Breaking Bad: Parameter Uncertainty Caused by Structural Breaks in Stocks

Most financial models that deal with portfolio allocation require some form of parameter input like expectations about an asset's future performance and risk (i.e. expected return, variances and covariances) in order to derive optimal portfolio weights. While this task seems straightforward, empirical evidence suggests otherwise. DeMiguel et al. (2009) for example, demonstrate that estimating reliable parameter inputs for portfolio optimization requires vast amounts of historical data and is notoriously difficult as the required length of historical observations is non-existent. Consequently, investors are required to estimate future returns, variances and covariances of assets with less data which increases estimation errors. Adding to that, portfolio selection models typically treat parameter inputs with absolute certainty, resulting in well-known issues related to mean-variance optimization such as poor out-of-sample performance, lacklustre diversification as well as unintuitive allocations (Michaud, 1989).

While there are approaches that try to mitigate or circumvent these issues, such as, for example, Kan and Zhou (2007) or DeMiguel and Nogales (2009), research shows that financial data is affected by outliers (i.e. Bali et al., 2016). Furthermore, Ang and Timmermann (2012) argue that the behaviour of financial markets can change abruptly and frequently which in turn dramatically alters the "mean, volatility and correlation patterns in stock returns" and increases the uncertainty of parameter estimates (p. 1057). Typically, such abrupt changes are caused by changes of technological, legislative or institutional nature as well as economic shocks or policy shifts (Pesaran et al., 2006). The presence of time-series breaks and regime shifts in financial data further complicates estimating/forecasting parameters and gives rise to two problems: First, using estimates based on long historical samples is no longer appropriate since these estimates are likely biased. Second, estimating input parameters with only few historical observations after a break has been detected is also highly problematic due to estimation errors (Smith and Timmermann, 2021). Consequently, it is reasonable to state that financial data is of a time-varying nature and in the presence of breaks, not all historical observations are suitable for estimation and prediction tasks (i.e. Dangl and Halling, 2012).

These findings suggest that parameter uncertainty is inherent to both, long- and short-term estimates and amplified by structural breaks. From the perspective of an investor, these regime shifts are not desirable as they further diminish the reliability of available data and ultimately increase parameter uncertainty. Garlappi et al. (2007) set out to investigate how investors react to parameter uncertainty and, according to Kan and Zhou (2007), discover that investors allocate less wealth to stocks with high parameter uncertainty. Based on this observation, they further deduce that investors are averse to parameter uncertainty which in turn renders stocks with high parameter uncertainty less attractive for investors relative to stocks with a lower degree of parameter uncertainty. This implies that investors with an aversion to parameter uncertainty (i.e. "ambiguity-aversion") withdraw their capital from stocks with high or increasing parameter uncertainty and re-allocate it to stocks with low or decreasing parameter uncertainty. On an aggregate market level, this ambiguity-aversion gives rise to a premium for parameter uncertainty as stocks with high (low) parameter uncertainty are avoided/sold (more attractive/bought).

Given these observations, we propose a novel measure called break-(adjusted stock-) age that proxies for parameter uncertainty and is based on detecting structural breaks in stock returns using unsupervised machine learning techniques. Although change point detection models are typically very technical in nature, associating breaks in stock returns with periods of elevated parameter uncertainty seems to be straightforward. This is especially true since breaks can hint at a change in the underlying return generating process which, in the worst case, implies that previous return observations are no longer reliable when it comes to analysing the risk-return dynamics of a stock, thereby increasing the uncertainty with which parameters have to be estimated. Over time, more and more post-break returns can be observed, increasing the reliability of estimated parameters from an investor's perspective. Based on this notion, the objective of our paper is to demonstrate (i) that there is a premium for parameter uncertainty in the cross-section and (ii) that the proposed measure which calculates stock-age on the basis of structural breaks (i.e. break-age) is a suitable proxy for parameter uncertainty as it captures the premium associated with it.

Our results indicate that there is a substantial premium for assets with recent breaks in their time-series regardless of the statistical test used to identify such breaks. Furthermore, we discover that this premium, which we link to parameter uncertainty, is strongest for breaks in variance and for breaks in the mean-variance relationship. Moreover, breaks that have been detected by parametric test statistics that assume Gaussian-distributed returns carry a higher premium than their non-Gaussian counterparts which indicates that investors implicitly assume that returns are normally distributed. Finally, the premium for break-age and, by extension, the premium for parameter uncertainty is more pronounced for smaller stocks. We argue that this is the case because smaller stocks are less well researched and have less media and analyst coverage thereby prolonging the resolution of parameter uncertainty following a structural break.