Abstract We study in detail the log-linear return approximation introduced by Campbell and Shiller (1988a). First, we derive an upper bound for the mean approximation error, given stationarity of the log dividend-price ratio. Next, we simulate various rational bubbles that have explosive conditional expectation, and we investigate the magnitude of the approximation error in those cases. We find that, surprisingly, the Campbell-Shiller approximation is very accurate even in the presence of large explosive bubbles. Only in very large samples do we find evidence that bubbles generate large approximation errors. Finally, we show that a bubble model in which expected returns are constant can explain the predictability of stock returns from the dividend-price ratio that many previous studies have documented.

Abstract Many financial markets, including electronic limit order markets, assign designated liquidity providers (LPs). We study the experience of the Stockholm Stock Exchange, where listed firms contract directly with LPs. Our analysis offers insights regarding situations where designated liquidity provision may be beneficial. In addition, we consider the form of liquidity provision contracts, including affirmative obligations required of the LP and compensation for LP services. We find that low current trading activity, wide spreads, and higher information asymmetry increase the attractiveness of contracted liquidity provision. The evidence indicates that LPs trade against market movements and in times of wide spreads. On balance, firms contracting with LPs experience a decreased cost of capital and significant improvements in market quality and price discovery.

We analyze the pitfalls involved in VAR based return decompositions. First, we show that recent criticism of such decompositions is misplaced and builds on invalid VAR models and erroneous interpretations. Second, we derive the requirements needed for VAR decompositions to be valid. A crucial – but often neglected – requirement is that the asset price needs to be included as a state variable in the VAR. In equity return decompositions this requirement is equivalent to including the dividend–price ratio in the VAR. Finally, we clarify the intriguing issue of the role of the residual component in return decompositions. In a properly specified first-order VAR, it makes no difference whether cash flow news or discount rate news is backed out residually, and it makes no difference whether both news components are computed directly or one of them is backed out residually.