1 Stock-Bond Correlation and Duration Risk Allocation Feb/2014 Edition Liu X, Fan H ABSTRACT Using weekly stock-bond correlations estimated with high-frequency data, the authors find that a lower (more negative) stock-bond correlation forecasts falling 10 -year interest rates over the coming weeks, and it also forecasts a falling 1-year interest rates over the next year. The reverse is true when the stock-bond correlation is higher (more positive). Therefore, investors, in particular those with long-term bond-like liabilities, should take greater duration risk when the recent stock-bond correlations are lower. The authors propose two possible explanations of such predictive power: (1) the markets and/or policymakers’ under-reaction to the changing economic conditions implied by the stock-bond correlation; and (2) the markets’ initial under-reaction to the long-term bonds’ safe-haven status.
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Stock-Bond Correlation and Duration Risk Allocation
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Stock-Bond Correlation and Duration Risk Allocation
Feb/2014 Edition
Liu X, Fan H
ABSTRACT
Using weekly stock-bond correlations estimated with high-frequency data, the
authors find that a lower (more negative) stock-bond correlation forecasts falling 10
-year interest rates over the coming weeks, and it also forecasts a falling 1-year
interest rates over the next year. The reverse is true when the stock-bond correlation is
higher (more positive). Therefore, investors, in particular those with long-term
bond-like liabilities, should take greater duration risk when the recent stock-bond
correlations are lower. The authors propose two possible explanations of such
predictive power: (1) the markets and/or policymakers’ under-reaction to the changing
economic conditions implied by the stock-bond correlation; and (2) the markets’
initial under-reaction to the long-term bonds’ safe-haven status.
2
Using weekly stock-bond correlations estimated with high-frequency (HF) data,
we find that a lower (more negative) stock-bond correlation (SB-Correl henceforth)
forecasts falling yield on the 10-year US Treasury bond (bond henceforth) over the
coming weeks, and falling yield on the 1-year bond over the next year. The reverse is
true when the SB-Correl is higher (more positive). For brevity, we refer to such
predictive power of the SB-Correl as the correlation effect. Investors, in particular
those with long-term bond-like liabilities, should take greater duration risk when the
recent SB-Correl is lower.
The empirical results are obtained using a simple decile portfolio that is based on
a straightforward ranking of the weekly HF SB-Correl on its historical values. The
procedure for our study is intentionally simple, transparent and easily replicable.
The contribution of our findings is threefold: (1) we document a strong predictive
power of HF SB-Correl over bond returns; (2) we document an inverse relationship
between the 10-year bond’s time-varying equity betas and returns, and; (3) we provide
possible explanations for the correlation effect, which include policymakers/markets’
under-reactions to the changing economic conditions, and markets’ initial
under-reaction to the changing safe-haven status of the long-term bond.
DATA AND METHODOLOGY
A number of researchers use daily price data to estimate SB-Correls. To study
both short-term and cyclical dynamics, we reduce information latency by using HF
futures price data.
From Jan/11/1985 till Sep/06/2013, we obtain one correlation estimate per week
with no overlapping window, totaling 1,496 weekly observations. The realized weekly
correlations are obtained using S&P 500 futures and 10-year Treasury bond futures
price data with 10-minute frequency from tickdata.com. A frequency of 10-minutes is
still of a low enough frequency to avoid microstructure noise. The futures delivery
months for all of the contracts are March, June, September and December. To ensure
good liquidity of the futures, we always use the contract closest to expiration,
switching to the next maturity contract five business days before expiration, and we
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use electronic futures contracts when they are liquid enough (Exhibit 1).
E X H I B I T 1
Futures Contacts Used
Date
from to
Futures contract used
for S&P 500
Futures contract used
for US 10-year Treasury
Jan/04/1985 Jan/02/2004
SP futures open outcry; CME
TY futures open outcry; CBOT
Jan/05/2004 Sep/06/2013
E-mini futures elec; CME Globex
ZN futures elec; CME Globex
We also obtain 3-month, 1-year, 2-year, 5-year and 10-year constant-maturity
Treasury yields from the FRED database of the Federal Reserve Bank of St. Louis.
Forward rates and zero coupon bond log returns are estimated by constructing
zero-coupon yield curves with a linear interpolation method. We obtain equity log
returns using S&P 500 Total Return Index (SPTR) and 3-month Treasury yields. From
1985 to 1987, SPTR is not available and the S&P 500 Index is used as the proxy.
As shown in Exhibit 2, HF SB-Correls exhibit a secular downward trend, which
are consistent with the low-frequency (LF) SB-Correls using daily data with a 1-year
rolling-window. HF SB-Correls are highly persistent in the short- and medium-term
and the sample autocorrelations are greater than 0.5 up to lag 80.
