The Sources Of Stock Marketplace Fluctuations

How much exercise dividend-growth vs. discount-rate shocks concern human relationship for stock toll variations?

An under-appreciated signal occurred to me piece preparing for my Coursera shape in addition to to comment on Daniel Greewald, Martin Lettau in addition to Sydney Ludvigsson's overnice newspaper "Origin of Stock Market Fluctuations" at the final NBER EFG meeting

The answer is, it depends the horizon in addition to the measure. 100% of the variance of price dividend ratios corresponds to expected provide (discount rate) shocks, in addition to none to dividend increase (cash flow) shocks.  50% of the variance of one-year returns corresponds to cashflow shocks. And 100% of long-run toll variation corresponds to from cashflow shocks, non expected provide shocks. These facts all coexist

I remember at that topographic point is some confusion on the point. If zip else, this makes for a expert occupation develop question.

The final signal is easiest to consider merely alongside a plot. Prices in addition to dividends are cointegrated. Prices fit to dividends in addition to expected returns. Dividends possess got a unit of measurement root, but expected returns are stationary. Over the long run prices volition non deviate far from dividends. So 100% of long-enough run toll variation must come upwards from dividend variation, non expected returns.

Ok, a lilliputian to a greater extent than carefully, alongside equations.

A quick review: 

The most basic VAR for property returns is \[ \Delta d_{t+1} = b_d \times dp_{t}+\varepsilon_{t+1}^{d} \] \[ dp_{t+1} = \phi \times dp_{t} +\varepsilon_{t+1}^{dp} \] Using solely dividend yields dp, dividend increase is basically unforecastable \( b_d \approx 0\) in addition to \( \phi\approx0.94 \) in addition to the shocks are conveniently uncorrelated. The behaviour of returns follows from the identity, that yous ask to a greater extent than dividends or a higher toll to instruct a return,  \[ r_{t+1}\approx-\rho dp_{t+1}+dp_{t}+\Delta d_{t+1}% \] (This is the Campbell-Shiller provide approximation, alongside \(\rho \approx 0.96\).) Thus, the implied regression of returns on dividend yields, \[ r_{t+1} = b_r \times dp_{t}+\varepsilon_{t+1}^{r} \] has \(b_r = (1-\rho\phi)+0 = 1-0.96\times0.94 = 0.1\) in addition to a stupor negatively correlated alongside dividend yield shocks in addition to positively correlated alongside dividend increase shocks.

The impulse response business office for this VAR naturally suggests "cashflow" (dividend) in addition to "expected return" shocks, (d/p). (Sorry for recycling onetime points, but non everyone may know this.)

Three propositions:

• The variance of p/d is 100% conduct chances premiums, 0% cashflow shocks

Iterate forwards the provide identity, to instruct \[ dp_{t} =\sum_{t=1}^{\infty}\rho^{j-1}r_{t+j}-\sum_{t=1}^{\infty}\rho ^{j-1}\Delta d_{t+j} \] multiply yesteryear \(dp_t\) in addition to possess got expectations (all variables are demeaned) \[\sigma^{2}\left( \log\frac{P_{t}}{D_{t}}\right) =\sigma^{2}\left( dp_{t}\right) =\sum_{t=1}^{\infty}\rho^{j-1}cov(dp_{t},r_{t+j})-\sum _{t=1}^{\infty}\rho^{j-1}cov(dp_{t},\Delta d_{t+j}), \] But \(b_d \approx 0 \), thus the dividend increase terms are all zero, in addition to 100% of the variance of price-dividend ratios corresponds to time-varying expected returns. (I know this volition bore people familiar alongside it in addition to befuddle those who are not. "Discount rates" has a chip to a greater extent than leisurely review in addition to citations)

 But

•  The variance of returns is 50% due to conduct chances premiums, 50% due to cashflows. 

\[ r_{t+1}=-\rho dp_{t+1}+dp_{t}+\Delta d_{t+1}% \] \[ \varepsilon_{t+1}^{r} =-\rho\varepsilon_{t+1}^{dp}+\varepsilon_{t+1}^{d} \] \[ \sigma^{2}\left( \varepsilon_{t+1}^{r}\right) =\rho^{2}\sigma^{2}\left( \varepsilon_{t+1}^{dp}\right) +\sigma^{2}\left( \varepsilon_{t+1}^{d}\right) \] The variance of the ii shocks comes out really to a greater extent than or less a 50/50 decomposition at an annual horizon. It's a lot to a greater extent than expected provide at a daily horizon, in addition to less at longer horizons. Here I purpose the fact that dividend increase in addition to dividend yield shocks are basically uncorrelated.

Why are returns in addition to p/d thus different?  Current cash menstruum shocks touching on returns. But a stupor to dividends, when prices ascent at the same time, does non touching on the dividend toll ratio. (This is the essence of the Campbell-Ammer return decomposition.)

The tertiary proffer is less familiar:

• The long-run variance of stock marketplace values (and returns) is 100% due to cash menstruum shocks in addition to none to expected provide or discount charge per unit of measurement shocks.

Here's why: \[ \Delta p_{t+1} =-dp_{t+1}+dp_{t}+\Delta d_{t+1} \] \[ p_{t+k}-p_{t} =-dp_{t+k}+dp_{t}+\sum_{j=1}^{k}\Delta d_{t+j} \] thus equally k gets big, \[ {var} (p_{t+k}-p_{t}) \rightarrow 2 {var}(dp_t) + k {var}(\Delta d_{t}) \] The start term approaches a constant, but the instant term keeps growing. As inwards a higher house the fundamental fact is that P in addition to D are cointegrated piece expected returns are stationary.

This is related to a signal made yesteryear Fama in addition to French inwards their Equity Premium paper. Long run average returns are driven yesteryear long run dividend increase  plus the average value of the dividend yield. The divergence inwards valuation -- higher prices for given develop of dividends -- tin touching on returns inwards a sample, equally higher prices for a given develop of dividends boost returns. But that machinery can't last. (Avdis in addition to Wachter possess got a overnice recent paper formalizing this point.)  It's related to a like signal made ofttimes yesteryear Bob Shiller: Long run investors should purchase stocks for the dividends.

H5N1 lilliputian to a greater extent than generality equally this is the novel bit.

\[ p_{t+k}-p_t = dp_{t+k}-dp_t + \sum_{j=1}^{k}\Delta d _{t+j} \] \[ p_{t+k}-p_t = (\phi^{k}-1)dp_t + \sum_{j=1}^{k}\phi^{k-j} \varepsilon^{dp}_{t+j} +  \sum_{j=1}^{k} \varepsilon^d _{t+j} \] \[ var(p_{t+k}-p_t) = \frac{(1-\phi^{k})^2}{1-\phi^2} \sigma^2(\varepsilon^{dp}) + \frac{(1-\phi^{2k})}{1-\phi^2}  \sigma^2(\varepsilon^{dp}) +  k\sigma^2(\varepsilon^d) \] \[var(p_{t+k}-p_t) = 2\frac{(1-\phi^{k})}{1-\phi^2} var(\varepsilon^{dp}_{t+1})  + k var(\varepsilon^d_{t+j})\] So yous tin consider the final chip takes over. It doesn't possess got over equally fast equally yous mightiness think. Here's a graph using sample values,

At a i twelvemonth horizon, it's merely nigh 50/50. The dividend shocks eventually possess got over, at charge per unit of measurement 1/k. But at fifty years, it's yet nigh 80/20.

Exercise for the interested reader/finance professor looking for occupation develop questions: Do the same affair for long horizon returns, \( r_{t+1}+r_{t+2}+...+r_{t+k} \) using \(r_{t+1} = -\rho dp_{t+1} + dp_t + \Delta d_ {t+1} \) It's non thus pretty, but yous tin instruct a unopen shape appear hither too, in addition to over again dividend shocks possess got over inwards the long run.

Be forewarned, the long run provide has all sorts of pathological properties. But nobody holds assets forever, without eating some of the dividends.

Disclaimer: Notice I possess got tried to nation "associated with" or "correspond to" in addition to non "caused by" here! This is merely nigh facts. The facts possess got merely equally tardily a "behavioral" interpretation nigh fads in addition to bubbles inwards prices equally they exercise a "rationalist" interpretation. Exercise 2: Write the "behavioralist" in addition to and thus "rationalist" introduction / interpretation of these facts. Hint: they opposite crusade in addition to final result nigh prices in addition to expected returns, in addition to whether people inwards the marketplace possess got rational expectations nigh expected returns.