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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. Rolling correlation
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.

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10Y notes are issued monthly by the US treasury. You should certainly be using the one-the-run notes, that it, compute the yield for each new note. Given the 1 month cycle, it seems natural to use 1-month SP500 returns. Note that newly issued treasure notes are 'special', they trade at a premium to off-the-run. You might look at GC (general collatoral) 10yr note yields. That removes the specialness. –  JayInNyc Aug 2 at 2:13
    
The best option is to use fitted par yields, which can be obtained either from the Fed (federalreserve.gov/pubs/feds/2006/200628/200628abs.html), the Treasury (treasury.gov/resource-center/data-chart-center/interest-rates/…), or banks. These yields are constant maturity, don't suffer from coupon effect, and don't have "rolls" (jumps resulting from new issues). The Fed's fitted yields also don't suffer from repo specialness since they're fitted to off-the-runs. –  haginile Aug 4 at 14:33

2 Answers 2

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.

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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. EWMA Volatility and Correlation paths

Or for the interval you specified in your charts. EWMA Volatility and Correlation paths

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