E X H I B I T 2
Stock-Bond Correlations Using High- and Low-Frequency Data
Let n=20 weeks for illustrative purpose. At the end of each week (𝑡), we rank the
last 20 weeks’ HF SB-Correls ( _ THF Correl , T = t -19, t -18, , t ), and determine
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the decile of _ tHF SB Correl , or tDecile20 accordingly. For convenience, we also
refer to week ( 𝑡 ) as week 0 and week ( 𝑡 + 1 ) as week 1. A simple HF
SB-Correl-sorted portfolio allocates greater weight to bond on week 1 when the decile
is lower on week 0:
1tweight20
1 2( 1) / (10 1)tDecile20 . (1)
Because of the secular downward trend of the SB-Correls, over the long run, the
weights above sum slightly above zero.
To show a more representative result, we may use different sampling windows
and use the averages of their deciles. As HF SB-Correls are highly persistent, we may
also use equally weighted moving averages of the deciles. A portfolio is hence defined
by its underlying asset, the object used for ranking, the sampling windows and
corresponding lag terms. For example:
[ Asset=Bond10Yr; RankObj=HF_Correl; n=20,40; Lag=1, 4] means
1
1
4
1
1
1 1 1
1 1
1
*
1 2 1 10 1
11, 4,
2
1, 2
4,
i
i
t t t
t t
t t t
t it t
t it
R Weight Return 10YrBond
Weight Decile
Decile Round MA Decile20 MA Decile40
MA Decile20 Decile 0 Decile20
MA Decile40 Decile40
(2)
For the resulting time series of returns, we calculate averages, standard deviations,
and Sharpe ratios. To test for the statistical significance of the Sharpe ratios and decile
differences, we use a bootstrap to estimate the t-values. For a given decile (or other
quantile) strategy, suppose we have a sample of T weekly observations of SB-Correl,
the associated contemporary excess return and the date (timestamp). To estimate the
P-value, we draw 10,000N samples of T observations (with replacement) from
the empirical distribution. For each boostrap sample, we sort the observations
sequentially according to their timestamps. We then use the defined SB-Correl-sorted
decile strategy, re-established a new portfolio and calculate the mean for each decile.
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The P-values are given by:
# (D10 D1) 0(D10 D1)
# (D10 Other D) 0(D10 Other D)
# (Sharpe ratio 0) 0(Sharperatio) .
meansP
N
meansP
N
meansP
N
(3)
And the t-values are given by: 1 1t N p , where N is the cdf of the normal
distribution with zero mean and unit variance.
We also use a simple regression to disentangle the correlation effect from other
factors. The factor portfolios are constructed in a similar manner to the SB-Correl-
sorted portfolio. We define curve spread as the difference between 10-year and 1-year
interest rates. In Equation (2), instead of letting RankObj=HF_Correl, we let
RankObj=Yield, RankObj= –Curve_Spread and RankObj= –VIX respectively. And the
resulting time series of returns _Yield Momentum
R , _Curve Spread
R and VIXR are used
as the yield momentum, curve spread and stock market uncertainty factors
respectively. By regressing the returns of the HF SB-Correl-sorted portfolio to the
underlying bond returns, the equity returns as well as these factors, we control for
possible systematic exposures to bond, equity and these factors:
_ _
_ _
Bond Bond Yield Momentum Yield Momentum
VIX VIX Equity EquityCurve Spread Curve Spread
R R R
R R R
(4)
where R is the returns of the HF SB-Correl-sorted portfolio; is the adjusted alpha;
BondR is the returns of the underlying bond; EquityR is the returns of the SPTR index;
and _Yield Momentum ,
_Curve Spread and VIX are the estimated factor exposures
1.
Following Ilmanen [2003], Campbell et al. [2013a], Johnson et al. [2013] and a
number of other studies, we consider a potential structural break in 1997. We also
consider another potential structural break in Dec/2008, when the fed funds rate is
essentially zero. The two potential structural breaks divide the sample into three
1 The t-value of the alpha and betas are estimated using the closed form formula for OLS instead of simulation.
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sub-periods, and we may study them separately depending on the context.
EMPIRICAL RESULTS
Exhibit 3 shows the correlation effect over the 10-year bond from 1997 to 2013.
