Choosing Windows and EWMA for Stock–Bond Correlation
Summary
The discussion considers how to measure comovement between equity returns and Treasury yields, emphasizing that the sampling horizon and rolling window should match the question. Short windows can serve short-term signal research, while longer windows can reveal secular patterns; short-term stock–bond correlations may be unstable. It also warns that rolling correlations built from returns that already overlap across rolling windows can obscure information.
As an alternative, it outlines an exponentially weighted moving average (EWMA) for updating each variance and the covariance from recent observations, then deriving a spot correlation. It cites different decay settings for daily and monthly data and mentions multivariate GARCH as a more involved option. The approach produces a responsive estimate rather than a uniquely correct horizon choice; the suitable window or decay depends on the intended use, and the reported historical stock–bond relationship can vary across eras and with whether yields or bond returns are measured.
Key ideas
- Choose the observation horizon and correlation window according to whether the goal is short-term signals or long-term trends.
- Short-term stock–bond correlations can be unstable.
- Overlapping rolling-return windows can smooth away useful information.
- EWMA updates variances and covariance from recent data to estimate a time-varying spot correlation.
- Multivariate GARCH is a more complex alternative when a richer volatility and correlation model is needed.
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Full text
# Correlation between S&P500 returns and 10y US Treasuries yields
# Correlation between S&P500 returns and 10y US Treasuries yields
I'd like to investigate the comovement of stock index returns with bond yields but I don't know which return's duration to use (1-year, 1-month or anything else) to get a better view of the relationship. Other than that, I'm unsure which is the better rolling correlation's duration in order to have a clear understanding.
I've elaborated these 4 alternatives:
1-year rolling correlation with 1-month returns and 10y bond yield. 3-years rolling correlation with 1-month returns and 10y bond yield.
1-year rolling correlation with 1-year returns and 10y bond yield. 3-years rolling correlation with 1-year returns and 10y bond yield. If I use 1y returns then the correlation has a max of 1, otherwise with 1m returns the maximum correlation is 0.6.
Thank you in advance for all your thoughts.
## Answer by Helin (score 1)
https://quant.stackexchange.com/a/14285
I don't think there is a correct answer to this question.
If you're trying to study short-term correlations (e.g., to construct short-term trading signals), then 1-month or 3-month rolling correlation of daily returns is a feasible option. These short-term stock/bond correlations are quite unstable though.
On the other hand, if you're studying long term secular trends, then you should use much longer windows. For example, Ilmanen showed charts of 5-year and 26-week stock-bond correlations when discussing the secular trends of bond risk premium. Here, the short story is that bond/stock correlation have been positive (positive in returns; negative if you use stock/yields) up until mid 1990s, and negative thereafter.
## Answer by Kiwiakos (score 1)
https://quant.stackexchange.com/a/14287
My opinion is that using rolling correlations of returns which themselves are computed over rolling windows is not reliable. Taking rolling windows smothers information.
Instead, I would specify a simple EWMA filter for the variances and the covariance, which would give me a value for the spot correlation. For example something like $$ \begin{align} \sigma^2_{SPX;t+1} &= \lambda\ \sigma^2_{SPX;t} + (1-\lambda)\ r^2_{SPX;t}\\ \sigma^2_{TBILL;t+1} &= \lambda\ \sigma^2_{TBILL;t} + (1-\lambda)\ r^2_{TBILL;t}\\ \sigma_{SPX,TBILL;t+1} &= \lambda\ \sigma_{SPX,TBILL;t} + (1-\lambda)\ r_{SPX;t}\ r_{TBILL;t}\\ \rho_{SPX,TBILL;t} &= \frac{\sigma_{SPX,TBILL;t}}{\sqrt{\sigma2_{SPX;t}\ \sigma^2_{TBILL;t}}} \end{align} $$
RiskMetric apply $\lambda=0.94$ for daily data and $\lambda=0.97$ for monthly data. If you are feeling more adventurous, you can specify a multivariate Garch model and estimate it. For $\lambda=0.94$ I get the volatility and correlation paths below. This are 'spot' (that is to say 'instantaneous') values. If you want a longer term correlation then you can project using some exponential decay.
Or for the interval you specified in your charts.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.