Correlation between S&P500 returns and 10y US Treasuries yields

Question

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 1

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 2

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.