We choose a geometric series: [20, 40, 80, 160] weeks as the sampling windows, such
that we are not showing too specific results of a sampling window2. We use no lag
terms for Exhibit 3. The 10-year bond underperforms exceptionally in the
bottom-deciles. Compared to other deciles, in D10, the 10-year bond returns are -20%
on week 1 and -7% per week on average from week 2 to week 8 (all returns are
annualized). The 10-year bond also outperforms considerably in the top-deciles. The
decile dispersion (D10–D1) is short-lived and is absent from week 9 to week 52. The
Sharpe ratio of a simple SB-Correl-sorted strategy is 0.66.
2 We find that the empirical results would be much weaker if we use only past-week Stock-Bond correlations or
other shorter windows. This is because there is too much noise with short windows. The exhibit will be available
online.
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E X H I B I T 3
Decile Dispersion for the 10-Year Bond
[ Asset=Bond10Yr; n=20,40,80,160; Lag=1,1,1,1 ], from Jan/1997 to Sep/2013
Decile Return Dispersion (per week)
No.
obs
Correl
Mean
Curve Spread
10Yr-1Yr
week 0 week 52
Return
week 1
Return
week
2 to 8
Return
week
9 to 52
Equity
beta
week 1
D1 80 -0.49 1.04 1.65 19% 11% 6% -0.14
D2 98 -0.40 1.30 1.61 9% 9% 6% -0.18
D3 84 -0.36 1.28 1.70 -1% 7% 5% -0.11
D4 85 -0.32 1.59 1.67 3% 9% 5% -0.02
D5 95 -0.30 1.32 1.53 10% 8% 4% -0.25
D6 83 -0.24 1.45 1.48 11% 4% 5% -0.09
D7 83 -0.31 1.60 1.53 14% 7% 4% -0.26
D8 84 -0.22 1.77 1.46 6% 4% 4% -0.26
D9 99 -0.17 1.65 1.31 -10% -8% 3% -0.02
D10 79 -0.07 1.80 1.19 -13% -2% 5% -0.05
D10–other D
-20% -7% 0%
(t-value)
[-2.3] [-1.4] [0.1]
D10–D1
-32% -13% -1%
(t-value)
[-3.5] [-1.7] [-0.4]
Strategy excess returns
3.9%
Standard deviation
5.9%
Sharpe ratio
0.66
(t-value)
[3.2]
Note: Assuming a hypothesized structural change in 1997, Exhibit 3 shows the result with the time series starting
from 1997 instead of 1985.
The short-lived outperformance of the long-term (10-year) bond in D1 is related
to a well documented “flight-to-safety” phenomenon. Gulko [2002] examines the
stock-bond decoupling induced by stock market crashes, and finds that during stock
market crashes the correlation between the returns of U.S. stocks and bonds switches
sign from positive to negative. Connolly et al. [2005] ascribes the negative SB-Correls
since 1997 to “flight-to-safety”, where increased stock market uncertainty (proxied by
VIX) induces investors to flee stocks in favor of the long-term bond, and realized
SB-Correls drop sharply.
On the flip side of “flight-to-safety”, we ascribe the short-lived underperformance
of bond in D10 to a “flight-from-safety” or “restore-to-normality” phenomenon,
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where investors bid down the price of the “safe” long-term bond, in times of increased
SB-Correls, inducing corresponding negative returns of the long-term bond.
For each decile in Exhibit 3, we estimate the equity market beta by regressing the
excess bond returns on week 1 to their contemporary equity market excess returns.
For D1 and D10, the stock betas are -0.14 and -0.05, while the excess bond returns are
19% and -13% annualized respectively. A plot of the results reported in Exhibit 3
would show that there is an inverse relationship between the 10-year bond’s equity
betas and returns. Hence, the 10-year bond has higher return when it is less risky, if
risk is measured by its time-varying equity betas.
As warned by Dopfel [2003], the various strategic and tactical asset allocation
issues related to a continued low correlation environment have an impact on investor
welfare in an asset-only and an asset-liability framework. For investors, a low
SB-Correl is beneficial in an asset-only context, but detrimental in the case of a
long-term bond-like liability context. From a surplus optimization perspective, a
lower SB-Correl increases surplus risk.
Our empirical findings suggest that lower SB-Correl (D1) is the best of both
worlds of return and risk for an asset only investor, especially if the investor is
allowed to take leveraged positions in bonds. Not only does a lower SB-Correl
indicate lower risk for a stock-bond portfolio, it also forecasts higher bond returns in
the short run. On the other hand, a lower SB-Correl is a double whammy for an
investor with a long-term bond-like liability, especially if the investor invests
aggressively in equities and not allowed to use leverage. From an asset-liability
perspective, not only does a lower SB-Correl increase surplus risk, it also forecasts an
increased liability in the short run. Therefore, investors should take greater duration
risk when the recent SB-Correls are lower, either with cash bond positions or with
derivatives if leverage is allowed.
