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Modelling the relation between volatility and returns

There is evidence for a double relation between volatility and returns in equity markets. Longer-term fluctuations of volatility mostly reflect risk premiums and hence establish a positive relation to returns. Short-term swings in volatility often indicate news effects and shocks to leverage, causing to a negative volatility-return relation. Distinguishing the two is important for using volatility as a predictor of returns.

Harvey, Andrew and Rutger-Jan Lange (2015), “Modeling the Interactions between Volatility and Returns”, Camebridge Papers in Economics 1518

On the difference between volatility and financial risk view post here.
On volatility, leverage shocks and collateral amplification view post here.
On the impact of volatility surprises view post here.

The below are excerpts from the paper. Headings, emphasis and cursive text have been added.

The two relations between volatility and returns

“Volatility of a stock may incur a risk premium, leading to a positive correlation between volatility and returns. On the other hand the leverage effect [or news effect], whereby negative returns increase volatility, acts in the opposite direction. “

“The leverage effect in finance suggests that volatility rises when the asset price falls. The rise in volatility following a fall in the asset price need not necessarily be due to leverage as such. For example the label ‘news impact curve’ is often used instead of leverage, reflecting the idea that a sharp fall in asset price may induce more uncertainty and hence higher variability.”

“A two-component model enables the researcher to distinguish between the effects of short and long-run volatility. Short-run volatility can lead to a [leverage effect or] news effect…that makes investors nervous of risk and so predicts a negative correlation between volatility and return. This negative relationship contrasts with the positive relationship between long-run volatility and return predicted by Mertons intertemporal capital asset pricing model (ICAPM). Failure to model both aspects of volatility has led to inconclusive results regarding the sign of the risk premium [in other research papers].”


“Returns may have an asymmetric effect on volatility [with negative returns pushing volatility up but positive returns not immediately pushing it down]. For example, considerations of leverage suggests that negative returns are associated with increased volatility…Indeed the term leverage is often loosely used to indicate any kind of asymmetry in the response of volatility to returns….it may be that an asymmetric response is confined to the short-run volatility component.”

How to distinguish the two relations between volatility and returns

“Here we show that a carefully specified two-component model…enables the researcher to investigate the possibility that when long-run volatility goes up it tends to be followed by an increasing level of returns, whereas an increase in short-run volatility leads to a fall.”

“We propose a reformulation and extension of the [exponential] ARCH in Mean model [EGARCH-M model].”

“N.B.: The econometrics the above term can roughly be explained as follows:

  • ARCH means “auto-regressive conditional heteroscedasticity” and simply describes a time series where tomorrow’s value (say return) depends on today’s value and a random disturbance. Importantly, the variance of this disturbance changes overtime as well and the size of tomorrow’s price move is seen as a function of the size of today’s price move. This changing variance corresponds to phases of high and low price volatility in financial markets.
  • A GARCH model is simply a generalized ARCH model that also uses moving averages. Specifically, the variance of price changes depends not only on past price changes but also on past estimated variances. This means that variances display smoother trends.
  • GARCH in Mean is a GARCH model, where tomorrow’s expected value (return) is a function of expected volatility. It is typically assumed that expected returns increase, when expected volatility is higher.
  • EGARCH (“exponential GARCH”) simply means that the logarithm of the variance, not the variance itself, is modelled. This implied that the actual variance increases exponentially in the event of shocks, as experienced in financial crises.”

“This EGARCH-M model is shown to be theoretically tractable as well as practically useful. By employing a two component extension we are able to distinguish between the long and short run effects of returns on volatility.”

“The standard way of incorporating leverage effects into GARCH models is by including a variable in which the squared [future returns] are multiplied by an indicator taking the value one for negative returns and zero otherwise.”

Empirical evidence for the double relationship

“The benefits of using the EGARCH-M are best illustrated with weekly data…in particular weekly NASDAQ excess returns from 8 Feb 1972 to 3 Nov 2014 (2,282 observations).”


“The long and short run volatility components are shown to have very different effects on returns, with the long-run component yielding the risk premium.”

“As regards the risk premium, our results…allow us to reject both a constant and a rapidly varying risk premium in favour of a risk premium that is associated with the slowly varying component of volatility. Whereas long-term volatility is associated with a higher return, the opposite appears to be the case with short-term volatility, presumably because increased uncertainty drives away nervous investors and less uncertainty has a calming effect.”

“Leverage effects are significant…While returns have a symmetric effect on volatility in the long-run, they have something approaching an anti-symmetric effect in the short-run.”



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