Plots of stock market and Treasury bond relationship over time [Not quite so quick anymore]

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lack_ey
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Plots of stock market and Treasury bond relationship over time [Not quite so quick anymore]

Post by lack_ey » Sat Apr 07, 2018 9:30 pm

Disclaimer: I expect this to be of little/zero use to most here, but I did a brief investigation for myself and am posting here as a quick recap and in case it's useful or at all interesting to anybody.
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Over the long run, the correlation between Treasury bonds and the stock market has averaged about zero. This point has been repeated many times on this forum. However, the relationship has been variable, changing over time.

The latter point is maybe less well established, so I wanted to do a quick check to make sure it seemed as right and as obvious as I thought it was.

I used the following data:
1) Stock market excess returns (Fama-French daily Mkt-RF series) above cash
2) 10-year Treasury bond yield constant maturity series, accessed via FRED

Then I computed the change in 10-year Treasury bond yield and used that rather than the actual level itself. The change in yield corresponds very closely to the inverse of (excess) bond returns over short periods of time. That's good enough for these purposes. I decided to resample to weekly data, as daily may be a bit noisy especially with bond yields.

Finally I computed rolling 52-week correlations between the excess stock market returns and the change in Treasury yields. For good measure I also looked at linear regressions between the two as well, also on 52-week windows.

As a point of reference, I took the same weekly data and generated four different random permutations of the same, breaking the pairs of data apart for the two series. These permutations have the same means and variances as the original, actual data, but the relationship between the two series has been scrambled. The resulting correlations should be zero.

Here are the resulting plots of the actual data and the random permutations. Click the images to see larger versions:

Image

Image

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edit: using a different model, described in a below post:
viewtopic.php?f=10&t=246444&p=3882010#p3882010
Image
===

Remember, positive correlation between yield changes and stock returns means negative correlation between stock and bond returns. Notice how the computed correlation coefficients for the randomly permuted data are bouncing around, mostly in the [0.2, 0.2] range but sometimes outside. That's what to expect when the true relationship is 0. Similar things can be said for the beta plots.

Any thoughts? I don't think there's too much to say here, and that's about it for me.

Actually, I'm thinking I probably should have used some longer-running bond yield series than DGS10. Is there anything with granularity on this scale that goes back further than 1962 or so?

For some reading on the relationship and how it might have been influenced, one relatively recent paper to check is "Stock-Bond Correlations, Macroeconomic Regimes and Monetary Policy" by Lieven Baele and Frederiek Van Holle (2017).
Last edited by lack_ey on Sun Apr 15, 2018 11:36 am, edited 3 times in total.

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siamond
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by siamond » Sat Apr 07, 2018 10:23 pm

Results are unsurprising, but the methodology is pretty cool. Thanks for sharing.

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SimpleGift
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by SimpleGift » Sat Apr 07, 2018 10:45 pm

lack_ey wrote:
Sat Apr 07, 2018 9:30 pm
Over the long run, the correlation between Treasury bonds and the stock market has averaged about zero. This point has been repeated many times on this forum. However, the relationship has been variable, changing over time.
Nice work. Of course, as buy-hold-rebalance investors, we have to just take the stock-bond correlations that we get.

Though it's a perilous undertaking to try explaining the causes of changes in asset correlations, the paper you linked to has an intriguing thesis: In periods of restrictive monetary policy, the correlation between stocks and bonds has been positive. In periods of low inflation and accommodative monetary policy, the stock-bond correlation has been negative (chart below).
  • Image
    NOTE: Negative monetary policy gaps = accommodating policy stance, and vice-versa.
    Source: Baele and Holle
The paper goes on to explain in detail the macroeconomic mechanisms for why they think this relationship has existed globally over the past few decades — but one can find a shorter executive summary of their research in this online article.
Last edited by SimpleGift on Sun Apr 15, 2018 11:46 am, edited 2 times in total.
Cordially, Todd

stlutz
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by stlutz » Sat Apr 07, 2018 11:24 pm

lack_ey (or siamond for that matter): If you have the information available to do so, would you be willing/able to test out a result I think I arrived at one time by looking at some of this data? One conclusion I had come to was that stock/bond correlations have been higher during periods of higher real interest rates and they've been lower when there was negative correlation.

That result would be consistent with a notion of there not being free lunches.

I haven't been willing to promote my conclusion as truth since it was just something I arrived at by messing around with a spreadsheet one evening. :D

lack_ey
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by lack_ey » Sat Apr 07, 2018 11:30 pm

stlutz wrote:
Sat Apr 07, 2018 11:24 pm
lack_ey (or siamond for that matter): If you have the information available to do so, would you be willing/able to test out a result I think I arrived at one time by looking at some of this data? One conclusion I had come to was that stock/bond correlations have been higher during periods of higher real interest rates and they've been lower when there was negative correlation.

That result would be consistent with a notion of there not being free lunches.

I haven't been willing to promote my conclusion as truth since it was just something I arrived at by messing around with a spreadsheet one evening. :D
What's your preferred measure of real interest rates prior to TIPS being available?

stlutz
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by stlutz » Sat Apr 07, 2018 11:39 pm

Maybe a 5 year treasury minus the inflation rate? (since the bond funds I invest in generally have about a 5 year average maturity.

lack_ey
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by lack_ey » Sun Apr 08, 2018 12:19 am

stlutz wrote:
Sat Apr 07, 2018 11:39 pm
Maybe a 5 year treasury minus the inflation rate? (since the bond funds I invest in generally have about a 5 year average maturity.
Previous trailing inflation or forward actual (realized) inflation or something else? I think what we want is expected inflation but that doesn't really come out of actual CPI figures. There are papers and methods on estimating expected inflation from other figures, I suppose, but that's a bit of a deep dive and still probably inexact.

Another way to partition the dataset would be to go by yield curve shape (more specifically, something like 10-year minus 2-year). That's at least a measure that's directly observable.

I may take a look tomorrow.

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siamond
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by siamond » Sun Apr 08, 2018 10:34 am

lack_ey wrote:
Sun Apr 08, 2018 12:19 am
I think what we want is expected inflation but that doesn't really come out of actual CPI figures. There are papers and methods on estimating expected inflation from other figures, I suppose, but that's a bit of a deep dive and still probably inexact.
I will be more blunt. When we took a deep dive on trying to model what would TIPS have returned before they were created, we looked at expected inflation models from the few well-known papers on the topic, and reached the conclusion that those models are just really bad, based on dubious assumptions, not matching actuals for the periods of time where actuals are known, etc. As to much simpler models (e.g. use the past average inflation to project the near future), this works as well as a random draw. So yes, I don't think there is any other way than using trailing inflation numbers to get to real interest rate numbers, but then the outcome might not be a terribly reliable input for further analysis.

PS. I would infer that the pricing of TIPS, where expected inflation should be the primary factor, is really a purely speculative guess from the market, and the 'wisdom of the crowd' is probably... not terribly wise... But I side-track.

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SimpleGift
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by SimpleGift » Sun Apr 08, 2018 11:29 am

stlutz wrote:
Sat Apr 07, 2018 11:39 pm
Maybe a 5 year treasury minus the inflation rate?
Just out of curiosity, I wanted to test whether the estimated real 5-year Treasury rate would be all that much different than the "monetary policy gap" identified by Baele and Holle in their paper from the OP. The chart below shows the 5-year Treasury minus core inflation (in blue), overlaid with the monetary policy gap (in gray) for the 1970-2013 period.
Though it's not a perfect match, there's quite a bit of overlap. When the Fed is tightening (positive policy gap), the real rate has generally been high and positive, and when the Fed is easing (negative policy gap), the real rate has often been low and/or negative.

So my thought is that we might expect the changes in the stock-bond correlations to look very similar, whether we are looking at real interest rates or Baele and Holle's monetary policy gap. Just my two cents.
Last edited by SimpleGift on Sun Apr 15, 2018 11:45 am, edited 1 time in total.
Cordially, Todd

lack_ey
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by lack_ey » Sun Apr 08, 2018 2:25 pm

I'm a bit skeptical of various ways to estimate expected inflation and also to some extent the conclusions in the paper because there really aren't enough economic cycles and regimes to look at. It could be something else causing both A and B, so to speak, or a combination of other things.

As suggested earlier, looking at yield curve steepness (via 10-year minus 1-year yields):

Image

Each point is a different year, plotting average yield differential against the correlation of [Mkt-RF] and [10-year yield change] evaluated over that year. So points are nonoverlapping, starting from 1962.

