt+1 ). • Estimate a vector autoregression for zt+1 as follows zt+1 = Azt + wt+1 • Note that Etzt+k = Akzt and in particular Etrt+1+j = e1’Aj+1zt where e1 is k-element vector with first element 1 and others 0.

Return variance decomposition • So rt+1 = (Et+1 - Et) jrt+1+j = e1’ jAjwt+1 =e1’A(I - A)-1wt+1 = ’wt+1 • Since rt+1 - Etrt+1 = rt+1 = e1’ wt+1 = dt+1 - rt+1 this gives dt+1 = (e1’ + ’)wt+1

Persistence measure • How long do shocks to expected returns persist? Shock to one-period ahead expected return = (Et+1 - Et)rt+2 = ut+1 = e1’Awt+1 • Define Pr = σ(rt+1)/ σ(ut+1) = σ(’wt+1)/ σ(e1’Awt+1)

Estimation • Estimate • VAR coefficients A • Variance matrix of innovations var(wt+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 Etrt+1 = Ett+1 • For example • Etrt+1 = constant  • Etrt+1 = EtΔlnCt+1 • Etrt+1 = Etrt+12 • 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- )} + Etj{-Δdt+1+j + } = (-k)/(1- )} + Etj{-Δdt+1+j} • If zt = [t–Δdt … ]’ then e1’zt = e2’Azt + e2’A2zt + e2’2A3zt … = e2’A(I - A)-1zt • To hold for all zt we must have e1’= e2’A(I - A)-1

Log-linear approximate present-value models