We replicate the analysis for a short-term (1-year) bond. Differently from Exhibit
3, we exclude data from 2009 onwards as the Fed funds rate has been kept close to
zero since 2009. Exhibit 4 shows the correlation effect for the 1-year bond from 1997
to 2008. The 1-year bond considerably outperforms in the top-deciles and
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underperforms in the bottom-deciles. Interestingly, decile dispersion (D10–D1)
persists from week 1 to week 52 by 1.4% annualized on average per week, which is
sizable given that the annualized volatility of the 1-year bond is 0.8%. The Sharpe
ratio of a simple SB-Correl-sorted strategy is 0.83.
E X H I B I T 4
Decile Dispersion for the 1-Year Bond,
[ Asset=Bond1Yr; n=20,40,80,160; Lag=1,1,1,1 ], from Jan/1997 to Dec/2008
Decile Return Dispersion (per week)
No.
obs
Correl
Mean
Curve Spread
10Yr-1Yr
week 0 week 52
Return
week 1
Retur
n
week
2 to 8
Return
week
9 to 52
Equity
beta*100
week 1
D1 63 -0.46 0.70 1.49 1.7% 2.0% 1.9% -0.8
D2 72 -0.35 0.84 1.38 2.2% 1.8% 1.6% -1.5
D3 61 -0.30 0.84 1.44 1.5% 2.5% 1.6% -1.0
D4 66 -0.28 1.33 1.55 0.8% 1.3% 1.3% -1.4
D5 78 -0.26 1.12 1.41 2.6% 1.2% 1.3% -1.8
D6 60 -0.18 1.09 1.28 0.7% 0.5% 1.3% -1.1
D7 54 -0.25 1.16 1.19 1.4% 1.2% 0.9% 0.0
D8 57 -0.14 1.45 1.16 0.0% 1.0% 0.6% -1.3
D9 63 -0.10 1.25 0.89 0.3% 0.2% 0.5% -0.8
D10 52 0.01 1.39 0.82 0.6% 0.3% 0.4% 0.6
D10 – other D
-0.74% -1.0% -0.79%
(t-value)
[-2.0] [-2.4] [-3.1]
D10 – D1
-1.1% -1.7% -1.4%
(t-value)
[-3.2] [-2.5] [-3.7]
Strategy excess returns
0.39%
Standard deviation
0.48%
Sharpe ratio
0.83
(t-value)
[3.7]
Note: Assuming hypothesized structural changes in 1997 and 2008, Exhibit 4 shows the result with the time series
from 1997 to 2008.
We study the different persistence and magnitude of the correlation effect over
the entire forward curve from 1985 to 2013. There are three sub-periods divided by
the two potential structural breaks (Jan/1997 and Dec/2008). D10–D1 is less
significant pre-1997, especially at the long end of the forward curve. Before Dec/2008,
D10 and D1 forecast rising and falling interest rates respectively at the short end of
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the forward curve for about 52 weeks. The results are available online.
As shown in Exhibit 4 and Exhibit 5, from Jan/1997 to Dec/2008, because of the
highly persistent short-term interest rate movements over the following 52 weeks, in
D1, the curve spread (10Yr-1Yr) steepens from 0.70% to 1.49% on average, while in
D10, it flattens from 1.39% to 0.82% on average in 52 weeks. Such decile dispersion
in Exhibit 3 is less pronounced, because it includes the period from Jan/2009 to
Sep/2013, when the Fed pledges to hold the key interest rate at close to zero for an
extended period of time.
E X H I B I T 5
Movement of Curve Spreads (10Yr-1Yr), from Jan/1997 to Dec/2008
Compared to previous research, our findings have a few distinct features. First,
instead of the transmission from stock market uncertainty to SB-Correls, we
document the transmission from SB-Correls to the bond returns. Second, we
document a pronounced “restore-to-normality” phenomenon in parallel to the
“flight-to-safety” phenomenon. Third, we document an inverse relationship between
the 10-year bond’s time-varying equity betas and returns. And finally, we document
that a low SB-Correl forecasts a more steepened yield curve over the next year.
CONTROLLING FOR OTHER EFFECTS
How does the correlation effect relate to other factors that have been documented
in the previous research? Could it be that the correlation effect simply captures the
yield momentum, curve spread and stock market uncertainties?
To answer the question, we construct a few factor portfolios constructed using the
same decile-based strategy. The factors are historical yield, curve spread and VIX.
Similarly to Equation (2), a factor portfolio is defined as: