Cochrane (2011b) for a review). In subsequent research, Campbell and Shiller (1987) showed that the excess volatility result remained even when the time series for prices and dividends were non-stationary (see LeRoy (2008)). Research has also shown that many other asset prices are much more volatile than the discounted value of the corresponding streams of dividend would suggest.<sup>18</sup> Moreover, asset prices appear to move away at times from their predicted fundamentalsbased values. This is not easy to confirm, however, because one does not know whether the mispricing represents a deviation from the “true” model or the use of a “mis-specified” model, including not knowing or using the “right” fundamentals. This can happen for individual assets (simply representing “arbitrage” opportunities) or for the market as a whole. Simple arbitrage opportunities are limited for most traded assets but markets may deviate at times from developments in fundamentals (Lo and MacKinlay (1999), Shiller (2000) and Akerlof and Shiller (2009, 2015)).

There is also ample evidence of stock price “bubbles”.<sup>19</sup> As an illustration, one can compare the aggregate price-earnings ratio, the dividend yield and the implied equity premium in 1999, just before the stock market peak in advanced economies, with their historical averages over the period 1980–99 (IMF (2000)). Such a comparison shows that, in the late 1990s, the valuations implied by equity prices were considerably higher than their historical averages in terms of price-earnings ratio but lower relative to dividend yields and implied equity risk premia. At the same time, real GDP growth was not very different from its historical average.<sup>20</sup> These comparisons suggest some overvaluation at the time, with markets indeed experiencing a major correction after mid-2000.

The high volatility of asset prices relative to their fundamental values appears to stem partly from the volatile nature of discount rates. Asset price volatility can be decomposed into two parts: the volatility of expected future cash flows (eg dividends) and the volatility of the discount rate applied to those cash flows. Campbell and Shiller (1988a, 1988b) developed a methodology for decomposing the variation in the dividend-to-price ratio into variations in expected dividends and discount rates. Their research and subsequent work suggest that the variation in expected dividends accounts for no more than one-fourth of stock market volatility whereas variation in the discount rate accounts for the bulk of volatility (Cochrane (2011b) for a review).

This relates to the finding that most asset pricing models, including the basic consumption-based model, cannot fully explain the magnitude of the risk premium actually

> 18 See Mankiw et al (1985, 1989), West (1988), Schwert (1989) and Barsky and De Long (1993).

> 19 A bubble can be defined as: “...the part of a grossly upward asset price movement that is unexplainable based on fundamentals” (Garber (2000)). Patterns of exuberant increases in asset prices, often followed by crashes, figure prominently in many accounts of financial instability for all types of economy and going back centuries (Claessens and Kose (2014)). See Brunnermeier and Oehmke (2013), Scherbina and Schlusche (2014) and Williams (2013) for recent surveys on asset price bubbles. Some argue that fully irrational asset bubbles are not necessarily harmful and could even be beneficial (Kocherlakota (2009)).

20 At the time, one explanation for the high level of stock prices was the strong labour productivity growth observed in the United States in the second half of the 1990s. This fuelled discussion of a “new economy” driven by information technology. However, later data revisions showed much lower labour productivity growth. The changes in (perceived or expected) productivity growth could well explain some of the run-up in prices and the following sharp corrections (Pastor and Veronesi (2006), Griffin et al (2011)). Lettau et al (2008) provide a summary of studies explaining persistently high stock market valuations, as observed in the late 1990s. They argue that a fall in macroeconomic risk or in economy-wide volatility can lead to such high stock prices.

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