There was a relationship but I don't really think it's necessarily causal and really this goes back to which periods had lower and higher correlation. This is a description of the past behavior, not a predictive model.

lack_ey
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Re: Quick plots of stock market and Treasury bond relationship over time

Post by lack_ey » Sun Apr 15, 2018 11:31 am

I tried analyzing the equity and bond (yield change) series using computational Bayesian statistical modeling. Effectively, given certain priors (weakly stated, here) and a model for the returns distributions, we treat the model parameters as unknown random variables and use Markov chain Monte Carlo sampling techniques to determine the likely distribution of the unknown parameter values, given the data. This answers the question of "what parameter values fit the data?"

However, the model I chose was oversimplified and is actually wrong, so don't take the figures very seriously. I model the alpha and beta of excess stock returns on 10-year Treasury yield changes as random walks over time and then express stock returns as Student t distributed with unknown mean, standard deviation, and degrees of freedom, with the mean being determined by the beta on 10-year Treasury yield changes plus the alpha term.

In other words, it's a model that expresses stock returns as linearly related to the yield changes, with the slope and intercept changing over time, and there being quite a bit of variability on top of that relationship. The slope of the relationship, the beta, is what we're interested in. We think of this as each week's market return drawing from a Student t distribution with a different mean (the mean determined by the relationship with bond yield changes at that time). This model is wrong in part because we know in practice the volatility should be time varying, and it doesn't account for that. At least it does account for the fat tails.

So here's what pops out, showing the mean of the posterior distribution (roughly, an estimate of what we think the beta is based on what fits the actual returns data and model) and the range between the 2.5 and 97.5 percentile of that distribution. This is analagous to but is not the same thing as a confidence interval.

Image

Remember, positive beta implies comovement between stocks and bond yields (so negative between stocks and bond returns).

Compared to just running rolling OLS regressions over time, the estimates are less jumpy, as it's assumed to be a random walk and random walks don't involve leaping around.

I hope that was at least mildly interesting. Any questions?

CULater
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Re: Plots of stock market and Treasury bond relationship over time [Not quite so quick anymore]

Post by CULater » Sun Apr 15, 2018 4:07 pm

Check out Dimson's work. He has studied the relationship between real interest rates and stock prices and finds there is a negative correlation: high or increasing real rates -> low real equity returns (e.g., the 1970s in U.S.); low or declining real rates -> high real equity returns. Nominal rates = the Money Illusion.
May you have the hindsight to know where you've been, The foresight to know where you're going, And the insight to know when you've gone too far. ~ Irish Blessing

lack_ey
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Re: Plots of stock market and Treasury bond relationship over time [Not quite so quick anymore]

Post by lack_ey » Sun Apr 15, 2018 6:17 pm

CULater wrote:
Sun Apr 15, 2018 4:07 pm
Check out Dimson's work. He has studied the relationship between real interest rates and stock prices and finds there is a negative correlation: high or increasing real rates -> low real equity returns (e.g., the 1970s in U.S.); low or declining real rates -> high real equity returns. Nominal rates = the Money Illusion.
There are a number of studies here that give explanations like that.

I think the conclusions are a little bit fraught given the paucity of data, there being relatively few cycles where we observe these things. It's hard to be sure of the causality. There are so many moving components. Some of it is likely or probably right, I suppose.

Here I mostly just want to establish what happened rather than why.

CULater
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Re: Plots of stock market and Treasury bond relationship over time [Not quite so quick anymore]

Post by CULater » Sun Apr 15, 2018 8:58 pm

lack_ey wrote:
Sun Apr 15, 2018 6:17 pm
CULater wrote:
Sun Apr 15, 2018 4:07 pm
Check out Dimson's work. He has studied the relationship between real interest rates and stock prices and finds there is a negative correlation: high or increasing real rates -> low real equity returns (e.g., the 1970s in U.S.); low or declining real rates -> high real equity returns. Nominal rates = the Money Illusion.
There are a number of studies here that give explanations like that.

I think the conclusions are a little bit fraught given the paucity of data, there being relatively few cycles where we observe these things. It's hard to be sure of the causality. There are so many moving components. Some of it is likely or probably right, I suppose.

Here I mostly just want to establish what happened rather than why.
I believe, but am not sure, that Dimson utilized data globally not just from the U.S. Inflation indexed bonds were available in other countries, such as the U.K., before they were in the U.S. So perhaps he had a larger data set.
May you have the hindsight to know where you've been, The foresight to know where you're going, And the insight to know when you've gone too far. ~ Irish Blessing

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